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So I want to discuss Grassmanns Vorrede and possibly go on from there into his Einleitung to the 1844 Ausdehnungslehre which I feel is about dynamic extendable magnitudes and the analysis and synthesis of these spatial dynamisms in an arithmetical symbolic way. That is, Grassmann defines, quantitatively what his subject is and then analyses and synthesises them.

http://www.drhertz.org/properties.pdf

This is a " complete list" of the axioms and definitions in the real numbers. This is what I was introduced to on my first analytical course at university, without any explanatory motivation. Of course I thought it was Algebra. Nobody told me it was philosophy in algebraic notation.

Now I know this too to be false. This was and is symbolic arithmetic. Algebra, for what it is worth is the Arabic name for " mind twisting rhetoric!" of course you would not put yourself through it if it did not have some use. The use is general and analogous thinking, dealing not with one specific calculation but the method or algorithm for all arithmetical calculations of that type.

However, this was a group theoretical presentation of the real numbers. It was not an investigation into the foundations of arithmetic. In the time of Grassmann, Justus his father and other pioneering group and ring theorists were attempting to rebuild mathematics from the natural and meta mathematical world in which they existed. Justus group of collaborators were early crystallographers, seeing in the dynamic world a geometrical order that produced crystals dynamically. They hoped by studying these crystal formations to understand how to construct a more applicable and relevant Arithmetic, or rather to extend the arithmetic of commerce into an arithmetic of natural processes.

Justus therefore analysed arithmetic and geometry and logic down to its nuts, bolts and washers and attempted to construct a mechano set of principles that built a sound mathematics.

He ran into difficulty with Legrndres redaction of Euclid, which everyone mistakenly assumes is about geometry, and with the concept of multiplication.

The multiplicative axioms are the problem. They unravel the whole of arithmetic because they do not logically come from considering geometrical forms!

Most of us have played with bricks.

When you set up an array of bricks say 2 by 4  we know we can set that array in any orientation on the ground.  I also know that you would immediately sy that there are 2 ways to construct this array.

Well then , imagine the bricks scattered across the room. To construct the array in the centre of the room you could collect each brick individually and return it to the centre and carefully place it in contact with the previous brick if there was one. Once you got to 4 you could then suddenly place the next brick on the top or quarter turn side of the first brick. The sequence is extended,by carefully returning and placing the bricks next to the previous one and in contact with the " row" of bricks below one brick to one brick. You woul stop at the end of the bottom or first row with your last brick in the second row on top of the last brick in the first row.

In so doing you have created and defined a row of 4 bricks and an array of 8 bricks consisting in 2 rows of 4

The combinatorial product of your effort can be defined as a 4 by 2 array, where the 4 signifies the count of bricks in a row and the 2 signifies how many rows., and  the word array means they are patterned not scattered, and indeed they are as contiguous as possible.

Well that was 2 products of a possible 3 using that particular method. The third variation is to start the second row below the first in contact with the ground. The second was to stack the second row on top of the first row.

Well of course there are more methods of forming this array. Now we know what it looks like we could save time by collecting 4 bricks in sequence before returning to the centre to place them carefully.. We save on the repeated foraging, but now at the centre we have to decide which sequence to place the bricks in, and of course we have 4! Choices now.

 There are many other scenarios that result in the same product but involve different choices in layout of the array. The "other" choice you may have considered traight away( why?) is where the trip involves collecting 2 bricks at a time and constructing the array 2 bricks at a time.

We still have several choices if we lay the bricks in pairs. We could for example lay the bricks in pairs in a row that totals four. But the one you are probably screaming out now is the laying of bricks in pairs, but carefully placing the pairs on top or to the topside on the ground or below the first pair on the ground so they are contiguous. The next pair is placed on top, above or below the previous pair until 4 pairs have been laid.

These are the pragmatic choices one is faced with when constructing or synthesising a mosaic array .

Now ,after all of this we are expected to note that the 4 by 2 array has become a 2 by 4 attay.. It should be clear that this is a rather meagre reduction of the combinatorial choices we have just touched upon. In addition we are supposed to assent to the term commutate as if it means nothing spaciometric ally. The fact is it is a symbolic , literal device referring only to marks or symbols on the page.

Giving our assent to it is all that is sought!  The methods and processes are distinct and distinguishable and do not say ab = ba!

To get to this statement we have to agree that row counts will precede column counts, thus 4 x2 means 4 in a row,2 rows.
2 x4 means 2 in a row 4 rows.

If I now apply this to the measurements of a single block say 3 cm laid out horizontally to me , left to right on the ground and 2 cm vertically on the ground, forward of me then I get
 
4(3) x2 for one method  and 2(3) x4 for the other arrangement

Equating these 2 formulae is equating the lengths of one side of the bricks! I have ignored the depths of the bricks.

Similarly I can write
4 x 2(2) and 2 x 4(2) to equate the other dimension.

However, now my notation has to be defined. The brackets are to signify the measurement of the fundamental brick in the direction of the row or column.

All this is usually finessed away. It is confusing to write it down in longhand and the student is not shown this instead they are expected to just pick this up from the demonstration and the exercises.

It is very tempting to ignore this detail, but it in fact is crucial to understanding multiple forms in symbolic notation. It was discovered or noticed by Hermann Grassmann, and it is the reason why logicians have fundamental difficulties with deriving multiplication

Let us use the full notation of this form now

4(3)x2(2)  can be symbolically equated to 4x2(3:2)

This format can now be defined to represent the foundation of multiplication of scalars.or ordinary arithmetical multiplication. The bracketed  pair are the dimensions of the brick and the notation relates them in a proportion or ratio format. The ratio format can now be defined as information about the single brick and representative of a single brick. The scalars now multiply to 8 the known numbers of bricks, and we have now captured the relevant information  for counting and identification purposes that should enable anyone to construct this mosaic array in any orientation in space.

2x4(3:2), and 4x2(2:3) and 2x4(2:3) are all different constructions, but the scalar commute in this precise sense. Not in the undefined sense hitherto taught.

Of course this is still a meagre representation of all the possible construction algorithms, and this is where Grassmann gradually realised that the stretch of these dimensions by an extensive method was very important to describing position in space. Of course he was not to know the Pythagoreans had already developed this idea.

http://www.youtube.com/watch?v=ScTJRrg8ZfU&feature=youtube_gdata_player
There are 2 important things to note here: by removing Arabic numerals the discussion becomes more geometric and a direct link to the Arithmoi is almost obvious. The arrays of 1 are geometric Arithmoi.

Secondly, because Norman has defined the arrays to be. Row arrays he misses the Law of transitive distributiyity in the associative law. He in fact illustrates it with the circles, but fails to name it the wedge product.

It's real significance is in crossing the operator imposed boundary. The so called operations are combinatorial arrangements so
2.(3.6) = (2.3).6 = 3.(2.6) but
In vector group algebra  the wedge (2+3) can be scaled  6(2+3) and this is the same wedge formation as (6,2 +3) or (2 +6.3) where the combinatorial+ joins the Strecken of different " lengths" into wedge forms which are rotated relative to each other.

The wedge form is a modern notation for the transitive distribution just pointed out. Because it is a proper multiplication it is confusing why modern notation writes it as a +

This is because of abelian group notation.

The young Abel was active at the same time as Justus Grassmann was formulating his ideas.
http://en.wikipedia.org/wiki/Niels_Henrik_Abel,
There is every opportunity that Justus and Abel were involved with the same or overlapping groups of mathematical researchers looking into extending and rationalising mathematics. Certainly Justus referred to his work in mathematics as Verbindungslehre, and looked at a combination of Zahlenlehre and Kombinationlehre. At the time, Galois and others were looking into the solution to polynomial equations and the combinatorial relationships  of roots figured heavily in that line of theory .

A year before Abel died, In 1827 Justus published a paper on his ideas that caught Hermanns imagination. It was a paper setting out how numbers and algebraic notation could be combined to extend notation for Arithmetic. It is likely that it drew upon the same vein of ideas initiated by Abel.

http://en.wikipedia.org/wiki/Abelian_group

Certainly much of what Abel laid out appears in Grassmanns Ausdehnungslehre. What distinguishes Hermanns work is clear: while others were looking at the general polynomial group, Hermann was looking at Geometry. He realised that die Begränzte Linie could stand as a symbol for many magnitudes measured in geometry. Only later did he realise it could symbolise motion and displacements in der Raume or 3d space.

He knew that he needed to have combinatorial actions that represented addition and multiplicatio, from his Fathers paper, but he does not seem to have known about closure in a group. This was fortunate. It meant his ideas were unbounded by convention, and eventually he realised that he could subsume nearly all of mathematics into his methodology of analysis and synthesis.

The Wikipedia article has improved somewhat since I started my research . It could be I understand it more, who knows, but I can recommend it as an overview.
http://en.wikipedia.org/wiki/Hermann_Grassmann

I still Have issues with some concepts  said to be Herrmnns, particularly the "linear " or vector references. The purest concept of Hermanns thinking is given in his Vorrede, as we shall see, but lineal means just that, a line! Ausdehnung is to reach or stretch out especially an arm as in pointing or exercising. This is therefore a theory based on stretching out lines.

Mathematicians often like to talk about extending a method in a certain way to make it more general or applicable, and I am positive Hermann was aware of this meaning, which is why he specifically puts lineale as the adjective.

Another way to think of it is as a tailor stretches out his tape measure to get the dimensions of a suit , or an architect does the same to measure out the positions of his footings , they perform a lineal Ausdehnung. They measure Ausdehnungs Größe and record them in a way that enables them to construct or synthesise some product in space. Hermann simply recognised this everyday procedure as a mathematical method of analysis and synthesis. The next thing he did which was down to his fathers interest was define the underlying group and ring structure for this method. He therefore intuitively recognised points and lines as fundamental elements. By trial and error he isolated the begränzte Linie as his fundamental mathematical object, which allowed him to see lines and pairs of points as duals of each other.

The third thing he always kept in mind was the dynamic nature of these entities. None of his terminology was ever meant to be spatially fixed, everything was meant to be variable and dynamic.

When Hermann rejected 3d space, he did not mean he was aware of some other higher alien esoteric spaces. He was saying that in der Raume the 3 dimensions have never been significant for proper construction. Every person knows that measurements have to be taken in any number of orientations to construct a form . Thus a form determines the dimensions most useful.

At one time he considered calling his method Formenlehre, because of the importance of form. The spanning of a space by a basis actually goes against his general idea. The basis was to be revealed as the kernel or the least structure that accounted for every point and which every vector could be reduced to, but this often involved using coefficients that were not convenient. If one insisted on unit coefficients or integer coefficients then there were a lot more bases to play with!

The diophantine constraints meant that certain bases could be used to solve or represent those types of mathematics, and other constraints led to other types of "spaces". The space where the coefficients are continuous or smooth is usually what is meant by a "vector" space, and this space reduces to a basis of 3 dimensions, but these dimensions are orthogonal by convention. It is possible to use more generalised reference frames.

It became possible to generalise this method outside of geometry into node or graph theory and category theory. The methodology has proved to be so robust that many computer programmers have been able to use it to structure their programming languages and concepts

There are several crucial passages in the Vorrede where my poor German meant I was confused by the language construction. But my heart and mind immediately apprehended what Hermann was saying. These crucially were the Law of 3 points and the Law of 4 points.

The law of 3 points involves only 2 Strecken. It was hard to rid my mind of some inept training in vectors. I did not know then that Strecken were run of the mill constructs, just line segments. I knew they were lines on the page, I knew they were constructed or construction lines, I knew they were symbolic, yet I still could not grasp  what they were!

Later I managed to gradually isolate their significance, and to appreciate their utter utter simplicity.
Then I found the definition Grassmann gave in the Einleitung, after an intense dreamlike passage about how things may be outwardly different but inwardly they were alike, that distinctness could be used to collect things into groups of likeness, and that likeness can be used to distinguish groups of entities, and this duality between likeness and difference was a powerful human experience that broke down barriers. He said that entities that had these properties could be represented by die Begränzte Linie( Strecke).,the line which was constrained between points is a Strecke.  It's not just a line segment, it is a symbol of some discrete magnitude,,and every discrete magnitude could be symbolised by a line. The lines themselves could be distinguished by orientation in space relative to each other.

If I have 2 points A,B they constrain a line AB that starts at A and ends at B . The points if fixed are distinct from the line. The line is that stretchy thingy that stretches betwween the two points from A to B. Needless to say the stretch from B to A is a related but entirely different stretch. The negative sign signified this at a time when the negative sign was not understood as a rotation operator. In fact more than rotation is occurring. It is an entirely different stretch. The notion of contra, coined by Hamilton is most useful in marking this distinction.

However Grassmann was constrained by the use at the time and he was learning about the negative quantities at the time. What he noted was that commuting the points on the page represented a different stretch in space. Commuting the points did not switch the points in space, it switched the mental focus of the observer. The negative sign does not quite capture that distinction Hamilton in his essay on conjugate functions or couples describes how the notatin conducts the mind of the observer on mental journeys, switches of points of view, rotations of orientation. This he called contra as in the contra step. It was as if a mirror reflected the stretch from a point behind to the poin in front, as the stretch from the point in front reaches out to the point behind the mirror.

So Hermann noticed AB could be an identity with –BA, but unfortunately  he used the = sign. This creates a stronger affinity with arithmetic than Grassmann eventually realised he needed. In part it was this notation that held him back, and also made communication with others slippery.  His rhetoric is clear, but for mathematicians trained in symbolic interpretation the redefinition of symbols often went unnoticed. More attention was given to the + and= sign than the rhetoric which mitigated their meaning.

The 2 Strecken in the law of 3 points A,B,C are AB and BC. AB combined at B with BC gives a stretch AC. Grassmnn noted this notation was" true"  in 2 situations.: when C was on the straight line extended from AB beyond B and when C was on the segment between A and B.,

He wished it was always true that the segment AC was always given by the combination of the other 2 segments . But then he relished if he omitted the segment AC then the 2 Strecken always represented a stretch from A to C via B. this was the simplest way that it could be true.

You may struggle to grasp this realisation. AC is not the line segment AC it is the stretch from A to B and then from B to C..

But then he could identify the line segmentn AC wih this unique pair of stretches thus he could once again use an equal sign, but of course it would have been clearer to use and identity sign.

However now he had a form of addition.

Again CA was the  contra of AC  and the negative switched the tems in the addition and commuted the letters within the terms!

It was the realisation of the law of 4 points that convinced Hermann he was onto something he could devote his life to.

The law of 4 points occurred to Hermann when he was studying some ideas of his dad about multiplication. As far as I know, Justus had difficulty placing multiplication in his logical and constructive " map"'of mathematics. In the end he resorted to a geometrical  definition of multiplication. Basically he said if you stare at the rectangle you will intuit that multiplication is soundly based!

I have explained the difficulty he had and why in the second post.  The difficulty arises from thinking numbers are real objects.  They are our response to a dynamic world and they are imaginary. The process of that response is called Logos, and it is a significant analytical concept discoverd by the Greeks , in particular the Pythagoreans
 
The Pythagoreans went further and compared logos with analogos, the basis of Eudoxus teachings in books 5 and 6 of the Stoikeia. However Justus had Legendres redaction of Euclid's Stoikeia and he felt therefore that Euclid was about geometry and his geometry was faulty logically .

Euclid's Stoikeia is not about geometry, but about philosophy, and it is an introductory course to Pythagorean philosophy. The real source of geometry was another subject called mechanics. Newton expressed this in his opening statements to the Principia.
Mthe artisan develops or discovers mechanical relations, tools that produce these mechanical objects. The skilled artisan refines these mechanical relations and objects in a meditative reflection called geometry .  This reflection abstracts and perfects these relations and records these ideas in symbolic notation.

The Pythagoreans went beyond this and looked at all ideas and forms in this abstract reflective way. This was and is the essence of philosophy. They uncovered invariant relationships at this level of abstraction and it is this discovery that intrigued Socrates. He playfully introduced the concept through the questions and discourses he had with his students amongs whom was Plato. Which one is reality? The sensory experiences or the meditative reflective experiences that reveal these invariant forms?

There is no answer. The listener must decide for themselves, but their decision will affect the whole course of their lives!

So Justus and many of his contemporaries who studied geometry through Legendres redaction found many logical inconsistencies not found in the Stoikeia, but introduced by Legendre 's redaction, which combined mechanics with the philosophical abstractions.

The problem with modern multiplication is that it was decided to base it on number bonds at some stage in the late 17th early 18th century. The renaissance men also failed to grasp multiplication from the quadrivium or Trivium in the seats of higher learning. The meaning of the Arithmoi was lost in multiple translations, and the consequence was the make do make piece called number.

So it was as if n a dream that Grassmann realised that the dynamic construction of a rectangle and more importantly the parallelogram  was the foundation of multiplication. He imagined the two required sides to be active and dynamic and they pushed forward past each other to create a parallelogram .,they also rotated relative to each other to generate different parallelograms. The crucial role of parallel lines in creating these multiple forms was clearly apparent.

 By constructing a parallelogram Hermann saw he could construct multiple copies that tessellated space!  That was a vision of an extensive measure that was fractal . The pattern of multiplication spreads out into space a one side by the other side.  It also resides in the single constructed parallelogram.

From a point A the two segments fan out to B and C. Through B a parallel line to AC is constructed, the through C a parallel line to AB is constructed, they meet at D and a form is made ABDC

AB x AC  = ABCD

Not very revealing until a third Strecke is introduced into the plane . For simplicity sake let us call it AD

Now the behaviour that makes this work is the proposition that says triangles on the same base and between the same parallel lines are DUAL( isos) . It was also generally known that parallelograms with the same constraints were dual. Hermann realised that if he made the combinatorial process of multiplication represent the construction of parallelograms then he could use this dual property with different looking parallelograms on the same Strecke !

So now considering say AB as dynamically extending along the same parallel line as he first constructed, he knew that knew that if AC was fixed then all these different forms were dual! And similarly for AC  . The point D might move further or closer so that the parallelogram at a went from acute to rectangular to obtuse, but ABCD would still be the same or dual.

It is important not to jump to the conclusion that we are talking about area. The area is not defined. This is the shape we will utilise as a Metron to count space. It is the notion of counting space that we have lost by losing contact with the Arithmoi.  The word area is derived from the notion of counting monads in an Arithmos.mthe original Monas is this constructed shape.

Now we are restricted if we think Arithmoi were just rectangles. They could be any pattern that tessellated, and even those that did not tessellated perfectly were used. The best concept I have found is to think of mosaics.

The third Strecke allowed Grassmann further insight he decided to use this to define the rules of addition and multiplication. In the parallelogram the Strecken were symbolic of every Strecken in the constructed tessellated space. Therefore a Strecke which was located at a point H say could be considered as brought into or projected into the same parallel Strecke in the defining parallelogram. Similarly what was demonstrated in one parallelogram was applicable to all .

A parallelogram construction must have two Strecken fanning out from one point. They do not converge to one point. That signifies construction so that AB. AC means construct a parallelogram whereas AB . CA is undefined.

The introduction of the contra concept allows this second one to be defined as well as BA.CA the converging case.

If a Strecken follows another through a common point then this is the law of three points situation, so the parallelogram contains two different law of three points that result in AD as the identity.

AC + CD does not = AB + BD, but they both have the same identifying Strecke . Grassmann overlooked this and thought he had commutativity. He clearly does not but at this stage I cannot quite explain what it is. It is certainly some kind of projective or reflective duality. I even think that Klein named it a glide reflection but I not certain

Emboldened Grassmann used the third Strecke to demonstrate distributivity of the new product and this is when he received the greatest shock?

AB = –BA
AC = AB + BC and CA = – AC = –( AB + BC)* = –(BC + AB) = –BC + –AB = CB + BA

Note the * on the bracket. Something else besides normal negation is going on and it seems to involve this anti commutativity through and through.

Finally AB can identify AD + DB and AC can identify AD + DC

The product AB.AC = (AD + DB).(AD + DC)

Assuming commutativity naively

AC.AB  = (AD + DC).(AD + DB)

These should all cancel but they do not!

AC.AB –AB.AC  = AD.DB – DB.AD + DC.AD–AD.DC

The first pair are legal constructions so they cancel, but the second 2 pairs are not legal constructions.
Converting them to legal constructions we get
0 = –DA.DB – DB.–DA + DC.–DA– –DA.DC

0 = –DA.DB + DB.DA – DC.DA +DA.DC
Carrying the signs through transitively creates the problem the right hand side reduces to 0 , but the constructions were changed to achieve this so
AD.DB= DB.AD = – DB.DA = BD.DA
AD.DB= –AD.BD = – BD.DA =DB.DA
AD.DB= –AD.BD = DA.BD = –DA.DB
The switching of symbols is confusing and misleading when mixed with the rules for signs. But he was determined to make it work so he eventually concluded that
AB.AC had to be made identical to –AC.AB

This ensures that everything coomutes to give zero but obscures what is really happening, and it is Cayley who pins it down in his matrix algebra and multiplication. Transposition is key to the difficulty, but by grasping the nettle and defining ab= –ba Grassmann captured an important part of the solution. The anti commutativity and the sign change crops up again and again.

Grassmanns product was therefore confused and that allowed him to make connections he might otherwise have steered away from.

The complexities of creating a intuitive and useful notation or terminology are hard to express, but it should be clear by now that it is a very messy and arcane business. Much is assumed, much is neglected, and much needs to be refined by applying the notation and ideas.  Grassmann took years to cobble together a utilitarian notation. It was not as smooth as it is portrayed now. There are many logical flaws and holes, but he had astrong framework in Arithmetic to guide him.

His notation can lead to inconsistencies. However he responded to this by setting bounds for applicability. Gradually what emerges is many methods specific to certain spatial problems , but all methods initiated in a common group and ring theoretic framework.

Thus he did not create a mathematics,,but rather he created a methodology of analysis and synthesis.. Certain tools were more useful and evident than others , some were strictly limited to a class of problems. Working in the subject of geometry he single handed lay created by his methods and by trial and error a set of handling labels and mathematical entities thst were very useful. This was the lineal stretchy things! The Strecken

He had a law of 2 and 3 Strecken which encapsulated summation and multiplication in an extensive sense. He had a difficulty which he resolved by defining Strecken multiplication as construction of a parallelogram, but it was hazy with regard to commutativity. He defined a property or commutativity to cope with most of the issues in that part of multiple firms of Strecken . He intuitively felt that direction of the constructed forms changed in commuting the factors, but he did not know why. It just seemed to be a recurring theme.

,but he went on to demonstrate some quite remarkable proofs and to simplify many presentation. His method was not flawless, but it had very real consequences in terms of computation.

This set of affairs evpxisted with the roots of unity, where the foundation was flimsy but a lot of great results came out of using what little was known certainly, even if it was" false".

The anti commutative rule hides many years of work trying to figure out the flaw in commutativity. Their are many accommodations to the simple rules initially laid down before it achieved a secure role in Grassmanns methodology.

Grassmann recognised he needed fellow researchers, and critics, because the task was too great for him.

It occurred to me, while trying to step out auseinander treten and ineinander treten with my feet, grasp what  Grassmann was saying  about the Senkrecht Strecke in the next important stage of his arithmetic, that the stossender Strecken may not just be fanning out from one corner as I imagined. These parallel lines would project out of every point of the defining Strecken , principally the three points that define the two Strecken A,B,C.

From A would project AB and AC . From C would project CD, and from B would project BD. these Strecken would thus be stossender, crashing into one another at the points of intersection, the 4 points A,B,C,D.

In addition we would have
AB//CD,AB=CD
AC//BD, AC = BD
And we could write AB.AC = CD.BD
And  AB.BD = CD.AC

Thus in the parallelogram direction and quantity of magnitude are inhered into the formal labels.
Going back to the law of 3 points I note Grassmann makes a point of apprehending the quantity of magnitude and the direction of each Strecke.

The equality sign is the misleading sign. The multiplication concept is not being extended it is being nalysed by Grassmann prior to extension.

If the forms above identify construction of a parallelogram, then the = should be replaced by —> to make a construction method.
AB.AC —>CD.BD

would then indicate that the sides are projected onto the following sides in that order by the parallel line projection.

Similarly
AB.BD —>CD.AC

Is a construction by parallel line projection. It is clear that the direction of the second projection has been reversed relative to the first construction method.

Commuting the 2 sides  should produce the same construction. We are not changing the orientation of the Strecke or their relationship in space. We are simply changing the order of notation

 
AC.AB —>BD.CD

It is clear that this projection maps divergent Strecken onto convergent Strecken

And

BD.AB —>AC.CD
This is glide deflection  mapped onto another glide deflection although the first structure is fractured by the commutation. But the non commuted structure mapped a glde deflection onto a fractured structural operation.

There is a sense in which the commutation has reversed some aspects of the construction process, but to define this Grassmann placed a third Strecke in the plane of the parallelogram set as structurally defined by these multiplication definitions..

Then he introduced the summation definition in the law of 3 points using the sides of the parallelograms to identify Strecken relationships.
To keep it simple  let AD be the third Strecke.

Now instead of construction forms being disallowed we can interpret them.

AC.AB –AB.AC  = AD.DB – DB.AD + DC.AD–AD.DC

The constructions give a pair of parallel lines, a parallelogram ith in the original and 4 half completed constructions, whichever commutation of the Strecken is chosen

This shows that in the parallelogram the combinatorial form of multiplication is commutative as before, but when we relate to the triangle we get a anticommutative result!

AD = AB + BC

AD.AD = (AB + BC)(AB + BC) = AB^2 + BC^2 + AB.BC  + BC.AB

Here we get 3 sets of parallel lines squared and 2 parallelograms.

Grassmann deliberately chose to make the parallelograms cancel! This was the beautiful result he was after, Pythagoras theorem. To get it he had to define anti commutativity,

I do not think he did it lightly , but knowingly and with trepidation. But as I have found it does not affect the commutativity of the parallelogram! By making this definition he had both the Pythagoras theorem, and the commutativity in the parallelogram where he needed it for calculation.

We can now distinguish the " fan" product from the " wedge" product. The fan product produces a closed construction of a parallelogram, the wedge product an open construction of a parallelogram , and these extend outside the original parallelogram. By adopting Grassmans definition we can swing the open ones inside the frame of the original so they are completed.

The fan product and the wedge product are isolated from the combinatorial product of constructing a parallelogram ABCD

Fan product
AB.AC

Wedge product
AB.BD

The fan product constructs the parallelogram by converging the parallel lines to point D in the projection by parallel lines. The fan product is defined only in parallelograms.

The wedge product constructs the parallelogram only by one of the projections being in the negative sense of the projecting parallel line this creates anti commutativity, but it has to be defined in the triangle form, where it creates the triangle as a Pythagorean form and cancels the parallelogram forms, by definition, not by overlapping. The wedge product can be defined for the triangle and for the parallelogram. In the parallelogram it does not affect commutativity, but it does introduce sign into the form.

This was not obvious to Grassmann all at once. By trial and error he refined his concept of summation and combinatorial product, and extended it to apply to 1 dimensional constructions and higher dimensional constructions. The wedge product suddenly defined a whole set of arithmetics of dimensional forms in a way that the fan pricy did not.

However the fan product came to have a role in Clifford algebras as we may get to later.

The next major development for his arithmetic was division.

He was working on tide calculations using trigonometric ratios, when he came across the hyperbolic functions. The revelation was that these functions were tabulated using the right triangle.

The relation of many functions to the right triangle is overlooked in the general introduction to graphing. In fact Descartes in his geometry gave no special position to the right triangle in analysing graphs, developing a generalised coordinate system. However the Greeks used the right triangle as a fundmental building block for relating curved figures to the circle, and rectilineal forms to the circle.

Bombelli is perhaps the first Algebraists to use the carpenters set square extensively to solve binomial and trinomial problems and then to extend to the quartile and quintic solutions, and sketching out how to find the roots of polynomials of degree six!

It was Wallis who simply fixed the axis orthogonally to gain immediate and permanent access to the Pythagorean formulary in deriving polynomial equations. He thereby showed the common form of the Parabola in its simplest expression, losing a lot of geometrical information but gaining the ability to formulate the conic equations.

The hyperbola therefore was defined as a function of the ratios in the right triangle, this meant that the exponential function could be mapped by right angle data points , and in so doing the link between the exponential and the hyperbolic functions was revealed. The ratios were similar to those of the circle and thus the terms Cosinus and Sinus were initially used . Eventually cosh and sinh was preferred

Grassmann realised the power of a vertical projection as opposed to a parallel projection.this gave him axes to pre calculated tables and meant he could always evaluate his algebraic foms. It also gave him access to stepwise rotation as the Strecken diverged by dynmic rotation. This he utilised in his Ebb and flow calculations, giving him access to swinging pendulums and other periodic motions. Finally he realised that by the Strecken projecting vertically onto each other or the Strecke extended he could define a new Strecke and the Strecken together could construct a smaller inner parallelogram, he had a n method of division and a set of tables to evaluate that division or fraction or ratio.

So he defined division of Strecken, particularly in an example to do with  hyperbolic  ratios. From this definition he was able to derive the square root of –1 as a division product of Strecken.

Grassmann now had a combinatorial ring based on these lineal Strecken and he used this confidence in the symbolic arithmetic to develop the lineal stretchy thingies as a powerful new geometrical language.

He needed help to further his research and development. He pleaded for help, but for -7 years non came. His book the Lineale Ausdehnungs Lehrer was eventually burned to provide warmth or damaged in storage in the ware house.

It was only rescued later by his brother Roberts driven intervention.

http://www.youtube.com/watch?v=mVd202X-2uM&feature=youtube_gdata_player

The Grassmann inner product is different to the usual inner product as we shall see.
Nomans establishment of division clearly shows the role of ratios in its description. However the real history of division is not as notmn has described it. This is a modern approach which does not recognise the fundamental role division played in the definition of the Arithmoi.

Flicking through an online copy I actually found some figures for the first time!
https://archive.org/stream/dielinealeausde00grasgoog#page/n315/mode/2up

http://www.youtube.com/watch?v=6XghF70fqkY&feature=youtube_gdata_player

So far, this is the best introduction to the fan and wedge products as I call them. The transitive distribution of the scalar over the wedge product was first brought to my attention here. Norman imply defines it and it looks different because of the wedge.. In fact it is a normal consequences of commutative  switching and associative bracketing.
2(AB)=A(2B) =(2A)B

It is the spaciometric significance of this switching which is fundamental, and the fact tha we have to define scalars and other entities to fully apprehend it,

Thus AB=0 is not a number product!

When the Grassmanns extended arithmetic to geometry symbolically they kept the arithmetical signage for the different entities. So the fan product and the wedge product look the same, but the geometry is different. Defining them to both produce the construction of a parallelogram is not to say that Grassmann ignored the differences. In fact the anti commutativity shows that he did not.

The fan product applied to three points and their 2 Strecken actually produces a parallelogram, but Strecken addition is defined by the wedge product. The wedge product produces a parallelogram only if the Strecken anti commute. This means that I can define the third Strecke by the wedge and then square the third Strecke .mthe combinatorial table produces combinations that have to be defined by the geometry and that's what gives anti commutativity. This does not happen with the fan product because no summation is defined for that configuration.

However, the fan structure has a use in Barycentric coordinates. A combinatorial form is defined by taking fractions of the fan Strecken such that they sum to 1. Effectively the fan structure is replaced by the wedge structure but only in the case where the vector scalars are these eigen values, hat is fractions that add to one (eigen meaning one single thing) this version of combinatorial Strecke picks out individual points on the line  given by the wedge sum –a + b.

By adhering to these strict formalisms Grassmann demonstrated a method of notation that was safe and meaningful. The many pitfalls of indiscriminate use of these symbols were well explored by Grassmann.

Most people feel that numbers are natural. They do not realise that numbers are a strict formalism which must be adhered to. Thus some procedures give the correct answer even when he formalism is completely shot to hell! It's a tough call when you are told your answer is wrong evn though numerically correct. The reason is the unsafe formalism used.

However I learned that Grassmann carefully distinguished his elements and how they sequence and combine, and then extended them to lineal combinations of products and summations!

Note the term lineal combinations has nothing to do ith linear polynomials. Linear polynomials have at most 2 terms. Libel combinations are combinations of lines( Strecken ) and can have any number. Count.

Because the fan product is so point specific, the Barycentric combination of the Strecken is not identified with another Strecke even though the word line is used.,strictly the Barycentric or fan structure focuses on points. The wedge product identifies or is identified by other Strecke.
http://youtu.be/2VhU7_R2gy8
Several definitions of signed area follow on from the wedge product. In particular the surprising last definition . The cyclical nature of the vectors , and the hidden fan definition make it work by negating each of the 3 vectors in each definition of the fan. This is again due to the restrictions in how we may define a vector component wise and the anticommutative rule in a triangle wedge product.
This is also a powerful link to rotation in terms of the cyclic direction.

All these formalisms Justus Grassmann implemented in his Maths course throughout Sczeczin.in 1820's.

http://www.youtube.com/watch?v=132amJvoLpU&feature=youtube_gdata_player
Now the word vector actually was coined by Hamilton in relation to his Quaternions. Grassmann kept the reality of what he was dealing with clear in his mind:Segmented lines in parallel formation. But the common meaning of stretch also he retained, and the dynamism of the stretch, however it is done, the difference between a point and a stretch from one point to another has to be kept clear.

The broader context of affine goemetry and projective geometry really need to be explained clearly, and Norman does that .

http://www.youtube.com/watch?v=XUfVl6cDuBo&feature=youtube_gdata_player

At last the definition of  Strecke in symbolic arithmetic. It is quite hard to explain how difficult it is to get a fundamental definition of the Begräntzte Linie!

The system Norman establishes in his Maths foundations series is highly recommended by me. It is not perfect, but is the most accurate system devised on arithmetic symbols. The fact that Notman defines a vector as a Strecke shows his intuitive understanding of Grassmanns thinking.

Because he thinks like Grassmann he makes some mistakes in his thinking, like Grassmann, but don't we all!

The fundamental difference between Norman and Eudoxus is Norman constructs the Arithmoi using a number metaphor, while Eudoxus starts with magnitudes and isolates them by ratios or logos. Then dealing with fundamental processes of division he analyses the structures and relations of the ratios. When this is done he can construct syntheses of these ratios to define multiple , subtraction and addition as combinatorial processes.

The Arithmoi come from these logos analogos relationships as being metaphors for forms.

Standardising form on the triangle and the sector, Eudoxus can build or synthesise complex mosaics and count space! Thus the Arithmoi are these various grid or graph paper patterns that tessellated space perfectly OR approximately. The overriding process was to count space.

Grassmanns Strecken allow him to follow Eudoxus synthesis of the Arithmoi. He did not recognise this all at once, but his tutor was Laplace and Lagrange, as well as versions of LeGendres Geometrucal redaction. His insights enabled him to develop a combinatorial method of analysis and synthesis with wide application. The method was under constant revision and refinement, and do Normans system is the latest in a long line of developments.

The commutative property , so called, of the vector sum is problematic

Suppose AC = AB + BC= AD + DC

Then 2AC = 2AB + 2BC =( AB+AD) + (BC+DC) under the rules for adding components. This clearly does not match up, so the second= sign is invalid.

We have to do a bit of jiggery pokery to get it to work, and that is why anti commutativity arises in the triangle product rules.

I do not see it as a major issue, but it is one to be aware of interns of how we make this method work best for us, especially if we want to use n dimensional bases.

The fundamental issue here is tackled in the imaginary "number " history. Bombelli recognised certain combinations as adjugate and others as conjugate. The adjugate combinations I have called wedge products and or summations, thus the fan products or summations are conjugate, according to Bombelli.

This subtle distinction accounts for much of the misuse of the – sign in elementary treatments of analysis of Spatz relationships. The drive to keep it simple runs rough shod over subtle complexities that show up in the analysis of rotation and reflection. These explain why we have to use 1/2 angles to describe rotations in 3 dimensions and n dimensions.

http://youtu.be/6Sgj8M5TJkk
Norman here defines the current formalism. It is based on only allowing the wedge sum in the fundamental definition of the addition concept. This you will recognise as you must go along before you can go up. This restriction hides the fundamental problem discussed above and seems to give a consistent Algrbra. However, things come unstuck when multiplication products are defined, and nobody tells you why?
The above is why!

Another way of covering over the issue is to make a distinction in vectors. So we define axial or basis vectors. We may even call them primitive or intuitive vectors. I just called them fundamental vectors. The point is tha we do not define are arithmetic in terms of these vectors but in terms of combinations of these vectors, called affine combinations..

This is analogous to Eudoxus not defining Arithmoi in terms of the magnitudes, but rather in terms of the ratio of magnitudes, or the logos. This is how we conceptualise real experiences as abstract entities: we compare. Them and it is the comparison we use to refer to them.
The philosophical issue is deep, but basically the experiences we have are subjective, the definiions on those experiences can never be communicated, and they are tautological, which is not a bad thing. However, it is better if two experiences are compared and contrasted, this means each party in a communication can hold his own experience in comparison to a common experience, the common experience is often considered to be objective, but that is merely a definitional convenience.

Now the common experience becomes the means of communication.

Simply if you have a bed and then I try to describe it to someone, you will be amazed at how difficult it is yo communicate your experience to someone who also has a bed, but not your bed! His comparison between the lingual reference and his experiential reference only partially helps. However if you both stand by the bed then the subjective experiences are different, but the comparison is made on the same object, even if from different perspectives( and that is another issue!).  The two can now agree the comparisons are the same, identical, congruent etc, even when they are subtly different!

However, the tautology now has 2 bases which allows for the referential process to gradually change and develop with experience while constraining how it develops.

Few consider the role of tautology in developing our logics, but  Vasil Penchev, a philosopher in Bulgaria drew my attention to this .

http://www.youtube.com/watch?v=bmEM75lTLes&feature=youtube_gdata_player

In this context Norman asks what is a circle. Turns out to be a proportion of 6 terms. This is twice the number of terms in a similar triangle proportion, because 2 specify the centre and the third the radius  in terms of quadrance. The first three terms are quadratic, that is quadrance forms.

To understand the full context you need to get book 11 of the Stoikeia.,then the definition of the meet and the join of points and then finally the quadrature of the circle..
http://youtu.be/ucczpxEySno
http://youtu.be/8rkrymhphMQ
http://youtu.be/B-5EVtEOSAU
http://youtu.be/huR1yKwx-DM
The discussion about the radians is a side issue. The quadrance and spread have  major applications in all geometries and that is its significance.

Contrasting the proportion for a line with that of a curve is highlighted by this video
http://youtu.be/eMt09NN1WOE
In which it is noted that circularity simplifies the descriptions, and emerges as the fundamental property of convex polygons.

Pythagoras theorem and Thales theorem are intimately connected.. The sphere is the goal of the introductory course called the Stoikeia of Euclid. Points and lines , planes and circles all derive from the interactions and relations of spherical surfaces.
Norman went forward to numbers to then go back to redact historical development of Maths. I went back to the Pythagoreans and found the mosaic that underlies all their modelling of space. From this they synthesised their Komodo , which was spherical. It was not just one sphere it was an infinite fractal of spheres in a regression of spheres. Apollonius was the greatest philosopher of the Pythagorean construction of the Kosmos.

It was Apollonius who redacted Euclids Alexandrian text to place greater emphasis on the circle and circle constructions in the early parts of the course. The seemeioon was the central concept of everything. It was where the compass placed , magically drew circles of every size in concentric splendour. It was also where the divider, placed marked off segments of a line which became centres for other circular constructions. And by these constructions circular lines of intersecting circles could be drawn and points of intersecting curves could be identified. These points of intersection defined linear relations some of which were good or straight. These were dual points from mutual circles  of radii that fanned out from two points or centres.

The equipment made the marks, but the marks were used to improve the equipment. So there was a symbiotic relationship between tools and constructions.

A true rule was constructed against the dual points, but for greater distances taut light string or tape is still used. Line of sight was used under the assumption of direct communication with the senses. Optiks and Data apprehended the sensory experience and drawings were made, but measurement was left as an individual choice.

The Arithmoi of the Pythagoreans were rich an varied.. But eventually the preparation of clay tiles standardised the pieces used in mosaic constructions. In space nets were constructed by intersecting string in a knots. These served as points for further nort constructions. It was clear that points of intersection and lines of taut cord were crucial to performing any gematrial or combinatorial reasoning.

Norman introduces the vector plane. This is a one to one correspondent with t is usually called the complex plane.
http://youtu.be/W9-NmJEE3ao

The vector plane removes the imaginary component completely and vectors simply do the job!
He goes on to define vector reflection.
http://youtu.be/HsQ0OSiAaow


Both videos highlight the simplification got by formalism and anti commutativity defined in the triangle product that mimics Pythagoras theorem .

What is missing in this development is the notion of a curve, later centre of curvature and arc lengths are reintroduced in the context of calculus. But I know curve is not fundamentally a product of calculus, rather calculus is a product of curve, as is continuity.

These are intuitively clear experiences that are hard to pin down. They are not meant to be defined by other easier concepts, although it is instructive to attempt to. They are fundamental prototypes that initiate new types of Arithmoi and schemes of arithmetic. Polar coordinates and Cartesian coordinates are an example of this point.

This overview of Grassmann thinking and difficulties really shows how a simple observation can lead to profound insight. The replacement of complex numbers by Strecke  is so fundamental that it is hard to apprehend why we do not start again, as Norman has done. In addition the fact that Strecke are different to points means that the points can remain fixed while the Strecke move ! The example of rotation is usually described as the whole plane moving. In fact it is a vector plane that lies coincident with every point that rotates relative to the fixed plane of points. And the vector plane can get screwed and twisted relative to the fixed point frame. This objectifies the vector plane, so that it is no longer referencing points in the reference frame, but vectors in its plane relate to each other as if the plane was some medium.

The vectors could be thought as attached to a rigid object . As the object moves relative to its initial position the vectors move with it.
If the medium is a fluid, as the fluid moves the vectors move with it .their relative positions describe the behaviour of the fluid.

At the heart of Grassmanns and Normans method of Analysis is an apprehension, conscious or otherwise , of the Logos  Analogos structure encapsulated by the mosaics , that is the Arithmoi.

I am going to explore in my blog the connections  between proportions , polynumbers and polynomials. I will provide a link when I am done.
http://my.opera.com/jehovajah/blog/2013/11/26/proportion-poly-numbers-and-polynomials


The theory of Ausdehnungs Größe is a method of studying forms. The Greeks did the same thing after the Pythagorean conception that space could be tiled in a mosaic. These were called Arithmoi. Using Arithmoi of all forms and complexions they could count space. We can count space.

But there is much more to the Arithmoi than counting space.

The polynumber theory in Normans math foundation are a fundmental shift to a logically consistent Algebra. It's background is explained at this link
http://my.opera.com/jehovajah/blog/2013/11/30/the-polynumbers

I initially recognised these polynumbers as Ausdehnungs Größe, but now I know they are not. The real difference is in anti commutativity. However everything else carries across between the 2 algebras. Arguably Norman's presentation is the better more fundamental exposition of both topics, and makes Clifford Algebras more accessible!
http://www.youtube.com/watch?v=UMWbb-WAPsk&feature=youtube_gdata_player

This and other videos in the series from video 55 actually, build the theory.
http://www.youtube.com/watch?v=KbNYsNWRtzs&feature=youtube_gdata_player

Norman makes a few mistakes and logical errors, but nothing serious, and in later videos he attempts to address the issue.

Of course I do not agree that he has corrected all of them, hence my blog post, but all that is needed is a clarification of the real origin of the ideas. In computing, the unseen element is the object oriented programming. This is as if it were a model of human subjective processing, and that is a breakthrough in apprehending what we are really doing with these things called " numbers"!

For this thread polynumbers are useful to grasp how Grassmann extended his arithmetic through the construction of the right triangle in notation, necessarily imposing anti commutativity to get what he prized: Pythagoras theorem. The consequence was that the commutative law in associative multiplication became " transitive" . In distributiyity the scalar is spread to the 2 elements of thr bracketed sum,
a(b+c)=ab+ac
But in associativity
A(BC)=(AB)C OR B(AC)

Ordinarily the OR would be an = sign but because of anti commutativity  the OR is replace by a =— ! This one change make polynumber arithmetic so called vector arithmetic.

Polynumbers in that regard found both matrix and vector algebras with the specified rule change. From this rule change the so called wedge product can be defined . This definition naturally relies on the combinatorial product of tables which can be cast in the role of matrices.

All these subtleties are tied together concretely and algorithmic ally in this computational Algrbra called the polynumbers.

http://www.youtube.com/watch?v=Sbet5GhqmyI&feature=youtube_gdata_player

This video was supposed to be the clincher. I have critiqued it in my blog. Nevertheless despite Norman finessing the fundamental computational structure behind this, I still would recommend it as indicative of the fundamental power of Normans approach to redefining" Algebra", and ultimately all of geometry and calculus.

The Trigonometry that introduces metric evaluation is not quite properly connected, but I will discuss how Grassmann discovered how to do this and why Clifford was so enamoured of Grassmanns method of Analysis.

The so called wedge product is a modern interpretation of a formalism that Hermann introduced when extending arithmetic to Dir Raume, or 3d space.

The product actually arises from the formalism of Justus Grassmann, and his attempt to place geometry, as exponded by Legendre on a logically consistent basis,  suitable for the primary education of the next generation of the Prussian intelligentsia. I won't go into the ramifications of the Humboldt reforms, but suffice it to say the Prussian Renaissance was pressingly overdue, and French expansionist goals had humiliated the once mighty Holy Romn Empire.

Hermann built on the Formalism of his dad. arithmetic was and is the greatest triumph of the Greek sciences and it was revered as a model of all systems of thought. Algebra was down graded by all but seen as a symbolic arithmetic that generalised certain results and methods allowing formulas nd algorithms to be expressed. Because of Descartes and DeFermat and Regiomontanus the metrication of other aspects of geometry was complete and formulae were in place for all kinds of situations. Consequently Synhetic geometry declined in importance academically placing it below algebra in most evaluations.

Justus therefore felt it was a suitable elementary subject for kids, if he could " iron out" a few logical inconsistencies! Further he could use it as a gateway to an understanding of Arithmetic and also logical thinking that up ports arithmetic.

However, his analysis ran into a subtle difficulty when it came to multiplication. In the end he found himself unsatisfactorily asserting it was obvious from geometry! Look at the geometry of the rectangle and multiplication and multiplication tables are an obvious consequence.

This may satisfy the casual reader, but it is logically tautological! And this bothered Justus a lot. It meant that the key operation in Arithmetic was fundamentally a geometric one, and not a logical or fomal concept. He had to have a prior geometry to define multiplication as it stood. It did not derive from the logical structures then taught in Prussian Higher institutes. No set theory could account for it , rather set theory required it to even exist in its higher forms.A x B where A and B are 2 sets is a clear mimicry of a geometric rectangular form, and Justus could not fathom it. It seemed one had to accept that logic was not the supreme source of all rational thought!

Hermann was thus used to a very very strict formalism which is why he noticed the letter patterns in the geometrical notation and the connection through the symbolic operators + and x.

The other aspect of the formalism was the growing, but still pioneering group theoretical approach to mathematical or rather algebraic analysis.

Hermanns law of 3 points hence the law of 2 Strecken  was the simplest summation form in geometry. 2 lines joined together. I all this the wedge join, just to confuse the issue! The term wedge here means to jam one hard into the other, and is not the same as the word wedge in the so called wedge product.

The summation of 2 Strecken in this specific way , precisely as defined by the order of the letter symbols for the points A,B,C allowed Hermann to later define a labelling identity

AC ~ AB +BC

While this looks simple and definite, once you allow a label to be an identity you can never stop!
So AB can be a label for 2 other Strecken and similarly  BC . The end result is a decomposition of the plane into nested  Strecken summations that sum to the above formula in some sense.

It is in defining that sense that Hermann developed his method of Strecken analysis, and the dynamic nature of Strecken.

The next defining moment was the realisation that Strecken multiply. This in itself is not earth shattering news, but in the formalism of his Father Hermann saw how this multiplication worked in a complex construction of a parallelogram. Unremarkably he could se in the labelling that  a fourth point was produced.
AB x AC = ABCD

This is what I called the fan product, and there was a wedge product version
AB x BC = ABCD

The fact that the D was in different relative positions was not lost on Hermann, but it did not seem to matter, until he combined the multiplication with a third Strecke in the plane. This was to demonstrate the associative rule in arithmetic.

AB x BC x EF. What did this produce?

Pretty soon Hermann was formulating how to construct this result using parallel lines. It is this formal construction tht underpinned his next discovery. By parallel line construction he could project any Strecke in the plane onto the original fan or wedge pair . Then, by the formal construction rules for parallelograms he could construct the various parallelograms in the product. Of course he ended up with a complex synthetic structure of overlapping parallelograms, but to do so he had to institute certain conventions. Using these conventions he attempted to combine parallelograms. It became apparent that these constructed forms would have to be signed if they were to cancel out in any way to substantiate the formulary.

When he moved on to the distributive rule everything started to fall into place and he saw a way to recover Pythagoras theorem from the notation alone. To do this he had to sacrifice commutativity at least in the triangle setting. That is AC x AC

When he did this identity expansion he realised that if he set AB x BC to —BC x AB  then he recovered the beautiful theorem.

At first he was shocked, because it seemed like he had to be wrong, but months of further work confirmed his decision. He not only worked out problems more simply using this rule, it also only made a difference in odd numbers of Strecken. Anti commutativity did not occur in the quadrilateral setting, but signed forms did..

So ther was no wedge product per se for Grassmann. There was just a realisation that Strecken multiplication was anti commutative to get a special result when a Strecke was produced with itself, and that Strecke could be constructed from other Strecke which could be set or defined as the basis Strecke.

The consequence of this was that AB could be arbitrarily used for numbers and Strecken multiplication. However the rules of the 2 types of magnitude had to be strictly adhered to, and not confused. Later mathematicians did confuse the 2 magnitudes and created some problems in interpretation.

These problems surface best when one looks at how Hermann came to the notion of " division".

The so called wedge product now has an interesting behaviour. In the triangle  or rather the law of 3 points
AB x BC can be multiplied by AC .

The construction is complex. What does it mean to construct a form from a parallelogram and a Strecke?
The consistent thing to do is to use parallel line projection. In the plane this results in overlapping forms , but in die Raume it results in a  parallelapioed if the third Strecke is not in the plane.

What one would ideally want is ( AB + BC)x AB x BC to construct the same forms as AC x AB x BC .

It is also consistent to define AB AB as 0 parallelogram. But one could also consider it as a degenerate parallelogram with lineal magnitude 2 AB .or – 2 AB ,

There are other combinations to define before the full import of this multiplication is understood.

As Hermann matured his ideas and notation he realised that the construction process was more simply identified as a lineal projection. That idea was in the air. Des Argues had pioneered a Grometry of perspective , with the aim of delivering realistic art into the 2 d frame. Now ponce let had revived interest in his work especially as it related to industrial and engineering design.

Hermann had entered a challenge to describe the ebb and flow of tides. He needed to pass this challenge in order to progress in his professional development. He read Laplace and Lagrange on celestial Mechanics. In it he came across the description of the half bow of a hyperbolic function. Much of it was written in unfamiliar mathematical,notation, but one idea shone out: the hyperbolic bow could be described by versions of the sin and cosine ratios. It was clear that these ratios , usually associated with angles and arcs were not tied to these measures. They were tied to the right triangle.

The idea was simple: model any curve by a collection of right triangles . Thus any curve could be tabulated like the sign and cosines by measuring or calculating the sine ratio for points on the curve. Equally by calculating the cosine ratio. In this case the ratios were eventually distinguishe by the nomenclature cosh and sinh.

Furthermore, the hypotenuse of the right triangle was a Strecke, and the sin and cos ratios were commonly seen as projections of this hypotenuse vertically( Senkrecht) or horizontally onto the right triangle sides. These projection lines were as much construction lines as parallel line projections , and later , perspective projections.

The nature of the product process was one truly about projection onto or into.


In passing, one projection remained unexplored and that is circular projection. This is where a point or a line is projected by a circle or a set of concentric circles.it is even possible to have a set of perspective circular projections where the circles are not concentric..

We continue with the direct or straight Strecke projections.

Two points A, B can form a product AB which is a projection by points joining A to B. it is either a line or a curve , the user defines its type.

Then we can define 2 lines AB, CD as forming a product that is a parallelogram. But they both are first projected onto a common point. This is called the meet, and then by parallel projection they project by each other to form the parallel sides of the parallelogram. By this means the parallelogram can be constructed anywhere in the plane.

With a third Strecke we can project by point, then by parallel lines. The Strecken act as projection directions before taking up their final position. Also. Parallel Strecken are sketched in where necessary to complete the form. The question is, How many are we allowed to sketch in? What is the final form?

So projection helped to clarify Hermanns thinking, especially about the product of Strecken.

He had made the fateful decision to let ab = – ba where a and b were Strecken acting as factors in a product. The product was constructed and it was defined as a parallelogram. The introduction of labelling and this Strecke sum or wedge combination led to a potential to recover the Pythagorean theorem, but he had to accept his new rule, mentioned before. Up until this point he had no division process and his method and Analysis was full of holes. He could use it to make incredible calculation streamlines and find general solutions, but only in specific cases, and in an as yet unorthodox manner. His arithmetic extention was not transferable to others.

The notion of a projection as a general construction process streamlined his understanding, and brought his ideas to an extended range of situations. He could now consider points as multiplying to project straight lines, and point sums he could properly define as the Schwerpunkt a point that counts connections to other points and Schwerpunkt.

The case of the right triangle was not general enough. It only really worked for rectangles.or Strecken at pi/2 radians to each other. There were 2 cosine laws that extended the Pythagoren theorem to general triangles . There was a direct connection to his decision to accept anti commutativity.

The term 2abcos( c ) resulted from the combinatorial product of adding a right triangle or subtracting a right tiangle to the general triangle to extend Pythagoras theorem to the side that is usually called the hypotenuse.in this case it was one of the Strecken determining the construction of a parallelogram. This triangle that was added to the general triangle was constructed by projection vertically. This is called in some texts dropping a perpendicular. This introduced the trigonometric ratios into the algebra and in so doing introduced ratios and ultimately division.

One important formalism that created this confused inight for Hermann: the horizontal layout on the page. AB x BC was laid out differently to BC x AB ! The first Strecke  was always the horizontal one! Consequently changing the order of the Strecken as factors changed the construction on the page. Subconsciously the constructions looked as if they were the negatives of each other in some way.

The projection representation of construction revealed that the real negative was associated with the trigonometric ratios and especially the cosine ratio. This removed some intransigent inconsistencies in his theory and Analysis thus far. The product could now be consistently associated to the trig ratios and the constructions guided by these ratios.  The anti commuting terms have to cancel, add positively or add negatively to be of most appropriate use, and this splits the method into 3 algebras. In one the anti commuting terms sum to 0, in the other 2 they sum to 2  or–2 or even 4 or –4. But the ratios provided a resolution which made the angle measurement unnecessary to resolve these decisions.

Hermann, in the law of 3 points or 2 Strecken recognised a summation of Strecken. In the law of 4 points or 3 Strecken he recognised the multiplication of Strecken and the distributive rule of multiplication, but also the rule of Strecken substitution or labelling. From this strong sentiment he actually concluded rashly that

ab = –ba.

This is the anti commutator , or at least its genesis. You might wonder why I say rashly, if you are used to the shock of this statement. Well that is because Hermann was deeply concerned by this conclusion. It put him off his dinner for several months!

Eventually he came to accept it, and this is why we have a so called vector calculus and algebra. However it is important not to get ahead of ourselves. Hemanns algebra differs from much of what is described as vector algebra, and that is because Hetmann never gave up on Strecken. In other words hemann was always trying to do synthetic constructive geometry in a more symbolic arithmetical way.

Later, hermnn came ino contact with LaGrange et al and projective Geometry. This refined nd corrected his views and lead him to redact his own ideas. The anti commutator came about When he realised that
If AC is set as the label for AB + BC then this sum can be substituted for AC in any multiplication.

In addition AC x AC is the product of 2 sums  or the binomial expansion of the substituted sums. This binomial expansion recovers the Pythgorean theorem precisely when AB x BC  = – BC x AB

But for the general triangle a more complicated version existed caled often the triangle coine law. This requires


AB x BC + BC x AC = 2 mn cos (ß)

Where ß is the metric of the angle at point B and m ,n are the metrics  of AB and BC .

This is the anticommutator in its general form written in Strecken notation.

Now you may be able to see how Grassmann intuitively made the correct decision .  When ß is pi/2 then AB x BC does equal –BC x AB! But otherwise it has the value of 2mn.cos(ß)!

The other trigonometric ratios also applied when projecting Strecken vertically( and horizontally).

The trig ratios might be thought to be exclusively about circle metrics, but his investigation of the hyperbolic function and curve opened his eyes to the more general, encodoing use of these ratios. We might think thar AC projected vertically onto another Strecke AB
 is AC cos(A) in the AB Strecke but that is not necessarily the case  it could be AC Cosh(æ) where æ is the area between the hyperbolic curve and the Strecke AC ! That is measured in some metric.

This projection   was dividing  the Strecke projected onto . Thus division was introduced into the arithmetic of Strecken in the form of trig and hyperbolic tables.

Few realise that Hermann introduced also the division or the product of 2 Strecken. This he called the inner product.

The Grassmann inner product is constructed by 2 vertical projections . One Strecke is projected onto the other and vice versa. The 2 divided segments of each Strecke are Strecken.,using only those projected Strecken which share a common origin a parallelogram is constructed WITHIN the larger parallelogram constructed from the initial Strecken.

The initial Strecken construct the exterior product, the projected Strecken construct the interior product.

The 2 products are related thusly

AB x BC  is the exterior product
AB x BC x cos ^2(ß) is the interior product.

They are in the ratio to the square of the trig ratios!

The significance of the last post perhaps cannot be overstated. However, if you have been following Normans redaction of mathematics you will recognise the square of the trig ratios in his definition of spread and quadrance.  What you may not know until you watch his universal hyperbolic Geometry videos is the quadrance and spread also define the hyperbolic trig ratios.

For this reason I recommend Normans work as the most fundamental implementation of the Grassmann method and analysis.

In none of these posts was it my intention to say Grassmann or Norman have got it completely correct, rather it is to point out that both of them enter into this research and work to set mathematics on the soundest footings they can construct, and they constantly attend to the foundations to make them secure and watertight.

Hermann never felt able to publish untill 1844 when he believed he had achieved something that was robust and secure. When that was assured his actual published aim was to call fellow researchers to his aid. The work was delicate and tricksy, and he could not do it all on his own.

What he did build  however has changed the paradigm of mathematical modelling in physics and may have influenced chemical notation.

I now propose to work on the text of the Vorrede and the Einleitung of the 1844 version, and would welcome discussion and meditations on those to be posted here, as well as any comment or question in the spirit of this enquiry or even good humoured comment. By god we need some! Lol!

I shall work from this text version
https://archive.org/details/dielinealeausde00grasgoog

Before I do I have to mention hermanns concept of a basis. Hermann, I think under his brother Roberts influence does not define a basi until 1862, when it is defined as an Rinheitung, a collection of monads or unit metrics. In 1844 the concept naturally arises from the law of 3 points. There we see the tru nature of a basis. The two Strecke in the law of 3 points are a journe broken into 2 stages. Thus these 2 Strecke can be used as analogies for any 2 stage process. In Cartesian geometry the ordered pair, or the sequenced pair or more tellingly the logs pair( ratio) are set to identify a unique point or position in the mosaic. They are said to be defined by the axes, but in fact this was an idea introduced by Wallis when he standardised the axes, and made his conic section equations take their iconic form!

Descartes simply took any 2 lines in the problem as fixed relative to each other, and defined all other points and lines relative to these 2 distinguished and fixed lines. In that regard Hermanns 2 Strecken were precisely the same system.

These 2 Strecken were the basis of the description of all other Strecken. Thus by working in ths way all evaluations depended on just 2 measurements.

The lines or Strecken were said to be linearly independent, geometrically this is obvious because the lines are in different orientations. The number of lines chosen depended really on choice and how much work you wanted to fo measuring.. The number of lines was never limited by Herrmann. He is united as saying no longer let 3 dimensions be the rule for space!

Because of real numbers I think I was given the notion that linear independence meant that no  " vector" in the basis could be expressed in terms of any other combination of the set of basis vectors. However Hermann was not that strict, and often worked with rational or integer reference frames not real. .

The basis Strecken then come from the notion of the law of 3 points. A representational third Strecke is added to hide the basis Strecken and these are treated as if they were Strecken themselves, not just representing a combination of 2 or a 2 step process.

This led to a distinction between basis vectors and generalised vectors, but hardly any one really explains the important distinction. The flowery set language actual obscures what it is talking about. Basis vectors are the metric vectors for evaluation ! Without them no representation can be pinned down to any place or value!

Representation theory also takes its lead from Grassmann!

Grassmann mentions in a later edition of the 1844 Ausdehnungslehre, that the heuristic process of constructing the lineal algebra was so instructive he would not change a word in the new edition. Instead he heavily annotated it.

What Grassmann showed was the way to use ring theoretical laws to construct an extension to arithmetic. In fact, any " object" that satisfied the laws of the AusdehnungsLehre , forming a ring group structure can be used as an arithmetic extension.

http://www.youtube.com/watch?v=fEWj93XjON0&feature=youtube_gdata_player

Reviewing Normans excellent introduction to the wedge product a v b I am aware that the treatment I have exposited so far has been confusing in the sense that one is always drawn back to the " dark side"! The wedge product is inherently based on the associatve laws of multiplication. That is precisely the discussion and the exposition. So why make these distinctions?

The distinctions answer the question: where does multiplication come from? The answer is it comes from the geometrical set up of Strecken when they are being used to construct a figure on the page or a form in space!

While, in the exposition it looks like we are mimicking associative multiplication, and using the terminology found in associative multiplication the point being made is: actually, stop and think. Divest your mind of what you think you know and think this way for a few minutes: What If the symbols used to describe associative multiplication really come from describing Strecken?

The representational theory which underpins this meditative thought process links back to the form of the mosaics of the Pythgoreans, and the format of the Logos Analogos.

When Norman writes the primitive or basis vectors in the form of a matrix and then isolates the scalar factors as coefficients, he is creating a coefficient matrix which is called a Cayley matrix or table. But much simpler than that or rather prior to Cayley this layout was done by the Pythagoreans on their mosaic patterns!

Revealing this prior history by placing a wedge between the symbols is not only a small symbolic sacrifice it is the key to linking the seemingly disparate processes in mathematics into a consistent and congruent whole based on the combinatorial nature of all synthesis. It reveals the fundamental role of sequence and sequences, of order and arrangement, positioning and relativity, Before we get to summation! Aggregation and disaggregation, combining into bundles or groups ,splitting into a multiple form, all of these processes precede counting and summation.

The wedge is a simple symbol to remind one not to ignore these underlying processes in our synthetic productions.

Setting a v a to 0 is but one choice. The algebra and the arithmetic derived from this choice happens to match our consensus arithmetic very well.
But what algebra and arithmetic do we get when we set a v a to 2a?
The dot product , so called, sets a v a to  a^2. This is the basis of the Clifford Algrbras.

The Clifford algebras construct or synthesise a set of algebras based on a very complex product! Because Bill Clifford wanted to combine Quaternions with Grassmanns Lineal algebra he used a synthesis he found in Grassmanns critical review of the Quaternions. In that review Grassmann sets out a product that is based on his notation that formed a 1 to 1 mapping with the Quaternions. Once he had done that he remarked that Quaternions had nothing more to teach him!

I cannot say whether it was arrogance or just plain fact, but I do know that Hamilton regarded Grassmanns methods as superior to his own, in the sense that he recognised the greater generality of Grassmanns approach.

When I first, ever came across Grassmanns method I regarded it as "cheating"! It was plain that you set the rules to get out what you wanted! It was and is like a game in which one sets the rules, plays according to the rules and gets the outcomes of those rules.

I no longer regard it as cheating but I still think it is a very useful attitude to have! The rules set out the way you want the elements to behave, clarifies everything. What is interesting is what rules work for everyday life? What rules work for science? What rules work for wave theory? What rules work for particle physics? The list goes on.

This is a great intellectual freedom, but alo a theoreticians nightmare. Dirac for example set out rules for quantum Mechanics. Subsequently he obtained results which implied that negative energy was a factor in quantum mechanics. This was not only counter intuitive but deeply upsetting to his colleagues Mach, and Planck. They could not fault his logic, so they browbeat him into discounting his theoretical findings. In any case it was the practice to discount all absurd results of mathematics including the so called imaginary ones.

However, they went further. They actually scoured Grassmann's analytical method for years to find a way to overturn this result by Dirac. They found a method of collapsing vectors to 0 which meant that negative vectors could be annihilated. Then they rewrote Dirac's findings in these terms and got rid of the negative energy. To work, they had to accept creation from 0. This later became zero point energy, and energy from the vacuum etc..

However, Dirac was proved correct, experimentally! There did indeed appear to be behaviours that matched his negative energy. Dirac, at the time preferred to remain in the club rather than walk away from these shenanigans and he lived to see these men impeached by empirical data.

The point of the story is: you can Create any Algebra you want, but the litmus test is does it work with empirical data?

The Clifford Algebras do not always have a physical interpretation but some do. The important thing is to lay out the rules clearly so that confusion does not lead to mistakes. I find that Clifford algebras are presented in a confused way because no one really understands Hermanns work or method.

How I build a Spaciometry is from the mutual spheres that share a displacement from their centres. The spheres intersect in curves called circles bounding curved regions I call Shunyasutras. The distinguished points on these Shunyasutras fom the dual points of the circles . Taking one arbitrary dual point I consider another sphere that is now mutual to two spheres . The intersections form smaller circular Shunyasutras

The displacement of the instrument can now vary the Displacement instruments to pick out all those dual points that form concentric circles relative to the initial arbitrary points that form a circular plane. Using this circular plane I can use a pair of compass instrument to identify all those dual points that lie on a straight line called the diameter of the circular plane.

So now I can start my planar geometry with the right triangle in a semi circle!

http://youtu.be/eMt09NN1WOE

Using Norman's definitions for planar forms  we now set the metric rules for our geometry. To do so se have to identify stretchy things or Strecken. Although a line has been defined , mentally we apprehend by stretches or extensions. This is how we measure!

Now the law of 3 points is applied to any 3 points on the circle . AC ~ AB + BC is representation theory for the sum of the 2 Strecken.

We will use this representation to define the general Strecken, making AB , BC  are basis Strecken .

Consequently we have to define AB x AB , BC x BC and AB x BC + BC x AB
Using representation theory we can set

AC ^2 ~c^2
AB ^2 ~ a^2
BC ^2~ b^2
AB x BC + BC x AB ~ 2 a.b cos (C)

This blogpost by Wolfram builds on the notions of the Grassmanns.
http://blog.wolfram.com/2013/11/13/something-very-big-is-coming-our-most-important-technology-project-yet/

The above representation map is not usually set out because it is not understood. I am very sure Hermann had it in the back of his mind as he progressed his analytical and synthetic modelling..

The model is a 2 dimensional map. It maps 2 dimensional objects into a 2 dimensional metric space or a trigonometric evaluation
 It works because it mas triangles into right triangles. It breaks down any triangle into right triangles and maps them to their appropriate trig evaluation.

The problem was that Hermann redefined= to mean this combinatorial mapping or identification. In his foreword and Einleitung he tries to explain these kinds of " identies" or correspondences. The word Gleichen attempts to carry this meaning. When I get to the passages I will attempt a better exposition. The German for me is not so easy but the intention shines through like a supernova!

The actual mapping is quadratic,

AC ^2~ (AB + BC)^2 —>c^2= a^2+b^2+2abcos(C)

This says that the general Strecken squared is to be thought of in terms of the general triangle formula extending Pythagoras rule. The small letters relate to the measurements of the Strecken what we now misleadingly call the magnitude, and the C relates to the measurement of the angle opposite AC which is in fact the point B!, that is the angle at that point between the Strecken AB  opening clockwise to BC .

There is an alternative form of this map which uses the Senkrecht to the Strecken AB stretched or cut which uses additional lineal measurements involved in the trig ratio for cos(C). This version highlights all the right angled triangles involved in the representational mapping.

From this mapping we can define the 3 exterior algebras and the interior products.

The first case is the primitive AB^2 that is AB x AB

This is a process of construction of a parallelogram by definition. The process is by projection parallel to the two Strecken .,however this is a degenerate case . The 2 projections only produce an extended lstreckn in the same orientation as AB. The construction produces or extends AB to 2 AB

By mapping that should represent a measure of a^2

However, it could be sensible to define this product as the 0 parallelogram. In that case we should define the metric as 0! That means we could define a^2 as 0

This makes little metric sense, but some group theoretical sense. It means that AC ^2 is 0 by definition and that AB x BC + BC x AB has to represent 2 abcos (Pi/2) to retain any sense. . But then if the Strecken are orthogonal, there is no requirement for the squares to be 0 as the 2abcos ( pi/2) eliminates the anti commuting terms and reveals Pythagoras theorem!

Instead of clarifying this twist in the rope, Bill Clifgord decided to create or intercalated the dot product from the quaternion theory, and leave the Grassmann product as apparently only defining a v a as 0!

However Grassmann was more subtle than that. Firstly because it was an extension of the associative laws of arithmetic Grassmann was able to see the representation as a Functional relationship.

By defining primitive units he was able to see the following
AC = ce3.  AC ^2= c^2e3e3
AB= ae2.  AB ^2 = a^2e2e2
BC = be1. BC ^2 = b^2e1e1

He could now define the units as forming what he called a bivector unit and these squares are base on the 0 bivector unit! The bivector unit is a separate notion to a production of Strecken. While the same constructional issue remains in that the unit Strecken construct a line 2 units long, the line is not a bivector by definition.

The product that picks out this 2 unit line can now be distinguished as a special product defined when Strecken are parallel only. This product for Strecken at an angle to each other is different to the product which is a bivector.

This special product is often called the inner product, but this is not the Grassmann inner product. It is the Bill Clifgord inner product. By these means he kept 2 distinct magnitudes in combination, a Strecken evaluation and a bivector. This was how he modelled Quaternions using Grassmanns method.

Grassmann defined it differently. He used a ratio of his outer product and his inner product to distinguish the cases! When the Strecken are parallel the ratio gives a trigonometric value, not a Strecken. The ratio can be set to go from 0 to 1 or from 1 to infinity, if allowed.

Now he arranged his algebras according to the anti commutator.

The valuation For the anti commutator  looks like this

AB x BC = abe2e1
BC x AB = abe1e2
Thus
ab(e2e1 + e1e2) ~ 2abcos(C)
When C is pi/2 then the 2 bivectors can be classed as negatives of each other and
e2e1= –e1e2
Because this sum is equated to a 0 bivector by the distributive law of multiplication. Which underpins bivector summation.
When C is 0 or pi these two sum to represent 2 or –2



There is one other question:what is the outer product? I have assumed it is the initiating vectors, but I am not sure. The outer product could be the parallelogram formed by using the vertical projectors, the Senkrecht construction or projection Strecken. It was mentioned by Grassmann in reference to his exploration of the hyperbolic functions. In addition, Norman Wildberger makes great use of it in his definition of spread.

There are still things to puzzle out, which is why I want to meditate on the text in a collaborative way.

The thought occurred:
Clifford algebras define vector multiplication in terms of
abcos ø + absinø the sum of the inner or dot product and the outer or exterior product. This may be an invention of Bill Clifford?

The contravariant and covariant tensors are defined in an analogous fashion, and this is definitely due to Ricci and Levi drawing on Peano's work using Grassmanns method .

I suspect that Hermann decided to use bivectors  bi Strecken to define the Strecken product
 Suppose A = ae1 and B = be2
Then AB is a bivector and it is defined as
  {abcos^2(ø) + absin^2(ø)}e1e2
Where e1e2 cos^2(ø) is the inner product ( Grassmann) of the Strecken e1 and e2 and the e1e2sim^2(ø) is the outer product and they are bivectors. That is bi Strecken.

The general Strecke a1e1 +a2e2 can be multiplied by b1e1 +b2e2

And factored into these inner and outer product forms
a1e1 b1e1+a2e2b1e1 + a1e1b2e2 +a2e2b2e2
a1b1e1e1+a2b1e2e1 + a1b2e1e2 +a2b2e2e2
(a1b1e1e1 +a2b2e2e2 )(sin^2(0)+ cos^2(0))+a2b1e2e1(sin^2(@)+ cos^2(@)) + a1b2e1e2(sin^2(-@)+ cos^2(-@))

(a1b1e1e1 +a2b2e2e2 )( cos^2(0))+a2b1e2e1(sin^2(@)) + a1b2e1e2(sin^2(-@)) +a2b1e2e1(cos^2(@)) + a1b2e1e2(cos^2(-@))

(a1b1e1e1 +a2b2e2e2 )( cos^2(0))+a2b1e2e1(sin^2(@)) – a1b2e2e1(sin^2(@)) +a2b1e2e1(cos^2(@)) – a1b2e2e1(cos^2(@))

(a1b1e1e1 +a2b2e2e2 )( cos^2(0))+(a2b1 – a1b2)e2e1(sin^2(@)) +(a2b1– a1b2)e2e1(cos^2(@))

Although the angle measurement direction does not affect the square of the trig functions it helps to explain why switching the unit Strecken is viewed as a negative of the initial Strecken position. It is a purely formal process rule.

The result is 2 types of inner product where  the  vectors are parallel and two bivector products one is an inner projection bivector product , the other an exterior projection bi vector product.
How we evaluate this is now up to our needs, but the coefficients, as Norman points out are "area count " coefficients and that is a useful result for all sorts of reasons, and that is why the anti commutator is retained.

The more I study these Strecken and bistrecken forms the more I see the ambiguity as flexibility . The law of 2 Strecken allows one to develop a representation theory with simple aggregation . The law of 4 points  including a third Strecken allows construction of forms and trigonometric multiplication. This allows an extensive counting of space if the bi Strecken are defined as producing parallelograms, or a calculation of areas of inner parallelograms.

The forms can be used ismply to layout the calculations inherent in the scalars or coefficients, in which case the Strecken and the bivectors do not have to have a geometric interpretation. . In certain cases bivectors degrnarate into straight line of a combined length of half a perimeter of a parallelogram. That may be geometrically useful but may not beed to be used.

Bi projection defines the inner and outer products and these can be used as a variety of given Strecken and used to form a reference frame connected to the initial Strecken.

It just seems to settle my intuition when I think that Hermann expected it to be a network of related building blocks which have to be chosen and put together with rules defined by the synthesiser.

You cannot think that Mathematics has some god given rules with Grassmann. Every mathematics is co structed by individuals or groups defining the rules of the game! That these rules can constrain a system that matches the complexity of existence is the intriguing part.

I now have to distinguish between the inner projection nd the outer projection.

Take a Strecke F the inner projection is a Strecke Fcos ø and the outer projection is a Strecke  F sin ø

These 2 Strecke are at right angles rotation to each other(pi/2).
They form a rectangular basis.
Call the inner projection a and the outer projection b
F~ a + b
ab is a bi Strecken
F^2 ~ a^2 + b^2 = (a + b)(a +b)
F is therefore a fundamental metric of the reference frame and the evaluation Strecke.
For example we could use a unit velocity vector to create a reference frame that allows changing velocity
And consequently develop a vector surface that we can use to construct
A manifold with a flip axis where the original line was placed.
The Inner Product is the bi Strecken abcos^2(pi/2)
The Outer Product is the bi Strecken absin^2(pi/2)

There are many trig relations that may be applied in this context of Strecken and bi Strecken, especially when the inner projection and the outer projection are not onto right angle axes.
http://www.sosmath.com/trig/Trig5/trig5/trig5.html
Amongst these you may recognise the anti commutator but in the context of the argument of a sin or cosine function .
http://en.wikipedia.org/wiki/List_of_trigonometric_identities

This video demonstrates the kinds of things Hermann explored in his mind regarding Strecken and bi Strecken
http://youtu.be/rz8A5l_yn34
http://www.youtube.com/watch?v=rz8A5l_yn34&feature=youtube_gdata_player

This is only one of the rich number of things Strecken have been used for.

This video is a solid introduction to some of Grassmann's thinking. Norman comes closer than anyone, but the commutativity of summation is where Gibbs parted from Grassmann!
The discussion above about the wedge product comes out of this distinction.
 In addition, studying Grassmann's figures reveals even greater differences in thought!
Unfortunately we do not see a full set of figures, so if anyone can post a copy I would be grateful!

http://youtu.be/132amJvoLpU
http://www.youtube.com/watch?v=132amJvoLpU&feature=youtube_gdata_player

I forgot about this lecture by Norman. Much much more careful presentation .

http://youtu.be/TuaInpbbsg4
http://www.youtube.com/watch?v=TuaInpbbsg4&feature=youtube_gdata_player

I do not recommend the following, but link them so you can see how some mathematicians complexify what is essentially a very simple notion,
http://youtu.be/jQ_SMijI9LM
http://youtu.be/K5d6FuQYyEU

I need to see copies of Grassmanns figures at the back of Ausdehnungslehre 1844, please, anyone.
What I have seen leads me to a nice concept of the circle and unit sphere trigonometric ratios underpinning Grassmann's initial notions., but I need to see the figures to meditate further on this point.

In the meantime I hope to start work translating very soon, and would appreciate all feedback on the translation.

It won't be an ordinary translation but a meditative one. Anyone who speaks, writes or thinks German can contribute!

https://archive.org/details/dielinealeausde00grasgoog

It seems some pages are missing so I recommend the following later reprint with annotations

http://www.uni-potsdam.de/u/philosophie/grassmann/Werke/Hermann/Ausdehnungslehre_1878.pdf






Some comments on applications to physics
http://my.ilstu.edu/~lmiones/Math-Phys-Seminar/M4PS2-EC-Introd.pdf

"Grassmann, Hermann Günther." Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com. (December 18, 2013). http://www.encyclopedia.com/doc/1G2-2830904887.html

Without annoying advert
http://www.encyclopedia.com/topic/Hermann_Gunther_Grassmann.aspx

part of the figures at the back

I have made several searches online to see if anyone has posted a copy of these geometric drawings to no avail.

http://www.forgottenbooks.org/
I had forgotten about this resource. If you type in Grassmann you will be able to find readable copies of Hermann and Robert's works.

You will notice that Robert published his own version of the Ausdehnunglehre after Hermann's death . You will also find the 1862 redaction of Hermann's ideas, and see it is a completely different book! It shows the intense mathematical rigour of his brother Robert. That is not to say that Hermann was not rigorous, but Hermann was certainly more innovative and flexible.

Roberts own philosophical viewpoint dominates the redacted version, which was done at a time when Hermann had come to the end of his tether and resigned himself to obscurity. So Robert was able to edit the new version with little real resistance from Hermann. It was only on the relative success of this second book, really only evident after Gauss and Riemann died, that Hermann's hope revived! He then reasserted his authorship by republishing his original book unaltered and heavily annotated with references to the redacted version, under his name.

I have concentrated on the 1844 version because I hate what mathematicians do to relatively simple ideas! We code it in symbols and make the symbols pretty. Then we try to convince others that this makes everything perfectly clear!

One cannot do mathematics without meditating, and one cannot explain mathematics without sitting beside the interested person and chatting and demonstrating.

Norman in effect does all this. His symbolic work is always behind him. To paraphrase  Newton, he is the main show, not his algebra!

This project is really of interest to me because I can read Grassmann's own words and grasp his thought process because he made it accessible.

I have emailed 2 resources to see if they can get me a copy of these geometric drawings.

I would appreciate it if anyone can get hold of a copy of the drawing on page 324 on a fold out sheet. Thanks, especially if we can publish it here as part of the research.

Plus anyone who can translate any part of the Ausdehnungslehre and publish it in this thread would be most welcome to.

I know that Kannenberg has done an excellent translation, but that is not what I am after. Hermann speaks authentically and philosophically not so much mathematically. His words resonate and radiate beyond mathematics. I would like to capture your response to those words after so long a time!

This quote is from the Encyclopedia of Scientific Biography link in a post above.

The whole article is worth a read. For example, Mõbius role in Grassmann's life is made clearer. In the past I have speculated on the role of Gauss and Riemann in influencing the progress of Hermann's work. I found documented evidence of Roberts influence, and this article gives evidence of several others. I find it hard to swallow the line that Grassmann is too difficult to understand! I think that documented evidence shows that established figures just "took agin" him because he would not conform! In today's spin doctoring speech he was bringing mathematics into disrepute!

That said, because of the times certain metric norms are incorporated in his Ausdehnungslehre that could usefully be revised , and by this I mean the sign rules. These rules are fundamental to calculus in that they underpin aggregation and disaggregation, but they also underpin half turns and whole turns around a centre of rotation.

The difference between lineal and linear is also intimated in this article, and that is a fundamental distinction. I also suspected Jakob Steiner was a pupil of Justus Grassmann and that seems increasingly likely.

The " militaristic" or structured nature of Prussian society also supports the concept of the definition of a mathematical function from a social metaphor.

Hermann, like his Father was a hard worker. To have produced so much is testament to the difference that avoiding television and video games can make to ones productive life. On the other hand, most people would not work in this way even in that era, so it is not conclusive that modern entertainment is cause.

AUSDEHBUNGSLEHRE

The second Impression, with regard to text not changed all over.

Print layout by Otto Wigand 1878
Leipzig

Foreword to the first Impression

If I designate the work , this first part which I am handing over to a public readership ,as a by-product of a new disciplin in Mathematics, then such a heading can only be justified by the way the work itself has to have been given to come into being)( justification of such a claim can only come to be given by the work itself )
In support of which, I myself every other justification therefore submit( hammer home!) in describing how I reached these conclusions laid out below. Step by step, I go over the way exactly as much as that is do-able , so that all that initiated this new discipline is presented.
(to bring  to the manifestation of it.)

Grappling with the negative in Geometry gave me the first spur onwards. I made it my habit to apprehend the Strecken ( line segments) AB and BA as magnitudes that sit in opposition to one another, from whence thus proceeds onward that, if A,B,C are three points of a direct line then it is always the case that AB + BC = AC! Wholly so even if AB and BC are drawn in the same mannere, as also when they are drawn in an opposing manner: that means when C lies between A and B. ( drawn in a manner set against one another)

Ausdehnungslehre

In the cases of the last two, AB and BC were not simply apprehended as lengths but their directions were applied as an equal attribute  to them. Through which even, Making it possible that they were perceivable as set against one another!

And thus the difference drew attention to itself ( urges itself on ones attention) the distinction between the sum of lengths and the sum of such  Strecken, in which the direction was firmly applied as of equal importance.!

From here out the uncritical support of these ideas itself gave  rise to notions that the last used terminology of the sum , the last used handles  was not just simply for the case of the last mentioned sum where the Strecken were directed either in the same manner or in a manner set against each other , but also for every Other case set down unswervingly. This can be demonstrated on the simplest cases in which the Law that  AB + BC = AC is true, then it may  be  vigorously held as true even if A, B,C are not laying in a direct Line !

Commentary:
I found this passage about the law of 2 Strecken and 3 points difficult to get a sense of the verb tenses, and more importantly what Grassmnn was thinking. It all depends on Förderung which was difficult to grasp by me, not having read a lot of pre 20th century  literature in German.

However I see clearly the mechnism of the terminology or rather the naive simplicity which is that for 3 points a stretch from A to B  followed by a stretch from B to C brings you to the same point C as a direct stretch from A to C.

This notion is intuitively clear, but it hardly seems mathematical or algebraic or numerical. This is why it was overlooked. Hamilton in his paper on the theory of couples missed this point. Or this stretch. It is the more remarkable because Hamilton caught the notion of a step. And he understood the necessity of assigning complex notations to accord these steps with the then arithmetical norms..

Assignments like this are fundamentally representative theory. But Hamilton missed the geometrical representation of steps by a single resolving line. He missed the general notion of a sum of any 2 Strecken, because he was fixated on Numbers not Geometry.

This particular passage , on the face of it looks like a definition. But logically speaking Förderung means it was purely a bit of wishful thinking! As such it stands naked and vulnerable to being dismissed as nonsense.

What it actuall represents is Grassmnn going ith a notion even when his normal thinking is screaming at him, this is nonsense. He has to vigorously defend his process not to his colleagues but to himself. This is why he defines a law or uses law terminology where none exists. . Here he creates a general law or rule by fiat, or of 2 cases where the "law" can be unequivocally demonstrated. It is the law that makes AC equal to the sum of other 2 Strecken when the 3 points are not collinear.

Importantly, he analyses why we might intuitively dismiss the law: because we do not regard the Strecken we regard the units of length!

Hermann learned to view the lines as symbols with 2 inherent but undefined properties: magnitude and direction.

In fact reviewing this conception which he fostered( Förderung) we find finer instinctive attributes of the line symbol. We also see the logical confusion possible by those who do not use force of will to promote this concept over the normal inculcated thinking about summation in geometry.

The notion of a Strecke as a symbol is also novel. It requires one to not see a Strecke as a construction line between 2 points, and thus of a fixed length period! It requires the concept of a dynamic experience bounded by these 2 points. The concept is enriched by more and more applications, so it becomes difficult to define it by a single example. However, it seems to me that Grassmann meditated on this Strecke and thereby found it was symbolic of so many experiences and processes.
Förderung is crucial to this development , because it describes his meditative approach to the symbolic nature of a well known element of geometry, the segmented line.

It has struck me how close the notions of Förderung and axiom are. The notions of postulation , a begging of privileges , belies the inherent coercion barely concealed in our systems of merit and advancement.

And this is all in the name of the gods truth. For in the past, truth was not known, except it be the actions or words of a god that were witnessed by those who observe. And for many, that a god or fate should survive a victor was trial enough!

But later more sophisticated ways were adopted, in which a contender willingly submit, in acknowledgement of the alpha male or female. Thus falsely assuring them of right and truth, only to be dismayed by a victorious contender!

When , in ancient Greece philosophers chose to debate publicly on many issues, the victor was he or she who won the heart of the crowd. Thus it became admissible to attack the debater's character, irrespective of the merit of his ideas! To avoid such castigations rules of logic were formed in the schools of rhetoric. Whether they were all adhered to I do not know, but ad Hominem was included in these rules as justifiable in the real situations of court hearings and trials of veracity.

It was however preferred to advance an idea systematically and based on clear and obvious connections, or previous convincing demonstrations. . The ground of such an approach was a Förderung, a postulate, a proposition in which one is begged to suspend judgment and be open to the arrangement set forth..

Quite often, these postulates would be persuasive analogies or metaphors, which mimicking an accepted experience would then be used to justify some undiscovered connection. The veracity of the connection was supposed to justify the means by which it was arrived at.

However these days, getting a correct answer does not necessarily justify the method. Other criteria are used to justify methods and much stricter protocols are used. Thus the Förderung has changed over time to make a more rigorous platform on which to develop theoretical and hypothetical notions into the category of "truth" or rather a " true theorem".

In the end it is we who decide by consensus which experiences of individuals we will value and which we will denigrate. Thus we do not do so on any intrinsic truth value but merely on our consensual considered opinion, our consensual Förderung!

Good news.

The New York Library has responded and i may soon have digital images of those Grassmann constructions and geometrical figures.

In the meantime , play with this in the new year, and have a good one!
http://home.comcast.net/~trochoid/TroWithMesh.html

https://mail.google.com/mail/u/0/s/?view=att&th=14349c87686811e9&attid=0.1&disp=attd&safe=1&zw

The geometric images have arrived today . Thanks NYPL !

Initial examination of the diagrams reveals the deep neatness formalism of my youth. The horizontal line was the initial reference. All angles, and thus Strecken at varying angles were measured relative to this initial horizontal. All measurements or compass displacements were measured along this initial horizontal. Thus to switch Strecken meant to switch which Strecke was the initial horizontal. Consequently the angle measure also reverses. What was measured anti clockwise would then have to be measure clockwise to maintain the same relative position. The negative sign in Grassmann's factorisation into biStrecken relates to this fundamental anti rotation.

It is initially not clear if Grassmann did perceive the inner product as I say. The diagrams show 2 types of projection( in fact 3 including the circular in figure 1) and those are parallel  and perspective. The vertical projection is evident as a vertical construction. For example in figure one I believe Grassmann starts with a Strecke ABE and draws the arc BCD. Then Using E I think he marks off the intersections C and D to construct a rhombus. The vertical line CD is a chord to the arc, the shorter diagonal of a rhombus which is orthogonal to the longer diagonal at F

This I think was an initial construction of an arbitrary parallelogram in which the angle at a is arbitrary, and the sides are in parallel pairs. Which can now be arbitrarily extended. This arbitrary Rhombus was perhaps the seed of his biStrecken concept.

A perusal of the index/ contents( inhalt) at the back reveals Grassmann conceived of 5 or 6 products! The inner and outer are only 2 of a much larger concet of combinatorial product. However, it seems Grassmann started with these initial ideas of the sum( which in group theoretic c terms he made closed by representation theory! ) and the combinatorial or constructional product which was not closed but in fact extended the algebra of the sum of Strecken to an exterior or enveloping algebra of the sum of BiStrecken. In this conception it becomes important that anti commutativity is accepted as a Förderung to keep the summation intuitive.

So e1e2 + e1e2 = 2(e1e2) = 2e1e2 = e12e2 = (e1e2)2
But e1e2 +  e2e1 = 0

This zero comes about through the anti rotations summing to 0. Relating this to figure 1 it is as if the construction of the Rhombus shows the 2 parts as triangles  which indeed have their altitude pointing in opposite directions.

Further thought intimates that the triangle is the fundamental construction and constructor. So although multiplication is most clear in constructing a parallelogram it is the triangular parts or pieces he is focussing on as primary constructors and primary Metrons.

Up until now I was focussing on the triangular summation and the parallelogram multiplication. But I feel that Grassmann was looking at the triangles in all cases!

Ausdehnungslehre Vorrede page Iv, new paragraph

Thus, with this ( axiom) the first step toward an Analytical method was taken, which in the process of  following I was journeying to  this new branch of Mathematics, which is layed out before ( you) here. But in no way whatsoever did I realise then what a fruitful and  rich field of study I had reached. Much more that result ( the axiom or law) appeared to me to be of little observational worth, until I combined it precisely as it is with a related idea. In the idea I specifically follow the labels ( handles) for products  in geometry how my father Had apprehended  them, and this made me realise that the Product of 2 adjacent sides was not only to be considered   as the rectangle, but over arching that was the  parallelogram, in which the 2 jostling against one another  sides are understood to generate the form in the same manner, even if one once again no longer focuses on the product of Lengths  but rather on both Strecken with their direction held in equal importance .

Ausdehnungslehre

Now, within this idea, this realisation, I brought together in combination the previous set out Sum ( rule idea) with these labels( handles) for the product : this itself gave rise to the most astonishing Harmony! Even if I specifically, replaced according to  the SENSE of the aforementioned rule the nominated( perceived) Sum of any two Strecken with a third Strecke, within the same plane, (but) within the sense of this (new) idea of multiplication in the plane.(in order to multiply in this idea of a plane  layout sense )
I multiply the pieces each individual with This same Strecke , and the products associated positive or negative values carefully observed i add them together,and so show that in both cases every time  the same result I have to obtain, that which I got to  before and that which I go toward from here.

Commentary/Kommentar


I have to say that this is the most important paragraphs in the Vorrede, and the one I have wrestled with the most.
Ausfallendste for example is more like serendipitous than surprising, because Hermann is clearly thinking geometrically, and this perhaps in meditation on his dad's paper opened in front of hom! SI what caught his eye and his imagination? How does he go from he rectangle as the product in geometry( what is that?) to the parallelogram as the generalised product to multiplication in the plane by planar Strecken?

And why was his unworthy insight into Strecken summation suddenly cataclysmically important?

I have meditated on this section many times in different ways, but now the diagrams give me guidance I so orely needed. Experience and research have helped me junk misconceptions, including" Algebra" , and focused me on symbolic arithmetic and it's geometrical justification. And what I learned and relearned, and learned when first introduced to geometry was the fundamental role of the triangle.

I think that Grassmann went from the product in a parallelogram to the products in a triangle!. But this was only logically possible if the sum of Strecken was considered to obey his previously unworthy axiom or rule for representation of Strecken summation.

Multiplication in the plane  had to be based on the sum of 2 ( basis )Strecken  in the plane. The 2 basis Strecken formed the fundamental basic unit of the plane, the bivector or biStrecken. And all Strecken were some linear combination of these basis Strecken as in a reference frame. Any enclosed planar form was now measured by these fundamental bi Strecken forms.

The mixture of linear variables with lineal Strecken was just there, on the page, provided you used the first rule of summation! The geometrical temptation is just so strong that one is forgiven or not properly analysing what one is tempted to do!

There was only one issue. A parallelogram consisted of 2 triangles essentially in opposition! How was this to be accommodated?

The point is Grassmann chose to accommodate this essential opposition rather than define it away.

What in fact was going on is analogous symbolic representation. But this time the objects are totally subjective! A line was used to symbolise some objective magnitude, but here Grassmann uses a line to symbolise a subjective experience. The basis Strecken symbolised a reference experience, the generalised Strecke was generated by these experiences, and the result was that the individual was now in a position to count or measure space or an analogous spatial experience.

Arithmetic was not being done as an variable calculus, it was being done as a subjective process of manipulating and synthesising space itself, that is as a subjective experience.

Somehow, these handles gave direct access to manipulating space itself , and this realisation just drops out of this way of thinking about construction; and summation is embedde in that synthesis.

Commentary

Grassmanns sentence structure is complex but this is because it is hypnagogic. Where he picked up this style is a matter of research, but it is a style that establishes" truth" where doubt could exist,, it is not propositional but coercive. It is not the style of a postulant either, but rather that of a sophist who places ideas into others minds!
Thus the statement about replacing the 2 Strecken by a third Strecke in the context of defining the geometric products has insistent commands: the 2 Strecken have to be selected according to the sense previously given to us as a sum of two Strecken .
Thus the Förderung regarding the rule for 3 points joined by 2 Strecken turns into a Law, not to be disobeyed.
But further, this law is applied to substaniatiate the sense of multiplying in the plane, thus the third Strecke Must lie in the plane of the 3 points.

This third Strecke was created or established to multiply! So he multiplies each individual of the pieces by it in the page layout sense , in the plane, but being careful to observe the positive or negative value of the pieces before adding.

This positive or negative value can only be defined by the 3 points! If the third Strecke is positive it means the other 2 Strecken are positive. But to multiply each Strecke has to push on the other to form the parallelogram. This is a Newtonian idea. If the Strecken are parallel they cannot be multiplied in this sense.so the Strecken have to be adjacent and pushing to form a parallelogram. In that case certain adjacent Strecken have to have there positive or negative values reversed in order to push out a parallelogram.

It is also possible to extend a Strecke so it pushes out a new "exterior" parallelogram . In fact by extending and jostling the whole plane can be tessellated by parallelograms exterior to the original, some of different sign value!

What remains unclear to me is precisely how Hermann chose any 2 Strecken and which pieces he individually multiplies to form the products whose sign values he so carefully observes before adding!

Ausdehnungslehre continued:


Now–this Harmony allows me to relate to all sorts of things: permits that I would close the gate onto a whole new field of Analysis which I submit here so that  it might lead to important Results! Yet, back then, this idea again remained quiet for a whole long time, as my responsibilities in other circles of professional activities drew me back down from there! And also the strange result initially deeply troubled me, because for this new method of products to work  with the rest of the usual rules of mathematical multiplication, and in particular for the relationship with addition to be retained on standby, One but only had to accept the interchange of the factors of the product switched the sign value of the product from+ to – and vice versa !
 
One  may commute the factors, but only if the foresigns  are "turning around" an equal amount ( + into  – transformed and so "turned around")

Footnote: compare J. G. Grasmmanns Raumlehre part 2  p. 164 and that of the Trigonometry p.10

Ausdehnungslehre continued
A worked exercise that I later had to take on the Theory of the Ebb and Flow of tides lead me to the Work and Ideas of La Grange his Mecanique Analytique, and those ideas brought back these ideas of Analysis. All the developments established in that work I used this new Principle of Analysis to establish again in the most simplest way! Often the Calculation was as little as one tenth of that in that original in the way it was lead through, and the result just fell out!

This encouraged me to apply this analytical method to the Difficult Theory of the Ebb and Flow of tides; there were many kinds of new handles to define and develop in order to suit the Analysis to the task! And specifically I was lead, by the label specific to the Exponential magnitudes relating to the swivelling round( of things) , to the Analysis of the angle/ arc and the trigonometric functions  analysis, etc, etc... And I had the Joy to see that the newly established extended method lead to highly simple fully symmetric Formulae, rather than the unsymmetrical and convoluted ones which are the basis of this Theory! And these formulae not only contained the old convoluted and unsymmetrical ones but also The method of development of the Formulae went to the  side ( of the page) of  the handle or label.

In the practice of advancing stepwise formulae from one formula to another things  not only go so easily, leading to the least wordy formulae, the development leads to greater symbolic handles , new expressions and the development of specific laws every time; but also the stepwise advance of formulae seems to lead to its own parallel going, consisting in labels, guided Tour .  Dark and obscure ideas shroud the development , and the Spirit of the idea is killed, by such an aforementioned practice. The introduction of arbitrary coordinates that do not relate to the things essential natures but obscures them shows this death dealing formula development!

In practice I could not only every formula which arose in the process of development clothe with the lightest of words, and I expressed then a new law( rule) in this way every time, but also every stepwise advance from one formula to another formula relentlesdly seemed like only the symbolic expression of a parallel going, guided tour consisting in labels or handles! Compared to this, Other usual Methods I showed that they, through the introduction of arbitrary Coordinates, those which had nothing to do with the essential idea of the thing in hand, the  idea shroud in darkness, and the calculation do in a mechanical (lifeless )formula development , the basic concept of which is definitely not presented, and which around that basic concept is formula development that is lifeless!


Footnotes
The nearest validation is found below

Compare with La Place Celestial Mechaniques book IV

Commentary

I think I might have got the gist of what Grassmann did in his the law of the third Strecke phase!
However I do not know if I can communicate it clearly!

As you may know Bill Clifford quoted a paraphrase of the scripture which said unless you are like a little child you cannot enter the kingdom of God, because the gate is so low!

In my wordy and voluminous thrashings about I have passed by this gate many times, but could not open it! And yet I was outside the gate and sometimes inside the gate and other times on either dude of it simultaneously!

I am raving like a lunatic, so forgive me. However vectors do not help you understand Grassmanns Analytical Praxis! They are a product of someone else's mind grafted onto Grassmanns straight forward but rigid even autistic thinking!

Everything has to be disassembled into pieces and constructed in the synthetic process precisely as Grassmann describes. It is tempting to leap ahead and think you are on the same path, but you are not likely to be. There are many many similar paths but they do not all end up where Grassmann goes. In particular, Grassmann builds new connections new paths but they rely upon his original path.

Are all these other paths wrong? Not necessarily, but they are not Grassmanns path and may not lead to his conclusions!

There are fixed conventions in geometry and trigonometry. No one really knows when they started or if they were ever really discoursed. It just seems that they were imposed as standard conventional form. These conventions of the plane are not discussed just enforced. All lines are drawn left to right , horizontally in the plane or page or slate. In some countries this is reversed and lines are drawn right to left horizontally. Even left and right are conventions of reference frames of a personal nature. A good proportion of us have difficulty with our personal left and right, and that extends to apprehending another persons left and right!

We do not seem to have a problem with forward and backward or up and down, other key aspects of our personal reference frames.

Back to the page and by extension the mental plane! The other fundamental orientation for a line or a line segment on the page is called vertical. This mental page vertical gets transposed into our 3 dimensional apprehension of height and depth, and yet these are entirely different experiences!

Finally on the page/plane/ mental page we can draw line segments at any orientation between the horizontal and vertical by several methods. To draw a direct line segment one needs a straight edge. The manufacture of these straight edges is of considerable importance!

When Newton begins his Principia he does so in the context of the dance between Mechanics and Geometry, each contributes to and impacts on the other. They firm a dancing pair, an iterating couple, by which we hone down our ideal forms to the greatest exactitudes until we become exhausted! We never achieve perfection we endlessly approach a mental or formal ideal.

Where does this ideal come from? Ah, both Plato and Socrates invented a game to play on that question! Poor Arustotle, however took it seriously and attempted to state what was logically true as opposed to what was logically not true. I make no bones about it: Aristotle went a little crazy in this regard!

So, returning to the page, we can use a manufactured straight edge or we can use a rigid property of fixedness or fixity namely the circle! In fact we use a combination of both in the plane which makes all our Geometry actually based on the spiral!( now there's a surprise!). One other convention: we specify our orientation relative to the horizontal which means we have 2 measures of orientation that are usually called supplementary.

This theme of couples runs all the way through geometry, or as I prefer Spaciometry. It is only in trigonometry where we distinguish them, and then promptly attempt to forget the other supplementary facts and details!.

We need all this conventional set up before we can even begin to talk about a point in a plane!
In fact the point, the line, the circle, the plane are all concepts that we have induced in our mental experience from mechanical interaction experiences. These we have re-applied to those mechanical experiences, iterating to abstracted mental ideals over time and experience.

What is a blank page or a clear plane is in fact a pregnant space waiting for us to pour out our mental conceptions from the full page or plane in our " minds". It is how we apprehend these mental conventions that Grassmann deals with in his masterpiece. It is the method or Praxis or Art that he discovered that he imparts in the Ausdehnungslehre. And it relies now on 3 arbitrary points and the 3 Strecken that join them in the context of all these prior conventions!.

When Hamilton invented a mathesis for complex numbers, he went step by step from one point moment  to couples of point moments. But he got stuck at 3 point moments because he was looking for something else. He was looking for rotation. Grassmann was not looking for anything in particular, but he was sensitive to patterns and interested in labels. He noticed the simplest things like a child. And the rigour 3 points placed on all notation was ignored by everyone because they were busy looking elsewhere!

Commentary

So starting with the first 2 points A,B he draws the horizontal Strecke AB . The concept of contra is introduced by a simple observation : retaining the points as fixed. He could have started with the horizontal Strecke BA.

Only the child knows the magic of what it has just done!

The statement AB = –BA introduces the contra sign as a negative sign, but the child knows it as a rotation symbol. Unkehren is to sweep round in a great arc and go in the opposing or contra orientation . That is to change direction.

Now draw the second Strecke BC . We could have drawn the second Strecke as AC but tht does not " flow"

The flow shows up in the statement AB + BC = AC

This is label poetry. .

The child delights in it!

This says something magical about the plane and 2 flowing Strecken in the plane: they create poetry. This is where the child starts from, this is the child's Förderung.

Now we come to trigonometry. Serious adult business! We do not want all the fuss about all the points and all the labelling. We adults want it ll neat and tidy out of the way.

So we start with the rectangle. Break it into all it's pieces(Stücke) and name them all and label them all. The vertices , the edges or sides, the diagonals, the area of the shape, the products of the sides or edges which are adjacent. Use lower case for the sides so we do not have to keep writing vertices or the points.

But then the bemused child sees an old friend, a reemerging of the poetry
 a + b = e the label for the diagonal
–a + d = f the label for the second diagonal
And look d + a = –f

Switching the adjacent sides in the sum and the contra on the a creates a contra on the diagonal f!
But multiplication is boring !
ad or ab or bc or ba or cb or cd or dc or da they are all the allowable products
So what about de or ed?
Now we can write  ed = ( a + b)d and thst is ab + bd
ab is the allowed product but what is bd? It is not allowed
So the product of the parallelogram is the allowed product of the rectangle  wit a disallowed product.

Suddenly the child sees the parallelogram has a simple product and it is the rectangle if disallowed products are ignored or set to 0

All parallelograms on the same base are equal!

But what about de which is da + db?

db is not allowed but da is
Is da the same as ad?

The poetry says no.they are contra!
The adult says poppycock!
The child curious goes back to peek. The labes drop away to reveal the Strecken and their points. The multiplication drops away to reveal the construction of a parallelogram compared with the construction of a rectangle.

In both the construction flows only one way! To interchange the constructed elements requires introducing contra construction processes.. These contra processes are not cancelled, thus they carry forward and make the whole process simply contra!
It is intuitively simple, but the adult is deeply troubled by it.
The child rejoices in the poetic harmony.

How to construct a parallelogram  ABCD( cyclic vertices) given AB is a and BC is b and CD is c  and parallel to AB and DA is d and parallel to b
Using a b as the product instructions that is construct the parallelogram using a and b only.
Draw a horizontally. Using a compass swing an arc radius b. mark off the required arc using C . Now bisect the arc to construct a rhombus , extend the side of the rhombus through C to meet D by marking off D  measured as c from C. Join D to A and check it measures as d
Thus ba as the product instructions: draw b horizontally.( already a major diagram shift! ) . Using a compass swing an arc radius a ( where is the centre for this arc? It cannot be at C as this violates the convention established. It cannot be at B because that violates the convention  a as AB ! To proceed we are forced to use not a but –a) . Correcting swing an arc radius -a. Mark of the required arc using A. Bisect the arc to construct a rhombus, extend the side of the rhombus through A to meet D by marking off Dmeasured as -d( because we are forced to use the contra convention) from A. Join D to C and check it measures as –c.

It is quite clear that ab is the contra of ba, and this is written as ab = –ba.

The question is can I cancel the constructions out?

The answer is no. These are allowed constructions.
I am tempted to say that we can arrange the coefficients to cancel, but the point is that the product is there, and the contra product is there.mthey are not annihilated in the notation, they are equilibriated!

While this may mean little to the child as it grows it will appreciate the value of dynamic and static equilibria in mechanical systems as well as physical ones.

Some say that geometry revealed these set ups in physical reality, but forget to point out the dual nature that runs through every relation we observe!

Commentary

I thought that if I let the last meditative transaction sink in I would gain some insight into why it was so fundamental and yet do underwhelming!

And bingo!
I awoke thinking about the Galilean transformation principle and how it could be analysed Grassmann style!
Then I thought of Miles Mathis protestations that Einstein got it wrong in his 1905 papers  regarding the principle of Galilean trsnsformation.

And then finally I realised how fixed Strecken are associated to one dynamic Strecke. But then I realised it was arbitrary which 2  are fixed and then only one might  be fixed. Finally what if no Strecke was fixed!?

That brought up the crucial role of a point and their fundamntal role in our referencing Strecken, and representing annihilation and creation of Strecken constructions.

And finally I realised it all starts with memory, or as Hamilton put it: moments in the progression of time.
But even time is a subjective relativity.

For exapmple a measure of force is a measure of pressure produced with a measure of area that is a measure of pressure producing a bistrecke.
A measure of pressure is by cross ratio a measure of force divided by a bistrecke for area.
Now a measure of force is a constant scalar product of a Strecke of acceleration; and a Strecke of acceleration is another constant scalar product of a Strecke of differential velocity; and subsequently a Strecke of differential velocity is a constant scalar product of differential displacement of position in sequential order Strecke.
And finally differential displacement Strecke are a representation of a lineal combination of unit basis or primitive Strecke with differential scalar coefficients which are functions of sequence progression or implicit iterated sequence or inductive sequence functions.

Thus underpinning any mathematical model of our experiential continuum is an arbitrary or generalised selection of unit Strecke that are accepted as a basis or primitive set of Strecke.

But what if those basis Strecke are dynamic? That can only mean: what if they rotate relative to each other interdependently?

Commentary
Grassmanns analytical method opens my eyes to one thing in particular: our arithmetical models are totally formal, and in our heads! Consequently the measurent schemes(metrics ) can not answer the question,"What is matter?" The models enable us to model behaviours only through applied iteration, carefully conceived.

What matter is, is the subject area of alchemy or modern chemistry and physics. But Alchemy cannot say what matter is but what ratios of distinguished matter appear constant or conserved.

Our conservation rules and sense are how we can move between scales, testing all the time. At a certain scale our conservation rules break down. At that scale we change alchemical description and say the fragments we measure or identify in other measuring schemes like interferometry , are new elements or particles. The smaller the scale the more fragments we identify.

But we know our mechanical concepts can be scaled down ad infinitum. So what prevents us from multiplying fragments ad infinitum?

Our conservation rules guide us to look for systems where they hold , and those scales then become the new fragment scales.

At the quantum level, then,  we are looking for conservation rules that hold. That then defines the scales of the new fragments.

How do we even measure on those sorts of scales?

Again we cannot, so we resorted to statistical methods. This does not tell us about particles at all, but about collections of statistical measures. If we get excellent agreement on these statistical samples, either as probability distributions or as statistical curves we infer that the measures bear some relation to an underlying assumption of fragment size.
This is modern quantum particle physics.

However, these measures are equally applicable to fluid elements, spatial density ratios, invisible energy density regions, Fractal regional boundaries etc. in other words the mathematics does not identify any substance. We assume, impart or infer substantive properties from sensory signal inputs processed by our proprioceptive mesh.

Our experience of reality is our reality, and it is beyond our powers to step outside of it in any sense or way at all. We simply reprocess the data to establish a particular sensory map. How useful that map is is how much behavioural control it gives us over local reality.

Commentary
One of the very real difficulties that defeated Newton in developing his fluid mechanics was the sheer complexity of the reference models.
While Descartrs system of coordinates were of a generalised nature, and it was Wallis who first placed them as rigid cross hairs, it still was difficult for Newton to relate roationally mobile local reference frames to a larger external reference frame with all the freedoms of movement in 3 dimensions. Hamilton's development of Quaternions made that possible for the first time, but it was fiendishly difficult to get an intuitive feel of what the axis were doing.

Even today in quantum physics theorists will not accept mere rotation as a description of these calculated state changes.

However, Grassmanns method makes it natural and intuitive to think of these local transformations as relative rotations, even though they can also be described as reflection in crossed lines.

It is better to include these rotational behaviours in the model, alongside the differential or integral calculus scalars for the Strecken.  The result is that rather than describing fields as spherical potential scalars, we need to describe them as vorticular Strecken scalar complexes. Using this model, it is easier to model fluid element behaviours as well as fluid dynamics as a whole.

While the curl of a vector field is an attempt to model this rotational behaviour it is too rigid to deal flexibly with vortex shedding , energy propagation along or between " reflective or refractive wave guides" etc. in particular the phenomena of reflection refraction and diffraction do not seem to be easily connected to the curl and or div of a vector field with integral or differential calculus scalars.

There is a better way is what Grassmann is saying.

Commentary.
It struck me this morning, that the train of thought inspired by Grassmanns thinking style only deals with the fundmental primitive or basis Strecken of any model. , that is to say that the importance of Grassmanns model is not just the basis Strecken in Lineal combination, but in fact that he could multiply any pair of properly configured Strecken!

The work he does on multiplication I have not yet studied because I wanted to get his Vorrede right. But already , after my experience with vectors and Clifford algebras I can see that many have lost the simplicity of his approach, as discussed here.

Yes Grassmann goes on to redact and to extend his method into more and more difficult analytical subjects, that is subjects that rely on symbolic arithmetic to collate and coordinate the theoretical system. And yes he does create new terminology and even new products, but what he comes back to is the essential simplicity of his 2 major initial insight: the representational sum, and it's use in defining the general geometric product.

The representational sum can support affine and projective geometries, and we have seen how it can support a dynamic mechanics. The general geometric product extends the method into the enclosures of the Strecken products.

What I mean by that is : when one learns Geometry the line and point are usually demoted to the role of boundary and boundary measure or parameter. The things that excite geometers are the forms or shapes starting with the triangle.These forms are the first intuitions of space and what it might be.

However , as I have pointed out, these forms are totally formal and in our heads, so cannot answer the question. However these forms are abstracted by us from alchemical substance relationships, that is matter behaves in ways that gives rise to these forms approximately. Rather than go down that arrogant route suffice to say that we abstract from reality these abstract ideal foms by a process of fractal distillation!  In other words, over time and experience we iterate by design to the ideal forms in our heads.

By reversing the iterative process we can " sum" or aggregate or integrate iteratively these foms into models of more natural forms , foms found in nature.. We have to reverse our arrogant taxonomical thinking to do this, which is why fractal geometry is such a subject for the people, the artisan, the artist, the builder and the mechanic, the engineer etc. and the little child can contribute as well and even lead us to new innovative thinking.

So now Grasmmann gives us a way to represent these forms as products of Strecken. Again, we need to avoid artogant assumptions .

Many who have read my earlier writings might remember my friends the spiders. They are the ndimenioners! it is a curious experience but once we get past about 6 dimensions in the plane we can no longer construct from these nets 3 dimensional solids that are closed. Thus multiple dimensions in the plane remain as planar forms. Hence spider webs which typically have between 20 and 30 dimensions are flat!

You see dimensions do not have to be orthogonal or mutually orthogonal! This one bit of classical nonsense has impaired understanding of the space in which we live!

We are so used to saying we live in 3 d we neglect to apprehend that is not a truth! It is a shorthand for a representational system based on orthogonal standards. These standards initiated by Wallis are one of many, and are not always convenient, for measurement, construction, engineering or technological process, or even Alchemical process..

Crystallographers and crystallography certainly would struggle with just 3 dimensions!. Grassmann made a point of stating this notion of 3 dimensions was a severe limitation to understanding. It turns out that the real numbers also supports this limitation in thinking!

Many physicists assume they need the real numbers to describe reality. But increasingly, quantum mechanics is showing that we can only really apprehend the natural numbers and their construct the rational numbers, if we need number at all!

As you may know I do not believe in mathematics as a subject or numbers as its primary objects. Here is why!
http://www.youtube.com/watch?v=w-I6XTVZXww
This is nonsense. But that is just my opinion, I hasten to add.

Fortunately for me I have had opportunity to apprehend the Arithmoi and Shunya, both are blessings beyond measure!

The motion of  a fixed local point when the local basis Strecke are gyres or Twistors of fixed radius

Things like this are why I refuse to believe infinity exists as anything other than a mathematical construct.  It simply does not work in the real world.

I get so many arguments from everywhere when I say such things that I prefer to just keep quiet about the fact that I hold the same opinion of Zero - it's another useful concept for math but it has no place in reality.

Thankyou and welcome to the tHread Sockratease.

I hope you will feel able to share many other points regarding the foundations of mathematics here!

Arguments are not the way forward. Disagreements are acceptable and welcome. If you or anyone wants to put an alternative view that is welcome, but ad Hominem or disrespect as tactics will not be tolerated.

I once made a comment on YouTube which requested a collaboration with prof Wildberger on research into Grassmann. It attracted the most virulent comment from another viewer that I reported it as spam! I was happy to see YouTube immediately took it down!

So why am I going on about these standards of civility? Because I want you and anyone else to feel safe to voice your opinion in and on this thread and know you will not get embroiled in some argument!

The video may appear to be a silly argument. I think it is a confused one. I am certain that most of us can see where the argument leaves the normal experience of additive process or aggregation. The question is, why are these men smiling?

Non verbally it is because we knw they are Ill at ease with what they are being forced to say! The reality is that they cannot explain it, all they can beg us to do is accept the process.
I have a choice, and I say no! Others may say yes.

This kind of dividing occurs all the time. We are divided into groups who accept certain things as standard. It is when one group attempts to coerce another group to conform that hostilities break out.

The summation of alternative opposites is classically undecideable if a process is endless. We found this out when our computers went into their first infinite loop!

So why decide on a 1/2?

You have heard of probability. This is where the half summation comes from. What the presenters probably do not know or fail to tell us is that the summation is no longer an arithmetic one, but a symbolic representational one.
The confusion lies in the lack of explicit definitions of the symbols on the page. Because of this lack of clear definition everything else is confused. The answer itself is also meaningless within the standard probability theory. Negative probabilities are not usually defined, although there is a great philosophical debate about their possible usefulness and interpretation!

I do not expect many to understand what I have just written, because the video is such obvious "non sense ", showing no common sense that it cannot be redeemed. Nevertheless  we humans do not think " mathematically" we think analogously. It is the analogies we use that give us our sense of meaning and contrariwise our sense of nonsense. It is for want of meaningful metaphor that many of our most innovative artisans alienate their constituencies. In such cases they revert to the " magical" paradigm, or the " mystery" paradigm.

There is a place for magic and myth as well as fact, as long as we realise that all 3 are totally human creations.

Commentary on Ausdehnungslehre continued.

The deep , conjugate and fundamental relation of Mechanics to Geometry is perhaps not appreciated by theoreticians even today.
 
In the ancient times the basis for geometrical knowledge was the practical and technological skills and experience of Artisans. Education, or rather academic education removed that connection by establishing qualifications based on written responses.  Education especially monastic based education was based on discipleship and apprenticeships. A student had a master and would gain practical and technical skills over time under the masters supervision.

Plato's academy, as a case in point, was discoursive in nature but required, demanded a level of practical expertise in Astronomy and Geometry. As time went on, and particularly when funding became an issue these entry requirements were modified.

Eventually academicians becme so snooty they would act and state that the Tekne or skilled artisan was beneath them socially and intellectually! The greatest historical Student of Mechnics and perhaps everything else was Archimedes. But I have heard that some accused him of " cheating" when he trisected the angle by neusis! That is he carefully moved a straight edge into position so that he could mark off the measurement required to trisect an angle. Later he demonstrated how a spiral could achieve the same exact measure.

Trigonometry is about applying metrics to space. It is called trigonometry because it relies on the trigonos . This is not the triangle ( ie the trilateral figure) but the actual 3 rotations at the corners of a trilateral.. However, these 3 rotations require a plane and a figure that is a closed " loop" in that plane.

There are many candidates for this role, but the do not arise from geometry! They come directly from Mechanics! Over time the best mrchanicsl archetypes turned out to be very strong rope or cord that could withstand high tension without stretching appreciably. But rigid objects of any description were also used.

It is perhaps excuseable to think that geometers start with a plane. In fact they start with a surface and mechanically produce or scrape out a plane!  The ideal surface and perhaps the easiest to produce is the spherical surface. Rotation around a fixed pole enabled these surfaces to be carved out. Some artisans became so skilled that they could do this by eye nd hand alone!

Returning to the scraped plane , the circle is the next easy fom to create, and the fundamental one. Choosing The Diameter as the greatest measure enabled a taut line to be defined as straight; enabled this to be confirmed by a curious property of a third point on the perimeter of a circle relative to the 2 diameter points. The rotation at that third point if a trilateral was formed by taut cords was always the same. It was called orthos!

Tekne were able to produce and reproduce highly curate models of that rotation! In fact it was so special tht it was used as one of the symbols of divine power for an Egyptian ruler or Pharaoh. It is the origin of our term " Ruler"!

The point is that our reference frames or Strecken are determined and constructed by artisans . The orthogonal reference frame was devised by artisans as a quick way to cut out an appropiately sized block for further preparation or construction..mthe orthogonal mutual rotations are placed into position by construction and measuring processing. At each stage measurements are taken to ensure accurate alignments. These constructs are our fundamental tools of reference and they exist in our heads not in nature.

As we construct using orthogonal frames usually, it is acceptable to use these as our ultimate basis for space, but using these and Grassmnn analytical method we can actually construct generalised reference frames based on Strecken in any rotational relationship.

The issue of dynamically rotating Strecken still requires us to reference these against a fixed fundmental set of basis Strecken and these may as well be orthogonal.  However, the point is, that building atop these fundamental orthogonal set we can have exotic dynamical basis Strecken in lineal combination. These are what are needed to describe fluid dynamics.

Einstein utilised this Grassmann geometric freedom to establish his relativity theory. Others, not understanding the freedom Grassmanns ideas give formally point out the fundamental "Euclidean" basis Strecken! The point is all these basis Strecken are in our heads not in Nature! In any case Ruclid was a consummate astronomer who used spherical trigonometry not plane trigonometry.

Commentary
Kegan Brill is one of the few artists on the web working intuitively nd accurately with vorticular spaces!
http://www.spiralodyssey.com/Spiral_Odyssey/INTRO.html

After Lazarus Plath he is the next intuitive genius whose work informs my conception of space and it's structural framework and dynamics..
http://www.spiralodyssey.com/Spiral_Odyssey/WORK_IN_PROGRESS.html

The structure of the fundamental or primitive reference frame , the standard orthogonal Strecken is not complete as a wire frame. The structure is best vsalised as a vortex shell. If the orthogonal trecken remain fixed relative to each other we fail to appreciate the shells which govern their relative motions. These shells , expanding or contracting model forces and motions in space and the complex behaviours of matter.

The basis Strecken for dynamic space spin out these shells like the webs of a spider, and necessarily make SpaceMatter fractal and regional at all scales.

However I realise from my above discussions that there is an unfathomsble gap between my formal apprehension of Strecken reference frames , dynamic or otherwise, so that there really is no way of framing an absolute reference frame in any decideable way. It is down to the individual. If they accept an absolute frame they think like Newton. If all frames are formal and personal they think like Einstein.

Commentary
It is amazing that Grassmann cut his teeth so to speak on a fluid dynamic problem. While the Scwenkung labels and the exponential handles guided his development of his own analytical labels for the swing, or pendulum like motion of the tidal flows, it was and is apparent that it would generalise to any periodic motion. Bill Clifford felt straight away that it would describe the undulatory motion of waves and in particular light and electromagnetism. Perhaps he was inspired also by Helmholtz and Kelvin, and possibly by Maxwell. Because of his early death we may not have enough documentary evidence to determine that, but we do know bill Clifford was a big fan of Grassmann and Hamilton.

Several references are made of river McCullagh and his rotary description of fluid mechanics . This played a major role on influencing Fitzgerald to redact Maxwells work. But Stokes is said to have demonstrated that angular momentum is not conserved in McCullaghs system. Why this should quite damn the work is still beyond me.
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html

The belief in conservation laws perhaps was taken to far in this case. Conservation of angular momentum is a clear impossibility in rotating systems. The reason is that angular velocity, as a concept is not mappable to a single valued translational momentum. Instead angular momentum as a transformation to lineal momentum is not a function as far as functions are defined. Thus any narrow , mathematical description of angular momentum is bound to fail.

Rotational momentum, as an analogue of lineal momentum is a nice idea but not physical!

Now I am clearly not talking about the normal definition of angular momentum, because if I was I would be dead wrong! I do not mind being dead wrong if it leads me to a better understanding.

If you consider a disc, for example , or even a stick it's angular momentum is defined fom the concept of torque in comparison with lineal momentum.

The mathematical facilitation is the placing of the mass in connection with the velocity or distinguished from the velocity of an object. However we have to back up to the notion of the velocity of an object.

Here Newton made a justified identity: the velocity of n object is the sum of the velocities of all it's parts. This is what is called a vector sum, by a man who invented geometrical vector Maths before Hamilton and Grassmann and Gibbs! Newton's third aw was a catch all that justified this approach once accepted. Geometrically it was based on Newtons understanding of the sphere nd the circle, as the quintessence of symmetry. Because of this symmetrical apprehension due to the sphere Newton was able to envisage a centre of symmetry collocates with a centre of mass and what Newton called a centre of Gravity.

This vector sum is set to zero at this precise point and an object is demonstrated to balance at this point. Barycentric coordinates apply.

However, if an object rotates about this point the vectors fir rotation are defined variously, but they am mount to instantaneous tangential ones. Or usually ignored nowadays arc or twistor ones!  I named these arc  or curved lines twistors. These do not sim to zero if an object is rotating . For any given circle or spherical surface thes arc vectors will sum to a specific value, but that value is quadratic lily determined or measured. The two measures are the rotational "velocity", an analogy with lineal velocity, and the radius  at which that velocity is measured.

For a real body the notion of angular momentum is tricky to justify!  The mass is distributed through space, and therefore is also a function of the radius at which the "velocity" is measured. The full concept of an analogy to momentum that describes rotation is therefore a double integral  over radius or an area integral or better still a volume integral. My point is I am not sure if it makes sense to discuss an "angular" momentum in this situation, and thus to expect a conservation of angular momentum. However it seems natural to identify a kinetic energy fir the object and to expect that to be conserved.

How dies this relate to the observed behaviour of rotational motion? It relates through density. As the density of the object increases the kinetic energy increases, but if the kinetic energy is constant then one would expect the greater density to impact on the rotational motion reducing it. However, if the density increases via an internal change in volume, because of the cube relation the expectation is that the rotational motion actually increases! This is because despite the density increase the system quantity of matter is constant!


When an object or a system is periodic the important conservation is that of kinetic energy. The notion of rotational kinetic energy compared against lineal kinetic energy is interesting. For a rotating mass that rotates around an axis , the rotational kinetic energy must be set to the lineal kinetic energy of that mass moving under its own rotation in contact with a non slip surface.
http://www.youtube.com/watch?v=s_R8d3isJDA

If we now transfer this to a fluid / viscous material one can equate the total kinetic energy of a vortex to the entire system after any vortex shedding has occurred.

The generation or dissipation of vortices is therefore of fundamental importance in describing fluid behaviours, and it is precisely here where Newton ran into difficulty, and stokes makes his non physical diagnosis,

Newton was able to describe how force transmits through a fluid around a shearing cylinder by means of his " calculus" method, but he left out the conservation of rotational kinetic energy. Thus a lot of expected rotational motion did not " appear" precisely because it went into driving shed vortices!

It is apparent that this vortex shedding was not understood or even apprehended. It has taken me a while to apprehend its ubiquitous and vital role in fluid dynamic description.

When I look at Newtons drawing of wave diffraction in water and by analogy on air. He represents it as short lines fanning out. The same kind of thinking carries over into his corpuscular theory of light, but this time he replaces the lines by corpuscles.. So in a fluid a fanning band of lines represented some notion of a wave front, and this epwould be a spreading " hill " or undulation of water. This was his concept of a wave. Later, this hill is almost universally represented by a sine function, this graphical image is burned so deeply into our brainwashed minds that we think it is a wave!

However, newtons corpuscles are perhaps better representations of the fluid behaviour than we are taught to believe.

Corpuscles are alchemical blobs of jelly. Later some regarded them as hard pea like objects or billiard balls. What they were precisely Newton never said. They behaved like projectiles and had a bodily attraction to certain materials, that apparently led them up and diverted them from their straight line course!

Huygens on the other hand looked at the corpuscle as not being a projectile, but a receiver and transmitter of an undulatory force!
http://www.youtube.com/watch?v=NeaNTbeEelo
This video gives a feel of how circles combine to propagate light spherically. the complexity of the motions hides the fact that these have been deliberately selected. imagine a physical process that selects out the wave phenomena of light:diffraction , refraction , reflection...

Huygens cascade model of light propagation was based on a more accurate understanding of Grimaldi's work. Unlike Newton he understood diffraction to be longitudinal not transverse. His own work on lenses showed him how light inhabits a material, making it glow. Therefore he surmised each corpuscle of matter absorbs and radiates the light "wave" like a pendulum. His model shows light propagating in circles from every corpuscle.  From this model he could isolate all the observed phenomenon simply by defining a kind of sum in a given direction, a vector sum!  
The wave fronts Newton drew as short lines rafting almost in parallel he replaced by circular arcs. Where these arcs crossed more light was generated than where they failed to cross.  I have not read enough of Huygens to apprehend if he regarded this as super positioning of hills or more likely affrefation of pressures in the plenum. In any case his wave front does not correspond to a sine wave , but rather to a spreading firm of curved rhombus. The wave front was made up of these rhombus like regions spreading out as the circles or spheres spread out and I filled by new sources of this rhomboidal form . Thus he had a fractal wave pattern which became more complex the further from the source you went.

That light dimmed the further away you were was not easily explained by Newton's ballistic  corpuscle model.

I mention all this to address the point that a wave is naturally observable as a rolling tube or vortex of water that rolls atop and within larger rolling tubes in a fractal pattern. Thus any physical wave description that does not recognise the rolling tube with an ellipsoidal or rhomboidal form as its fundamental description will mislead. And in addition the conservation of kinetic rotational energy is all thst is necessary to maintain Wave propagation in space matter.
http://www.britannica.com/EBchecked/topic/187240/conservation-of-energy

Commentary.

This paper attempts to give a straight forward overview of Grassmanns ideas and developments of them.
http://faculty.luther.edu/~macdonal/GA&GC.pdf
Especially at this point in the translation it is crucial to recognise that much of what gets physicists excited seems to be the development of his Ebb and Flow paper in which he made considerable innovative effort to extend and apply his initial insights.

Set down here against this, where the idea is not obscured by extraneous methods! Moreover  through the formulae  i set it down and thoroughly made it shine in full clarity,  also for every  formula development  the basic concept of the forward development of the idea was  labelled.

Commentary

By this stage it is clear tha Grassmann did not like obscure formula development!

From the outset he cannot help expressing his discoveries as Laws or rules. The analysis he was doing helped him to see clearly but symbolically!  He felt pleased to see symmetry instead of unsymmetrical formulae. He also liked neatness on the page, with position on the page meaning something in the step by step development of these formulae.

He felt that it was important that the labels communicated the basic concepts, and contrasted his style with other usual arbitrary styles, which he showed obscured the idea and the basic concepts.

Much of this attitude reflects an autistic trait which was probably passed down by Justus his father. However, like a master craftsman it meant he could construct a set of formulae which clearly showed the basic concepts and illuminated the development of those concepts into the greater formula.

Formulae tended to become simpler , symmetrical and less wordy, and more general in scope.

How ironic, that what was as clear as daylight to Grassmann was as dark as mud to his contemporaries! gauss is recorded to have said that the new terminology or labels were so busy on the page that he did not have time to learn what they all meant! Grassmann, unfairly got a reputation of being Obscure!

I have to accept that Grassmanns, dad and 2 sons were onto something when they set down formal rules for everything. However, academics usually prize their academic freedom over all else.

Commentary
As I read and absorb Grassmanns words, it is clear that he is using religious or clerical metaphors derived from the theosophy or christology of his times.

Philosophers in Christian Europe were mainly employed in the church as ministers or ecclesiastical functionaries. The churches role in Education especially public education was fundamental. It was the church and church schools that taught most students to read and therefore to think! This form of good works was commandeered by the political powers in an uneasy trial of strength. Historically the pagan Romney empire converted to the holy roman empire on the field of battle! The victorious forces of the roman emperor were immediately converted to the catholic faith, because the emperor had seen a cross in a vision and this had assured him of success in battle!

Thus the church naturally fell into the subsidiary role of educating the state as a political entity and as a demographic region in the faith of Christianity.

This did not mean reading and writing, but over time it became clear that both church and state needed an educated source of functionaries to keep the system going.

The licence to teach or preach was granted through the ecclesiastical arm of the state. In Prussia, every philosopher or teacher was officially a cleric in the church..Kant for example had his license withdrawn for a while because his philosophy was not in accordance with the presiding religious consensus.

I say consensus, but in fact the Theosophical and philosophical scene was riven by heretical theologies! Clerics of one theology were arguing against clerics of another theology. In these arguments pagan philosophers were often cited. These philosopher had to be used with care, because they were deemed pagan! But theosophy claimed to have a sanitizer version of their teachings, one purged of all pagan, anti Trinitarian anti Christian theology. However some classicl scholars( also clerics) objected to this kind of censorship and secretly tried to understand the world in an alternative and frequently alchemical and material way.

However, the thought control was so deep and thorough that they could not express hems elves in any other words, metaphors and paradigms than the prevailing religious ones.

Thus we can see often terms hich have a clear theological meaning, applied theosophically or even philosophically to clear mechanical and material scenarios. More interestingly is their application in dialectic or logical discourse. This old often result in readers bring totally confused! Was this statement an mpirically observation or a religious deduction or a statement of faith?

In many cases it was all three, nd some curious logical inconsistencies arose, some which could never be straightened out by the prevailing view.

Returning to Grassmann, much of his style of writing therefore reflect sermon writing. His long sentences with many clauses and subclasses, adjectival and adverbial phrasing follows a speech pattern rather than a written pattern. The connection between clauses in speech is carried by intonation and emphasis, things which are difficult to convey in writing if a person is unfamiliar with them in speech.


While it is arguable tha Grassmann had only his analytical jsystem in mind, and Arithmetic on the brain, this is not sustainable in my opinion. His seeking for laws is almost a religious affirmation that his method was revealing the secrets of God and his angels! As such they carried not only a purely mathematical interpretation, but also a guide to how to apprehend god's thought process in natural events.

This would be a later development as his method gave him more and more insight into the relationship in space between Geometry, Mechnics and Optics.

Many of his explosive iights derive their power from a crossover from religious expressions . The spirit is lifeless is an example or the spirit performs nothing is another. This is tantamount to demonising a particular spirit which is a powerful condemnation.

Grassmann is therefore making powerful statements which are lost if the translator takes a narrow view.  At the same time they obscure the mathmatical or geometrical reference they relate to!

Really?

Yes if you take the view the Ausdehnungslehre 1844 is a mathmatical text book, but no if you understand that like Newton, Grassmann was writing a metaphysics and a metamathmatical treatise, from which to derive his formulations.

The 1862 redaction is a more mathematical treatment in which the symbols play the role described in this section. However, without the metaphysics and metamathematics these long combinations of products and the construction of labels from them seems whimsical and abstract or as mathematicians like to say, completely general or arbitrary!

They are not.

Ausdehnungslehre Vorrede p. vi

Through these pursuits i held myself to the hope by right, that i had the unique, the nature measured method  discovered in this new Analytical Method, by following which all branches of mathematics applied to Nature must advance, and which the Geometry , by its identities would be treated if they would lead to general and richly fruited revelatory results!

I raised in myself, therefore the emphatic decision to dedicate my life to the presentation, the enfurthering and the application of this analytical method, as my life's work! Consequently I undividedly applied my spare time any kind of  way that filled the gaps, to these contents, which the the earlier laid aside projects had left.

Specifically it revealed the wisdom , and with the modifications how I presented them in the work itself, I was able to set out  how to apprehend the sum of multiple points as the Schwerpunkt, and the product of points as follows: 2 points:- the joining line; three points:- the enclosed flat figure space(fletch, a kind of arrow feather); four point :- the pyramid or tetrahedron( the body or 3d space)

Ausdehnungslehre

The apprehension of Schwerpunkt ( weight point) I was intrigued to compare with the title of a calculus of Möbius., the Barycentric calculus. I knew it in Title only down until that time  so I was not a little overjoyed to discover Möbius used the same notational labels For the summation of points which the process of development  had lead me to.
"Also sprach Zarathustra!"
And it was so!
 with the first (part),

but I lectured the succeeding parts;

and also measured the unique points of tangential contact, which the new method of Analysis already with the  differently understood way could present!

While digesting that calculus(method) certainly there was no label or handle of Product Of Points  forthcoming, but in this labelling of the new Analysis, in which the Sum stepped in combination with it , it begins first  the   unfolding revelations  of the new Analysis. So could I from that( Möbius) start No more extensive  basic assumption,  more fundamental mindset(Förderung) , more inclusive Axiom,  more wider postulate envisage!

Commentary

This video by Norman is relevant, but it is the more refined treatment  alluded to by Grassmnn, who literally was initially caught between 2 schools of thought: his own And that of Möbius. It seems that  he realised he had developed a successor to Möbius treatment because Möbius point of view was fundamentally different to Hermann's. They did not share the same Förderung he lrealised. Invention and development did not seem possible using Möbius mindset!
http://www.youtube.com/watch?v=aN5qULIFuBs

Commentary

(http://nocache-nocookies.digitalgott.com/gallery/9/410_12_11_11_11_26_38.png)

The theological reference kept chiming until I recognised it! I substituted Zarasthustra because that is also a famous German poem, and I think Hermann would perhaps have used that allusion for Möbius rather than God, but I might be wrong. In any case I feel the allusion is conveyed. Even the Berührungspunkte seems to be an allusion to the Sabbath day! Ruhe means rest while rühren means to stir or contact with some energy, the opposite of resting!

The Schwerpunkt I left as a distinct label. It has been associated with he centre of gravity or mass, but I think it is confusing to think of it that way. It is an aggregation point, and it has magnitude, which is why it fits in the Strecken scheme of things. Magnitude with no direction? It would initially seem so until you realise this point moves with each new point added to the aggregation. It can therefore be represented by a sequence of lines.

It is all deep thinking by Hermann.

The Berührungspunkte deserve to be highlighted. The Barycentric  is a point derived by his new method which is general, and represents a point of symmetry! Thus it can be taken as a centre of symmetry, mass or gravity! Also a centre for rotational and translational kinetic energy, and a centre of force, momentum and velocity and displacement!
The Berührungspunkte are associated tangential contact points.

It is a big deal!

Commentary

This is a meditative translation so I would encourage you, if you are following to return to see corrections, not only to typos!

My praxis is to read the text and give an impressionistic free spirited translation/ interpretation. This is verY exciting, and generates lots of ideas and connections.

I am surprised some of ou ngerman speaking members have not joined in, because all are welcome, and it is a lot of fun!

Then I come back to the text with some ideas in mind after I have let it stew for a while, and look up words I do not know the meaning or significance of. I try to pay attention to the tense, the case and the prepositions and conjunctions. That is when I find how far off the mark I was, and I correct the translation.

Sometimes I might do 2 translations of the same text , it depends.

So , please if you want to, engage with the text. I have posted a link to it. It would be nicer if you do to do your own translation rather than critiquing someone else's, but if you must criticise criticise mine. I can take it!

I want everyone to feel they can contribute without getting their Feelings hurt unnecessarily. Even if it is necessary to correct a bad translation, it is better to do the correct translation yourself than to offend another member unwittingly.

An open thread can receive all on topic contributions and become a resource for further thought.

Commentary
Norman's video clarifies and raises a point about points.
From a arbitrary origin. Point vectors , rather Strecken can be drawn. These are nodal points and uniquely defined by the displacement and direction of these lines.

I called these multicompass vectors in earlier posts in the fractal foundation of mathematics, a different thread. It is as if the line is more fundamental than the point!

In fact this duality of point and line is fundamental to get right. We cannot envisage one without the other.

I usually start with a scatter of points as an arbitrary starting frame, but I cannot process the frame without establishing any arbitrary sequence  and as soon as I sequence I can use a line to represent that process.

There is a notion of contiguity and continuity. They are both continuous states but one captures only smooth things while contiguity models everything.

Straight lines are special, and in fact much of Grassmans method applies to curved Arcs too!

Whatever we use we must admit that this Förderung is not connected to reality, but Is a  tool for measuring and counting space.
http://www.youtube.com/watch?v=Nu-YPJSNFpE

Commentary.

Now here is a real revelation to me and a resetting of my Förderung!

Until just a few minutes ago I thought that the Schwerpunkt concept was an Anomally. Grassmann, I thought included it in his method because he could, and it produced the marvellous Berührungspunkte.

But then it suddenly dawned on me that Hermann had said that the product in combination with the sum was the first concept he understood from the beginning! Yes A x B was defined as the joining line between the points(2) A and B, yes yes ... And I have seen that before.( in Norman's presentation) Yes and it is straight forward but what does that have to do with the Schwerpunkt? And why did he say mehrer punkte?

Slowly it dawned that the law of 3 points, the law of 2 Strecken was not what I had teased out of the confusion created by vector math! It was something altogether more straight forward!

I had come to Hermann hoping to understand what vectors really are. I had not been disappointed. When I realised that vectors are an alien word to Grassmann and that he dealt simply with geometrical terms, the line segment a lot of BS fell away!

I just finished grasping that the fundamental object of our geometrical processing was the Begränzte Linie, a line segment with 2 points marking the start and the finish. I said that normally the point is the fundmental object of our Logical Processing, and our dynamic modelling( after Newton) . I knew the difficulty of going from points to lines. Within that transformation lies Zeno and Parmenides lurking with their paradox traps! But the Pythagoreans avoided these by taking what is called "the method of exhaustion " route. In other words they avoided exactitude generally, and worked with approximating the best they could!

Newton used the notion of time or absolute divine time to overstep these logical difficulties. Until Huygens time was not accurate in any sense and so despite the water clocks of Archimedes and Timaeus, time was not universal or divine. It was a local measurement of duration. Time as in day night etc was a celestial track record of star and planet motion and day And night light variation.

When Huygens clocks were shown to be accurate enough to guide ships across the boiling oceans, then clock time became commercially important and so philosophically important. By creating absolute time as a concept to philosophise by, Newton established the basis for making universal claims.

We miss this fact that Tyme is not universal, and is a construct of a measuring tool called a time piece. It as an entity does not exist , but as an experience we can group together many many related experiences around the ticking of a clock, or the swinging of pendula or fundamentally the motions of bodies in the heavens!

Motion and relative motions exist as identifiable experiences. Time does not. It is a subjective consequence of experiencing these relative motions.

So there were several ways of pragmatically voiding Zeno's and Parmnides mind games, but it is hard to over state how many scientists still struggle with these doubts introduced purposely into logic by these 2 philosophers.

There reason was simple. Man's reason is not sufficient to apprehend reality! At the end of the day we have to accept reality is what it is, and I experience it how I experience it. I have to build my apprehensions on that basis not on some handed down, unchallenged axioms or postulates! I have to think for myself and believe for myself, and expect for myself from my own experience.

So Hermann did the following

He took 3 points A,B,C and realised that AB + BC = AC was a sum. He then, in the simplest way possible mentally set this as

                       The sum of 3 points.

Now I would have written A + B + C as the sum of 3 points, and in fact you find this in the Barycentric literature, where the coefficients are set to 1

So what was AB ?

This was the product of 2 points!

And so was BC and so was AC !

Within the " law" of the sum of 3 points Grassmann saw the product of 2 points!

So now he had to modify his Förderung.

The sum which you can construct from 3 points involves first forming the products of any 2 points!

Thus to illustrate for 4 points A,B,C,D there are 12 products of pairs of points
AB,AC,AD
BA,bC,BD
CA,CB,CD
DA,DB,DC

Of these 12,  6 are the lineal negatives!
Considering the points as FACTORS in their respective products, automatically means that you have to accept that switching or commuting the factors introduces a negative sign!

So now for any four points in a plane I can justifiably replace any 3 points by a representative product of points providing the 3 points are in a triangular relation in the figure drawn by connecting the points into a perimeter!!

That difficult passage that I struggled with? This is precisely what Hermann was alluding to. The products, the multiplication, the observation of sign value the pieces of the figure , all come together in this insight!

In addition we can now define parallel and perpendicular in this new analytical sense of point products and point product sums!

These product sums occur everywhere in mathematics, arithmetic, algebra ,Combinatorics trigonometry...do they always refer to point products?

The answer is no. With the scheme laid ot by Grassmann just now we can extend this concept into lineal,planal and spatial forms!

Of course to do this you have to select the correct points relative to one another.

By specifying all the points of a form ,like a crystal or a geometric solid these product sums can be specified analytically and subsequent manipulation and analysis and synthesis done.

The lineal Algebra is but one method derived from his general method of space analysis and synthesis.

I hope you can let go of years of incorrect training and begin to apprehend Grassmans style of spatial analysi.

Commentary

I really think that this is a modification of Grassmanns initial impulse. So he did not come to this point view until he started his dedicated work on what until then had been a put downable hobby.

The gaps he filled covered all the deep questions of trigonometry and geometry. And that means tacking the issue of what is a geometrical object?

Sometimes you cannot define something without a tautology. So far I have not found a definition of point, so these I assume are axioms. But then he defines line and then a plane in terms of a product of points.what he does define is a class of points called Schwerpunkt, the sum of these primitive points.

The story of the Schwerpunkt is not finished., but when he started it he was in awe of Möbius, but as he progressed he realised that his simplest concept for three points being summed had an extra point associated with each point. Instead of a point with a fractional coefficient he saw a point multiplied with a neighbouring point, and this product was the verbindungsstrecke, just as the coefficient gave a fraction of this Strecke to a single point.

But then what was a Strecke multiplied by another Strecke?

This was defined as a parallelogram between these Strecke. .

His ideas were still forming?

http://journals.cambridge.org/download.php?file=%2FPEM%2FPEM10%2FS0013091500030923a.pdf&code=abc71143ee5fa3d47ed3eff3d6c22b2f#xml=http://journals.cambridge.org/data/userPdf/

http://www.youtube.com/watch?v=BC1jft03k6M

Affine combinations or lineal combinations

http://www.youtube.com/watch?v=FdgMmHIXA_s

Note the practice in Barycentric coordinates makes little sense, whereas the Strecken approach makes sense..

Converting to Möbius notation is one of those backward steps that mathematicians impose on their students. This turns the Fórderung into a kind of "trick" , while disguising the fact that Barycentric coordinates are a confusing convention

Commentary

The Schwerpunkt episode reveals a source of inspiration for Hermanns work.

It is clear that in 1827 Möbius published his Barycentric coordinate method. This is the year Justus published his treatise on trigonometry and and later on space theory. It is clear that Möbius, a professor in Leipsic would have contributed to the Humboldt reform discussions and collaborations. The fact that he published in the Crelle journal also meant that the young Grassmann would come into personal contact with him, as Hemann published and worked on the editorial board of that magazine.

So both Justus and Hemnn were directly influenced by Möbius, Justus more so. And Justus introduced his ideas on sign into the curriculum of his district . So it was that Hermann came to be acquainting himself with negatives in the first place !

AB + BA = 0 is a pure equation introduced by Möbius!

The Schwerpunkt means the weight point, and is a conception Möbius used to motivate his geometrical proportioning of a line. In a sense a Schwerpunkt becomes a smeared out point !

Now Hermann read many of the great mathematical works in his study to qualify to become a teacher, but he says he only knew of the title of Möbius work. It seems that he absorbed Möbius's ideas by osmosis then, only to find he had come to the same conclusions.

However he makes a bold claim in his 1844 Vorrede to have switched from the student to the lecturer! By this time Mobius had been publishing many articles on various topics and extended his calculus considerably.

This was a bold if not arrogant claim by a primary school teacher who by then had yet to qualify to teach at secondary level. We know that he applied to teach at university level on the strength of this belief, after publishing the Ausdehnungslehre. It was this text and a district commissioners  judgement of it that in fact denied him a Lecturer post at that stage!

It must be said that Möbius recognised the text as an advance on his ideas, but could not apprehend in what way. He suggested sening it to Some academicians he knew including Gauss, and they basically judged it as poorly executed communication of some interesting ideas.

While it seems harsh now, there was a system of accreditation, and clarity of thought and communication was a requirement to teach. Because of the huge scope of his ideas , and the " work in progress"  nature of his presentation. Mathematicians prefer theorems to conjecture, demonstrated proofs to heuristic discussions of how to prove or demonstrate something, and generally a standard format of presentation. Papers were designed to answer specific questions, not to rewrite the whoe of mathematics and science from the ground up!

Newton did it, but from the position of a recognised academic genius asked to solve a problem of orbits by an astronomer, Halley.

As far as anyone knew no one had asked for mathematics to be uprooted, or had a specific problem requiring this kind of solution,( afterwards people began to recollect Leibniz posing this problem). Nor had anyone commissioned an obscure teacher in  to do it! Later Gauss got Riemann to ask for such general solutions in his Habilitation speech in 1854. He asked this of a wider audience than Mathematicians. His scope included physicist, chemists and mechanical engineers .

Gauss felt Geometry had lost its way, specifically because it could not solve the the 5 th postulate Problem.

It is only in this climate that Grassmann's fellow Prussians began  to apprehend what he had singlehandedly achieved, and especially after Riemman died!

Robert his brothers redaction of his work also now had an audience receptive to these kinds of enquiries.
This is when he was rewarded for his life's work , by recognition of his advancement on Möbius' ideas.

Commentary

I have completed the survey of Möbius Barycentric calculus, and observe that Hermann drank deeply of its elixir!

As Hermann expressed it , he almost conceded to the master!  If it were not for his concept of product of points his work would have been done at this point.

Möbius could not understand how you could multiply 2 points!. His calculus relied on length as a measurement and sign as a lineal direction in a given line. But his real focus was on the coefficients of the lines. He therefore reduced the line through a point  A that extended to a related point A', as long as it was parallel to a line through a point B that extended to a related point B', to the point A and similarly to the point B.

This was a notational decision, based on notational laziness. The line was crucial to his geometrical proof of his results! But then he cast it away to focus on the points. In this sense he relied on the concept of number and particularly rational fractions as they contain the fundamental notion of Logos Analogos.

Möbius calculus is pure symbolic arithmetic. It looks like algebra but the Greek symbols represent numbers. The points A,B,C etc represent the only reference to space!

Herrmann's introduction of the product therefore needs some explanation, but it's simplification in communicating and organising the labels or notation on the page is demonstrable and understandable!

Commentary

This is , perhaps the clearest section so far in which Grassmann directs a course of study! We must study Möbius Barycentric calculus as the entry level preparation into his own Barycentric  method!

This is of interest to me, because Möbius is an old school geometer. He has developed along the lines set down by Legendre in his redaction of the Stoikeia of Euclid. Thus, like DesArgues, Poncelet and Chevas, and Steiner he rejected the coordinate geometry of DesCartes!

For us, brought up on Cartesian geometry and mathematical descriptions of lines and curves, we have no apprehension of any other way to describe space! We certainly lose sight of what the Arithmoi are , and how they measure and count space in a mosaic. This becomes an alternative description of surfaces and edges.

What distinguishes Möbius is his use of sign. His use of sign is mathematical, in that it is applied according to the formulary by inspection. The notion of a direction is what Grassmann uses, which encompasses orientation and travel in an orientation. Contra comes out of this process as a specific reversal of direction Of travel.

Studying Möbius I suddenly realised that length and sign were of equal importance to him, and this attitude of mind was reflected by Hermann in terms of length and direction. Suddenly AB + BC  must actually be "equal" to AC . These 2 directions were equivalent to the third direction. These 2 displacements were equivalent to the third displacement. These were " equivalence" relations not numerical sums.

However, Möbius formulae were numerical sums! These numerical sums encoded a mentality to space. That is we can count its quantity and sum its magnitudes by area or volume. These sums recorded the process by which they were obtained, and this was by traversing or tracing out a surface or line.

Commentary

I never thought I would ever criticise numbers and elementary arithmetic, but my research has found nothing but practices that need to be criticised!

The unfortunate thing is that some critics have not gone deep enough, and so their attempts at a correction lead only to a new misdrection.

Why did Hermann intuitively look for the product or multiplication of points?

Some would say that the aggregation of points naturally leads to the multiplication of points, but this reasonable line of thought is in fact counter productive.

There is no fundamenta physicall multiplication!

The fundamental physical actions are aggregation and disaggregtion. It is within disaggregation that we locate the notion of multiple form.

Thus a single form divides into a multiplicity of forms, and we recognise these as factors of the single form.

So why do we not locate multiple form in aggregation? We do, but an aggregating form has no goal form, so we do not sense a multiple form is aggregating to anything in particular. It is that sense that aggregation has a goal which gives us the sense the sum has a solution or an answer or a definite form!

So we develop our mathematics in the division of a single form. In that division we can do aggregation and multiplication of aggregates , division and subtraction all with the sense they have an ultimate " meaning".

  Given this nursery approach, we then extend it to aggregation of forms where the numbers as forms are potentially limitless!

Thus the Arithmoi cover these forms as shards of space, that is a large chunk that has been disintegrated, and we are somehow putting it back together again.

Number bonds in this context are cloely linked to block mosaic patterning, or tessellation of space. The full geometry of the Arithmos is then evident.

The relation of the Arithmoi is precisely that parallel displacement of forms , in order to calculate centres of symmetry  and gravity. The proportioning( logos analogos work)  is precisely descriptive coordination of the observer interacting with the form and its environment to measure it and manipulate it , in order to help synthesise it Abstractly, in our heads, and physically in substance.

Start with the bricks, the pebbles, the apples and herd animals, and learn the knots, the cusinaire rods, the sticks, and the schematic or sketch patterns. Declare their names and dance rhythmically with them, singing and swinging, twirling and hopping. Balance them on your arms and with the balances of commerce, and look at their reflections in the mirror, understanding what is yours and what belongs to some other.
Pay what you owe and collect what is owed to you.
Measure what is your land and know your boundary and your neighbours. Know how to travel to and fro , to what is yours and from what is yours, learn how to see these things through your neighbours eyes.

Hopefully you will find that these processes are ingrained in your soul and your spirit, your very being.
Maybe then you will apprehend that Shunya is everything!

Maybe the mystery that all is one or that dynamically all things balance may cast light into the darkness of "Nothing"! For it is a profound ignorance, not easily got rid of!

If you have ears to hear, listen to what I am saying to you.

Shunya is everything!

Commentary

Barycentric apparently means Schwerpunkt! Baryos is Greek for weight( which is a force!) but is usually understood as mass( which is a quantity of matter). Centris is Greek for a unique point in every circle! So this calculus is based on rotational dynamics.
Schwerpunkt means heavy or difficult to handle, so it is a good synonym for baryos.

This Schwerpunkt is usually translated as a weighted point, especially in statistics and analysis. In a physical sense, a mass is hung from a point , creating a moment of force. This is why Möbius dropped the extended lines, and focused on the coefficient and the point.

However his geometrical proof implied that the connecting Strecken had weight or a mass attracting weight. Thus this mass considered as uniformly spread along the line can be proportioned. In this way the point actually smears it's weight out along the Strecken!

This is of course physical nonsense, but what is physical is that thin wires and rods behave in this way, and that is sufficient justification. Thus the abstract nonsense is a limit case for physical dimensions!

We have here both the concept of a stretched out point as well as an extended, elastic magnitude called a Strecke. The stretchy things in this threads title!

Commentary
Normsn's discussion of Affine Geometry explains Barycentric Coordinates simply. But you see immediately the disconnect . The parallel lines are discarded and replaced by point notation! This is precisely what Möbius did, but what Hermann chose to keep!
Further Hermann regarded the parallel Strecken as a product of points!

The reference to coordinates is lost in the computation.

A point in Cartesin geometry is given by 2 lines called axes and 2 parallel lines that intersect. In Barycentric coordinates we use two parallel lines that do not intersect as axes! We thn use a common line that intersects both as our line of interest . To find any point on that line of interest we use ordinates on the axes. This then enables us to draw a line that intersects the line of interest in a unique point!

Barycentric coordinates therefore give us a unique point. By varying the ordinates on the axis in a proportion that remains as a fraction sum equivalent to 1, we can traverse the intersecting line through all points on the line of interest.

If the proportion sum exceeds 1 or is less than one then a point of intersection on the line of focus does not exist. Using negative or harmonic proportions we can extend the line of focus.

Now by changing the points of the line of focus we can change the position of this point in the plane. This is the Source  of Bezier curves .

In 3 dimensions we trace out positions of points in a surface in space.

These coordinates become very general algorithms used in animation or vector graphic applications  to animate or deform space .

http://www.youtube.com/watch?v=wa-RJJYnwCE

"Schwerpunkt" is literally and figuratively the center of mass.

Thanks Kram1032.

If and when you have time , I would welcome your input into the thread, particularly your version of the translation.

The etymology of the word Schwer, takes me to many ideas, and so I consider them in the commentary. For example besides heavy Schwer has the sense of difficult  or dense.  The idea of a complex point or a dense point derived from nodal connections to all other points in a system suggests itself from the term Schwerpunkt.

Newtons practice of reducing a body to a poit mass is precisely the notion of Schwerpunkt.

But there is another connection. Newton attempted to introduce an explanation of his Fluxions method on the basis of a notion of the moment of a rectangle. This derivation recounted in the foreword to the Principia gave rise to many misunderstandings. Yet it is almost precisely the Barycentric method!

aX + bY,  for a rectangle with sides X,Y, with variations a,b

Moment (http://books.google.co.uk/books?id=OydcAAAAQAAJ&pg=PA45&lpg=PA45&dq=the+moment+of+a+rectangle+Newton&source=bl&ots=B5fBSAcdQF&sig=brU3_bk6wcpVo0dxT2wxqfARaPo&hl=en&sa=X&ei=D0f0Usy2AY-y7Abx64DQBw&ved=0CDoQ6AEwBQ#v=onepage&q=the%20moment%20of%20a%20rectangle%20Newton&f=false)
This formulation of the derivative as a Barycentric equation is more precisely Hermanns notation than Möbius, because the capital letters refer to the Strecken of the rectangle not the points. Yet it is clear that Möbius acknowledged the contribution of the  Strecken of the parallelogram to determining the Barycentric coordinates of the centre.

It is therefore significant that this point plays such a vital role in the analytical methods of Möbius and Newton and Grassmann.

Further we see how it underpins Lagrange's mastery of the multivatiable calculus.

I sense there is more to this point than meets the eye!

Commentary

The spline or Bezier curve is truly remarkable, especially when it is computed real time by a modern ray tracing package/ app! Often called vector graphics it is based squarely on Möbius calculus and shows what a grasp of space curves is possible by the proportional variation method fundamental to all measurement, relation and analysis.

http://www.youtube.com/watch?v=ct_uGOSPtok

And the calculus connection

http://www.youtube.com/watch?v=LFFPbBe7aAs

This really is the fundamental nature of Möbius Barycentric calculus, and thus the fundamental nature of Hermanns Ausdehnungslehre!

Commentary

There is a direct connection between the Schwerpunkt and the so called complex point.

The easiest way to realise this is to use the Barycentric calculus to trisect an arc  of a circle. Trisecting an angle is supposed to be impossible by straight edge and compass. Bur Archimedes demonstrated it was possible. He is often accused by some purists of using Neusis to achieve the result, but in fact he probably used a form of the Barycentric calculus.

For a symmetrical Barycentric Bezier Casteljau system setting t = 1/3 gives a trisection of the arc, and so the required points for the trisection of an angle.

A ruler can be constructed using a straight edge alone let alone having a compass pair available!

The Barycentric notation conceals a reference frame based on pairs of arallel axes. The affine geometry is a coordinate Grometry that makes this clear. Using this geometrical reference frame together with a compass enables oe to establish a point at 1/3 rd circular arc.

But this was demonstrated to be an impossible construction for a so called real value. Gauss of course showed that a complex valued solution was possible. The difficulty was knowing what that meant!

It became fashionable to talk about Argand diagrams and eventually the complex plane, where the solution could be directly constructed,. However few realised that the Barycentric calculus gave a real solution to this problem, mostly because they were not looking for a frame of reference beyond the Cartesian as established by Wallis.

anti trisection bias (http://www.geom.uiuc.edu/docs/forum/angtri/)


Jim Loy (http://www.geom.uiuc.edu/docs/forum/angtri/)

Thus the direct relationship between the Schwerpunkt and the complex point either as a polar coordinate or a complex plane coordinate was missed. The direct connection to vectors was also hard to accept, but over time it has tended to come together, through he transformation rules.

Later we will see how Hermann derives the as the constant required to describe Strecken associated with the circle in the plane uder an exponential trig relationship.

Commentary

A lot of French École teachers develop the Barycentric method quite nicely, but it is noticeable that they use Hermanns Strecken notation or early .möbius notation.

As I watch the examples unfold, and the discussion proceeds it is noticeable how finding ones way to the Barycentric is a process of directions and proportions. Thus the Strecken notation is ideal for positioning a proportion in a direction. One part of the lineal proportion, therefore is commonly described as a vector, but the complete picture is actually a lineal proportion.

The specification of a Barycentric has become formalised, and do the handles or labels are noticeably dependant on who is teaching you! This makes the understanding dependent on a certain labelling, for many. It is important to get behind the labels, because the calculus is very general, and seems to crop up everywhere where physical systems are being described. The Keplerian laws are for example based on the Barycentric of a planet and it's moons. In that regard, the elliptical error in gravity, where the second focus of an ellipse is often in so called empty space , clearly reveals the geometric nature of the description. That is, the so called gravitational force is a geometrical construction, and implies therefore " action at a distance" !

The dual rotation of planet and moon systems around the barycentre is a result that still needs to be explained!

Newton's centrifugal  AND centripetal acceleration model is based on a fixed rod tether. Bothe accelerations are therefore necessary to explain apparently fixed dual orbits. The question Newton asked is where does this force pair originate?

Because og magnetic force he suggested as did Gilbert and Biyle that it was some form of magnetism, but he did not know enough about the electro Thermo magneto complex to possibly form or frame a hypothesis. However, he, Bernoulli Ruler etc were able to discuss many rinciples of resistive media( ie the  indiscernable  aether) that would account for certain observed motions, but whether it was so or not they would not be drawn. Aristotle on the other hand would have said" it was so! " which is a fallacious argument he committed often apparently!

Newyon was open go the notion of impulse generated forces, that is a subtle variation on a collision model, and followed La Sages work with great interest, but no public support!

His attempt at a fluid dynamic explanation rested on a description of a fluid as a resistive medium. However his attempts to match the observations to the geometrical principles and calculations failed to give satisfactory agreement. His interest in the Cotes Euler formula came too late and with the unfortunate early demise of Cotes, for hom to apply his impaired intellect to a redaction of his model.

Today we have no such excuses. We have a lineal Algebra, an  identity, a formulation of vorticity and numerous other principles of fluid dynamics to draw upon. The extensive abilities of the computational sciences now mean we can do what even Newton struggled to do computationally, even if we do it with less wit and sagacity.

It may be that this is what Einstein has done, but those who follow Einstein and those who follow Planck, seem curiously at odds. Special and General relativity versus Quantum Physics seems to be the status quo with us left in the middle not knowing who or what to believe!

Believe no one! Instead figure it out for yourself.

I'd really like to give more feedback on all your work here. I just have to say, your collections are rather overwhelming. It's impossible to keep up with it all.
And often it's a bit hard to see what your point actually is. You seem to assume fairly high standards and you take all these topics in a deeply philosophical manner.

I think that, while there is nothing wrong with a philosophical stance, and sometimes it can even be helpful, if you take it too far, it is more confusing than anything else. And I fear you are taking things too far at times.

(* Hestenes, after all, began as philosopher interested in truth and then switched to math because he thought that that's where a good, rigorous notion of truth lies, and that eventually lead to the very nice, clean nature of Geometric Algebra)

Can you, perhaps, start at the beginning, fully explain your points not only on a philosophical level but also on a mathematical one and then slowly (slowly being the keyword) build the explanation of your views from there? - There is nothing wrong with hinting at eventual philosophical or mathematical implications, of course. But don't get derailed. Just give one or two sentences of future outlook and come back to the ground. Take your time to lay out more of the basics, then more people will be able to understand your points.

The other thing you are doing is putting a lot of value into the precise history of how things developed. I agree that this, also, can be both interesting and enlightening. Especially, in hindsight and through views and tools informed by our modern understanding of math, we probably can significantly improve some old historic flaws if we understand how they came to be in the first place.
Once again a good example for this is Geometric Algebra which is just a big combination of historic ideas that initially were perceived to be very different approaches to the same problem.
Another, as far as I can tell, seems to be Homotopy Type Theory, combining three completely separate notions into one coherent one that gives a formulation of the foundation of maths on a level that promises to be both the most rigorous and the most intuitive one yet.

In that way, history does matter. However, while there certainly are historical oversights that may or may not have slowed down progress for years and years(**), for the most part, new developments are there for good reasons, well understood and straight forward to apply and, unless you want to point out such a historically grown problem, there is often little use in going back into the past and solving some problem that nowadays is hard not to solve in less than five steps that might once have taken twenty or more steps of deep insight, just because the theory behind the techniques wasn't all too well understood yet.

(** All the various formulations of geometric spaces as used most commonly in physics today and also as taught in schools and universities, vector-, matrix-, tensor-, spinor-, twistor-, or quaternion formulations, are a confusing mess that often seems highly arbitrary and the notation hides a lot of the inherent internal structure of a given problem. Something that can be avoided with the grand unification of geometric spaces found in Geometric Algebra which makes working with all those things crystal clear and uniform. Had the people who once decided this already known that, they surely would have put more value into the teaching of Clifford- rather than Vector Algebra and we might live in a different world today. That's where historic retrospection is important.)


And while I'm at it: "Strecke" simply means line segment. In some rare cases it might mean "distance" but really, "distance" is just the length of a line segment, so this still indirectly refers to a line segment. - That's another thing. It would be advantageous if you kept your explanations to the common words of the language you use to write your explanations in. As far as I know (and I might be wrong) it just so happens that either the largest or the second largest group of people in here are from a German speaking country (the other one, from what I gathered, seemingly being from English nations, probably mostly from the US), but not everybody will understand a German word if there is a perfectly good English word to be used instead.

One of the rather rare exceptions to this is "Eigen-" which is a German term commonly used in Math and Physics and refers to something that is special to a particular mathematical object and, in some sense, gives that object its most natural description. The English word for that would be "Own-" but historically, "Eigen-" got used instead.

However, for things like "Schwerpunkt" or "Strecke" or other such words, there are perfectly good English words that you can use just as well or, since you are using English to write all your posts, even better.

Hallo Jehovajah, I hope this post will give some help on the use of german expressions.

For translations between english and german expression
I use the internet page leo which, for my purposes does an excellent job:
http://www.leo.org/index_de.html (http://www.leo.org/index_de.html)

For Schwerpunkt I get the following results:
http://dict.leo.org/?lp=ende&from=fx3&search=Schwerpunkt (http://dict.leo.org/?lp=ende&from=fx3&search=Schwerpunkt)

Here we have "Förderung"
http://dict.leo.org/ende/index_de.html#/search=F%C3%B6rderung&searchLoc=0&resultOrder=basic&multiwordShowSingle=on (http://dict.leo.org/ende/index_de.html#/search=F%C3%B6rderung&searchLoc=0&resultOrder=basic&multiwordShowSingle=on)

Here we have "Strecke"
http://dict.leo.org/ende/index_de.html#/search=Strecke&searchLoc=0&resultOrder=basic&multiwordShowSingle=on (http://dict.leo.org/ende/index_de.html#/search=Strecke&searchLoc=0&resultOrder=basic&multiwordShowSingle=on)

Does this translations meet your meaning or would you like to express something different?

I own the book "Lehrbuch der Experimental, Bergmann Schaefer"
It has a "Fachwörterverzeichnis" that contains translations into englisch.
Schwerpunkt - center of mass
Strecke        - distance

Before the internet age I used the dictonary: "The comprehensive dictionary of current german and english - The Collins German"
Containing 450 000 translations.
This book is mirrored by the Klettverlag in Germany as PONS Deutsch-Englisch Englisch-Deutsch
This book did always a good job for me.
I have bought it for 16.50 pounds in Manchester.

Hermann

Commentary

The Berührungspunkte is the next idea mentioned by Hemann as his discovery.

http://YouTube.com/watch?v=CizogTmSju4

http://www.ifi.unicamp.br/~assis/Archimedes-2nd-edition.pdf

http://www.cs.uiuc.edu/class/sp06/cs418/notes/10-MoreSplines.pdf

I think he has made a point that has been overlooked. The research into the barycentres, based around Möbius Barycentric calculus has revealed some taken for granted assumptions, not the least of which is the static barycentres of a system!

The law of levers is used in some calculations of the orbits of planets, and it's validity and valid use is not often checked by those who teach it.

The biggest discovery for me is the impact it has on rotational motion ! Because orbital motion is ellipsoidal it is clear that the barycentres cannot be stationary! A dynamic barycentres is not discussed very much anywhere, although it is known.

The barycentres of a system is not the same as the centre of symmetry, because the barycentres for a dynamic system is dynamic. Thus it is ok to calculate a fixed barycentres for a fixed mass, but it is misleading to calculate it for a dynamic system without mentioning it is dynamic.

The Relation between the Schwerpunkt and the Berührungspunkte is an. Indication of this dynamic nature. Hermanns insight not only meant he could determine it for " static" systems, but also for dynamic ones. This is a much more general idea than the principle of levers ! It reveals something about the transmission of force in space and action at a distance.

When it was first determined that a centre of mass can lie outside a body, some light bulbs should have switched on. Instead, the focus was on making the rotation uses look like the translation ones, and obscuring this physical difference.  Certain shapes can only be moved translatioally if a force or pressure applies equally to points or areas around the barycentres , otherwise rotation results!

The concept of angular momentum is flawed by this kind of analysis, without caveats regarding the Barycentre. Discounting the Barycentre has made the gyroscope into a modern mystery, whereas including it not only explains the gyroscopic behaviour but reveals how "empty" space can influence the dynamics of a system.

Newton's third law, the third stage in setting up his compete dynamic equilibrium or dynamic inertial frame, obscures a lot of internal physical interaction, that is body force interactions . We now commonly use a fluid dynamic model for body force interactions, but we use strain and stress parameters, both surface and onto to model these interactions. The Barycentric calculations are not to be found, because they are simply not included.

Yet, consider as a strain propagates through a body the Barycentre of that body also is dynamically altered. This effect means that for a given body , wave propagation of strain will induce rotational oscillation!

The Berührungspunkte with the Schwerpunkt by definition mean this is an inherent property of space, no matter how dense or vacuous! This is action at a distance, and it applies to any measurable magnitude. Thus any energy field, fluctuating will have a dynamic Barycentre that also oscillates. The significance is, that in such fields rotation will be induced as a matter of course for some fractal regional systems.

Typically a point in geometry has a value of position only. But physically, especially in Newtons point masses, motive must be attributed. The one motive that is ignored in a physical point mass is rotation, but it must also be included to make complete sense of rotational dynamics.  Berührungspunkte and the Schwerpunkt rotate, and thus the null vector hides the rotation of the barycentric point.

For a Schwerpunkt outside a physical mass, I may concede that point has no physical rotation, but in fact I do not know if that is the case, although it seems likely.

The doubt arises only because the transmission of any strain wave into a space outside the mass will inevitably obey the principles for that spatial density. It may thus severely dampen the incident strain wave resulting in null rotation. On the other hand it may increase the transmission of the strain wave resulting in higher rotation, if the density of the space is greater! In such a case it would take more time to speed up the rotation of. A denser medium, as we know.

On Wikipedia an section on barycentric coordinates can be found.
It contains alsosome very interesting animations on that issue.
http://en.wikipedia.org/wiki/Barycentric_coordinates_%28astronomy%29 (http://en.wikipedia.org/wiki/Barycentric_coordinates_%28astronomy%29)

http://spaceplace.nasa.gov/barycenter/en/ (http://spaceplace.nasa.gov/barycenter/en/)

German - English translation
http://www.linguee.de/englisch-deutsch/uebersetzung/barycentre.html (http://www.linguee.de/englisch-deutsch/uebersetzung/barycentre.html)

I started to develop some JavaScript to let the planets move. But its not ready for publishing:
http://www.wackerart.de/Animationen/Animation1.html (http://www.wackerart.de/Animationen/Animation1.html)
The page requires several minutes to get loaded.

Hermann

Hallo kram1032,

I have the same problems as you following Jehovajahs thoughts. I think its important to have him write his idears down.
He has given a lot of excellent and understandable background information in the form of YouTupe Videos and papers.

Specialy the excellent lessons of:

• Norman Wildberger
• Leonhard Susskind
• DrPhysiks

At the end of last year I spent much time to see the videos which gave me a deep view in physics and mathematics.
I discovered that Jehovajah was writing about mathematics and physiks I am personaly very deep interested in.
I also have the problem to understand all his idears. So I started to write down my own thoughts.
By doing so many shapes of Jehovajahs thoughts become visible.

For me the the main problem is, that I do not have enough time to work on the subject.

Hermann

Kram1032 great post thanks!

All I can say is I will try, perhaps in another thread.

Please if you could, work with the text of the Ausdehnungslehre 1844 and translate it yourself and post it to the thread!
Please!

See I have asked you twice because for me the meditative understanding of ths text can only be enhanced by different views.

My views and transactions are as stated. Sometimes I come back and change them as I gain further insight. It is a philosophical adventure!

You may know that Robert made precisely the same point to Hermann when his book failed to get the attention it deserved. Hence the redaction in the 1862 version!

Because this is a meditative translation I am not restricting myself to just a one to one correspondence, but looking Los at how the idea or word was perceived or morphed through time and historical forces, and human nature!

Keep on contributing kram1032 and Hermann. It is lovely to have such interaction. Constructive criticism is also welcome!

From the german Wikipedia I learned that Hermann Grassmann wrote two main versions of his "Ausdehnungslehre"
One  "Ausdehnungslehre von 1844" and a second Version form (1861 - 1862?)
Both seems to be completly different.
Also from these to basic books there seems to be different versions.
http://de.wikipedia.org/wiki/Hermann_Gra%C3%9Fmann (http://de.wikipedia.org/wiki/Hermann_Gra%C3%9Fmann)

Lloyd Kannenberg had written an englisch translation of Ausdehnungslehre
http://www.amazon.com/New-Branch-Mathematics-Ausdehnungslehre-Other/dp/0812692764#reader_0812692764 (http://www.amazon.com/New-Branch-Mathematics-Ausdehnungslehre-Other/dp/0812692764#reader_0812692764)

Wouldn't this be a starting point to combare the german and the english text?

When I look at the following text of the 1844 Ausdehnungslehre I discover that the first Pages of "Vorrede" are missing. With this scan.
https://archive.org/details/dielinealeausde00grasgoog (https://archive.org/details/dielinealeausde00grasgoog)

The same problem with the following scan:
http://www.uni-potsdam.de/u/philosophie/grassmann/Werke/Hermann/Ausdehnungslehre_1844.pdf (http://www.uni-potsdam.de/u/philosophie/grassmann/Werke/Hermann/Ausdehnungslehre_1844.pdf)

The "Vorrede" in this scan seems to be complete. Know days a book has a "Vorwort" interesting how language changes over 150 years,
But it is the 1877 issue
https://archive.org/stream/dieausdehnungsl00grasgoog#page/n7/mode/2up (https://archive.org/stream/dieausdehnungsl00grasgoog#page/n7/mode/2up)

This I think this is the Kannenberg translation.
http://books.google.de/books?id=yeGPeaPVLKoC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false (http://books.google.de/books?id=yeGPeaPVLKoC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)

Hermann
P.S The word "Ausdehnungslehre" I would translate to "Expansion Theorie", I don't know if this translation meets the spirit of the book.

As said, you are moving too fast to really keep up with.
I'll try my best to from time to time contribute though.

As of "Berührungspunkte" - this is plural. The singular would be "Berührungspunkt".
"berühren" means "to touch", "Punkt" means dot (.) or, in this context, "point" or "vertex"  (point is more of a topological thing and can be any particular element of a set, while vertex is a point on some kind of mesh, for instance a corner of a cube represented by 8 vertices, 12 edges and 6 faces)

If things touch, they are considered tangential.

Thus, a "Berührungspunkt" is a tangential point.

As far as a translation for Ausdehnungslehre goes, it's written in really tough, old-ish, overly long-winded German. It stems from a time when academics in general wrote like this. There was quite some elitism in that style of writing, making it overly complicated to prove that only a certain class of people could even get this. There still is some of this in the modern day and age but today's papers and scientific literature is written vastly more accessibly.
A close familiar of this is legalese, where TOSs and T&Cs will be 300-page-documents written in a way that makes no sense at all.
Here is a science spoof example. This is from an April Fool's episode and it's actually a remake of an older joke-campaign that first parodied this practice:
http://www.youtube.com/watch?v=wwmdf5m9khg

Despite this, I can at least try to translate some of it. Now that I know a bit more about the underlying subject matter, perhaps I could actually be more successful with that.
However, Hermann's suggested books may also be valuable.

@hermann: I'm completely with you: There is quite some value in jehovajah's threads, but they are incoherent collections of thoughts that are hard to untangle, and also to keep up with. A lot of it could be helped with by a good dose of focus though.

Thanks both of you.

I have looked  again at the text and Hermann wrote Berührungspunkte. Your suggestion of tangential points is interesting and gives me something to review the translation with.

While I welcome your comments and corrections, please do not feel you have to keep up with me! I have tried to clearly mark my translations and my meditative comments. This format I hope allows any one to translate any piece in the Vorrede, just giving the page reference. While commentaries may become a bit intertwined if we agree to identify commentary on the text as such , I think it will be clear by author who is commenting on what. Then general discussion like this need not be distinguished unnecessarily.

I am the first to admit my German is bad. But I find it more than rewarding to engage with he text rather than a translator!

However if anyone wants to use Kannenberg I am happy with that, and it is a fine translation.

Finally there are many versions of Die Ausdehnungslehre authored by Robert, not Hermann, because Robert saw it as a duty to expound his Fathers ideas as did Hermann. However it is Hetmann who actually corrected a supposed flaw in his Father's theory. And it is Hetmann who laid the philosophical groundwork for the later 1862 redaction mostly edited by Robert, in collaboration with Hermann.

And of course it is Hetmann ho singlehandedly set out the "vector" or Strecken  algebra as an example of the power of his analytical and synthetical method.

Thankyou so much Hermann and kram1032!
First for contributing to the thread and secondly for empowering me or anyone to engage with the text directly themselves!
 
The Berührungspunkt is a case in point. The tangential contact points or points determined by tangential contact now refer  me to incircle and exterior circle constructs!

Schwerpunkt as weightpoint  ( a point where weight acts, or a weighted point) seems to suggest itself as appropriate. Even point mass( Newton) or point weight carry the notion succinctly. It is a physical point even in " empty" space.

I am reviewing the translation in this light, and will amend some commentary accordingly, but without obscuring my tracks I hope!

A weighting point with its associated weight would typically be a Gewichtspunkt with its Gewicht.
That being said, some people may refer to it as Schwerpunkt.
Still, normally, Schwerpunkt should refer to center of mass. - And barycentric coordinates typically indeed are based on a center of mass.
In the simplest form, you take the three corner-vertices of a triangle and replace them with just two coordinates that describe the distance from the center of mass - the average location of the three points.
This coordinate system is great for procedural texturing the surface of an object. Though with more vertices (e.g. if you are dealing with quadrangles or other n-gons with n>3), there is no longer a single, simple, natural, well-behaved barycentric coordinate system along the surface. There are two reasons for this:
First of all, in case your n-gon is actually flat, e.g. all the vertices lie in the same plane, and convex, e.g. there is no "dent" in the n-gon, then only 3 points are needed to determine a fully valid barycentric coordinate system.
So for n>3, anything more complex than a triangle (which notably always is planar and convex in euclidean geometry), the problem is over-determined and you get a valid instance of barycentric coordinates for each combination of three of the n-gon's vertices. None of them, however, will have the nice property of being constrained to a nice, compact interval anymore. Usually, you define your coordinates such that to go from one point to another, you need to go from 0 to 1, e.g. one of the vertices will have the coordinates (0,0), one will be at (0,1) and the last one will be at (1,0).
For an n-gon with n>3 you can't easily have all the points lie on such notable, "nice" coordinates.

And in case you look at a bent or concave shape, things get even worse. Instead of being over-determined, the problem suddenly naively is unsolvable.
You'll need to think of more complex ways to define yourself a nice, "well behaved" barycentric coordinate system.
Ideally you'd want such a system to:

• describe all corner points, e.g. all vertices with simple coordinates like [0,0], [0,1] or [-1,1]
• be smooth (e.g. arbitrary derivatives at either the center point (the center of mass of the vertices) or along the edges and vertices should always remain continuous)
• deal with bent or concave shapes
• be reasonably fast to calculate, ideally in a closed-form expression
• be smooth in another sense (I don't know how this property is actually called but basically you don't want your coordinates to wildly oscillate - in a perfect world, the second derivative would already be 0 everywhere)
• be unique - e.g. no point inside your n-gon should have more than a single coordinate associated with it (this is another problem for n-gons that doesn't happen for triangles)

I believe some of those are mutually exclusive and I'm not sure if all of them even are possible for all n-gons, so which coordinate system you end up using partially depends on the things you want to accomplish.
Finding "the perfect barycentric coordinate system for arbitrary n-gons" is still an open problem and a lot of research goes into this.

And all I said above also applies to volumes, in which case the natural shape that has no problems is the tetrahedron, while any volume with more vertices necessarily has some problems. This is especially important for finite elements methods which have become one of the most important tools for simulating complex physical systems like car crashes or  other strongly deforming problems.

Thanks Kram1032.

The more I research the more "applied" the Barycentric method seems to be! Still the tangential contact points are only vaguely associated in my searches, but I placed a link to the most direct I could find.

My interest in such a coordinat system is not just for smooth curves or surfaces. I am more inclined toward contiguous surfaces or curves.

However, in this thread I am seeking a global understanding of Grassmanns thinking.  As I translate I refine and review my wild conjectures closer to what the text probably means.  I do it this way to avoid being too narrowly constrained, but as I go along backward constraints necessarily apply.

I do not think at this stage Grassmann got it all right. He certainly freely recounts modifying his concepts.. I am ok with that because it is heuristic and natural. Yes I agree it is confusing, but it provides me with insights and sensitivities that keep me far from any dogmatism I hope.

Nice survey, by the way! Very informative and clearly presented.

Commentary

The Clifford algebra even the Geometric algebra derive from this analysis and synthesis. But I have to say it is more accessible to the student in Hemanns format than it appears to be in modern presentations.
Hermann reveals his basic product rules in geometric descriptions. He also highlights the role of the product of points, and finally he relates the summation format with the inclusive point products as the principal and simplest law or rule.

In every discussion of the Barycentric method I have seen the ' vector " approach has been the fundamental superstructure of the presentation. In that regard it has been more like Grassmann than late and lazier Möbius notation.

The product rules laid out in this section, plus the summation format of those products is what begins the meaningful relationship of this method to geometry. Starting with a product of points gives me a line segment or Strecke. But the Line segment is bounded by those 2 points.

A sum of 2 points gives me a weightpoint or a Schwerpunkt. Now I know it is almost exactly early Möbius Barycentric calculus. That is the 2 points are extended to 2 other points by parallel lines to form a parallelogram with the product of the 2 points being one diagonal and the product of the 2 new points being the other diagonal for equally weighted or equal weightpoints!

As the weighted points are independently varied the parallelogram becomes a quadrilateral with only one pair of parallel sides and the Barycentre of the system of weighted points moves along the originating diagonal.

So contrary to Möbius instinct, dropping the construction lines for the proof of his Barycentric theorem actually obscures how the system works, not clarifies or simplifies it!

I can now see how in Hermanns system, a point added to a line segment ( Strecke) actually gives a weightpoint or Schwerpunkt dependent on the weighting of the added point, and the direction associated with that "construction line" weight. Ideally to work it has to be parallel to the line segment in the sum.

For any 3 arbitrary points we can work out the Schwerpunkt or weight point, the sum of a point and a product of the other 2 or the sum of any pairs of produced points given the fundamental rule for the sum of 2 Strecken in a 3 point system.

This generalises to 4 points in the combinatorial way , based on the products of points. A point in this system is clearly a Schwerpunkt with a weighting of 1.

For me the confusion starts with the imposition of the notion of a vector space. This abstract entity is written in set notatin so RxRx...represented by Rⁿ is the n tuple vector notation for an axes system of n orthogonal axes generally( but this is not clear).
The direct sum of n R sets R + R +... Would seem more in keeping, but the internal products in Grassmans method preclude this set structure.

Thus we do not naturally reflect the lineal combination of basis " vectors" in this set structure. In fact , due to the heavy number line use of R the lineal basis of a Dedekind cut for example is lost and we look for abstract numbers.
http://www.youtube.com/watch?v=wwmdf5m9khg
Applies, I think!

Normans simple device of square and round brackets emphasises this subtle distinction . But in addition his primitives are natural numbers or extended to rational numbers, which are coordinated by the underlying axes geometry , and then distinguished as numbers or vectors by bracket notation.

The sum of vectors then becomes clear as a geometrical type constructed from differentials in the Rⁿ space.

To convey all this set up it is usual to say let Rⁿ be a vector space. Then we define an element of it as a lineal combination of basis vector primitives.

All very neat, but somehow it cuts me off from the actual mechanical space that gives rise to geometrical intuition! Also points are not clear in this set up. In fact Rⁿ is used to represent both!

The dot product and the wedge product and the general product also create a mental strain, especially if one is not really comfortable with them as a product or multiplication!

The generalising of product into a complex aggregation process is best compared to long multiplication! At least then a student might grasp that a product is essentially a sum of sub products!

Further, the arithmetical nature of these set ups is crucial to communicate. In that regard Algebra confuses the issue by purporting to be something other than symbolic arithmetic!

Looking at the reference Hermann gave to a Kannenberg translation of the 1862 redaction, it is clear from the get go that this is all about arithmetical summation of multiples as sub products. By this stage in his thinking, the lineal aspect is lost in the number theoretical approach. This I guess is down to Roberts influence.

While not called such the system is reminiscent of the p-adic number systems in which relatively prime numbers replace the prime powers. In Euclid these are called protoi Arithmoi.
That being said the application of Grassmans insights derived from his own Schwerpunkt calculus, which coincided with Möbius Barycentric calculus have still to be explored in his own treatment.

Commentary

The Schwerpunkt Kalkül that Grassmann developed applies equally to arc segments as it does to line segments . In fact a mixture of line and arc segments gives many naturally trochoidal curve locii and surfaces. The flexibility of this method and it's power has much to recommend it. But it is only the technologists and the mechanical engineers who exploit it to the fullest potential in their manufacturing processes..

Again, Laz Plath at qqazxxsw on youTube showcases the extraordinary potential of trochoids, which are a Barycentric calculus of incredible potential for curved surfaces as well as lineal ones.

Commentary

http://www.cut-the-knot.org/Generalization/ceva.shtml
Chevas theorem may relate to Hermann's comment on the tangential points

Kalkül = Calculus

So Schwerpunkt Kalkül probably is the calculus of the center of mass or, perhaps, something like the barycentric calculus you mentioned before

Commentary
The Schwerpunkt in Grassmann's method naturally associates to Möbius' Barycentric calculus Hermann was overjoyed to see. This was because his Schwerpunkt concept used the same notation as Möbius. But, something changed! Hermann seems to say that the roles switched! He was no longer the student of Möbius but the teacher of Möbius in the succeeding episodes!

As far as I can tell Möbius published his method in 1827. But continued to develop and refine it over his lifetime. He published in the Crelle, a journal that Hermann published in and even for a time was on the editorial board, until he fell out with the political line the journal started to take.

The only real change Möbius made was in notation. It seemed minor and insignificant to him. He replaced the constructing Strecke with the the initial point label.

What he had done, in German, was replace a Strecke with a Punkt!
In English he replaced a line segment with a point!

Now I called this laziness in notation, but there is a powerful precedent in Newton, who replaced a corporial mass(Körper) with a point mass. However the significance is entirely different. In his establishment of the Barycentric principle, möbius does not use a mass at all, he uses 2 parallel Strecken passing through the points A and B.

Thus , note bene!, these are point Strecken!  They were labelled AA' and BB'. The length of these 2 Parallel Strecken corresponded to the mass or more accurately the mass weight hung on that point. By a principle of Archimedes, the Baryos, or mass of the system was balanced at the fulcrum, the barycentre. This fulcrum was represented by a point

Hermann therefore understood the Schwerpunkt as a point Strecke!

A point line segment does not make it any clearer, because the intuitive gut idea of Hermann's Förderung was that line segments could represent quantities! Thus the weightpoint is the point at which the mass is attached and through which the associated weight force acts. But geometrically this was represented by a line segment.

Being a line segment it has a direction. While it is tempting to draw both line segments as acting in the same direction, ie as a force vector , Möbius equation specifies the line segment must have contra direction!

This is geometrically justifiable, because we wish to construct a parallelogram in which the second diagonal cut the focus diagonal at the point of balance.

However, the parallelogram is not general enough for differing masses. The constructed figure is actually a trapezium in general. The Barycentric calculus is actually based on trapezoidal geometry.

Now the point is that Hermann's Schwerpunkt is his own conception, and his own meaning.mwhatever it translates as, hermanns concept was lineal and thus a directed line segment.

Returning to the concept of a point vector, as it is commonly called, Norman, using point vectors describes how the Barycentric coordinates may be found, by given proportions. However Hesse point vectors come from an arbitrary point origin. Using a concept of a point at infinity, common now in geometry, we can see that we recover the parallel lines that Möbius and Hemann based their calculii on. The contra Line Segments can be marked off on these parallel lines and the trapezium constructed, which determines the Barycentre.

The new idea, developed it seems with this point vector notion is that these line segments do not need to be parallel! It is the proportions of similar triangles that was key. The Logos Analogos of Eudoxus underpins this method , and a Schwerpunkt can also be identified as a point line segment, or a point Strecke.


OMG!
http://mathworld.wolfram.com/Trapezium.html

I find from this quote that Möbius was a student of Archimedes?
http://books.google.co.uk/books?id=AXl9j4n2iP4C&pg=PA29&lpg=PA29&dq=trapezium+barycentre&source=bl&ots=WfSfujRmmn&sig=FsxOwJf-5UGijS5jeqhjtd4rte8&hl=en&sa=X&ei=A6X9Uu2hItGRhQfW3IFA&ved=0CDYQ6AEwAw#v=onepage&q=trapezium%20barycentre&f=false
http://uk.answers.yahoo.com/question/index?qid=20070422093416AAVaU4s

Note in these answers no mention of Möbius method!i

There is also one subtle point usually finessed by referring to Archimedes fulcrum: the coordinates switch the line segment relationship to the point. Using the point vector approach the correct relation is maintained. The finesse overlooks the important anti commutativity relations in space.

Commentary
The barycenttic method again
http://books.google.co.uk/books?id=AnPl88uF_XEC&pg=PA1269&lpg=PA1269&dq=trapezium+barycentre&source=bl&ots=kiVPtjnSvU&sig=8NFTdsR_3_MOxrW543tQDGf4oYE&hl=en&sa=X&ei=j7L9Uv_fOor07Aal4YGIAQ&ved=0CDEQ6AEwAjgK#v=onepage&q=trapezium%20barycentre&f=false

This discussion of the centroid is the nearest I have found to the allusion to the weightpoint and the tangential contact points.

http://math.tutorvista.com/geometry/centroid.html?view=simple

Wow! "The theory of stretchy thingys" topic seem to be a "brother" of the topic "light" and same long
 :peacock: :wow:

Ausdehnungslehre Vorrede p vi

Footnote : in practice it usually shows itself quickly how through this  method of analysis the difference between the analytical and synthetic treatments of Geometry are completely removed.

Back to text: p vi and vii.

Commentary
An earlier passage in the light of the Barycentric calculus now makes more sense!


" I did not realise what a rich and fruitful field of study I had mapped out, and thought it not worthy of consideration until I combined it with a related idea, found in the notation of geometrical products as presented by my father!

In following this notation, as my father would apprehend it , it dawned on me that not only the rectangle, but more importantly the parallelogram, should be considered the same  as the product of 2 adjacent sides, jostling the one against the other, even  if One specifically   Once again apprehended the product not as of lengths but as of both   line segments with their length and direction firmly combined !
In which thought I now brought into combination the aforementioned notation of the sum and this notation of the product, and thus it was revealing itself the most stumbled upon Harmony, specifically, even if I

• replaced any  perceived sum of pairs of line segments (in the sense already given to you)  by a third, lying in the Same plane line segment, in the planar constructed sense to multiplying;
• multiply the individual pieces with this same line segment;
•and add the products with the appropriate careful observation of their positive or negative values;
Then it shows itself that in both cases every time the same result has to be coming about and has to proceed onward"

Thus for 3 points A,B,C 3 Strecken / line segments have direction and length AB length c, BC length a and AC length b.
In combination
cAB + aBC=bAC.
Combining the length and the direction.

Thus replacing
cAB + aBC by bAC and multiplying it by itself gives
(bAC)² = bAC(cAB + aBC)
—>bcACAB + baACBC

This is in the planar or 2 dimensional sense attributed to multiplying lengths in the plane to calculate areas. However, the usual convention was to use rectangles and Pythagoras theorem for line segments not rectangular. Hermann innovates on convention by allowing any adjacent line segments to multiply(ie no overt use of Pythagoras theorem).

But now look what happens:

c(cAB + aBC )AB  + a(cAB + aBC )BC  –> (cAB)² +caBCAB + acABBC+ (aBC)²

Pythagoras appears !

It can be set to 0 revealing again the anti commutativity.

Although Hermann does not state this , extrapolating on the concept of lineal summation, we can consider the product of a line segment with itself as lineal " multiplication". Thus planar multiplication can then be thought of as the product of any two general Strecken ( in parallelogram formation and adjacent) . These necessarily lie in the same plane.

Thus using the perceived sum in this way reveals Pythagoras as lineal multiplications and 2 planar multiplications or planar products. , by zeroing Pythagoras we get lineal products are zero planar multiplications, and planar multiplications are anti commutative!


This planar construction to multiply quantities is not properly understood. Justus got stuck here in his analysis of Mathematics and it's foundation in arithmetic. He got stuck because arithmetic is founded in geometry, which in turn is founded in and refined by mechanics,

I was Newton's opinion that the two worked hand in glove to perfect each other, but at the end of the day for me Mechanics and interaction with spatial objects or regions founds and refines our Metrons, which , providing we use the concept of Monad, establishes our arithmetic.

However we do not need to use Monad, Monas is just the concept of 1( one). Many of us do not even realise it is a concept!

If we do not use this concept, we may still use our concepts of comparison: greater, lesser, dual. This leads to Logos Analogos thinking, often called proportioning, but less well recognised as reason, or rational thought. Within this praxis systems of reasonings often called logic developed, but that is another story.

Thus planar multiplication is not well understood, nor is multiplication in general.

The Pythagorean school started not with multiplication, but division into factors and or fragments or parts. From this the concept of a whole as a sum of parts is crucial. This gives a singular purpose to summation or aggregation.

Within this summation context a Metron as an entity representing Monas becomes psychologically important. With a Metron we can count by measuring the Metron against the focus object. The focus object or any entity can be determined as the whole , the parts of this whole may then be counted as varying parts, or a standard part can be used as a Metron to measure and count all other parts.

These parts are factors of the whole. Nevertheless, with a standard Metron, various factors can be defined as counts of this Metron.

Recognising the whole and a lesser entity as a Metron, allows the conception of the whole as a multiple form. This simply means the whole is factored from many parts or many repetitions of the Metron.

The form of the whole may be geometrical, or spaciometrical. That is to say it may be set in the ground plane or freestanding in space.

The implications of multiple forms is down to each individual to determine, but most learn factor tables for standard units. These are misleadingly called multiplication tables. They are factorisation tables or factor product tables.

The geometrical representation of these multiple forms, usually set in a mosaic, led to various geometrical standard shapes being formed or created and measured by standard Metrons. These measurements are the basis of the planar sense of multiplication.

The various formulas for the areas of the standard geometrical forms further define the planar sense of multiplying. Their derivation however, were strictly by geometric proofs. Thus any geometrical manipulation in a sense is part of the process of establishing a formula to " multiply".

Commentary

The lineal product is not the same as scalar multiplication.

If we enter Grassmanns promoted geometrical view, his Förderung, we consider length and direction as bound together. Notation ally or rather how we get a label or Handel on this is by using lowercase letters for lengths and uppercase letters for points. Yen a line segment " opposite" a point is labelled in its length with the lowercase letter to the letter labelling the point.

This was common practice when I learned trigonometry, and the advantage was cyclic or rotational symmetry!

This frequently meant, tht if you had proved something to hold for the letters upper and lowercase in one interpretation, you had proved it for all interpretations in the cycle of the letters..

In addition to this labelling system the labels were directly combined!

Thus cAB represents a line segment between points A and B whose length ( in some arbitrary metric) is c. This length and this line progresses in the direction A to B!

This notation is none other than the notation of a basic vector concept! The length c is a quantified magnitude, and AB is the designated direction. This product represented the notion of a magnitude with a direction.

In the early lessons of physics I remember being introduced to a vector in these terms, bu without these labels or notation!

Now, this being the fundamental idea Hermann had, expressed notationlly by letter labels in this way what are the implications?

First of all the length of a line segment: the dimensional units are arbitrary the fundamental thing is the numeral names the count of some Metron . What is this Metron? It turns out to be a standard piece of matter , usually rigid, used to compare.
But we can and do represent this standard unit by a straight line segment!, we do not move away from the line segment, but rather gravitate to it  as if drawn to it psychologically..

Thus 3 cm means the unit line segment is 1 cm long . What is nowhere to be seen is its direction! This 3cm AB specifies it completely.

Nw conventionally we emphasise the length unit but not it's direction points , but when that is done I hope you can see that 3 AB cm specifies the unit direction leaving the numeral free and unrncumbered..

We can now apply this to the number line concept as shown in this video.
http://www.youtube.com/watch?v=t0aHtXud9r4

The thing to note, is that when we do linel multiplication we do not mention the units! We just use the numerals nd add in appropriate units at the end!

Well, now Hermanns notation allows us to define precisely what we are doing, and how precisely units fundamentally work in the lineal space!

Leaving Sade the sums and products of signed rational numbers I focus just on the product of the line segments
Thus AB x AB  is AB². Now as a product it is a line segment considered as 2 parallel and point wise collinear line segments. It would seem consistent to define it as just AB , in this special case of lineal multiplication. However it can also be seen as a special product of planar multiplication, in which case it would be more consistent to define it as 2AB.

But again in the planar sense it is a collapsed parallelogram  and therefore a null or 0 planar form. It is completely lineal and lineal projections within that space are defineable which again is consistent with a line segment defined as  2AB.

We would then define lineal products as 2 times the coefficient product in the direction AB.

By a similar projective pricess
AB ⁿ would be defined consistently as nAB.

When we use a number line concept to define multiplication in a lineal sense we ignore this analysis and just use ABⁿ = AB.

As a notation it leads to minor inconsistencies which we plainly just ignore.

If we do not ignore it we can actually use the collapse of space to model an instantaneous impulse in the line of collapse, even beyond the square of the coefficients for the planar case.

There are other products that Hermann explored in the planar setting, so the lineal case has not received much attention.

The other thing we do is we use a concept of scalars! They are not numbers on the number line they are counts in our heads. Using these we do not get the expansion of space on multiplication, instead we get multiple copies of a Metron.

Commentary

I have to say , research shows me why Grassmann was not distinguished as today. The ideas were not as sharply defined as they are today, wth respect to Cartesian coordinates!

I have remarked before that what we call Cartesian have their fixed form by Wallis!  Descartes in fact presented a generalised coordinate system! We can pass through De Fermat and DesCartes to Bombelli and others who acknowledged the Coordinate systems of the Greeks, particularly Archimedes and Apollonius.

Even in DesCartes time DesArgues continued the Pythagorean and Greek systems in his projective geometry of the Conics. DesCartes, somewhat archly corresponded with DesArgues chiding him for retaining this " old fashioned" system!

In any case Hermanns innovation must have seemed just another fancy notational sstem( which it was) doing the same thing as Cartesian coordinates( which it was) . However, it was more flexible and more applicable to more general situations than Cartrsian coordinates. Also it was more general than DesArgues system which was developed for Conics. In that regard it was more general than Apollonius who developed coordinate systems for each shape. Hermanns one system could cope with all rectilinear shapes and coordinate systems.

He did remark that he had more work to do to apply it to circular geometry, but he had started in that area with his work on the ebb and flow of tides. He just needed to clarify his Förderung in that arena.

The principal of lineal spaces and planar spaces, linked to either one of his combined labels or the perceived sum of 2 of his combined labels is the subtle advance, for it sets out the basis of a space! Following through we would look for 3 of his combined labels to describe volume space, and 4 for some other type of space etc.

However, his contact with the Barycentric calculus of Möbius redefined his concepts in terms of the Schwerpunkt or weight point, or if you like point mass of Newton.

The story of his developing ideas continues.

String theoretical 2d model

http://youtube.com/watch?v=dv3MzIawR1w

Ausdehnungslehre

Therefore In relation to which I was proceeding  now thereby (my lifes work), to continue producing the so found results co dependently and as from the start until now. So that I myself also thought on not one thing in  relation to a single one of them( the results) , to convene branches of mathematics to vet established Record!

and so it revealed itself, that this method of analysis discovered by me was not as it first seemed to me restricted to the fields of geometry,but I confirmed also quickly, that I was measuring the new field of a body of expertise , from which Geometry appeared as a special application.

Already for a considerable ( time) it was enlightening me that in no way could Geometry in the sense  like arithmetic, or combination theory be considered as. simply a branch of mathematics, rather I was Much more already relating itself as given on a thing in Nature( specifically the space) , and therefore it should give rise to a branch of Mathemtics that was pure and abstract in the way related laws could originate from itself, and how these laws appeared to be bound in the geometry of space!

Through this analytical method the possibility had been given to build out of it a pure abstract branch of Mathematics! Yes this method of Analysis was the body of expertise itself, as soon as it was being developed, without some copy to publish already elsewhere in advance  an evidently tested reocord, and it was restricting itself to the purely abstract.

The  essential fundamental advantage, which through this apprehension would be outwardly achieved, was the Form of it, that now all the fundamental statements by which the spatial manifestation or view of space was expressed, were completely pushed out of the way, and so an unhindered , straightforward beginning would be established, just like in Arithmetic, rather the contents of it , so the restriction of 3 dimensions could be put out of the way!
First in this line would be the rules treated as " the light" , in their unrestricted and most general form, setting out there their essentially natural co dependency, with many a principle, which either still are not available in 3 dimensions or only available in a masked way , in this fully general way will unfold in full clarity!

Commentary
This incredible lecture is worth repeated watching!

http://youtube.com/watch?v=dIEYch4yI08

Difficulty level about 10 out of 10, but breath taking none the less!

Commentary

Already it is becoming apparent that Hermann was losing touch with any potential audience. Time and agin he says his general approach and method would make everything clear: clear as light, clear as starlight, sparklingly clear!

When the text was presented, none of his peers could see the clarity he was describing!

This testifies to the revelatory and personal nature of his work, and why I use expertise rather than science in translation, quite apart from science being a word whose connotation has changed since Grassmanns time.

Hermann spent some 17 years composing this work. For 17 years he thought his ideas were completely translucent! He endured a further 17 years finding out they were not, at least to his pears!

In those 17 years he reworked and developed certain ideas, but it was Robert his brother who drove him to redact and republish the work.in his print business, which was quite successful.

Commentary
Hermann's vision of an abstract ,pure mathematics, unmcumbered by constraints of 3 dimension is largely the current agenda of mathematics. But in that regard he was ahead of his time, because mathematics was derived from mechanics through geometry. Most intuition is therefore rooted in 3 dimensionl experience.

This is how we embed ourselves in reality. But Hermann believed there was another level, a spiritual one that did not mix with reality. This was the experience he was discovering and bringing his method into being for!

To him this was clarity, this was god peering into the darkness and saying  "Licht an!"

Of course, the religious metaphor was completely in keeping with his cultural and social mileu. But many arguments about the nature of this reality existed at the time , a huge European debate about this philosophical question existed, a huge and bitter controversy. Kant was responsible for brokering a kind of peace in his time, but it was a fractured one.

Hermanns method was a solution ahead of its time, and not sought for in his time until Riemann's Habilitation speech!

This video touches on this theme but without recognising Grassmann's contribution.
http://youtube.com/watch?v=kkKTW-Q0qkc

Commentary

Grassmann states that the usual distinctions fall by the wayside in his method. Thus whatever reference frame we use in 3 dimensions will make no difference in his analytical method. This means we have to put these distinctions in at the synthesis stage!

In this regard, the point becomes the only concept left to cling onto, that is to label distinctly. Evenso it is not the usual idea of a point, whatever that might be. The Schwerpunkt therefore is not a simple point , but an idea of a space measure!

Within the Schwerpunkt all space is enfolded. It is truly an equivalent idea to Shunya, but not as a single poit in space, or in time. It is a mental experiential point in our psyche, or rather distinct such " points". How we synthesise such points I have yet to read, but where they come from is the method of Analydis so far alluded to.

One clue on synthesis is given by the product rules for 2,3,4 points and the summation rules for multiple points, the Barycentric calculus a la Grassmann, or Grassmann style.

Let us see how Grassmann teases us further in what is only an introduction after all.

Why didn't you add that video to the thread on Homotopy Type Theory? :)

Commentary

Those that engage space, design and model it have a different view to those who revere number. For too long, the Mechanical basis of geometry and therefore mathematics so called has been treated as second class. And yet Newton highly revered mechanics!

This video provides some back ground or motivation to terms scattered throughout mathematics.
http://youtube.com/watch?v=JuNDS4R-OwI

Commentary

This is not restricted to geometry, Hermann said!

While this video restricts it to vectors , makes the point that it is a sum of products!

Hermann realised those products could be anything! Specifically any 2 things could be produced in the sum, including geometric point notation, A, B, C!

http://youtube.com/watch?v=A3PvLqLzdcw

Commentary

Hermann and most modern mathematicians like to disarmingly say:

"Well now I can throw away the picture and just do this Algebraiclly! I could just define this product or sum algebraically ".

Well of course we could, but what happens then is one psychologically starts to look for "meaning" in all this " algebra". One tries to " interpret" what one is doing!

Algebra in the vernacular Arabic means " mind fluff! ", or less venally "mind contortions!". It has taken me a while to see the joke Al Khwarzimi has played on us, in succeeding generations! The Indian system of numerical notation is based on cycles and Gunas and Ganitas. The modulo or clock arithmetics form the fundamentals of their system position and order carry significance in terms of size, thus the decimal system of the Hindis reduced calculation, aggregation and dis aggregation to a mechanical whirring and rotation and twisting of the clock or dial indicators!

This subtle replacement of one to one counters, beads, pebbles etc by a mechanical clock system where Shunya meaning " full" tipped the mechanism into the next cycle or circuit for accounting, is what we call the decimal placeholder system. In this system Shunya takes on a dual role" full/empty".

Shunya can never mean empty or nothing, it always means full or everything. But in this system, once a pot is full ( Shunya) it is moved into a different cycle and sn empty pot replaces it. The empty pot has no name! Shunya is not its name. That is the name of the completion of the filling process.

Given this mechanization of calculation by the Hindis, Brahmans Harappians and other Dravidian or Indus Valley peoples, the Arabs found this constant cycling very disconcerting. Thus the term Al Jibr was used, which, as I say has a vernacular meaning that is apt!

Thus we deal not with mythical Algrbra but actually with Greek geometrical mosaics. These space filling forms were counted, identified and related meaningfully to space and spatio temporal events.

The greatest mystery was our interaction with space . Psychologically we needed to have things or experiences called seemeioon ( singular) or seemeia( plural). These terms are usually translated as point or points, but they are more related to signal or signals. Thus space attracts our attention or focus in some way that directs us to investigate, to understand, to find meaning. Seemeia are all those signals that define a region of interest.

The Pythagoreans over time aggregated these geometrical regions into standard measuring patterns or mosaics called Arithmoi. It is these Arithmoi that we inhabit and name when we count.

Lest we forget the Indian contribution to this process I coined the term Shunyasutras.. These are the same Arithmoi but without the rectilinear bias one picks up from being taught legendary geometry books 1 to 7.

Of course Euclids Stoikeia is all supposed to be introductory and in that sense elementary, but after book 7 Arithmoi theory predominates, untill the treatment of solids  and spheres is introduced. By this time a solid reference frame experience applied flexibly to measurement and proportion should have been ncountered by the student.

If the whole course was taught now as Euclid laid it out, I doubt if it could be done in less than 2 years!

This was and is sn undergraduate course in Pythagorean philosophy! That it has been mistaken for a mathematical text book is due to a misuse of the degree status "Mathmatikos!"  Mathematikos is a qualification in Astrology. One would need to study more thn the Stoikeia to obtain it.

Both Plato and Aristotle were studying for this qualification. I do not know if Plato ever achieved it, but Aristotle certainly never did. In fact he rejected the qualification on several key points to do with Arithmoi!

Dissent was not uncommon, so it was an unfortunate political situation that prevented Arstole from perhaps resolving his difficulties with some Pythagoren concepts through further study in the Academy. Certainly both Eudoxus and Euclid obtained the qualification, and it would seem that both Archimedes and Apollonius obtained it also.

Aristotles Lyceum granted a rival qualification, for a while, until geopolitics again disrupted his school of thought, transferring it into the hands of Islamic scholars, who were much persuaded by it!

At the end of history, the clear, spatial mosaics of the Pythagoreans were obscured by the later interpretations of succeeding generations who did not always get it, and often covered true descriptions by mistaken ones.

Whatever we claim to be doing Algebraically is in fact being done arithmetically, by the methods derived for the Arithmoi. Symbolic arithmetic is a very powerful and cogent interaction ith space.

In this regard all Grassmann did was to not leave out orientation, but further to make it explicit and in combination with the quantity of the magnitude.

We do not use magnitude in its Grrek sense much, nowadays, so the distinction between quantum and magnitude is not clear. A quantum is a heap or a lump of some magnitude. A magnitude is an extensive experience.of something.

Thus the magnitude of space is an experience of its extensivenesses. But a quantum of space is a defined regional and bounded amount of space. These bounds do not necessarily have to be physical, as one can mentally bound or envisage a boundary to a region.

So
When Rⁿ is used to specify a set of ordered real numbers, it's use is I'll defined at the level of "number". Confusingly it is used to specify points in a reference frame And vectors in that reference frame. The reference frame is not defined.
Grassmann's method starts with the primitives that define a reference frame, and these primitives are lines!

Usually hey are termed unit lines, and for that reason the concept of a unit has to be added to the fundamental definition.. Because these primitives are lines we can in fact model the primitive reference frame by the unit sphere in hich any radial represents a primitive line of unit length.

To define a unit length one radial is distinguished as the fundamental Scalar of the whole system or reference frame. Which one it is is a matter of choice, but some conventions exist around the use of this choice.

Thus we can hold a primitive sphere in our minds as modelling not only the primitive reference frame but any reference frame. We can then constrain the elements according to our design purpose.

Once you have a model of a primitive reference frame, one can understnd definitions of points as vertices of parallelapioeds or spherical pyramids  with apexes at the centre.

In this regard, the dot product can distinguish angles between radials in the same great circle.

http://youtube.com/watch?v=k5hqdAkPYzg

Commentary
This is a fun representation of a reference frame. This machine projects spherical surface points into space describing planar geometric forms. The whole experience is the model of the reference frame primitive. What is missing is the unit scale reference. But suppose you set that as a ray length to a specific point on a wall, then relative to that point you will see forms scaled accordingly!


http://youtube.com/watch?v=udP6ak5StKY

Commentary

We can begin to see how the product sums become layered and constructed, renamed and reinterpreted by different constraints and different mentalities.

At the primitive level it is Grassmanns analytical method tht underpins.
The vector cross product over powere the simpler so called wedge product. This innovation derives from Gibbs reinterpretation of Grassmanns method and his algebras. It is retained because engineers and others like to know where a normal is pointing! However that information is contained within the surface of an object. So in terms of manipulation it is not needed and is a hindrance. However, at the end it can be put in to indicate the normal direction to those who like to process surfaces in this way.

http://youtube.com/watch?v=ZpvlrvwcXk4

One very important point. Do not be misled. There are no defined vector nulls arising from vector products! What  are defined as such must be an identity with a geometric point. Thus since a bivector null, trivector null etc cannot be a geometric point! They cannot be vector nulls.. They are not even the same type of product as a vector null. A vector null is a scalar product.

Here is Peano's take on Grassmanns ideas. His work lead Levi and Ricci to develop Tensors. A bit over complicated maybe, but they work, apparently.

http://youtube.com/watch?v=UdFl_4xWDlk

This indeed sounds very funny. Do you have a reference on that, because I cannot seem to find it. Translations I can find are things like "restoration", "completion", "reunion" or "reunion of broken parts".

This is the original reference I stumbled on.

http://www.argyrou.eclipse.co.uk/GreekEtymology.htm

And this was the blog post it inspired!

http://my.opera.com/jehovajah/blog/2012/03/04/algebra-always-has-hurt.

When I read the translations such as you quoted and compare the evidence which is still observable today in students reactions I have to think: is that a good translation? The answer is as usual yes and no! For about 95% of the human race my allusion makes sense! For the other 5% they would object to the vernacular as disreputable! Lol!

However, when I tell mathematicians they are not actually making things clearer by jargon and symbology they do not believe me!

In that regard, Grassmanns assertions about clarity and light are strongly idiosyncratic in my experience, but I do appreciate that some notations are more reader friendly than others, and actually can aid meditation. However I agree with Newton, algebra is too personal to be an aesthetic medium of communication.

The machintions of ones mind and the whirring of the cogs can be too complex to notate clearly, but the tested conclusion often can be presented aesthetically and elegantly. The analysis leads to a profound synthesis that is often beautiful in its appearance. De Analyse was Newtons attempt to place his conclusions in that light.

Grassmnns use of the word Satz I am still mulling over, but it seems to be referring to the idea of " a public record"  or record in the sense of "it is on the record". In that regard he seems to me to be saying he did not think his ideas neede scrutiny because they were confirmed by well known " facts" or by an accepted statement of facts. While this was deeply encouraging, it missed the point about communicating with ones peers!

So that is the point about algebra in a nutshell!

Historically Al Khwarzimi ( his nickname)  was guided by the Greek rhetorical tradition. This may or may not have made his exposition clearer, but the gradual and increasing tendency to symbolisation, snd the attitude that conciseness of symbols is somehow elegant ( calligraphically this is true) misses the aspect of communication! This is not to downplay the necessity for meditation in mathematics: the so called " stare"! However, it is to recognise the difficulty presented to students by mathematicians who cannot see they are not communicating!

I could go on, but I will stop here!

Commentary

I actually came to note that the term "geometrical product" as used by Grassmann has a foot noted explanation. It refers to trigonometry and space theory.

http://my.opera.com/jehovajah/blog/2012/02/25/the-complex-numbers-arise-out-of-3-mistakes

This link is not one of my clearest blog posts, but reflects the meditative digestion of a lot of new information at the time. The basic thesis was all of mathematics is based on trigonometry nd trigonometric ratios!

While I have not read the referred to text, I skimmed through a German text of that era dealing with space theory in the style of Grassmann! I can not say which Grassmann it was alluding to, but I guess it was Hemnn.

Space theory ( Raumelehre) seems to be equivalent to the concept of topological or metric spaces. The essential idea of a metric space, cutting through the jargon is : you and or me can measure our relative positions! Well, when you look at it like that one can realise that trigonometry , classical spherical trigonometry was devised by Hipparchus I think to do precisely that!

Measure theory, Lesbesgues measures etc deal exactly with this same issue that in a real sense spherical trigonometry solved millennia ago!

So the term geometrical product, in Grassmanns mind was a reference to the products common in trigonometry!

Most of us learn the basic rules for calculating areas without ever being told these are geometric products. If you get on with trigonometry and some do not, you may realise that these products have a deep trigonometric relation, due to the right triangle. Again you would not have been told, these are geometric products! I just call them trigonometry.

So it is easy to see why the notions of geometric products seem so alien and yet give such familiar results!

Actually, as I have cleared my mind from imposed distinctions, a process Grassmann deeply refers to in this text, I realise more snd more that this text is about basic arithmetic and fundamental trigonometry.

Why does it appear so revolutionary? And why does Hermann give it such a big build up?

Well my guess is that when you cut through all the BS jargon and find it is really a few principles used iteratively , that is over and over agin at different levels, it is a big deal!

For me at this stage, just thinking about a line segment as the basis for all this analytical superstructure( and I have not even got to chapter 1 yet!) is liberating. Die Begränzte Linie! Die Strecken. It creases me up when I think : is that all it is? Lol!

Aside from the deep symbolism of the line segment, an ancient Greek Notion, Pythaoreans certainly, the inevitability of a line in my mental analysis of space is unavoidabl. I do wonder how a blind person conceptualises space, and in that regard, symbolic algebras take on a new and practical sense.

Braille and musical dynamics provide an alternative symbolism I imagine for what must be a different proprioceptive experience of space.

Oh by the way, memory and consciousness is another fundamental duality like line and point or region and space.

Simply put : Grassmanns method of analysis and synthesis applies trigonometric products and sums to the space in and of which we have our very being. As such it is a powerful but limited description of our experience unless fractally iterated!

Thanks for the references! I do understand the more popular things (like meaning of words) are mostly shallow and not always true or complete. This "pain of mind" reminds me now of some explanation of Arthur Benjamin ( see https://www.youtube.com/watch?v=M4vqr3_ROIk (https://www.youtube.com/watch?v=M4vqr3_ROIk) from around 12:30). My math is above average (lol, just image average), but I still count myself at the 95% you write about.

This really sounds fractal ...

Thanks for that aswell!  ^-^
You can be very long in your writings.

When math and programming overlap, I do find myself struggeling between using (greek) symbols and semi-full names. The first keeps the code "clean" and easy (faster) to write down. The second results in far more text and symbols, but do keep it on a more understandable level. Seems you prefer to stay away from much symbols ...

The remainder was revealing itself in due course, that with the appropriate specifications how they themselves will be found  in the work , the intersection point of two lines, the intersection line of two planes and the intersection point of three planes could be apprehended as products of these lines or of these planes, then next  identically a highly simple and general theory of curves, was revealing itself from this.

Thereupon I now went over to the wider application  and  the fundamental ground of it, which I have in mind for a second volume, where I specifically have critiqued all those  things as a group  , which fundamental ground presupposes all similar ways to the handles or labels of pivoting( swivelling round) or angle,
 There in the second volume , which will complete the work , it should appear later in print for the first time, so the need  seems to me for the general view of the Whole work, which is to here signpost to some exact form the appropriate revelations.

Youhn,  "windbag " could be my middle name! Lol! ;D

In my defense, I have to say I have an autistic trait, one of several, called polylogia. I either say too little and so become obscure, or too much obscuring what I am saying!

A lot of my writings are effectively machine dumps!

I hope some find something useful in them, but if not I do not or cannot care, because I have to get them out of my processing system to move on!

With regard to your programming style: I opt for meaning over symbol. Machine processing today is so powerful it hardly makes a difference. But for a human legibility is crucial . Commenting is an alternative strategy, but I find the use of symbols with commenting is a compromise with little machine benefit.

Because I prefer this style I write shorter function or procedure calls and call them in main almost like a translation. Thus, in c or c++ I used to write a main that looked like an explanation of what it was doing!

I am not up to date with current programming practice, but I did like the object oriented concept, and I loved pointers. These gave me greater freedom of expression than function calls.

I never did any real machine code programming, because I was not a technician, but I appreciated just how hardware dependent it was when I devised my own pointer calls for Qbasic, using peek, poke and LRecord.. It was fascinating, but I almost lost my wife while I was lost in that deep world!

Arthur Benjamin very entertaining! Lol!
In the Indian subcontinent Vedic mathematicians demonstrate such skills, and certsinly our mathematical greats had this ability: Newton, Wallis Cotes, Kepler,Euler, to name but a few.

While it was mysterious in the past it should no longer be mysterious to us. This is a neurological processing trait that some of us have. We probably all have some ability to do this but not as efficiently as others.

The neurological programming starts with counting using fingers or digits usually. Most of thus are then bullied into dropping this neurological association and forced to use something usually called, but never explained , " rote learning! "

This rote learning is actually quite brutal! It consists of continually disciplining the processing centres until they respond in only one way.. For about 90% of us this is where we learn to hate mathematics ! On the other hand most of us love counting and singing and dancing in one form or another! Neurological processing is so much fun!

Of course we won't all be able to calculate at that speed, but we would all be better at calculating and all happier!

Commentary

Grassmann here lays out the summary of the current work and then describes the plan for the second volume!
The work here he saw as revelatory and progressive in the way he felt it clarified and simplified much of geometry and Mechanics. His second volume would be more stringent , formal and exact. However the underlying principle to him was the dynamic rotation and the angle!

This is a familiar analysis to me. One could rephrase this as " it's all about trigonometry! ".

I came to the same conclusion by the end of my research in the Fractal Foundations thread. I appreciated, but only to some extent, how vectors, so called were a form of trigonometric summation involving the constant i, but I had very little clarity on that constant at the time. I have a settled opinion about it now, after researching Bombelli, Hamilton and the Greek forefathers.

I know thst Grassmann read widely and deeply including Lagrange. Lagrange adopted most of Newton's approaches, including the use of i, as did Cotes through De Moivre. As Mechanics the i had the significance of an arc measure. Cotes said as much in some work he was doing on integrals. Unfortunately he died before he could write his Harmonium Mensurarsm, which I think would have revealed the Concepts Euler put forward some  decades later.

In this case Grassmann is alluding to the dynamic and rotational nature of the Barycentric calculus, as a foundation to all the results he was revealing. None of his Strecken were conceived as stationary. They always pivoted about some fulcrum! Thus he could transform from one coordinate system to another, simplifying any description by an appropriate choice of reference frame.

The use of the product rules in addition meant he could work with points, convert collections of points into lines, planes or volumes, product these to get planes , lines or points, and so move surely through a complex space description symbolically.

The theory of curves he refers to has piqued my interest. I wonder if it is a version of the Bezier et al curves?

" which fundmental ground presupposes all the similar ways to the handles for pivoting or angle".

I tried not to presume too much, but was constrained initially by the concept of vector, and its issues. Meanwhile I kept reading of a great freedom in text . At first I thought it was to be found in the rectilinear foms and constructions. I could not understand how Clifford Algebra brought trig ratios into the labels.
Then I found the inner product relied on the cosine tables squared to evaluate.

Eventually I started to explore the trig ratios in relation to Grassmanns statents.

This sentence explains all, to me. Trigonometry is assumed because it is in fact fundsmental to the whole method. On top of that trig was deliberately left out of these first presentations to make the revelation more dramatic! The exact formulary would be presented unequivocally in the second volume..

Clifford therefore brought the trig ratios out of the cupboard ad made them plain, but lost the "purity" of line and point product.. Instead he replaced it with an actual product of lengths and trig ratios.  Then the dot product, the outer product, the combined product all tend to be confusingly presented. Terminology is changed and the concept of ponit ,line and plane product seems to get overtaken by some other concepts.

Now I know that the trig ratios are key everything starts to fall into place, especially the intimate link to Laz Plaths Trochoids for the Dynmical pivoting Grassmnn clearly had in mind.

The work is clearly focused on cells or regions that tessellate. The work of Escher show how iterated small changes can result in detailed descriptions .
http://en.wikipedia.org/wiki/File:Escher,_Metamorphosis_III.jpg

Commentary
Now, when I was a lad!

In primary school I was taught to draw a line horizontally and mark 2 points at a given displacement from each other, the beginning and end of a length on a ruler, even to name them A and B. Then I was taught to use a semi circular protractor to mark a third point , which was a construction point. This point was approximately at the radius of the semi circle, but was never measured. The aim here was to construct an angle, so a ruler was used to try and draw a line through A and this construction point.

Well, we were told to use a sharpened pencil, to look very carefully in a certain position to the ruler , to hold the ruler still, checking all the while that the construction line went through the 2 points. This was definitely a skill honed by practice.

Length and angle were the first geometric constructions taught and eventually mastered. In the process I absorbed a reference frame for the page called horizontal and vertical.. Angle was used rather than direction or orientation.

The next skill was the compass. This was a trickier skill for drawing arcs and circles. The paper had to be kept still, the compass rigid, the point stationary and not ripping the paper and the pencil fixed in its screwed holder and sharp!

Very tricky!

Later I did some technical drawing and was introduced to the costly equipment designed to minimise all these issues, but by then I already had fixed ideas about Geometry. Using a compass was somehow better and more accurate than measuring with just a ruler , even though you had to use a ruler to set the displacement in the first place!

Then I learned the parts of various figures , and how to construct them with a ruler and compass on the page. No 3d constructions were ever done! I did woodwork instead and used carpenters tools to construct face edges and planed faces.

All in all a neurological understanding of constructing forms in space, based on a hodge podge of geometrical ideas and theorems, especially Pythagoras. Everything was applied but never connected by any theory I knew. It was just a collection of skills and methods I learned to call upon to construct forms.

In primary school I learned by rote that area was a calculated thing. I had no idea what area was, just how to calculate it. Come to think of it I had no idea what length was or what angle measure was. I had tools . These tools defined length, angle and later area when I was shown how graph paper divided surface into little squares!

Graph paper, ruler and protractor were my fundamental tools with a pair of compasses to draw arcs. Arcs could not be accurately measured I was told, that was why we needed to calculate pi!

Formulae were introduced. Sum of angles, perimeters of forms , another sum. And areas of forms, these were called products.
So we had sums and products for these geometrical forms. I just had to learn area products which initially were just base times height, 2 lengths producted as long as they were perpendicular. Perpendicularity was the concept of relative orientation , direction still did not enter the geometrical world I had developed until we started to do navigation problems!
Base times height covered the are of all 4 sided figures with parallel lines. 1/2 base times height covered all triangles.
Complex shapes were solved for area by a sum of products! Usually two lengths producted and summed with another 2 lengths producted . The complex form was broken into forms accessible to the formulae , usually triangles and parallelograms!

This was not connected in any consistent theoretical structure to my knowledge.

Curiosities were brought in such as Herons formula, and just learned but never really understood . Finally the trigonometric rules were learned and the relationship between the circular geometry and the internal cyclic quadrilaterals and triangles . This was presented as an overall set of rules, but no general theoretical structure was alluded to with the proofs..

From this survey I hope you can see that the components of Hermanns Ausdehnungslehre, the Ausdehnungsgroesse are common in Trigonometry, even the sum of products. However, here the Barycentric calculus and the Grassmann analytical method advance the subject  by providing a unifying framework to the methodology!

In so doing Grassmann introduces a " fancy" or a Förderung, that he promotes. This idea is that the product and sum concepts extend to all the different types of magnitude, namely point, line ,plane, volume. We really do not have n dimensional magnitude concepts or experiences, and yet this powerful conception organises our thinking even in that " inexperienced, imaginary " realm.

To show his idea and to make it work he had to focus on how labels were and are defined.

The second thing he did was quite obvious but overlooked: he kept all information on the page in symbolic form, and specifically as a product form he kept length and associated direction  ( orientation) tigether(Festhalten).

Using this he picks out the fundamental group and ring structures required to make this work as an algebra. In so doing he generates not one but many algebras!

Now he has a consistent method of Analysis he can use that to synthesise or construct these algebras in detail. Each detail is different, but the whole benefits from a consistent notational concept and the group and ring theoretic structures.

Now in mathematics we can pretty much agree that it is about aggregating things, and the principle aggregation methods are summation and production! Hat we do not always realise is what production is, and where it comes from.

Production comes from division or disaggregation. This seems counter intuitive, and it is. Most philosophical mathematicians would opt for production as repeated aggregation. However, nature shows that production is based on division. Once division has occurred aggregation follows. This aggregatin is regionalised, which is why division is crucial as a first step to production!

In book 7 of the Stoikeia Euclid lays this out as best as he can unfortunately translators have confused multiple and multiple form with multiplication. Hamilton in his paper the theory of couples or the science of Pure time makes this distinction clear, but it is lost on the reader who believes multiplication is as taught through multiplication tables! These really should be called factorisation tables.

Thus Grassmann corrects this fault in the process of developing his analytical methods and the product is freed to be a multiple form, once again!

For example, finding the area of a rectangle is finding the multiple form that covers the rectangle precisely. The form can be little squares or little rectangles or little triangles. The area is the count of these little forms.

Hang on a minute! How do we get the area of these little forms?!!

That dear reader, is the reason why all trigonometry is fractal! In the past we stated that we defined these area to be Monas or unit, that is 1. Today we look past this statement without realising that ultimately our models are not based on any ultimate standard, but on an agreed convention of a standard! If god or nature does not agree with us, it is up to us to change our definitions accordingly!

Euclid and the Greeks knew this, and so they framed their philosophy so that it applied to whatever standards or conventions we could ever make. Today this is called " gauge invariance" or even gauge symmetry. Conservation laws particularly are looked for wherever gauge symmetry can be identified. Hermann called this " Keine Abweichung". More importantly, this was invariance achieved in a dynamic rotational and translational setting!

Commentary

Now the initial whimsical fancy Grassmnn started off with .
AB = –BA

This makes geometrical and trigonometric sense, but no product sense as yet.

Then  AB + BC = AC .

This makes trigonometric sense in only 2 cases. It makes no geometric sense outside these 2 cases. Why define it as a general rule?

There is evidence in the figures that Grassmann considered this from the poit of view of relative rotation about the common point. In that case the statement is trigonometric ally true if the lines are free to rotte into a straight line. Thus he might have from the outset considered these commonly joined Strecken as rotating hands of a clock evaluated in a straight line . His notation at this stage wold then imply rotation, with a stretchy elastic bnd AC joining the ends .

This is a fanciful notion indeed, and of little trigonometric value beyond implying that Pythagoras theorem would need to add or subtract a term to make this true.

In the second stage, while looking at trigonometric products he realised that in the general quadrilateral, to find the area one needs to divide the shape ino triangles. This is a procedural step, purely intuitive and not trigonometric in any way. But if one looks at the rule it suggests that 2 joined sides should be replaced by the third line segment joining their end points!

Thus the fancy suggests a technique of aslysis: divide complex shapes into triangles by using the appropriate directed line!

Again in the trigonometric product for area, two directed lines in a fan shape suggest a product for a parallelogram .

Thus the notation is gradually becoming suggestive or representational of how to proceed with an analysis!

Then when he saw the trigonometric notation where the points of a form are joined by lines to which a lowercase letter is assigned to represent displacement measures, it suggested that the 2 points ths elves were producing to give a line segment!
Finally ombining the measure with the point product gave him a symbolic representation of ny line segment.

But further support for his himsical notion, the idea he promoted comes from the use of coordinate systems. This was a later realisation that modified his thinking. Using the orthogonal axes coordinate system he could represent the three points by projections onto the two axes in the plane.
Thus every point was a vertex of a rectangle. But for every point the measurements were projected into straight lines. In that case his equation was true in those straight lines! Thus the combination of those projections wold also be true, that is they would validate that the displacement between the points would sum like the general rule. The actual trigonometric length would use Pythagoras to calculate it using the information stored in the points, or point representation!

This whimsical idea turned out to be the way to record the subjective processes we humans use to analyse and synthesise a spatil description!

In that regard it is a remarkable demonstration of belief in an intuition, against all odds! Grassmann had notated a simple way to include or engage our human software in the best practice for analysing spatial descriptions. He had found a heuristic methodology that was consistent and consistently optimal! Moreover, he persued it with fortitude and conviction to demonstrate his claim that it was an expertise that would give rise to,new branches of Mathematics. To prove his point, he single handely created one! The lineal Algebra.

Bearing in mind that Norman bases everything on natural numbers, remember that for Grassman the line and its length are locked together. Thus a number is actually a length of line/ line segment

Now in this case , using orthogonal coordinates as primitive lines, at right angles to each other. These then product to form rectangles or parallelapioeds. Everything else applies as Grassmann conceived it.

http://youtube.com/watch?v=dVk3CpjHR4Y
Don't believe the misrepresentation of Euclid either!

http://youtube.com/watch?v=TuaInpbbsg4
Actually that product rule is now, by Grassmanns modifications misleading! The product of 2 lines is by now a point! So what we now have is the product of 2 points being the primitive x axis displacement and the primitive y axis displacement( 2 orthogonal line segments) . Then these 2 line segments are projected by each other onto parallel lines through the end points. These lines then product to give the required point of the rectangle.

Thus in Grassmnns thinking projection is another fundamental process. There are 3 types of projection: parallel, point and circular. Both circular and parallel require 2 initial points( one line segment) while point requires 2 line segments with a common point.

The projection line will be what was traditionally called a construction line or arc.

So far straight line segments have been catered for by Grassmans point product notion. Nothing has yet been defined for the arc, yet, but it was definitely under consideration under the Schwenkung and angle notation and research. The labels for swivelling have been mentioned twice so far.

Note we cannot escape from the point product, except by 0 length! Thus in many descriptions one factor of the point products may be " known " or given, the other has to be found by a series of product and summation manipultions.

By this Geometry is reduced to a computational framework.

Heap sorts, binary tree( binomial structure)

http://youtube.com/watch?v=GreOd9DzZn0

There is a deep structural connection between the binomial expansion and all of computational mathematics. Similarly Grassmanns scheme would relate to a deep connection between the trinomial expansion and his insights.

Few, perhaps realise the trinomial can be modelled by a 3 dimensional or tetrahedral structure. The symmetries therefore can be quite profound.

Normans treatment of polynumbers uses this heap structure as a polynomial structure for the concept of an arbitrary magnitude. Consequently the terms variables, unknowns  can be rep,aced by a data structure based on the binomial heap. Note the polynomial structure of the levels. .

The important point is the binomial expansion encodes or includes the choices of combinations for the product factors somewhere in its structure.

Commentary
Grassmann advanced his ideas and his Förderung. He was not creating lineal algebra, he was creating a sound method of analysis and synthesis.

As a student learning mathematics he developed, as we do, his own method of understnding nd memorising the material. We all have our own idiosyncratic memory techniques, mnemonics, states of mind, lucky items etc. for rote learning we all have our little lins and hooks that get us to the next line or through the next stanza. We are actively engaged in how we learn and how we remember.

Unfortunately this is all schooled out of us. We are forced to conform to some standard that reflects the teachers understanding, not ours! We learn pretty quick through praise and rebuke what the teacher expects. Unfortunately what the teacher expects and gets more often thn not, is not necessarily consistent with our experience. This is where confusion sets in.

How can there be more than one right answer? Well there usually is, but some teachers do not let you know that. It is worse in mathematics! Some pedagogues actually boast that they like mathematics teaching because an answer is either right or wrong! They present a digital view of mathematical correctness.

Mst of us in our bones know this is not " true"! However we learn to dread this draconian power the Maths teacher has over our developing understnding of counting and measurement. For most of us this purposely grnerated nicety is too much! We learn to hate the subject.

What if some child came up with an alternative way of counting, using different names rhythms and images? Would a Maths teacher celebrate? I suspect not! But the music teacher probably would, the dance teacher might admire the creativity, maybe the cultural aficionados might hail it as a major breakthrough!  All these possibilities are so easily crushed with a cross by some myopic Maths teacher.

Thus it require intestinal fortitude on the part of a child to believe in nd promote its own vision of how to o things over the consensus. Much hardship has to be endured, much misrepresentation and misunderstanding.

Hermann saw things in a slightly if fervent way. He did not alter trigonometry or space measurement, he simply relabelled it. He relabelled trigonometry in such a way hat he did not need to write down reams of notation or keep looking up to orient his description according to a diagram. All the information was recorded in the symbols he set down on the page.

By inspecting the symbols he had all the information he could ever need and more!  By manipulating the symbolic positions he could identify which lines he needed to focus on. Once that set up was done he could write out the tigonmetric um that gve the result!

Yes this seemed to be a quicker way to o these calculations. Hermann did not say that. He said it was less wordy and more compact! Thus he could write a proof in a few lines!

Have you ever been fooled by those ads that say build a " whatever" in just minutes? Well that is what Hermann is doing here! Write the celestial Mechnics of LaPlace on the back of a postage stamp!  Of course the meditative processing that is involved with analysing these symbols to see which should be combined could take hours days or even weeks!

The ,ore familiar you are with ny system, the quicker your expertise shows itself. Grassmanns methodology for his analysis and synthesis does not exclude the work of set up and adjustment to the new description. However, once that is done and agreed new results consistently follow!

Grassmann methods apply geometry and trigonometry in a new consistent notation which vntually leads to sound computational results related to mechanical reality.

Cn we replace  empirical experimental and observational enquiry with mathmatical Anlysis and synthesis?
No!

But we can use the 2 in tandem to build better models, that compute accurately what we have soundly described geometrically and trigonometrically.

We have to first get the mechsnics right o get the geometry right to get the correct trigonometry. Once that is done, we have a sound model we can explore theoretically as long as we like. This model is not " the Truth", however. It is jut oe model of many we could and can now build using computers! The human battle for dominance and truth takes us out into the big bad world of religion and geopolitics!

http://youtube.com/watch?v=GUvoVvXwoOQ

CONTINUE

To this end, I have  given to you firstly the results which had  alteady revealed themselves before the co dependent processing( melded reworking) . I have flatly shown  how the product of a  pair of line segments can be considered to be the parallelogram, even if  specificlly as everywhere happens here, the direction of the line segment is  firmly fixed to it. But then how this product is distinguished through this (requirement) : the sign of the product has to switch as the factors are switched,  while at the same time it is apparent The product for a pair of collinear parallel line segments in the same direction is null.

To This labelling is related another set of labels at the side of this one, which likewise is related to the line segments with a direction firmly attached . Speciically, if I project one line segment into the other, I put an arithmetic  product in the line segment projected onto. Likewise the product of those kinds of line segments have this arithmetic product in, in this case , for this reason also the multiplicative relationship for the addition applies. However this product is now of a different type to the first, in that switching the factors does not switch the sign , and these kinds of line segments have a null product when they are directly perpendicular to one another .

I named the first kind of product the outer product, and the last kind of product of line segments resulting from onto projections the inner product,  and so for the outer line segments, because only by stepping farther apart in directions; and so  for these inner line segments only by  stepping closer to each other in directions,they had an assignable Value
 that means  
•to evaluate by ratioed amongst one another products always had a  value that could be found( in a table).
•There is always a value between the 2 products when they are put in ratio or divided.
•By partially being nested


Through my working through  the Mecanique Analytique I  already had pointed out to me  these kind of inner products as Essential! At the same time it lead to the label of the absolute length.

Footnote
Compare with the copy of chapter 3
Compare with the same chapter.

Commentary

It is instructive to look at the sums and products Grassmann has introduced by his Förderung!

Ths sum of 2 joined Strecken .
AB + BC  = AC

This sum relies on the joint point B so that the line segments can rotate relative to one another Ito a like directed or oppositely directed line segment pair. Only in these 2 cases is the sum correct for the lengths of the line segments AB, BC, AC .

However this sum is the main plank of his Förderung and he defined or set it as a rule that is always the case in a specific sense: that of relative displacement. It is a quite different rule to the trigonometric rules Sally associated ith geometry.

The first product is AB = – BA

I am not sure that Grassmann initially saw this as a product. Apart from the minus sign it is standard geometrical notation for describing a line. I am not even sure if it was used to describe direction apart from the specific act of construction when the direction of producing a line( extending it) . Certainly it is a convenient shirt hand, but as written it was meaningless until Möbius set out his rules.

In this sense it is an invention of Möbius not Grassmann , and similarly the um of 2 Strecken idea owes a lot to Möbius. However it is clear Möbius did not have this precise summation rule for Streken.

Although this is a product rule I think Grassmann states he did not realise until he had his revelation regarding the trigonometric product formulae.

The second summation rule comes later after the point product realisation and is the Schwerpunkt or Barycentric summation rule
mAB + nBC = pAC

m,n,p are lengt factors, A, B, C are point factors

In this way he combined lengt and direction into the notation for a line segment. After this, whenever he uses Strecke or line segment he means this triple product combination, with a sign switch included if the point factors change position, but not if the length factor changes position.

Initially Grassmann introduced the line segment product. This was to be considered as generating a parallelogram . Later however he introduces the point products. Now 3 points produce a flat space or flat figure.mit is called a Fläche not a parallelogram. Those same three points generate 6 Strecken of 3 contra direction pairs( binomial coefficient), thus 3 positive signe parallelograms and 3 negative signed ones,ndepending on whether the fan product AB . AC  or the wedge product AB . BC  was used to prod cue the parallelogram.

So the point products produce a triangular form in space, while the Strecken products produce parallelogram forms in space.

This is not yet clear to me but it is clearly set out regarding the products of points, so I was surprised to see him seemingly revert back to a Strecken form until I realised it was my understanding which was at fault.

For example the intersection points are not defined in terms of Strecken but in terms of lines and planes. In fact the intersection products are defined in terms of primitive concepts for which as yet Grassmann has given no product definition! There is no product definition of a plane or a line or a point. These are primitives on which he build some product definitions.

Now to this collection of products he adds the inner and outer product for 2 Strecken by this stage it is important to note that a Strecke while being a line segment now has a complex notational form!

So cAB = –cBA is the basic product label for a single line segment.

The outer product for a parallelogram would be, say for a fan product

cAB.bAC = cbABAC

Swapping any of the point factors around would introduce a sign switch. It is yet to be revealed what AABC produces! All I know is that it is negative to the parallelogram. Of course BAAC is a wedge product parallelogram and again negative in sign.

The inner product is different AC is vertically projected onto AB . AB might have to be produced to e to meet the foot of this projection. But that is the same as increasing the length factor of AB so AE = qAB say.
In addition the length q is an arithmetical product of the length of AC = b! The full descriptor being bAC

q is set to bcosø the angle measure between the line segments. Thus the line segments project this arithmetical product into each other. This is an important distinction in the types of line segments!

Grassmann states that the two types should be written besides each other. I am not sure if this is a recommendation to save time or part of his method.

For example cAB projected onto aBC produced give 2 more associated line segments  c cosøBC and acosøBA.

If BA has to be produced to project BC onto it then cosø will actually have to be negative. The 2 negatives convert to a positive. These types of Strecken can be negative or positive depending on the angle between them, but the product of these vectors, the inner product is always positive

The product of these vertically projected line segments is cacos²øBCBA

Looking at this product one can see how switching the Strecken factors change also the direction of the angle measure! This change causes the Strecken sign switch to be switched back.,also the projected Strecken when at right angles give a null product.

Grassmanns inner product is different to the dot product or the Euclidesn dot product,

Also the outer product is proportional to the inner Productt in the ratio 1:cos²Ø

Kram1032 found this resource:

Grassmann Algebra in Game Development (http://www.terathon.com/gdc14_lengyel.pdf) (slides from a talk at GDC'14

Commentary
Grassnann identifies Two types of related Strecken products, the outer or spreading hands of a clock product, and the inner or closing hands of a clock product of Strecken generated by projection.

The trigonometric ratios are unavoidable in the second kind of line segment. But that makese their behaviour completely " normal" . The distributive rule of multiplication over addition applies to these; the product never goes negative, but vanishes when these line segments vanish, when the outer projecting line segments are at right angles.

Bearing in mind that these geometrical products are just trigonometric formulae for areas , perimeters and volumes what does Grassmanns method allow us to do?

His method allows us to first of all think mechanically. Then to describe geometrically and at that stage to refine the mechanical model or observation. Finally to think trigonometrically and again to refine models at that stage. However, his notation means that we do not have to write every single jot and tittle down while we are in this meditative process. The broad strokes get us to a general solution.

Then, providing we understand the rules and what products apply where , we can write out a detailed computation, with all the trigonometric ratios in the correct positions.

Then we do the calculation often by looking up values in tables and we get the result!

However, we do not always want just the result. Sometimes we want insight into how our model is behaving. We can get this at a certain level as trigonometric and or geometric relationships.

Thus for undulatory behaviours we would expect certain ratios to appear. Higher analyis leads us to expect certain exponentials to appear.

My interest has been not in these basic ratios , which are crucial, but in the higher derivations of these ratios. Thus the exponential function derived in terms of these ratios , as linear or Fourier combinations: plus the exponential forms of the rotational measure i.

Beyond that I have an interest in general rotation which will be specified by lineal combinations of these higher analytical forms..

The question is and was, what ties all these analytical forms together?

The solution was and is this method of analysis and synthesis promoted by Grassmann!

In this regard, vectors miss the real point. The relationship depends only on these curious products Grassmann created to describe fully a line segment.

http://youtube.com/watch?v=uw6bpPldp2A

Focussing on Euler to determine the higher arithmetic for the trigonometric ratios. The use of infinite series  reveal the iterative nature of these evaluations. Interpolation and extrapolation immediately reveal the approximate nature of sll evaluation, and the indeterminacy of our calculus or number measurement.

This is how it should be, because Panta Rhei! Everything moves.

Consider Eulers derivation of the cosine series, and the sine series. Note how he uses these contra dynamic limits. Soon Grassmann will show how i derives naturally from his set up.

http://youtube.com/watch?v=wcjknbmTYOI

I love the tent constructions of Frei Otto

Of course ,Hermann!
Catenaties, splines , Barycentric points all displayed in a functional construction!

The thing about such an image is that it is n dimensional!  Thus however many stay lines a and supports he has to utilise, this number characterises n.

Hermann and indeed we would characterise the construction in this way naturally. Thus a system of Ausdehnungsgroesse could be set up to solve for the precise shapes he wanted or designed!

This system could be programmed into a computer to draw the solution points and compute the Beziers Catelau curves, but I am certain any good vector graphic programme now makes this easy to design.

My friends the spiders showed me these kinds of n dimensional spaces a few years ago, while I was writing the fractal foundations thread.

Trochoids and roullettes are the next plateau I am aiming for! I have been on that trek for a long while. Who knows I might get there before I transform into this universe of dynamic equilibria!

Panta Rhei.

This book excerpt is just a note
http://books.google.co.uk/books?id=UdGBy8iLpocC&pg=PA182&lpg=PA182&dq=euler+and+circular+function+introducciones&source=bl&ots=RWmA35JAvc&sig=LB1l93LC_O2MOU_x6XUOTkB83uQ&hl=en&sa=X&ei=HhwaU4KZM4rMhAf_nIHgDg&ved=0CDIQ6AEwAw#v=onepage&q=euler%20and%20circular%20function%20introducciones&f=false

I am looking for a particular lectureNorman gave and can't find it, but on the search I ploughed through much topological discussion. My view of topology changes as I encounter Normans treatment, but my question is : how does it relate to Grassmann?

The real difficulty is te charisma given to these topics? They seem hardly mechanical and only remotely relevant, but I believe this is down to the cult of hero worship that surrounds the teaching of these concepts.

Bernoulli defines a function . You will see that it relates directly to a Grassmann Ausdehnungsgröße. The concept of a formula and a function are identical in Grassmanns method.

This is because Grassmann deals directly with the subjective process of describing anything by measures! Instead of using the ill defined concept of a variable he uses the well defined line as a symbol, specifically the line segment. This means no matter how abstract the thought processing is and can be ,mgrassmanns method can always construct a drawn geometrical model.

I mentioned that there is some confusion between a point set and a vector set, or better a line segment set. There does not have to be, if students are taught correctly.

I gave a geometrical light projector in action as a thought model. This model was dynamic and interactive with space. The light rays represent the line segments the projected timages on surfaces the points at the other end of the line segment. The points centred at the projecting lense naturally sum to a Schwerpunkt!

Because the line segment has 2 points , they are by default entangled. If I arrange all line segments so they originate from a spherical centre that centre, under Grassmann, now must be considered as a Schwerpunkt. But also it is considered as the sum of ll the points at the other end of each line segment.

In the past, we have simply ignored these geometrical ntities, especially points. We have chosen whatever we wanted fisregarding these interrelationships. Because Grassmann demonstrated that there was a benefit in keeping all this information everyone of his day would have characterised it as unnecessary. However, the mosaics on which the Pythagoreans developed their concepts do not have points to throw away! The disc in the plane constructed by bricks laid ot radially would all key from the same central brick which might have a circular form. But every count of the bricks in a radial include that central brick. That central brick is process wise carrying a heavy role, but it was discounted. However when it came to building domes and arches the actual weight it was carrying became apparent! Those discounted bricks have to be put back in!

This is a natural phenomenon in the universe: the centre always piles up and down. You see it in ornados and in the centre of galaxies. You see it in the x ray and synchrotron emissions.

We discounted it because we did not realise. Grassmann counted it because , like a child, he saw no physical reason to change his accounting behaviours.

The infinitesimal calculus is not Newtonin Fluxions. Newton also did not discount things in analysis. . Infinitesimals were used to justify subjective processing ith out evidence. Newton always used evidence. Thus like him Euler based his thoughts on evidential processes.

1/1–x = 1 + x² + x³ + ....

Is an identity that is evidenced. For a certain range of values.  Algebraically it is always true!

But this is the fallacy: hat is meant by algebraic truth?

Limiting our description to symbolic arithmetic it is clear that if x is a variable quantity of magnitude that the infinite sum has either no mening or the negative " numbers" behave in strange ways!  And thn positive "numbers " end up summing to a negative fraction!

What is going on, and can we rely on these algebraic identities?

The context is our subjective processing. The process discounts important mechanical details, and so we get a contradictory puzzling result.

Our computers rely on this polynomial behaviour, and this is why glitches are end,ic within computational systems. We avoid the severe proms by establishing rules or safety recommendations.

For example
1/1–x = 1 + x fails at the x² term. It fails because the finite expansion generates an  x² and we have no prior assumption that cancels it. It is not an identity because of this. Neither hold the infinite sum be an identity, but we justify this claim by secretly employing a demon to continue the subtraction so we cab say " it is an identity! " it is therefore a matter of faith.

However Newton used the same reasoning to develop his binomial series.!

The justification is mechanical. If these things are very small quantities( or we have a demon) then the LHS is a good approximation o the RHS. As long as we are guided by approximations .

I am quite surprised at how emotional this subject of still is to me!

Partly it is due to how much effort it took to uncover the absolute confusion surrounding the number theory of this geometrical process! But mostly it is due to the realisation that pedagogues have been misinforming students for centuries about space and our interaction with it.

To be simple and I hope clear, the Greeks developed from or through Pythagoras school the symbolic code of the mosaic. These mosaics in this context are called Arithmoi and they are a decomposition of space. We might now say they are a topological decomposition of space. However they are entirely subjective. The concepts of space is entirely a conjugation of my experiences into a focus region and a defocussed region which interacts with memory through conscious and unconscious processes.

By conjugating experiences in this way and accepting tautological processes I can reference a constructed map of my behaviours contingent on an accepted and stable construction of a self.

Without elaborating further on that ( which I leave for your own meditations) this map I have called the set FS and what it maps the set notFS.

The decomposition of notFS occurs in the set FS and nowhere else. By sharing symbols in common the constructed we can encode and specify some details of our subjective maps. We can agree on an external shared symbol to reference a subjective process.

Language and alphabets etc arise out of these kinds of hared processes. Eventually specialised language structures encode processes of dynamic change and how we respond to them

2 fundamental responses , linked indivisible , but yet we still separate them, are vocalisations and extension of limbs. The combinations of these result often in musical choreography, of which a special form is the measurement Dance!  The words of the song and the application of the Metron are usually called counting and measuring. It is this decomposition of our subjective experience called space which is encoded in the mosaics called the Arithmoi.

This is a rich and fruitful cultural activity, and I do not intend to reduce it to a few words! But what is of relevance is rhetorical style. In such styling various details are foreshortened or even left out. So it was that the rotation of the Arithmoi in space were neglected!

The introduction by Brahmagupta of the concept of balance of everything, mistaken as the introduction of negativity resulted in great trepidation! Brahmagupta was interpreted as saying certain quantities bring misfortune!  As an astrologer his words were taken quite seriously and misapplied. The hatred and fear and confusion around certain numerical processes generates from this time.

While Indian merchants took to his teachings, understanding it in terms of debt snd credit, engineers and temple builders did not understand , nor did they want to bring misfortune onto their constructions.

It took time and imperial upheavals to overcome people's natural astrologically  based fear of so called negative numbers, and to appreciate their use in calculation. Unfortunately the basis of Brahmaguptas teaching, the Sanskrit Shunya was completely misunderstood. It became a symbol for darkness and emptiness!

Brahma would laugh his socks off!

In any case the negative symbols being accepted as signifying ome process of rebalancing what was owed filtered down to western engineers as having a mysterious geometrical significance. However this was obscured by the Arsbic term Al Jebr.

Symmetry reflection, debt , reversal were all implicated in the negative symbolic process. But when Cardsno and Tartaglia both came upon the symbolic process tex]\sqr(-1)[/tex]  they had no explanation outside their secretive algebraic manipultions. It was Bombelli who let the cat out of the bag for. Engineers. Suddenly in his work on Algrbra he showed how these symbols obeyed rules set Down by Brahmagupta. To these he added his rules for this new symbolic process.

All well and good until Descartes decided to call them imaginary quantities! What was a symbolic process was now confused with a previously understood concept called Arithmos. Not only was Arithmos now turned on its head , no longer a geometrical decomposition of space as a mosaic, but now some Arabic notation ,also this process of calculation in symbolic form was confused with the same Arsbic notion!

These symbolic processes were divorced from their geometric base and could not therefore be understood. It was only by returning to the earlier concepts of geometers that some resolution gradually became possible.

The first person on the track after Bombelli, was John wallis. Using the Descartes Fremat model of coordinate geometry, he established the practice of using fixed orthogonal coordinate lines. While Descartes developed the process of using fixed lines they did not have to be orthogonal. Wallis showed that by using this type of reference axes he could get what is now the canonical forms for the Conics as well as many quadratic and cubic curves in the plane. In so doing he clearly considered the values of –1 on the circle. It was his opinion that ince he extended Descartes fixed lines into measuring lines orthogonal to each other, and since he could place on his measuring lines every known quantity of his time that the quantity that tex]\sqr(-1)[/tex]  might be should appear in the plane somewhere!

While Euler definitively uses i to symbolise this magnitude and also an infinite magnitude I have only his word that he regarded it as a quantity though which other ways of processing solutions became possible! Thus I do not think he fell into the trap of looking for a value on Wallis's measuring line. Newton, De Moivre and Cotes actually remained classicl with regard to this symbol. To them it was merely a processing symbol not a quantity. It was a magnitude, hich clealy went against Descartes description of it as imaginary. Because of Newton, all three knew it as a magnitude associated with the unit sphere and the ages long computations of the sines and the more recent logarithms.

It was Cotes who suggested that i was a magnitude of arc. Newton waited with bated breath to find out what Cotes meant beyond hat he nd De Moivre leafy knew. Unfortunately Cotes died, and De Moivre had bern too busy on his probability project to closely follow Cotes extensive research. He was unable to fathom much beyond the logarithmic formula for the magnitude of the tex]\sqr(-1)[/tex]

However, decades later Euler puts it all together clearly and convincingly and writes it in the familiar exponential form. In both formulae the essential idea is that i is a magnitude of arc. The idea was later revisited by Wessel in terms of magnitudes of orientation or direction, while Cauchy and Argand were using the Cartesian plane to map properties of these magnitudes. Into the mix blunders Gauss , not to be left behind, showing his mature thinking on this subject, but also indicating his severe doubt of its validity!

This is where Grassmann enters quietly and unseen with the solution!

One other major difference between modern numbers and classical quantities and magnitudes is logos analogos.. While we are taught fractions to the exclusion of proportions , fractions in the 16 th and 17th centuries were proportions and ratios!

1/1–x  is a proportion or a ratio.
To equate it to 1 +  x + x² +.... Is to write a proportion .

The nature of proportion is scale so the RHS is a scaled version of the LHS.

With a proportion we are not confined to the page, so we can imagine a unit compared to a heap characterised by 1- x. X as long as it is a magnitude can be removed from 1 . But of course x being large leaves the situation with a negative heap! This may physically represent as a hole! . But now the infinite series must form an infinite heap, that is it must fill everywhere including the hole. Comparing that to a unit shows disproportion on an incomprehensible scale..
This makes little sense of the identity. However when we actually relate the process to space something remarkable happens. After the first x term the other products vanish as actual spatial forms! This is because orientation matters. The bivectors vanish, taking with them the endless computation

This is not obvious . But when you take all things into account in the process , as Grassmanns method does you  begin to see the importance of the rotation that was left out.

Most of Eulers early days were spent perusing the rich mine of Arabic texts available to his mercantile family. He learned to love the Indian masters repeated fractions, but he also learned how they made the aggregations make sense., casting out 9s or whatever scale they were using. The modulo arithmetics were ways of dealing with these infinite series sensibly.
http://youtube.com/watch?v=XXRwlo_MHnI

There is one other aspect of our interaction with space tht makes the current notion of number a nonsense, and which goes to the heart of Berkeley's attack on Newtons Fluxions. Leaving aside the human irascibility and religious nd national sentiments berkley pointed out the obvious: numbers do not exist in our formal sense, especially in dynamic descriptions.

Newton described his entities very well and promoted them very well . He really showed that if we accept a formalism for the concept of quantity then that clashes with the glib notion of a varable quantity of magnitude. His Fluxions therefore were symbols of a dynamic experience of quantity which we only sample at each instant in time.

Of course to a cleric numbers were god given! They exited by fiat, out of the mouth of god. Newton's philosophy of uantities placed doubt on that belief system. At the same time the other notion of infinitesimals, very small quantities which never vanish was also not favoured by Newton. His idea was to use ratios to identify proportional relations as these Fluxions varied and ultimately vanished. The consequence was that his symbols identified a process whereby the Fluxions compared diverted the attention to a quantity in ratio with one. It was this quantity that was then utilised for the next stage of any process . It was called the fluent. It's existence was not as a number but as an identified experience of a constancy.

It was not a constant but an approximate constant, this a fluent.,in today's terms, when we film motion at a trillion frames a second the fluent appears as rigid as if it was a still image..

If it was a constant, nothing more could be achieved by taking faster framed videos, but since it is a fluent a second rate of frame speed can be utilised to capture these changes. So it is like taking the shot at 2 trillion frames a second to capture some other Change.

We know fluents exist as Newyon conceived them. We do not know infinitesimal " numbers" exist! However we can omalise any concept we like concerning what symbols we will use and the consensus meaning of them. The trick is not to fall into the belief that they exist!

The entities that exist, we can measure. Our measurements however do not confer existence on an entity! Thus we may decide to measure using a very fast film technique, only to find what we thought was there actually was not! As an example : certain streams of water which appear continuous actually film as individual spherical drops, not even contiguous!

Eulers identies therefore have to be examined dynamically. First the proportion they establish in notation has to have a referent secondly the more compex RHS proportion has to be explicitly constructed. That is to say, usually the construction is n infinite process. Thus. One is invited to finesse too an impossible conclusion. Parmenides and Zeno pointed this reasoning flaw out millennia ago. Bearing it in mind therefore we may reasonably ask what is the equating that is being done?

While the expression may be unravelled, in the unravelling it becomes clear that several processes have been unjustifiably combined. That these combinations are not usually justified is no exude. At some stage all process has to be justified and it's use certified. It turns out in my opinion that these issues are created by the false conception of the reality of " number".

There is no cure! The concept of number has been created wholly flawed! However, space and region and ,option and counting are evidently true experiences. It is these mosaic patterns that the Arithmoi deal with, and deal with effectively.

I recommend Euclids book 7 if you want to explore these issues further, but again you have to read it in the Greek to understand how far the number concept has drifted away from the Arithmoi and why Newton called these dynamically changing patterns of mosaics Fluxions and fluents.

We would better understand computation if we got rid of the concept of number as an entity and multiplication as an operation.

The concept of a product is adequate for our needs and we could then appreciate ratio and proportion in due context..

The significant role of disaggregation would be restored and we would not lose all hope of understanding the Spaciometry of our experiences.

The primitive product notion exists in Euclids Stoikea book 2 where it is introduced without explanation; in book 5 where it is introduced with explanation and book 7 where it is elaborated on.

In book 2 any line segment is arbitrarily sectioned or cut . These cuts are points of relative rotation and contiguous continuity. The first product is the adjugation of the 2 parts of the line segment. The
2 parts are conjugates. The line has been conjugated. If more than 1 point is introduced this is conjugation of a conjugate! Thus the process of division is recursive from the outset.

However having separated the sections we may now aggregate them , again iteratively, and in this process they become adjugates.

These products are aggregation products.

Now if one section is rotated relative to its conjugate section , this is a product, by rotation that is a form. The form is completed in this book 2 by parallel lines, identifying in general a parallelogram product . This is by relative rotation of conjugate sections of a line segment.

These primitive products will later be used to count, and thus they will be identified as a Monas or a unit for counting and doing so by contiguous placements in a comparison..

In book 5 Eudoxus identifies one line segment as the greater of 2 line segments in comparison, making its conjugate the lesser..

Into this comparison we, our subjective processing, introduces the concept Monas or unit. It is also a Metron, that is a unit to be used to count with in a comparison.. The lesser is often taken as the Monas or unit. Now when it is placed on the greater and this is done repeatedly and continuously we count as we do so. . We then describe the greater as a count of the lesser. The greater is now apprehended as a pollaplasios a multiple form of the lesser.. This is the second level of primitive product and it combines our subjective process with a spatial reference object, which we then define as  Monas and use as a Metron.

In symbolic labele we write if A>B then A=nB where n is our verbalised count, and A and B represent real spatial forms. nB is thus a symbolic mixture that points to internal processes(n count) and external spatial object(A is some block of wood,say, and B some other object or even the same type of object)

These primitive products are representable by some sketch of drawn lines, which are rotationally connected at vertices. It is this sketch which resides in our formal system as a symbol of our real experiences.. It is these drawn lineal symbols which are our most enduring and powerful symbolic representation of our experience of entities.

In book 7 the formal Arithmoi are introduced, in words . Their symbolic representation having been established in books 5 and 6.

There is no multiplication in these texts, only counting and counting of multiple forms. The highest common factor algorithm formalises these different styles of counting and embeds this counting within the context of comparison and multiple form identification.

The fundamental rotations of lines and forms in this counting process and thus product realisation are contained within the practice. When one divorces from the practice, these essential rotational cues are lost and ignored.

Commentary
Besides each form there lies a shadow. It is these shadow Strecken that form the inner product.

The perpendicular projection forms a shadow Strecken in the adjacent Strecken. But not all projections are perpendicular. The shadow Strecken take the method into a new realm of calculation and relationship, beyond the right triangle into the cosine laws for triangles. In fact all trigonometric laws are now opened up to the method of analysis and synthesis,

1862 1844 pix

Thus, plainly, the geometrical exponential magnitude had already revealed itself to me by the reworking of the theory of the Ebb and Flow. Specifically, even if  a represents a line segment( with orientation and/or direction firmly attaced) and represents a corner angle( with the swivel plane firmly attached), so, on purely internal grounds, the explanation of which would  yet lead me way off track, the line segment whose swing arises out of a ,creating the corner angle , revealed itself to mean   a•, where e can be apprehended as the  base number for the Natural logarithmic system.
That is a• means the line segment a swung round  the corner angle

Further, even if , where expresses a corner angle  in the geometric sense(by its amount of turn), reminds one of the same number ( value) as cos ā where ā, which to relate to the corner angle as expressed, has to mean the ā obtained through the Radius measured arc ( radian measure) , so immediately out of that label of  the  exponential magnitude follows


,


that is.**[footnote: in figure 1 of the diagrams at the back; In practice if AB is the originating line segment, and the same is swung into the position AC around a corner angle and note into the position AD around a corner angle – and one fully completes the parallelogram ACDE then
The sum of the line segments AE is  AC+AD  and the bisection AF of this sum is the Cosinus( cosh) of the angle corner .]

Thus, plainly even if reminds one of the  quantity with which the line segment (for ) is multiplied , which line segment changes around in its direction, 90° relative to the swinging arm of the corner angle , and immediately its absolute length varies on the like manner as sinā, so it reveals



And the equation arises therefrom


Belonging to  all equations which betray the most striking  Analogy with the known Imaginary expressions.

These labels had revealed so much,  much sooner.

Footnote
also this label with the presupposition of swivelling relates to  the second volume.

Commentary
This passage officially confirms Grassmann as mad as I am! And of course I love it!
There is no teacher of Maths on this earth who would tell you this passage is correct! In fact Robert, Hermann's own brother made sure not to copy this passage into the 1862 version of Hermanns Ausdehnungslehre(redacted!) .

Robert strongly steered Hermann away from these kinds of "mathematical" identifications, onto s strictly formal system acceptable to most mathematicians. This stuff was definitely not for Möbius or Gauss. The formulae Grassmanns method gives rise too were undeniably correct or well known, but the method of derivation was highly questionable, highly mystical and highly unusual!

The reputation for obscurity derives from passages like this.

I disagree that they are obscure. In every instance Grassmann tries to make it crystall clear. But they fail to communicate to those taught to think only in certain ways, or more accurately to write their conclusions in a formal set up.

It is interesting to see how tentatively Mathematicians finesse this piece of historical lore in the development of the hyperbolic descriptors.

http://youtube.com/watch?v=er_tQOBgo-I

There are many other brief introductions into the lore, but no one really goes into it too deeply!  It is presented as a confusing mnemonic scheme. However all trigonometric ratios are usually taught heavily using a mnemonic structure. SOHCAHTOA is just one mnemonic for the sine ,cosine and tangent ratios..

What is really going on here?
For centuries, at least from the 5 th century AD astrologers had been calculating improving and recalculating the Sine trigonometric ratio for the Bow or the Indian half Bow. The Persians made great headway based on the Greek research into the Babylonian records, but the Indians made the greatest innovations based on the Greek astrological ideas, introducing the great simplification of the radius and the half chord forming the half bow.

However it was the Arabs who collated all the best work together nd set in motion a centuries long calculation of the sines to many figures of accuracy.

Without these tables and the ongoing research, the differential formulae for interpolation would not have been invented. The ratios of the diminishing proportion would not have been investigated by Napier leading to his logarithms of the sines and mercators conundrum for map projections would not have been solved logarithmically.
Finally the evident applicability of the method of Fluxions would not have suggested itself in formulaic terms to Newton, Wallis et al , nor would Newtons mastery of the Multinomials establishing his development of the binary series expansion have been so sweetly done!

In fact De Moivre who took time to study every page of Newton often astonished fellow scientists by his ease in solving Multinomials with large factors!  Both he and therefore Newton knew the numbers were in the sine tables, the Multinomials in the differential formulae for interpolation, and the calculus in them both!

Because of the binomial series and the coefficient, de Moivre made probability his life's work, again relying on the resource of the sine and logarithm tables.

However, the Cotes deMoivre theorems for the roots of unity was perhaps his greatest uncontested achievement!

Based on principles and ideas laid down by Newton, de Moivre solved many difficult Multinomials using the . In particular he factorised the cosine and sine sum to find roots for the circle equation.

While he found many useful answers in this way it was Sir Roger Cotes who , faced with the problem of calculating the rhumb line for navigators , on inquiring of De Moivre found a remarkable and important use for these methods of De Moivre.

Together they collaborated to definitively expound on this trigonometric solution to the circle equations. The Cotes De Moivre theorem is a remarkable piece of calculation based on the sine and haver sine tables, or the cosine tables that were beginning to appear.

Cotes took it one stage further and made the discovery that the constant magnitude was indivisibly associated with the quarter arc! He devised an alternative measurement method for the unit circle based on the ratio between the arc length and the radius. This meant he could write down a length, associate it to an arc and then tabulate the sine or cosine for that arc. No right triangles were necessary, but of course they are indivisible from the circle in any case..

With this method he could now rewrite the logarithm of sines in terms of arc lengths. Using Newtons logarithmic series he was able to demonstrate the remarkable equation or identity



Essentially the arc length is related logarithmically to the sum of the trig ratios in the form written where i is eulers definition .

He died before he could explain this to Newton, and before De Moivre to get up to speed with his research.

He had written an important identity derived from an equation with infinite terms!
The whole idea was preposterous. Parmenides and Zeno were invoked! But the fact of the matter is, ignoring the equation , the identity related very different ratios together in a consistent and robust way. This was an implicit formulation. Actual values could not be sensibly equated, but actual values were sensibly calculated!

What was going on?

The answer lies in our subjective processing.
We can and routinely do calculate in parallel. To survive , many parallel calculations occur simultaneously, and results are applied to the solution by the various subsystems that require it.

However, most mathematicians are not trained to calculate in this way, and indeed there was no notation, logarithm or method explicitly designed to emphasise this, except long division!

Most of us who still do long division organise our calculations after the Indian methods. In so doing we do multiple calculations in parallel and in a highly spatially organised way, on the page.

We are not taught to see these long divisions as linear combinations of products! Neither are we taught to recognise the spatial disposition as separate operations.

Writing out a long division as a combination of products would organise them into a series of products arranged geometrically. In the case of division the connectors or conjunctions would be mainly minus and the coefficients would be the figures usually written atop the division box.

Thus all divisions are some form of geometric series, that is a large sum, and contrariwise all products are some form of geometric series, that is a large sum!

So returning to votes, his identy reflects the fact tht our calculations are the results of large summations. The fact that the summations may be infinite is not an issue if one can identify when to stop! This is a practical decision. It can of course be ndlessly debated, but pragmatics has the answer to Parmenides and Zeno.

So, realising this astrologers used Cotes method. They examined the hyperbolic curve using the sine tables. The sine tables were and are a reliable resource. If a curve can be described using them, then many techniques if analysis are available to the astrologer.

When the curves were examined it became clear they had to be flipped by a quarter turn, because the sine goes from 0 to 1' and in addition 1 had to be added to the values.

Well Napier showed how the logarithm of sines could be calculated on diminishing proportions from 1 and the hyperbolic require increasing proportions from 1. By this time everyone knew the exponential growth calculation for compound interest! Both Citrs and Newton and DeMoivre calculated this logarithmic base ratio to about 20 decimal places. It was 2.718...they knew this ratio, they calculated the logarithms of this ratio after the manner of Napier, that is why they are called Napierian logs or natural logs ( to distinguish from the logarithm of one's, and the later logarithms of Briggs) and they were used mainly by actuaries in the calculations of life insurance etc.

Decades later Euler sets out the identities in all clarity, but by then many did not recognise the summation nature of all products, even when Euler was pushing their faces into it.

Grassmanns foible now makes historical and radical sense.

Commentary

I promise you, the next translation is totally worth the wait!

There is so much to comment on in just this section alone, but I am going to leave it until I have done the next translation! This crazy brain swivelling notation gets you everywhere! ::) :dink: ;D
 
I have a lot of corrections to make in my exposition, but hey that is the nature of the game! :embarrass:

AUSDEHNUNGSLEHRE px-xi
As I now seek to drive also these labels to their more general form, firstly the label of the inner products extends itself in the way it is inter communicant, precisely how I have indicated this above  for the outer product in relation to the intersectioning of lines and of planes; so then I can immediately follow with the label of the  quotient of differently oriented line segments, and represent as


where a and b represent differently directed line segments from the same Length( of line segment), the Quantity which every line segment lying in this same plane varies around this corner angle intersection point ba ,in the direction from b to a , so that in the process , as it must be,


stands.
And then out of this reveals itself the label for the case when a and b are of unequal length, without difficulty.

But now that simple label became the source of a series of interesting relationships!

Firstly, out of it instantly revealed itself a new style of multiplication, which is directly inter communicant with  Division, and which through this itself inherently differs to all earlier ( multiplication) that the product of this new style could only be 0 if one of the factors was 0 while the factors remained commutative. In short, a multiplication which is exactly the arithmetic  analog of the usual multiplication in all its rules, and and thus easily goes on to establish the label of the same things even if I gradually multiply the line segment with  different kinds of such quotients, and which then I considered as a single quotient, which could be substituted for these gradual factors.
Thus now, according to the definition , if ab means the intersection point of the corner angle of both line segments, which are the same length:

Stands.
Thus one has



Furthermore, if ab is  the m-th part of  the corner angle with the point  of intersection ac ,
Then one has



Because specifically, if a line segment has m times gradually " suffered" the swivel

, then in total it has completed the swivel

.

Also  if the corner angle wth intersection point ab is also half the size of that with ac then


Stands, and also one has

.

Let specifically be the swivel of a right angle, then equals the one related to 2 right angles ,let it be thus. There c = –a
Thus also

stands

that means a line segment multiplied by the expression alters its direction around 90°, depending on  whichever single line segment, but then always to the same side.
This beautiful meaning of the Imaginary magnitudes further completed itself in thai it revealed, that

and
signpost the same magnitude  if is a corner angle but means the related arc divided by the radius.( radian measure)
Then in practice



Finds itself related, and plainly also
.

Formulas which also have a clear geometrical meaning , in that


means the swivel around a corner angle , whose measurement in radians( arc divided by the radius) is x.

From hereafter all imaginary expressions achieved a pure  geometrical mening, allowing themselves to be presented as constructions.

Immediately the corner angle was chisen to represent the Logarithm of the Quotient , then also the infinite number of its magnitudes for each arm position!

( page xii)

Now, so plainly shown turned round, how one utilising  the meaning of the imaginary magnitudes found in this manner can then derive the rules of the analytical method within the plane . On the other hand it is  not more possible, utilising  the imaginary magnitudes also then  to derive the spatial analytical method rules!
Also the difficulties of observing the corner angle in space are placed prominently against this, regarding which the all-sides "Looseness"  has not yet become a sufficient Muse to me!

Commentary
A remarkble section giving insight into the mind developing the method of analysis and synthesis which is the subject of this book.

The next section concentrates upon a number of philosophical and academic issues relating to mans perception of space and his relationship to it. Hermann gives a reason why this first volume is more philosophical, and why his plan of presentation was decided upon.

But in these 2 translations he carefully reveals part of the process of deriving his label methods.

The Muse plays a large role in the artists life. Most scientists will perhaps not recognise the Muse, or if they do they may not consider it scientific.

Most mathematicians will probably be lost round about now!

There is  one mathematician/philosopher, Polya, who wrote a small pamphlet called "How to Solve it". In this book he sets out a heuristic approach to solving mathematical problems. If you have not read it, do so, because the term heuristic is partly defined there for mathematicians.
http://en.wikipedia.org/wiki/How_to_Solve_It

There is another term that requires your understanding and that is sophistry. To apprehend it properly you need to contextualise it in analogous thinking. Analogous thinking is evidenced or documented as far back as the early Babylonian / Sumerian civilisations. It represents mans use of symbols both tautologically and iteratively in the process of interacting with space and its inhabitants.

To restrict it to symbology is to miss the point. For human consciousness everything is symbolic! Thus for every advance in this regard I do not discover a reality as much as I inherit a cultural reality which I explore and manipulate. I may break out of the cultural wraps I inherit and develop new and mysterious symbolic interaction, which then eventually transform my cultural environs, or I may be imprisoned by them. In any case I am rarely if ever interacting directly with an objective entity sometimes called space.

The sad thing is, most scientists , especially empiricists believe that they are!

The symbolic set that they have been told allows them to do this is called Mathematics! Unfortunately it does very little in that regard, and is not likely to. Mathematics is completely formal.

Now Mechanics is an interesting activity, it usually acts directly on a symbolic space. In so doing it reveals properties that ground any formal definition. . Such properties are usually mused upon into some symbolic entity called geometry.. This then feeds back to direct and inform mechanical processes.

From these mechanical interactions we can develop models, usually geometric, that behave pretty closely to the space interaction" modelled". The advent of computers and computer graphics and fractal geometry has made this match pretty darn close!

We have gone from analogous thinking to computer graphical representation. Where does sophistry fit in? The kind of expertise that human consciousness brings to make all these models happen and Work requires a great synaesthesia of much data and information. The synaesthetic results in special symbols that allow a human expert to think, to mentally manipulate, to guess , and to extrapolate and interpolate with a high degree of certainty. This set is called sophistry.

The set of symbols used by a sophist have a powerful unifying effect. While experiencing one event, the sophist gains insight into how an analogues system or system set should, would or may behave.. By taking this insight through a careful empirical testing sequence the sophist verifies, denies or modifies his intuitive insight. This results in no change or a slight change in the symbol set, or precisely how it is applied.

Grassmann is perhaps the author who put this all together most consistently. The amazing thing is that no one had actually done this before, publicly.
 Grassmann's strategy is clear by now: read widely ingest deeply, strip back to geometrical magnitudes and arithmetic " operations", be inspired by labels that support the geometrical relationships and remind you of the essential dynamic of the geometry.

This strategy requires hard work reprocessing the accepted layout and format, the invention of new suggestive labels, the setting down of symmetrical formulas and progressive formula development. The result is a clear, simplified notation that mnemonically guides the expert to the correct formulation for the calculation.

But what happened in 1844? Grassmann was stumped! He had no inspiration from his muse! In fact his muse had published in 1843, but Grassmann had not had time to hear or read of it. Hamilton published his major work on Quaternions. Later, in 1870 Grassmann is to write a paper on "the place of Hamilton's Quaternions" in his system.

http://youtube.com/watch?v=Q2FCMjlI9Fc

If anyone can get me a copy of that paper I wold be grateful.

In 1853 Hamilton gets to read Grassmann's Ausdehnungslehre 1844. He recognises Grassmann as his master, and strives to rewrite his Quaternions Grassmann style!

In the Einleitung Grassmann goes into detail about how he constructs his labels, and why but for now i am going to comment on his application in these 2 translations

Commentary
The inner and outer product of line segments are " inter communicant" . That is they inter communicate , one with the other. The usual word I suspect is "corresponding", but I feel it needs a shake up of the mind!

Both Hamilton in 1831 , in his paper on conjugate functions, or couples; and Grassmann in 1844 were particularly keen to explore the activity of the mind in mathematical or rather algebraic contexts. At the time arithmetic and nalysis or higher arithmetic reigned supreme. Algebra was hardly a subject. Geometry and Analysis after the Cartesian model was the vogue.

However the issue of number created an opportunity for certain philosophers to make a case for a rhetorical discourse which made the human mind clearer to students. If one is being kind one might say that Al Khwarizimi was intending to do this all the while! However we probably have to wait for Bombelli to find an educator serious about Algebra in the rhetorical style with his own inventions. Descartes used algebra but did not really promote it. Consequently we arrive at John Wallis as a serious contributor to the idea of it being a topic. Much of the modern symbology derives from his work.

It is not until Hamilton Grassmann and Boole that a serious subject boundary war developed a niche for Algebra in mathematics, but as I say this was really around the concept of the mathematical mind. Logic, up until this point was the prevailing subject dealing with reasoning, but the increasing use of symbols lead to a confluence with Algebra. Boole attempted to rigorously and mathematically reveal the laws of the mind and how it reasoned. This together with Hamilton's work , also  called A Science of Pure Time, and Grassmanns work constituted a full body of work to study, alongside logic, such as that espoused by De Morgan.

The human mind takes a person cantered point of view. In that regard everything has to be relative to the individual. The individual connects to space by choosing points and joining to itself by mental lines. These lines are thus extensions of ones connection to surrounding or interior space.

These extensions are characterised by bounds and orientation and direction and by how they swivel.
But there is a different but allied characterisation: the projections of these line segments onto each other defines a set of projected line segments. The differences are utilised by Grassmnn to create many labelled products. These products, used as specified by him, simplify the process of calculation considerably.

Commentary
A line segment is the product of 2 points.
The product process imparts a quantity to this magnitude, and the magnitude itself imparts roll, pitch and yaw to the line segment, thus general attitude and orientation in space . To this the observer adds an arrow denoting direction of displacement. For any line segment or arc segment there are only 2 directions of displacement. Firmly attaching this to a line segment creates 2 distinct line segments.
The labelling for all this is
aBC = –aCB
Where B and C are the generating points.
For 3 points we may write
cAB + aBC —> bAC where —> means "is linked to"

The = sign Grassmann used has a complex meaning and it is not numerical equality. Geometrical duality is a possible meaning, and this would mimic the Euclidean common judgement that dual things fit atop each other.

The geometrical space which is the product of the 3 points, is a triangular space. This supports the duality of the side bAC to the other 2 .

Now the lines through these 3 points extend out providing anchor points for all other lines in that specific plane. Within this plane are special lines parallel to the line segments cAB, aBC  all these lines identify another point D in the plane that gives a geometric product called parallelogram ABCD. That is to say that the intersection product of cAB x aBC is a point D.

This is crazy talk, because usually onli 1 specific pair of parallel lines intersect at the point D! However either by design or failure to think it through rigorously Grassmann defined all line segments parallel to AB with the same length c , the same fixed direction as " equal" or dual. The effect of this definition in his mind was to allow substitution in the symbolic representation. Thus every line segment pair that meets these criteria intersects in a point D .

This notation or labelling is therefore inherently confusing, but wonderfully flexible. It means that universal statements can be made and local behaviours extended to any parallel region anywhere in space or spacetime.

We have no way of knowing if this is how space behaves in all parts of the universe, but this bare labelling convention means if it does we can symbolise it very concisely.

However the major observation that overlooks this odd feature is that when one goes through the process of solving a geometrical problem trigonometrically, mentally one is only focussing on identities, congrnces and similarities. Thus wherever these parts are in a geometrical sketch is irrelevant to the solving of its length and and angles. Typically one reinserts the calculated values into the actual or real world oriented design. Thus the design, sketch or plan remains unchanged, one just mentally twists and shreds it to derive a metrical solution for its defined parts.

We have used the same labelling to get a handle on 2 things: the geometric product as an area, or rather an Arithmos, a multiple form;; and the intersection product of 2 lines or line segments as a point identified by parallel lines. The second product encapsulates the idea of projection by parallel lines, but it is more general than that, because it allows any 2 non parallel lines to intersect regardless of what they individually are parallel to.

I make the distinction that in a parallelogram ABCD , cyclically arranged points, that AB+ BC is not equal to AD + DC, but Grassmann does not at this stage deal with this distinction. He de facto states they are the same because they can represent the same displacement AC . This is useful iteratively because one can collapse an iterated universe of these relations onto a single displacement!  However you lose the distinction of what is mechanically happening at a precise moment in time. On the other hand, one can introduce statistical or probability models into the description to give one mentally a model that can deal with arbitrary non discernible even hidden events.

This is in fact what has been done in Quantum Mechnics. The events are usually beyond our ability to discern. However, because of this extensibility or collapsibility to an end result we can model it as a 2 state process with indeterminate intermediary steps! We usually ascribe this idea to La Grange, which is where Grassmann derived most of his ideas for labelling from.

So for me the difficulty is in the use of the = sign and I typically do not use it in this context. It can lead to procedural errors and misunderstandings. However the point is that Grasmann was setting up a mnemonic system of great complexity and sophistication, so to remove it on grounds of incorrect or unsupported identity is to remove also it's mnemonic role. Often what Grassmann writes down is total nonsense, just like SOHCAHTOA! But it is an an extremely helpful aide de memorie in tackling certain classes of problems.

Having established the full labelling for a line segment, Grassmann to avoid Tedium drops writing its full description out. Instead he focusses on the part he is seeking to derive further. Now when Möbius did this with his Barycentric coordinates I pointed it out, and so do so here. The main difference is that Möbius dropped the labelling in order to forget them. Grassmann forgets nothing, but also often gives two meanings to the same labelling. For him, as the developer this was not a problem, but for you and me as the reader we have to be extremely alert to his thinking! The context helps, but also small seemingly insignificant adjectives or adverbs become crucially important.

Commentary

The parallelogram product is Grassmanns first product process.. It was dynamic from the start, representing a generalisation of a geometric rectangle construction. Why di it become the outer product?

It has to do with how the product morphs as the line segments rotate relative to each other. Initially the product , on an anticlockwise relative rotation went from no constructible area inhabiting form to a form( a rectangle) with the maximum area inhabited. Then it decreases again. This he linked to a value scale that goes from0 to maximum and then back to 0.

To get it to work computationally for him he had to accept that factors were interchangeable only as the sign changes to its opposite. Geometrically it represented a switch in the direction arrow fixed to one of the line segments. Hardly anyone uses this interpretation nowadays. They prefer to use perimeter travel direction ( clockwise or anti clockwise). Whichever interpretation used the calculator still has to imagine the products as cancelling. In other word the process constructs 2 forms each the signed opposite of the other. The observer has to interpret their significance.

The right angled triangle is crucial to all metric results, but to include it in his method he had to recognise the vertical projection and horizontal construction lines as line segments. In so doing he introduced an arithmetic trigonometric product into the length of a line segment.

aBC could be rewritten as b cosøBC where ø is the angle ( in radians) between bAC and aBC.mdirections of line segments and angle measurement are important.. In fact the rewriting is a bit more complicated for the general triangle and involves the cosine law, but I give the case for a right triangle fulfilling b²= a² + c²

These line segments vary contrariwise to the initiating ones and do Grassmann called the product they produce an inner product, because they get larger the nearer the initial line segments are together rotationally. The parallelogram they produce is always inside the one produced by the initial line segments, but it is there rotating outwards or apart that caused Grassmann to call them outer products.

Essentially then we have 2 algebraic or symbolic descriptions of the same thing. The different and contrasting multiplication calculus leads to different insights and a way of handling a single form more intuitively in symbolic form. For example if line segments are perpendicular the inner product is 0 at the same time as the parallelogram product is at its maximum. Similarly line segments ar arallel when the inner product is maximal and the parallelogram product is clearly nonexistent!

So now this swivelling behaviour was what Grassmann wanted to set down fully in volume 2 , but he also wanted to give an overview in this volume , because the work was great and he did not think he could achieve it all in detail.
We now move to his overview of his labels for swivelling line segments!

Commentary.

We now enter Grassmnns crazy mnemonic world! All the same it is crazy cool!
For reasons presumably explained in the body of the text  figure 1 is labelled by •.

Figure 1 is a rhombus. There is no connection between any metric on the rhombus and .
However, the diagonals of a rhombus do intersect at right angles. Thus the sides project Onto the diagonals as and .
Now let remind you of !
But then has to be measured in radians.

These radians are a geometrical circle arc either side of the diagonal ,forming one continuous arc from to

But then we have swung or swivelled one side of the rhombus directly onto the other.

So, again, imagine having the same numerical values as where now is the same angle measured in radians.....

Well then I can write
=
to represent the projection!

Stare at that a minute and notice that is the swivel in the positive direction by the radian of the unit line segment and is a swing by the same radian arc in the negative direction.

Now consider these 2 line segments as labelled by the exponential. Then we can sum line segments in a parallelogram, Grassmann style. Consequently we get the formula for the , Grassmann style!

This is an analogy with the actual, trigonometric analysis and synthesis. It makes no metric sense whatsoever, but it is a powerful mnemonic aide memoire for the hyperbolic functions.

To get the sinh the same figure must be used but the swivelling sides are now in the corner adjacent which represents a 90° rotation in the swivel plane. But because the labelling of points does not change the diagonal is now a subtraction sum, Grassmann style.

Adding the 2 hyperbolic formulae gives the asserted form for a swivelling line segment. But it is a tautology because I pointed it out as being in the formula to start with.

There is nothing wrong with tautology, by the way, except if you do not recognise it as such. It is a useful tautology because it grounds a symbolic label in a dynamic geometrical experience.

However, the underlying trigonometric ratios which are the basis of the analogy do not sum to any exponential function, but they come as close as you like!

We know these functions do not sum to the trigonometric function equivalents because
and
But within a defined range the hyperbolics are approximated closely by the trigonometric radian functions.

So it is not so crazy after all! It is crazy cool!

Not only is the mnemonic useful, but the analogy is a useful approximation. This means, thinking geometrically, that a swivelling line segment within a rhombus form is a useful interpretation of the asserted label.

We will see how Grassmann goes on to ground the so called imaginary magnitudes in the common circular arc!

Commentary
Firstly Hermann observes that this result depends on his analysis of the inner product! The swivelling line segment is projecting vertically onto the diagonal of the rhombus. In fact it is clear that he is dealing with the circular arc projections onto the radius in Euler's style. Following Euler gives him all the results he asserts. However, where Ruler uses an infinite series argument , Hermann devises a geometric mnemonic that makes sense of the summation of infinite series in terms of arc segments and line segments.

I do not know if anyone else drew this kind of conclusion from Eulers work on the imaginaries, based on The Cotes De Moivre analysis, and Cotes suggestion, but it would seem not. Wessels seminal paper and the work of Argand , Cauchy and eventually Gauss strongly support this interpretation, but no one, not even Hamilton lays it out as clearly as this!

The reason again is because there are serious procedural difficulties in demonstrating this conclusion. However, using the style of analogous and Mnemonic analysis, set out in helpful labels, the conclusion is almost obvious!

The danger with this style of demonstration is the hidden or unaccounted for parameters. Grassmann has pre digested a lot of material to come up with these labels. But what if he missed ome thing?

Hermann was alert to this difficulty, which is why he required collaborators to check and correct any errors, analyse further and advance the generality of his methods and labels..

So I have come up with a name to distinguish the 2 styles of line segments! The ordinary line segment, that is aBC and the trig line segment , that is bcosøBC, in the context of a t least a pair of line segments intersecting at a point with an angle ø between them and the projecting line segment being of length b.. Ths for bAc and aBC this hold if the angle between them is ø.
The ordinary line segment in the context of a plane figure has a trig line segment compatriot! A line segment has 2 descriptions, an ordinary one and a trig one.

Amazingly the rules of combination and computation are different, especially in the product. Thus Grassmann coined( just as lame as mine I might add. Lol! ) the names outer product and inner product.
Now we move onto the swivelling trig line segments and their labels! Panta Rhei!

Commentary
We have to work on the labels in a certain way. Grassmann took the labels of the ordinary line segments to an extraordinary generality. Once again, I remind us that these are nonsense terms; they label only the fanciful thinking of a whimsical mind. However, they form a powerful mnemonic superstructure for just about all mathematical models in mechanics. It is hard not to credit them with some validity. They do have a validity, but not in mathematics per se. They firm what I call a process arithmetic, written in symbols. Or, if you like, a programming language with its own rules and syntax.

So the development that Grassmann treats us to is a development based on trig line segments.

But first a result from his ordinary line segment reworking assists. This is the labelling for the intersection product of 2 lines( or line segments).
If a and b are labels for 2 lines then the product ab is the parallelogram product or the intersection point product. In this instance we choose the intersection point product. And, since the trig and ordinary line segments are " inter communicant" we can let a and b be trig line segments and the product for the point of intersection remains identical. However the parallelogram product now becomes the inner product. We use the observation that the inner product has virtually the same rules as the usual arithmetic to develop the labels.

Firstly we set a• to a•

This means that the trig line segment a swings from the position or orientation of a to the position of b around the intersection point  of the 2 trig line segments.

It should be noted that the trig line segments are rays and so identify a corner angle between them specifically. The other thing to note is the notation result relies on the quantity of angle being 2, that is from – to + . Somewhere along the line we neglect the doubling of the actual angle when analysing rotation!noften, Euler angles "go wrong" because of this oversight.

Setting a•= b expresses that a swings and becomes b
Thus =b/a
That is the swivel between 2 trig line segments in the same plane is the Quotient or division or rationation of the trig line segments.

It is unusual today to think of dividing a magnitude by another magnitude. We think that numbers are the only thing! A quotient is properly a process of comparison and the result is precisely a ratio not a number. In this case the ratio and the exponential are identities. That is we can replace or substitute on for the other.. The concept of comparing the 2 trig line segments focuses on the swivel magnitude between them.

There are other magnitudes to compare like length etc but this labelling is comparing orientation.
From this Grassmann moves whimsically to the logarithmic form
ab=

Thus he declares the point of intersection to be the logarithm of the Quotient of 2  trig line segments. He points out that the logarithm will have infinitely many values because the product of two trig line segments has infinitely many solutions for the angle between them, in terms of multiples of 2+ the angle,

However, thinking purely geometrically, the point of intersection of the corner angle, as a point will take infinitely many angles. To specify one angle the containing rays are necessary, alongside the radian measure.

Commentary

So why do I keep pointing out the lack of logical necessity and sufficiency of this method?
Really it goes to the heart of logic. Today we use logic not dialectic, the comparison of two arguments. Today we mistakenly think logic is undefeatably correct. Logic has never made this claim., if it is correct to anthropomorphise logic. The humans that devised logic, including Aristotle, sought only to battle bloodlessly using words, concepts and common intuition. The whole paraphernalia of logic is not often viewed, but for the mst part it involves rendering your debating opponent speechless. Tactics include ad Hominem attacks as well as misdirected attacks or spurious argumentation, ie deliberately misrepresenting your opponents views. The debate held in public would continue until one was unable to answer, or until the crowd decided the victor. You may well imagine, debates were not well attended! If there was no blood sport, not many could spare the time!

Philosophers however refined the force of proportioning evidenced in an argument into a system of syllogisms and dependent clauses with validity proportional to the preceding statements. Deduction and induction were specified and universal logical operators. Thus if a set of statements had this proportional characteristic, it was described as logical.

But not everything we experience can fit into this ordering of presentation and rhetoric. Analogy for example does not follow this system, and yet is the most important form of reasoning we humans use to apprehend the world. Analogous thinking is more often treated as suspect or not dependable, and particularly not proportional to external verification.

To justify an analogy one must compare both the experience and the so called nalogues of an experience. It is thus a subjective determination.

Virtually all of Grassmanns method is based on deep and cogent analogical thinking and insight. However it is hard to justify it by any other means than directly using it as instructed! From the outset Grassmann states this, and this is why his Vorrede takes us on his journey of discovery.

I do not fault the work. Rather it casts deep and piercing light into the way we subjectively process our interaction with space and labels. To see that line segments have a firmly fixed direction, nd that trig line segments also in addition have a firmly fixed plane of swivelling, and that really the logarithm of a quotient of 2 trig line segments is a point with infinite values, a kind of logarithmic Schwerpunkt, testifies to the inviting nature of this system toward creative thought and heuristic exploration.

It is alive, and invigorating, and it leads to deep insights, but then much hard work and rejigging and modification of labels has to be carried out to ensure it works seamlessly.

Grassmann got stuck on the 3 d rotation problem. Rodrigues had a solution in 1840, Hamilton in 1843. Both of these, had he read them would have been sufficient Muse for him to have set out his labels with that case in mind. His system already pointed out the importance of the half angle!

Commentary

Reworking part of the earlier transaction to draw out the set up for the second translation on the imaginary magnitudes but more specifically the trig line segment "Multiplication" system, which is really a quotient arithmetic.

The uses of is twofold. The first is just as an identifier. The second however is in its geometrical sense as a measurement of an amount of turn. The measurement scheme or metric chosen is the radian measure..

To make the analogy work the radian measure has to be used both for and ā. Hermanns delivery of this point is casual, because he knows he is referring to the radian measure, but the reader may not be certain what he means by " the geometrical sense".

The other point is that in the rhombus only the angle us used to derive the magnitudes. Thus the swinging arm for cosh has to change to the adjacent corner to be the swinging arm for sinh.. This is all by analogy with cos and sin.

The closeness of the values of sinh and sine in this radian measure is remarkable., but it only works modulo /4

Commentary
Hermnn has looked into the heart of darkness and like me found all our calculus or metrication depends on the trig ratios. But what I did not know was the close relationship between the hyperbolic and the quadratic forms of these trig ratios.
From Napier snd De Moivre I knew of the close link between the binomial expressions for the sine ratio nd the exponential ratio; that one curved inside the boundary of the unit circle snd the other spiralled outside the boundary. But I never realised until Euler pointed it out there was a link, an intertwining that hovers around the circle boundary passing in turns inside and outside the disc.

Having used the analogy one way, that is to establish a mnemonic for the swing of an ordinary line segment, Hermann now uses the analogy the other way, that is to establish the swing of a trig line segment. At the same time he establishes a quotient calculus. This completes the so called 4 operations of addition and subtraction, multiplication and division of line segments. His analogous arithmetic of line segments he called a lineal algebra. It was a crazy giraffe of a creation. Already it had about 7 different products, and 3 different outcomes for a product!, that is the products went to zero or not according to 3 different rule sets. Also ani commutativity was necessary in one of the product processes. Very strange and initially very troubling!

I suppose we all have to grow up sometime! Interacting with space has never bern the Pabulum we get fed in elementary school. Consequently we struggle when we encounter the real processes we have to engage in. However, Hermann deliberately chose the clearest labelling practices to couch his mnemonics in. They were both apt and supportive of intuitive development.
Starting with the quotient of 2 trig line segments of the same length swinging through an angle he was able to immediately write this as n exponential form.. This identity between the quotient and the exporntial immediatel meant all summation would be logarithmic. But logarithmic summation is the labelling for multiplication! So now he had another form of product( multiplication) which was entirely arithmetical.

The rest relies on definition and consistency in definition.  If we do not think too hard it all flows swimmingly! Do not think what the square root of a quotient of 2 trig line segments might be., just realise the special case where the swing is radians and thus half the swing is the square root of the quotient!

So how does this quotient get into the exponent of the exponential?

I have given this a lot of time and have come to realise that Grassmann has no explanation for this. It is pure Euler!

Euler arrives at this by expanding the exponential into a series . Having done that for he replaces it with a complex magnitude. In so doing he obtains the cosine and sine series in a complex form.
http://mathsforeurope.digibel.be/Euler.html
http://www.songho.ca/math/euler/euler.html

Thus Grassmann is saying that this shows and to be identical geometrically: they both mean a line segment is swung through an arc about a centre point!

Finally using the rhombus again, but this time to remind you of the sine and the cosine ratios he immediately writes down the form. However he places the also before the sine because the sine line segment is a rotation of the cosine one.

Commentary
Grassmann establishes the Magnitude of turn. Thus a/b is the label for the experience of swivelling or rotating in the direction from b to a. That could be clockwise or anticlockwise, but by convention it is anti clockwise. The rays( line segments with direction arrows) pick out a specific corner of the 4.  Thus it is not ambiguous in Hermanns labelling system, but nevertheless we use the convention, mostly!

This experience of swivelling has an identified centre of rotation, the intersection point of the 2 line segments. Further, all line segments that lie in this plane and pass through that intersection point will experience that swivel or rotation. Thus Hermann is stating that the notation applies to the rotation of the whole plane!
The quotient label is thus a label expressing plane rotation and specifying the centre and direction of rotation.

Now Grassmann moves on to the Quantity of rotation . The magnitude becomes a quantity when bounds are identified. In this case Grassmann specifies the bounds of the swivel as the line segments themselves. Clearly this specifies a unique swivel from b to a. Thus when it is applied to b the result is a. The general result is just the symbolic form.

Now it strikes me that Clifford Algebra has this magnitude differently( and confusingly) expressed. Check this out.

http://youtube.com/watch?v=sHjXccDAIzw

You will note, hopefully, the discussion of sin, sinh, cos and cosh reduced to this so called Clifford geometric product! However it is not explained and the only motivation is to multiply it out!
It differs substantially from Grassmanns treatment because it obscures everything to avoid seeming logically inept! Grassmann on the other hand wants everything out in the open, explained with crystal clarity. Thus he suffers the charge of being fanciful, but gains my respect for pointing out precisely what is experienced. Thus I do not fault him or his method. It is an analogous way of thinking and analyis and synthesis that has demonstrable mnemonic and intuitive value.

This exercise has been very valuable to me. Shaking off the misconceptions about Grassmanns work and praxis reveals a clear picture. The AusdehnungsLehre is simply put part of a programme analysing how humans think. It's real topic is how humans think about solving problems, that is how they think about thinking!

Mathematics was " chosen" because it was and still is considered to be the supreme intellectual product of the human mind whilst thinking about space and our interrelationships with it. However, essentially Grassmann fell into this topic of heuristic thought while doing what most mathematicians do: wonder how the so called geniuses solved the mathmatical problems of their day.

Basically Grassmann studied the great minds of his day, at least all those he could lay his hands on, which in Prussia were mostly European and French scholars. As he read and learned their ideas he made " crib" sheets, brief notes and summaries to remind him of their methods, advice or examples of how to tackle particular issues. It is these study notes that form the basis of his Analytical and synthetical system.

If you are presented with a problem how do you solve it?

Grassmann adduced the key ideas required to do just that for a vast range of mathmatical problems. Whereas most of us might learn a direct solution or a general principle of solution or two, Grasdmann constructed a system which was originally intended to be split into 2 volumes. The first volume he called Ausdehnungslehre, and this was really all about line segments. The second volume , I guess he might. Have called Schwenkungslehre, because it was his intention to deal with solutions to rotational problems. His concept in overview was to bring everything together in an accessible form in that volume, providing an easy reference to all results he had discovered and formulated in the process of his research.

The two volumes, therefore would have provided the serious student of them with considerable expertise over a wide range of the then known mathematical landscape. Beyond that it would enhance the ability of all engineers and mechanics to formulate solutions to their everyday problems and support them in their research into ground breaking applications to mechanical problems.

In this age of steam, electricity and early chemical and industrial engineering the need for a skilled and intellectually ready work force of engineers was paramount.

So in essence Grassmann created nothing new in terms of applications of thought to spatial interactions, but what was new was his exposition of how to think about solving mathematical and mechanical problems.

Without a broad and deep understanding of the mathematics of his day Grassmnn would not have been able to develop his system of analogous thinking and mnemonic hooks.

It may well be that when they fully publish Euler's works that they find that he has written on this topic also, because Euler was a master of clear thinking. However, as far as is known, Hermann Grassmann is the first author to put together such an extensive exposition of human thought process in the mathematical arts.
http://mathsforeurope.digibel.be/Euler.html

We can characterise the scheme of his approach by commonly used geometrical terms: translation, rotation and reflection. In fact these are less general than his conception, but I mention them to connect his work to the Erlangen movement inspired by Klein as a student of an academic who seriously studied Hetmanns work.

What happened to the Schwenkungslehre?

I have no idea, because Robert tore up Hemanns planned progression and forced a redaction of the work as previewed by Hermann. In 1861 much was written in a new format that appealed more to mathematicians, of which Robert was no mean slouch, and thus much was redistributed throughout this second version. However the paper on Quaternions perhaps would be most helpful in answering the question.

In passing I note that because of Euler's work, Grassmann had the solution to how to deal with a swinging arm in 3d, he just could not see it in 1844 or even until he read Hamiltons Quaternions. It turned out that his exponential form, and his firmly fixed plane of swivelling was all he needed to construct the quaternion solution! When he realised and demonstrated that, he wrote that Quaternions had no more to teach him! As with Möbius he believed his system now enabled him to instruct those who used Quaternions.

This may sound arrogant, but as I pointed out, Hamilton on reading the Ausdehnungslehre 1844 recognised this immediately! It took Grasmann another nearly 20 years from 1853 when Hamilton first encountered Grassmnns work to come to the same conclusion!

Hamilton derived a solution for the 3 dimensional swivelling situation. The following is a treatment of that solution which shows a similar approach to Grassmann with regard to the quotient operator, but which does not recognise the value of ordinary line segments and trig line segments.
http://www.songho.ca/math/quaternion/quaternion.html

I want to tackle the rule of anti commutativity. Reviewing Grassmanns introduction to it, I can find no geometrical model that matches it precisely beyond the line segment.AB=–BA

I have puzzled extensively over this rule for a long time, under the misconception that Grassmann saw something in the geometry to inspire his insight. However, understanding his praxis better leads me to conclude that this too is a bit of Grassmann analogising!

In terms of the history of complex numbers Bombelli stands out as the only one to "define" a negative area. Using neusis and his carpenters square he drew conjugate forms as solutions to some of the quadratic equations. In fact he defined the terms conjugate and adjugate for these geometrical forms. In effect, therefore the conjugate of a figure was its " negative" . It allowed for the solution of say x^2 +1 =0. One figure would be above a line, the other would be below that line. Thus x+ y and x- y would form 2 figures conjugate to one another( that is as linear forms in their own right. Set them both =0 and draw the lines on graph paper.) Thus x+iy and x–iy made the same sense to Bombelli, but he was on his own with this!  Today we can consider these as vectors along the 2 lines , and the rotation angle between them to be pi/2 a quarter turn rather than a full half turn implied by the concept of negative.

Especially in the the completing of squares, this conjugate representation was helpful in representing the " negative" rectangle or square required to solve the square problem.

Geometrically this was always harder to represent than " algebraically" or in symbols. This is because in the page you have no way of distinguishing a negative value of area! A negative value of volume can be represented, but area , being a surface in a Surface has no orientation to support a negative sense. Bombelli elected to draw a line and draw positive area above it and negative area below it. However he called thise arrangement conjugate, and did not assign a negative sign to the whole form.

The defining of negative in the plane is not so clear cut. Many positive and solvable equations also have squares or quadrature that has the " conjugate" form, in terms of the difference of 2 squares. These Bombelli called adjugate numbers.  There was some property of the conjugate numbers he could not apprehend. That property is rotation!

So now, from the outset Grassmann has to pin don this " negative" property.

Switching the order of the symbols in a line segment , and then translating or displacing in that " direction" seems to make sense of the notion of " contra" or opposite. This notion is often simplified to negative, and a single symbol represents it. However it is clear that contra is not simple at all.

The orientation of a line segment is assumed or given. If this is not so, then it literally becomes impossible to assign" contra" any meaning.

Usually orientation in the page is agreed by convention to be horizontal and vertical. The principal orientation is horizontal.

The first line segment is drawn or constructed in the principal orientation. This is so ingrained in us after a while that we will actually rotate a drawing on paper into this orientation.

For any line segment in this orientation we may choose 2 points to mark the beginning and end of the Direction we propose to set as principal or positive. Thus we have a principal orientation followed by a principal direction.

We now move into Relative orientations of line segments. We usually establish a relative orientation by rotating away from( Aus treten!) the principal orientation. But we have 2 rotation options in the plane. By convention and imposition anti clockwise is the principal rotation direction.

Using these 3 principals we should be able to communicate a precise drawing plan, or construction instruction to another person.

The problem arises with the introduction of contra.

To construct a contra line we first have to construct a defining principal line segment or set of relative principal line segments. Contra is defined relative to these principal line segments. What is usually missed is the definition of a " contra" area or form.

In both the trig ratios and coordinate geometry it is natural to define regions of negativity, but these regions themselves are not defined as negative! The usual patter is that a line measurement( axial) goes negative or a function value goes negative in a specified region. The region itself is not considered in these definitions.
Bombelli could define 2 regions in this coordinate system as negative and the other 2 as positive. They are vertically opposite geometrically, contrariness is to the " side" of the positive regions not diametrically opposite!

In the light of all this Grassmanns notion of firmly fixing directions to line segments has consequences.
Firstly having defined the principal Directions how do you define the product of these line segments?
For example if AB and BC are principal and relative principal directions what is AB x BC compared to AB x CB?

Arithmetically one should be contra the other, but geometrically what has altered?

In the drawing nothing really. The difference is in the observers procedural behaviour relative to the drawing. This is neatly encapsulated in the symbols, but obscure in the diagram!

So how do I proceed? Grassmann states he carefully multiplied all line segments bearing fully in mind their " relative" directions. But how do we determine relative direction?

Strictly speaking we should do this relative to the conventions, but almost always we lose connection to these principal directions as we swing the drawing round to complete the construction. Then we question the validity and usefulness of the convention!

So, specifically Grassmann states that symbol interchange accompanies sign switch, that is the factors of a product of ordinary line segments ( the parallelogram) are the line segments that actively construct that form. Switching those line segments results in a contra form being constructed.

The problem is when you switch the line segments the only conventional meaning for that is that the principal line segment and the relative line segment change. When you do that construction you do not get a clear " negative" in the sense of a region separate and distinct from the initial region, you get a differently " oriented" form that overlaps. It is in fact a rotation out of the plane and back into the plane about a diagonal axis between the two. It is in fact a conjugate reflection in this axis.

That is fine, if that is what is meant by contra geometrically, but arithmetically the 2 forms are defined as cancelling each other. While the overlap would fit this geometric description, the remaining parts in the Gmomon or conjugate are vertical opposites either side of this gnomon diagonal. What is the geometric significance of that, and are we obscuring a geometrical or mechanical phenomenon by reducing this to 0?

Commentary
Clarifying what anti commutativity is not geometrically leaves one possible solution as to what Grassmann was observing.

First a brief introduction to cyclic quadrilaterals.

http://youtube.com/watch?v=ylXdx2qv_FQ

Now parallelograms as a rule are not cyclic, so my point is not about the theorem properties of a cyclic quadrilateral, but about how we label a quadrilateral in particular.

Underlying all math is the trig ratios. These ratios are extended by the CAST rules into all quadrants of the circle. While not every quadrilateral has 4 points on a circle , we still tend to label the points cyclically..

The next point is to understand vertauschen and umkehren in relation to trigonometric conventions.

Simply put these ideas combine in the practice of cyclic order interchange. This is a basic "symmetry " process used to prove similar results in specific circumstances. In fact it relies solely on the observer realising their freedom to label corners cyclically starting wherever they choose, certain demonstrations are independent of the geometry at a locale because they rely on the consequences of procedures not on specific measurements..

Given that we have principal orientation, principal directions and principal rotation it is usual, once a figure is constructed, to leave the figure untouched in the plane. In this way the figure itself serves as a mark or reminder of these Principals.

For a drawn or constructed parallelogram ABCD what can we interchange? Nothing of the lineal symbols but everything of the labels for the points!
The question is how should we interchange the points?
It is usually but not always specified that we change the point labelling cyclically.
Thus the same ,fixed  oriented parallelogram in the page may no be labelled DABC, or CDAB, or BCDA. This is an anti clockwise cyclic order inter change. The principal orientation has not change, nor has the relative principal rotation for adjacent sides. What has changed is the relative principal direction of at least one of the sides at each interchange .

This effect only applies precisely to parallelograms.

Nothing but the labels have moved, yet according to the strict formalism, what was a positive product now becomes a negative product!

In this case we are introducing a negative sign, not to show that the figures principals have changed, but that we have changed our labelling of its principals!

This has 2 effects: one we appear to cancel something which has not changed, and contrariwise what is fixed appears to rotate round its perimeter! If you can imagine the track of an early world war 1 tank you will have the movie of what I refer to.

Dynamically then, and Grassmann was from the outset dynamic, the label interchanging cyclically is the only process that I understand that introduces a negative sign into one of the factors, thus supporting his assertion ab= –ba.

If a was in the principal orientation with the principal direction and b in the relative principal rotation orientation with the principal direction, specifically in a parallelogram interchanging cyclically in the principal rotation would bring symbol a into the relative principal direction position and b into the principal orientation with the principal direction now pointing in the contra direction. The product of these 2 factors must now be deemed negative to the initial product.

As you can see this is a purely symbolic device . It's power is to arithmeticalky ignore certain products but also to encode rotational information of a special kind.

Of course, as soon as you introduce the trig line segments, this kind of interpretation has to be ignored because the explicit trig functions or ratios introduce their conventional sign rules as in CAST.

Finally, using trig line segments but with swivelling of these line segments relative to the principal orientation allowed, the rules change again back to normal arithmetic. The swivelling itself introduces principal relative rotation as a presupposition, before any relative rotation against ( relative to) the swinging arms!
http://youtube.com/watch?v=PFb4lgdB2aI
Note that cyclic order interchange does not give a negative area! This is because in a triangle no lines are in the contra direction! That is the principal orientation is not utilised in any other part of the figure.
From this it is clear that only figures which are constructible from parallelograms will have anti commutativity!

Note also how shifting the paper does not change the principals, rather it indicates the mental processing of the observer. That mental processing per force involves rotation relative to the observer.
http://youtube.com/watch?v=sd10yVm6y_E

This is the missing part of any notational or labelling system which Grassmann managed to capture in his labels, and Hamilton only captured by coordinate transformation!

Because Grassmann could notate rotation by the correct choice of labelling( ie cyclic ordering) his work was label wise much simpler and cleaner.

As misconceptions fall away I ask " What is the advantage of Grassmans method over traditional Trigonometry? "

Grassmann after all does not claim to remove trigonometry, only to attain its ends more quickly and earlier. In the meantime he also offers some insights into the nature of certain Practices and processes.
 For example, what advantage is AB + BC  = AC?

I suppose that as one explores  Grassmans enterprise, this gradually becomes clearer. As it is, in this Vorrede , it offers tantalising glimpses of something!

For example: could the cosine and sine laws be derived using Grassmans " system?"

It is unlikely, because Grassmann relies on fundamentals like these to define his system. But it is tantalising that  AC ^2  as ( AB + BC)^2 is almost the cosine rule except for no trig values. But then, instead of keeping the anticommutative terms, they are cancelled! This means that the nearest possible identify is Pythagoras theorem.

For several reasons this is not easily obtained by this new parallelogram product. Instead some have used Grassmans trig line segments as primitives and used these to define the dot product. It is almost the same as the inner product but there is some confusion..

Whatever the result, it seems, the ideas have been yet again reworked by certain others to make their own consistent algebras.

This begs the question; what did Grassmann actually "teach", and was it understood by admiring students who passed it on faithfully, or was it just baudlerised for its inspiration and technique. But never fully presented?
Consider this video in which Norman uses the trig line elements and the inner product properties.

http://youtube.com/watcg?v=iUSbqBeaXpk

Normans nition of quadrance obscures the constructional issue for the parallelogram product. Also it introduces the dot product as a expression of entities called points which are rename trig line segments arranged in a grid.

While stating that we do not want to presuppose Pythagoras, while being motivated by his theorem, he nevertheless establishes a grid in which it is only possible to prove Pythagoras theorem by algebraic manipulation.

Again, this is the game algebraists play, when they hand wave and say: we do not need pictures, just clear algebraic rules!

The way I put it is geometry for the blind!

We sighted people know we need both.

Is this a demonstration of Grassmanns method? In fact it is not. It is however a presentation full of supposed Grassman terminology, but very little is solid Grassmann labelling. It is a reworking in his style.

Now I find Norman quite logical and systematic. I have pointed out how Grassmann appears whimsical and fanciful. This element of Grassmanns method is the one that seems to be rooted out by " serious" mathematicians. In 1861 his brother published his redacted version of Hermanns work.

Commentary
The use of gleich in German mathematical and philosophical texts is symptomatic. It reveals the context and the motivation of the author, for it is the authors choice to use gleich or gleich ung to denote some situation or expression.

Like "like" the word has many subtle meanings, because like is a fundamental comparison. In fact we often as students to compare( that is look for like things) and contrast( that is look for differences).

When considering analogous thinking, Simile , metaphor, likeness and analogy cover the same ground as does duality. Thus the humble= sign can be overworked.

Terms like polynomial, expression, equation, formulae, identity, constraint are often ill defined references to the same set of symbols.
As of yet I have not read the work of LaGrange, but I am aware that in his system, an expression of a constraint was written in some equation form. This tied together those symbols whose range and meaning were bounded by each other.

As an example, Möbius would constrain his subjects in the Barycentric calculus by writing or deriving an equation that expressed the interrelationship of his subjects parameters. He wrote AB +BA = 0 to define the concept of negative in his line segment descriptions. Thus his statement expresses is " line segment AB combined with line segment BA is set to 0"

Note it expresses nothing about length or orientation. Thus it conveys direction along the line segment can be opposed. The 0 is symbolic because it expresses Shunya, meaning everything. The constraint implies a sinle line segment universe!

We can regard some of Grassmanns symbolic expressions in this way, that is as constraints.

Commentary

Misconceptions fall by the wayside?

On every pair of parallel lines it is usual to put arrows indicating parallelism!

Suppose , now those arrows indicate direction. Then for each pair of parallel line segments we have an indicated direction. As we move these parallel pairs round cyclically we note , in the parallelogram, that the direction of poining( the principal direction) poits at some stage in the contra direction. Thus we have in a parallelogram the evidence for these pairs of parallel lines turning round and going the other way. This means, in the product of 2 adjacent line segments the value must switch sign as they cyclically interchange round the parallelogram!

The use of Grassmanns style pervades modern mathematics particularly Normans. This video shows a mixture of styles and how they are used to communicate mathematics and notions of space, especially trigonometric ones. We cannot step back from coordinate geometry or mechanics, but we can use the reference frame more intuitively and heuristically. The cyclical interchange  of points also connects the point notation to permutation and combinatorial theory.
http://youtube.com/watch?v=jHm3C1UCR5o

I think it is important to note that the product of 2 line segments is a parallelogram. The introduction of the dot product was to formalise the trig line segment projection. It should not be made somehow " mysterious! " I note it takes Norman until lecture 28 in his wild Lin alg series to introduce the dot product! And that is not because he has not used it in his earlier lectures. The misconceptions around this method of analysis nd synthesis are many and confusing. In the end , Grassmann is all about trigonometric Analysis and solutions to issues.
The cyclical interchange of points?

http://youtube.com/watch?v=2VhU7_R2gy8
Note also the clear definition of a negative area! This is down to Grassmann and is perhaps his major historical contribution! Until his book not even Bombelli had a clear definition of negative product in the plane. It is per force associated with rotation

Commentary

The obvious point of these translations, but perhaps one that escapes notice is what I call " observer as animator" .

To calculate, the procee involves animation at all levels. This the observer/ calculator must do. They must interact with space, in 3 d thus may involve building a model or sculpting a form. In 2d this may involve drawing a sketch consisting in line segments( Skesis or Schemata). But then the solution involves manipulating that form, either mentally or physically until it gives a known ratio to a given Metron. Mentally the form is rotated, translated reflected , topologically bent, covered, shrunk, expanded and otherwise distorted to give up its secrets. This the observer has to do. This the observer often does not realise or have a clue how to proceed doing this!

Grassmanns method provides a set of standard heuristic moves to bring closer the solution. The first is the parallelogram product, but in dynamic form. The second is the summation constraint that identifies the diagonals of the Parallegram. The third is the cyclic interchange of points which facilitates cancelling and elimination , the fift is the trig projections and their constraints. The swivelling line segments are a sixth .mthere are many other Grassmann only touches upon in the Vorrede. The Barycentric calculus, the tangential contact points, the absolute length labels etc etc. all tools that make for a faster heuristic approach to solving problems.

Returning to the product of a line segment with itself it is now apparent that AB^2 is a line segment of twice the length of AB , that is 2 AB . But what we now do, because it is a double line segment by parallel projection or translation of each parallel line( collinear line if you will allow) relative to each other, is treat it as a segmented line segment and bend or rotate one segment relative to the other!

Our language tells us that is what we do, for we say the square on the line! Because we cannot do it by translation it has to be constructed by rotation. So the finl step of the product

AC ^2 is to rotate one of the lengths to form perpendicular line segments AC and us this to product a square.

On the face of it we would perhaps do it in this situation to uphold Pythagoras theorem. But in fact, allowing this rotational construction from line segments is directly linked to the content of Book 2 of the Stoikeia.

Commentary
 It occurs to me somewhat whimsically that we can now use Grassmanns swivelling operator or quotient Algebra  in the following way
Given (aBC)^2 => a²BC^2=>a¹(BC + BC)--nb. The a now has to be distributed by the projection/ translation of BC by BC  usual in the construction of a parallelogram.
Since the second part is now free to swivel:

=> aBC  + aBC•e ^å => aBC  + aBC•e ^iå where i is and å is a radian measure.
For the so called "complex numbers" we can write

=> aBC  + aBC•i

And we can say this refers to the diagonal of a square formed by rotating the second line segment by /2 radians.
Whence
a(BC +BC•i). Revealing the distribution of the factor over associativity rather than summation, and also describing square production as a rotation by an imaginary operator or quotient on a bisection of a line segment.

The whimsical development can be extended without hindrance to the general "complex number".
aBC + bCA•i .

What does this mean?

The interpretation of "complex numbers" in the Grassmann system is intuitive and geometrical and perhaps suffers by being pigeon holed as number!

The distributiyity over association occurs in this whimsical fashion .

aBC  + aBC•i is the diagonal of a square parallelogram.

Thus the parallelogram product of this square is
a²BC•BC•i

BC•i is a line segment rotated at the corner C of the square into a positin CD which is perpendicular to BC .

Thus I can rewrite the product as
a²BC•CD

Where BC•CD is a unit square.

Distributiyity over associativity is one of the features of the Grassmann parallelogram product and it has geometrical meaning. I discuss this in the earlier post about building with blocks!

Normally it is completely passed over or ignored in the notion of commutativity, but in older lists of arithmetic rules it used to appear in the section on distributive rules.
• here is used in both senses as arithmetic product and the now ubiquitous dot product! But it only takes the first product meaning in the quotient algebra. In the exterior algebra it just means the parallelogram product!

Again, I have to draw your attention to how whimsical this alternative way of thinking is! It is completely idiosyncratic and in one sense nonsensical!

However it is really quite radical because it is totally inclusive of the subjective and objective processing involved in the human mind.

Most scientists and mathematicians are taught to exclude the observer and to promote the observed! Here Grassmann by default includes all and attempts to clarify all! Every twist and turn, wart and blemish in our thinking s recorded and annotated in his method. Therefore it reveals precisely where we fudge things to get sn idealised form, here we reason by analogy, where we derive by ly logistic logic, where deductive and inductive reasons are used to support a claim, and where pure " fancy" has obscured a deeper intuitive significance or meaning.

It can be uncomfortable for some mathematicians to see that what they hold as true is only analogically so, or what they claim as " wrong" is actually a doorway to a deeper relationship, a metaphysical and or subjective processing one!

It is this which has made Grassmanns method ultimately triumphant!

Ausdehnungslehre p xii

This roughly is the content, which I have reserved to myself  for the second and last volume, at least as far as They have been processed by myself up till now, with it  the work will be closed !
The time when this second volume will appear I cannot utter, in relation to which,  by the diverse roles in which  my current office involves me,to me  it is impossible  to find those ( topics) the peace, which is necessary for  the processing of the same.
Thus  this first volume also builds  itself into a unique whole established for itself and complete in itself, and I keep it for a multi-purposed event  ,  in order to give permission to   publis:  together, this first volume  with the applications related to it;  as both volumes together, and  separated from the applications !

In practice it is according to the unavoidable, necessary presentation of a new expertise, therewith its rank and its significance becomes correctly understood,   to show like "bang!" its application and its relationship  to related content! The Introduction should also identically  serve this (purpose). This introduction is the nature of the matter, according to a more philosophical nature, and, even if I separated out this introduction out of the melding of the complete work, still this happens:  they recoil in horror through its philosophical form!
Around mathematicians   not like "bang!"; yet still, Specifically there is ruling  among mathmatician, and to a certain degree not with injustice, a conscious Dread of the philosophical characterisations of mathematical and physical content , and in practice  the most deep examinations of this Style, how it is promulgated  from Hegel  and his school , suffers a disclarity and an  arbitrariness which annihilated all the fruit of such deep examinations!
Pxiii
Regardless of which  things ( just mentioned) I believed it, the matter , to be  guilty,   by which  (in recompense impelled by a Muse) the new expertise   must be allocated its  place in the field of Wisdom, and therefore so as  to satisfy both promotions,   setting before you all an introduction , which without the fine grasp of the whole essentially causing loss( of interest), can be(come) thoroughly worked over, strenuously evaluated.

Commentary
Well that was fun!

For a digest of Hegel I recommend chadafrican YouTube site. Not only is it a good review but it is alternative to Grassmnn's assessment!

I also recommend searching for "analytical mechanics lagrange" on YouTube  to see how Newton Euler Lagrange and Hamilton shaped modern Mechnics while Grassmann was right on the outskirts of this group because of his isolation in the Prussian empire, and his low academic status.

We see a reason for his lack of advancement in his conviction that mathematicians in particular needed a good strong dose of Philosophy! He was not alone in thinking this in Prussia where Kant and Hegel Schiller and others were of the same mind. However Gauss was only slowly forming this opinion and only later encouraged Riemann to explore this issue.

Consequently Grassmann deliberately set his first volume the task of being the cat among the pigeons. Later he was to write: " that work was more for the reader of classical thought because of its more philosophical form".. He took on the mathematical establishment and lost. Consequently the 1861 version is a radicl redaction of the first volume, with it's overviews and applications

Commentary
Although I regard this as a Gross simplification, nevertheless it may be helpful to some readers.

Hegel's philosophy I characterise as mindfulness, in which the observer pays attention to both conscious and unconscious processes within their subjective processing. Consequently, it starts with tautology ( self referencing notions) and ends with tautology.

In this process, entirely subjective, the final tautology represents a " movement" without moving, that is a new understanding of the original tautology.

This process is called dialectic, and it is also called Platonic or Socratic dialectic or discourse.  It is a mark of the influence of Hegel that it is also called Hegelian Logic.

Because dialectic deals with notions in a process, it is noetic. This means it encompasses all aspects of the subjective experience. Therefore it has a place for intuitive thought and experience within its process.

It posits that all human experience is entirely subjective, and we only experience objectivity by denying part of that subjective experience. As a a consequence it can analyse all kinds of human experience as a result of this denial or suppression of " half" of the whole. The whole however cannot be determined because of an assumption of indeterminacy right at the start of Hegel's analysis! This essential tautological assumption  reveals " movement without moving" and " inversion" as fundamental notions of subjective processing.

While I fully acknowledge Grassmanns characterisation of the Mathematicians dread of philosophical discussion, I am still unsure of my translation of "nicht sogleich".

However for a mathematician Grassmann is attempting to show that if you fully label every aspect of your thought process you in fact make analogous thinking much easier and more creative. In addition you show the vanity of " abstract" thought and " algebraic" representation, because they are simply tautologies of analogous thinking.

The " strangeness" of Grassmanns logic is thus my experience of his thoroughly dialectic approach!

Commentary
This video explains the trig line segments algebraically, that is symbolically using some extra ideas like a function . While it may not seem related to the swivelling arm of  Grassmanns quotient operator, the generalisation is in fact a rational version of it.

http://youtube.com/watch?v=iDIYUw1QcDk

---

AUSDEHNUNGSLEHRE pxiii1844continued

---

Also I note that likewise among the applications, those which are themselves related by contents related to Nature( physics, laws of Crystals), they can be thoroughly evaluated;  without that, the progress of the whole development would be disturbed thereby.
Through these applications to physics I especially believe the  importance , yes the indispensability of the new expertise and the analysis presented within it has been demonstrated.

That the same applications in their Concrete form , that means in  their transference into  Geometry, an admirable teaching content might provide,  of which an elementary treatment is definitely a thing  doable. Once again I am hopeful  to be able to provide supporting evidence in a part time way , in relation to which, to such a verification, according to its appropriateness in the work itself  I am able to bring into being as found not a single place!

Specifically it is an elementary  treatment of Statics, even if it should go forth in the same manifest and genera( also by construction presented!) results, unavoidably necessary, the labels of the Sum and the products of line segments to be taken on board, and the principal laws for these to be developed, and I am certain that no matter who has vehemently sought once again the taken on board bits of this labelling, it never again  will be given as extra insight!


Even if I have thusly completely adjudicated its Right in favour of the new expertise , (which at least partially its processing is published now), and in favour of it the  demands, which  in the field of wisdom it can make,  I will not cut it short in any way,therethrough i believe  to myself not to  draw close to the accusation of the pretense! Because then the truth calls out for its right; it is not the work of things which it brings to the conscious awareness and to  acceptance by you ; it has its nature and its therebeing is of itself; and to curtail its right out of  false modesty,it  is a betrayal of the Truth. But  I must demand the more forbearance of  hindsight  for all of everything which is what is due to my work on the expertise!

http://www.gaalop.de/wp-content/uploads/134-1061-Zamora.pdf
This paper located by Kram1032 makes for some useful practice in identifying the Grassmann forms or labels / handles. The presentation is elliptical, that is a lot of the detailed work is left out. The Clifford algebras are presented clearly and concisely and the fundamental forms or "objects" of the Algrbra are identified. These fundamental forms are the surfaces that most easily conform to the generalised products. Thus in 2d the fundamental forms are parallelograms and do conic curves . In 3d tetrahedrons and thus conic surfaces. The number of dimensional line segments Gives greater rotational freedom, but the " view" is selected by which products are set to -1,0,1.

It is hard to connect the dots! We start with points and line segments and then we get various products and finally a simple theory of Curves, here generalised to curved surfaces!

To find ot how you really have to read the rest of the Ausdehnungslehre!

I am still working on the above translations, but the point of this whole thread was to grasp Hermann's thinking. It turns out that it is very very radical for his time, but it's subject of meditation is just trigonometry and the geometric processes like extreme and mean etc that evaluate or proportion geometry. Thus it is useful and instructive to put aside the notions of vector and to pick up an old traditional course in trigonometry, especially one where the right angled triangle ratios are emphasised over the circular functions!  While the circular functions are useful, they are Eulers construction of the right angled triangle indexer or ratios. These ratios do not require the arc to be useful! They were extended to the arc measure by the Indian and Arabic scholars at a much later date. In many medieval images you will see that astronomers used a square sextant to measure the position of stars, not an arc. The square gave right angled triangle measurements directly!

Later use of the arc , minutes and seconds relied upon a accurate machining of a disc into equal degrees. 360 was chosen to accomodation Babylonian data, but in fact Babylonians still used the chord to diameter proportions!

The arc has one natural measure and that is the diameter of the circle to which it conforms. Having made a true disc b
Y mechanical means the rolling of the disc could only be measured by how far it turned to move through its diameter! Thus pi was always a symbol of a whole rotation of a disc. Euler threw a spanner in the works when he associated  pi to a hemi circle arc!

Thales geometry of the circle is the foundation of Trigonometry. Here the arc, the chord, the radius , the diameter all meet and they reveal themselves to be incommensurable! And yet this is the simplest of all curves, but the hardest to machine. It is special and makes straight lines special and planes special. Cones and cylinders also become special . Their specialness is in our human conscious perfecting of them. They do not occur naturally. They occur by a pragmatism whose inflexibility leads to ideal forms. They are thus totally subjective creations of our subjective processing.

Because they are subjective, the incommensurability is either to be exxpected( human error) or unrxpected( design specifications are fixed and simple). By them we learn that we interact with a changeable reality we cannot apprehend! We must snatch at ephemera and make them real to conceive of " reality"! Reality is what we make it, what we can make( fact) or what we can describe( myth) as processes by analogy. Both are vital to our sanity.

Panta Rhei.

I am currently researching Napiers Wonderful Logarithmic Tables.

We have seen how Grassmann introduces the logarithm in the quotient operator algebra, that is the quotient product for trig line segments in distinction to ordinary line segments. Tha analogy is exact.

Moreover, the life's work of Napier has the identical goal to that of Grassmann in as much as they both wanted to make mathematical processes easier on their fellows by carefully constructing notation or labels that encsulate a notion of process!

These articles make very clear how. Napier worked, although some historians are confused by his kinematic motivation. Do not trouble yourself over this, because his motivation was circular motion. In fact his logarithms of sines are a particular correspondence between the sines and the tangents of a angle measured. From the vertical direction. It seems confusing therefore that he should refer to the sine not the cosine, but in fact it is not. For a projected radius the measurement horizontally was often called the Haversine. In fact nautical tables preferred this layout to the one we get introduced to at school. The cosine table was quite a late introduction into the trigonometric toolkit, as most mathematicians actually used symmetry notions to determine the dine. Thus a notion that a projected radius splits a line into its cosine with the remaining part being the sine , although in fact inaccurate , was close enough to be a useful rule of thumb. The haver sine was in fact the correct notion.

Commentary

So both Grassmann and Napier recognised the logarithm as a correspondence notationlly. The indexial arithmetic is a contrived convention based on the geometric means in series with infinite terms, the fact that we say infinite terms shows how we have disconnected from the geometry. Napiers construction and velocity constraints initiate that the process woul never end, but would always be regular. Like the famous paradox by Zeno the faster runner never actually completes his race before the tortoise! However, pragmatically the faster runner passes the tortoise at a given time. Zeno's paradox only confuses when a notion of time is left out of consideration.

Later, Newton uses this same concept to frame his deductions in the Principia. Time, whatever it was( and it is only a comparison of motions) was crucial to avoid the argues using Zeno to refute infinite process deductions. This is not to say that such deductions should not be rigorously scrutinised, because many strange ideas have crept in under that particular line of reasoning!

This is a post i did on the proportioning based on my translation of the relevant actual text of Mirici Logarithmus.

In point of fact it is not at all difficult to conceive of a rotary machine ith just these properties!
A wheel has a straight arm extended radially beyond its perimeter to intersect with a scale set at right angles to the diameter. At the same time a rod drops from the point on the rim from where the armature extends, onto the diameter. The length of this rod is the sine of the angle measured anticlockwise from the diameter. This degree scale is set against the rotating wheel.

Because this rod is attached to the rim of the wheel the sine can be measured directly along the diameter from the rim, once the length is marked and the wheel turned to the diameter line , or alternatively the rod is swung to the radius of the wheel and marked off there.

Commentary,
I want to look at the concept of function in terms of 2 notions: purpose and process.

The concept of function has its origin in the Prussian Holy Foman Empire, where it's organisational role was paramount in social and militaristic settings. That the idea should be carried over into the structure of mathematics is uite natural . Determined and deterministic behaviours are the bedrock of mathematical consistency. Thus the purpose adoration of a function highlights it's labelling role. Such. A role encapsulates a notion of order, structure and relationship to a whole. It also acts as a characterisation and a definite distinguished or distinction.

The process role is precisely the duty and actions the function is tasked to fulfill. Consequently any expression of those duties is in fact a process or procedural statement. Mathematics is composed of procedural statements, formulae algorithms etc, and in order for these procedures to be about something numbers are defined in an objective sense. This distorts the underlying concept of numbers which are the Arithmoi. Indeed Arithmoi are spatial concepts, but entirely general and formal. They are the products of procedures acting on subjective interpretations, evaluations and distinctions. The du jective nature of mathematics has been obscured by misunderstanding of the role of an ideal, or form. It's function is to allow iterative interaction with space.

Because a function has a purpose it is labelled to identify that purpose. Many of the labels Grassmann creates have this function role also.

Commentary.

Ok so I have given the background to logarithms some thought, and the claim that they are a link between arithmetic and geometric series.  For me they represent a functional relationship. The notation covers over a very great deal of subjective processing and picks out only certain aspects of the total information.

Napier reputedly performed 10 million calculations, this was so his colleagues would not have to do so.

In fact Napier used the binomial expansion to structure his work, and lessen the  calculation burden. The second diagonal of Pascals triangle tells him which sines to pick for each logarithm. Gradually as the index increases more of the diagonals may need to be consulted to see if they affect the 7 th significant figure, but 10^14 is a very large number, and implies the 14th significant figure would have to be factored by a coefficient greater than 10^6 in the binomial expansion.

In addition, the proportions are not equated. They are calculated and associated with a count of the number of terms factored together. Thus Napier and we have complete freedom to associate the proportions  or geometrical terms to any regular dynamic or arithmetic progression. Usually these progressions represent arithmetical forms, hence the name Arithmoi in his title.

Napier associated a geometrical series of values from the sine table to an infinite straight line. However he could also, less clearly associated them to infinite sectors in a circle. Thus as each proportion is calculated it is associated to a fixed arc length. In this way the proportions turn the wheel. The value of the proportions is not necessarily connected to the rate of turn. In fact Napier specifies constant speed.

When Briggs came he was able to demonstrate that any geometrical series can be constructed by Napiers method. In fact, as this was before Descartes, DeFermat and Wallis, he could not show the now familiar log curve in Cartesian coordinate geometrical form using the logarithms as the x axis, and the proportionals as the y axis values.

The base, as we came to call it can be arbitrary, but it must have a product defined in terms of itself. In the case of the initial value for Napier it was sin pi/2 But then 1 small length had to be subtracted. If a small length element was added to sin pi/2 it would cease to be sine. It could be consecant or secant,or tangent. Instead it was given a new name as e the exponential base derived from the binomial series , over time.

We can apply logarithmic analyss and synthesis to any product algebra . Grassmann applied it to his quotient product. As far as I can tell he did it according to the rules of Napier. The fact that the products are not proportionals is of no import. If a product is defined then a logarithm function can be designed.

Commentary

While I cannot quite frame it yet I have the notion of the logarithm of rotations. By this I mean that rotations may be described as exponents of the exponential function, and in fact must therefore be complex or quaternionic logarithms. Whether these can be extended to n-dimensional Tensors or reference frames I do not know.

The structure of Grassmanns algebras allows for sums and products of these exponential forms as well as quotients. While Grassmann was at a loss until he researched Hamiltons Qaternions( only then realising he had solved his "looseness" problem for swivelling arms in 3d without realising) he later set a task for himself to do the incumbent processing to continue his planned development. However he died before he could make much more headway

The tantalising snippet from his proposed "Schwenkunglehre" whets the appetite and the imagination for more detail. The logarithm of rotations is precisely his idea of the log of a quotient. There he considers it to be the point of intersection of 2 lines forming the angle of swing. My notion is more related to Napier's logarithms using the sines.

By the way, it seems clear that this name is misleading. These proportions are found in the sine tables, but they are true geometric terms, the angle is insignificant to the calculation in fact  the geometric mean between sin90 and sin30 is sin 45 not sin60.

The logarithms do not index angles uniformly, nor indeed the tangent line. The concordance however is fairly accurate up to tan45, by then over 4 million calculations of the proportion had been done. To get to tan60 so as to calculate sin30 2 million more calculations needed to be done.

It is not necessary to perform the calculations with these constraints as Brigg showed, but in doing so Napier reveals how logarithms can be shaped to any scale or form, as they are only indices not measures. Setting them out in a measured way allows a calculation process to trace out a form by the logarithms.

As an example, placing the logarithms uniformly on a circle drives a rotation around the circle by an infinite iteration,

Napier  also placed the proportions on a line parallel to the logarithm line. He could have placed them on the radius of a circle, starting at the centre. As the sine decreased so the angle of the radius decreases and the point on the radius traces out a small semi ctcle, if it is rotated by the sine itself. But if it is rotated by the evenly spaced logarithms of the calculations it will trace out an inward spiral.

The relationship between the trig ratios is extremely convoluted. It requires the subjective process to account for orientation, direction and direction of rotation. It has multiple concordances between ratios and many surprising and convoluted algorithmic identities. Accordingly it is a rich field for meditative exploration, and a school for concepts of rotation, reflection , translation, rotational and reflective symmetries and computation.

It is the computation or arithmetic which is the odd one out! The introduction of quantity into a magnitude is simply so we can make a song and dance about it, literally. The experience of magnitude is entirely subjectiv; to communicate to an external other we have to specify and bound a region, this quantifies it and we can then communicate that specific region to an other. Depending on the conscious process of that other, they may understand the quantity on face value or as a label for the experience of magnitude .

The quantification of a magnitude always introduces a difficulty. The form or magnitude in a form is as is. The quantity we introduce is totally subjective and arbitrary. Thus as we compare,count , distinguish  thus generating a logos or language model of the activity of comparison, the dynamics of it, we have no way of predicting if the comparison will be artios or perisos , perfect fit or approximate fit ( even or odd, which I hope you can see is now inane!) . Consequently we have subjectively moved from an indeterminate whole without anxiety to an indeterminate multiple form with anxiety! Will the quantity specified fit?

These specific quantities are called Metria a single ( Monas) one (en) is called a Metron. The idea of singling "one" out ( ekateros) is fundamental to book 7 of the Stoikeia of Euclid. This Monas becomes the standard monad or unit for a process of covering (sugkeime) which is done by placing the monad down(kata) onto the form/ magnitude to be measured/ quantified/ compared, and counting( Katametresee). This count is literally a cultural song and dance, by which we interact with and order space.

The form so covered by contiguous( edge joined) Metrons as monads are experienced as multiple forms( pollaplasios). But in fact they are also experienced as epipedoi or floor coverings. We came to call these things mosaics . Archeologists finding these patterns on the floors in Mousaion, houses for the Muses coined this term.

The mathematical significance of Mosaics is a fundamental and continuing analysis of the Pythsgorean school of philosophy. Indeed no Pythagorean astrologer could qualify as an Astrologer( Mathematikos) without a deep muse inspired intuition of these forms.

These mosaics did not consist of standard Metrons, ie a cube tile or a hexagonal tile, but of a mixture of tile or block forms that continuously tiled or blocked the space being compared. Thus the spaces were Topologically described and counted. Area as a standard concept of counting only standard monads is a much later idea and of a different school of thought.
Mosaics were aesthetically designed to inspire, and thus often depicted scenes as well as just abstracted patterns. Such patterns were often traces of shadow dynamics throughout the yearly cycles.

By introducing standardised Metrons, a standardised approach to topology was introduced before we came to realise how limiting that was. Also anxiety was increased because one form as a Metron does not fit all!. The proclivity for perfect fitting forms drives aspects of mathematics today, but it is perisos or approximate fits that these mathematicians see as monsters! These standardised multiple forms are called Arithmoi. Thus all Arithmoi are mosaics but not all mosaics are Arithmoi. The counting of these standard forms eventually became confusingly modified into the notion of number.

Engineers and architects however love these perisoi! These approximate fits are pragmatically engineered to construct or sculpt real objects and structures. Pragmatics chooses the best approximation for the task, and filling and smoothing gives the final fom. It is artisans and engineers who apply forms iteratively in construction projects which are grand mosaics! We live and have our conscious bring in these grand mosaical structures of our own hands and minds. And we continue to process the experiences around and in us in this way.

As much as this is formal and subjective it is also our experience that magnitude is regionalised. The very deepest meaning of this we encapsulate in the perfected magnitude, a formal creation, called the sphere.
There are 2 other formal creations which result from the deep processing subjectively of the sphere itself, These are the plane and the later straight line. Nether exist as magnitudes in our experience. We formally construct these notions from regions, that is from plane segments or line segments. In fact it is clear that the sphere is a formal construction from an iterative process of construction requiring infinitesimal regions.

The complexity of the notion has fascinated ever since it was first conceived and continues to this day. The sphere encapsulates all our notions of analysis and synthesis, all our methods or processes of calculus both differential,integral and logarithmic. All our conceptions of topology and finally all our concept of spatial mosaics.

Because we quantify and thus introduce perisos anxiety it is not surprising, after so long a time of philosophising about it that we should find some counts of seemingly unit magnitudes should involve an endless process. In fact Zeno and Parmenides drew pointed attention to this. The pragmatist had no problem identifying the solution, as do engineers. You embrace approximation!

At some stage you simply decide enough is enough! This is essentially the principle of Exhaustion! Motivating such a principle is not only tiredness but also a notion of cyclical count. This count, as a record of planetary positions became known as Time and is dynamically measured, by dynamically cyclical objects in motion. Such measures are called Metronomes!

As you can see the Metron concept underpins all our measurement, including dynamic ones.

Dynamic measures answer the Zeno Parmenides conundrum. An infinite subjective process of analysis occurs like everything else within a dynamic cycle. Thus unless we actually extend the analytical process into infinite cycles, we can stop at any cycle by design or exhaustion! In particular. We can note that Dynmical systems traverse these infinite process measurements in cyclically finite ways! That is to say I can count a number of cycles and while thus distracted a dynamic object would have traversed a magnitude I was unable to determine by an infinite process!

The issue therefore is pragmatic. Is the infinite process necessary ?  The answer is no, but to be able to be as " accurate " as desired or needed is necessary. The use of the term accuracy and exact is misleading. Simply we can choose the form as the standard and then it is exactly and accurately 1 !

The underlying process is a comparison. The count is to determine a ratio. The ratio is to be reapplied pragmatically and iteratively in some construction or synthesis process. We only need to be using ratios that do not " fall over", crumble or shatter under stress and vibration! In addition, if we can we want to use ratios that are aesthetically pleasing. And we want to do most of this within the dynamic cycles of a lifetime! Pragmatics and aesthetics govern many of our most fundamental processes.

Before I finish, it is good to observe that iteration of the pragmatically generated ideal forms is fundamental to our experience of change. Having devised these forms and reapplied them to interacting with space we are reminded of why we had to devise them in the first place! Everything moves! Panta Rhei!! Our anxieties drive us to try to keep things still, but in so doing we lose contact with real life experience. We  often kill the thing that caught our interest and so inspired us in the first place. However a compromise is to abstract by analogy a form and then use it iteratively to identify the dynamic experience. This is precisely how are neural networks work!

A good example is found in film or video capture. Each frame captures an analogous form to the real life object. As the cycles continue the analogous foms change and we thereby capture change by iterative analogous forms synthesised into a contiguous mosaic.of frames.

Grassmann in his analysis and synthesis intuitively understood that these forms found our notions of everything, and their mosaic combinations are the stuff of our Musings. He therefore worked very hard to establish a labelling system that made this very clear, and rediscovered a deep and abiding connection to the philosophical enquiries, observations and formulations of the Pythagoreans!

The heuristic, mnemonic and whimsical approach is actually psychologically consistent with the way we subjectively process our interactions with space.

The modern number concepts, devoid of this rich association actually obscure the natural human processes involved in the logos analogos response: how we language our experience of real life , and thus synthesise a language model of our subjective Kosmos!

Commentary
Lol! Norman mentions the engineer versus the pure mathematician.

While this is called the dot product Grassmann dealt with this under his quotient product.

http://youtube.com/watch?v=2ZeP_Gn1x7M

In Norman's treatment the numbers have to be recovered or extracted from the vector. However Grassmann simply defines these line segments as what I have called trig line segments. The projection is an " arithmetic product" he says. Of course he means the product of the hypotenuse and the cosine, but in ratio form. Thus h x a/h = a where h is the hypotenuse and a the adjacent side . Thinking of the quotient a/h  we see it is Grassmanns quotient operator, but restricted to the right angled triangle. Grassmann generalises this for ant line segments  of any magnitude by a scaling factor.

Again the advantage of Grassmans labelling is that we can think beyond just finding a length of an adjacent side! As Norman demonstrates in this series. Lines, reflection rotation and transformations of the plane by the quotient Algebra.

I have rejigged some of the translations in the light of these later translations, and better resources for translating.

I think
AB + BC= AC  is as important as any of Eulers fundamental equations!

It takes a mathematician and a geometer and makes them into mechanics of Nature!

Commentary.

The reflection of a reflection is the original; but the reflection of a reflection in a reflected mirror is a rotation of the original!

This experience highlights 2 things: reflections are reflections through a point around which point a singular rotation occurs for each point matched to each Pliny in the mirror plane. ; the rotation around one single point in the mirror plane requires an intersecting mirror precisely at that point.

By obscuring the reflection from the hidden part of a mirror plane by another mirror, we provide our subjective processing centre with additional information that allows it to factor rotation into its computations. The result is a complex Jaleidoscope image, tha subjectively regionalises all the information showing all possible results!

This superposition of solutions is regionalised because real life results occupy a region, but the multiplicity of results indicates the ability to parallel process and blend and resolve a computational result..

How do we know the result is not real? Only by comparison with other surroundings and situations, otherwise we are per force so constructed as to respond to the complex image as real!.

An insect with a compound eye has the same experience but not the ability to blend the images bifocally into a
a single depth image without visible joins. In reality we have the same bisul field experience, but we merge the boundaries after the parallel processing computes the results. The mirrors force our processing algorithm to remove the blending step in that region.

The rotation we compute in the reflected mirror is not introduced by computation. The rotation is first physically done by rotating the mirror. The computation then uses the given constraints to compute solutions.

As an academic fact geometry disregarded rotation in favour of the form. Mechanics however utilised the form to model rotation. Rotation is actually the fundamental motion in and of space, in my opinion. From it we can derive all else.

Thus when mathematicians were asked to establish a method of computing rotations, it was the engineers and Astronomers that came up with the system. Geometers were too far removed from real life to even understand the request!

Grassmann therefore arrives at an interesting time. Before Riemann asked for the help of Physicists and engineers to solve the applicability problem of academic Maths, Grassmann was having an experience that gave an answer. While Gauss was muttering over the failure of geometry and instructing Riemann to find a way round these past failures, Grassmann had sent him a copy of the Ausdehnungslehre1844 for comment. It is my suspicion that Gauss got more from perusing that draft than he realised. His busy schedule probably meant he did not give it the time it deserved, given the philosophical challenge Grassmann placed before Mathematicians. Indeed he was in danger of being called arrogant! However, gauss who did not necessarily mince his words if he felt he had preeminence was quite mild and witty in his response. He recommended that Grassmann not multiply I many new and extraordinary terms, if he wanted his work to be seriously studied!  To me this indicates he read it after a fashion, but like Möbius could not make much sense of it.

However, if my experience is anything to go by , the work affects you intuitively!

Thus despite the progress on rotation in Gauss's time it is Grassmann who nails it tight in his overview. Later Hamilton acknowledges his preeminence.

Euler" s analysis of Rotation while fundamental was flawed in that he did not realise the issue of the doubling of the angle. This was caused by his arbitrary assignment of pi to the semi circle, when up to that point it was always assigned to the full circle..

The concept of rotation is still confusingly taught today, with students being given the impression that circular forces do not exist, only lineal ones do !

Both Newton and Grassmann, and many others believed in circular forces. Their labels were devised to best capture them, not to obviate them!

The analysis of rotation is quite subtle,

From Archimedes we get the notion of progressive orientation adjustments. From wallis a Cartrsian point version for all the Conics, and from the indian mathematicians an association with the sine and cosine ratios. Formerly it was the chord diameter ratio, and it is the chord diameter ratio that best describes rotation.

While Wallis derived his equations via Pythagoras theorem, and the work of Thales and Apollonius , Newton resolved to analyse rotation in terms of the tangent and the radially directed forces, the centrifugal and the centripetal. Combining all methods he explored rotation as a projection onto the orthogonal axes  of the force that acted tangentially to the circular path or course of the force, and thus always at right angles to the attractive and repulsive forces.

This mechanical analysis is in fact the description of the electro magnetic combination of forces, but that was not clear to him or anyone else for that matter.

Both Newton abd De Moivre used tHe as a constant in the unit circle to define many secret relationships between the sine / cosine and solutions to Multinomials equations of degree 2 and it's powers.mthe culmination of their work with Cotes was the Cotes De Moivre theorems of the roots of unity.
Alongside these roots was the constraints I have called the zeroes of space. The logarithms of these adjugate forms sum to 0.

It was votes who realised that by using the circle perimeter to record the logarithms he could write down the logarithm of the factors of 1 that is (cosx+isinx)(cosx–isinx)

ix = ln(cosx + isinx)

Later Euler fully derives the exponential form of Cotes discovery. This was the most celebrated formula dealing with rotation, and it made it plain rotation involves both sine and cosine. However mathematicians all over Europe were choking on the diet of negative numbers. They were positively manic over imaginary numbers! There was an intense and bloody resistance to these developments, with the preferred choice being traditional geometrical descriptions using sines and cosines. It was not untill Cayley reinterpreted the imaginary combinatorics as matrix algebras that mathematicians were able to begin studying rotation properly.

Grassmann had an alternative solution to that which involved imaginary values. In fact he restated the  concepts in terms of geometrical constructions. However the rotation is handled notationlly it requires orthogonal measures , arc measures and double angles!

The biggest error in misunderstanding rotation comes to the so called wave mechanics. Kelvin, De Broglie and others mistakenly  took the Eulerian sinf function as the model of a wave! The wave model was simplified from the vorticular dynamics of Helmholtz and Kelvin. The vortices were quietly dropped in favour of corpuscles and do the combination of cos and sine was lost. It returned later as physicists struggled to describe rotation of these corpuscles in free space. Wave or undulatory mechanics is just rotational mechanics. The forms in space are properly, vortices, and the rotations are more properly trochoids.

There is one last fundamental aspect to rotation and that is dilation. Dilation refers to the Newtonian centripetal and centrifugal forces. We are schooled to think of rotation as circular. In general it is better to think of it as trochoidal.

When an object is rotating it experiences forces radially. Such forces are the forces of dilation. In most treatment of rotation this dilation is normalised, that is the rotation is made to conform to our ideal! In fluids the normalisation actually leads to error in the description. A rotating space dilates as a function of its environmental constraints and the energy type of the rotation. In general rotations are outwardly dilating or inwardly dilating gyres. A purely circular rotation is an equilibrium between the two dynamics.

If we can accept this as endemic within fluid rotation we can better model fluid behaviours and thus rotational mechanics.  This relates then to the electro thermo magneto complexes of plasmas.

Commentary

I was on the bus when I finally grasped Grassmanns presentation of the quotient operator, which is an exponential operator. Crucially it elains why figure 1 in his Ausdehnungslehre is the construction of a rhombus. It is well to remember that at this stage Grassmann did not have a "solution" (Lösung) to the " looseness"(Lösung) problem of a trig line segment swinging freely in space. He only felt he had a notion for it swinging in a firmly attached (!) plane.

Having revisited the text to revise the translation of geltenden Werth and theilweise, I also realised that the complex sentence refers to " potable" values.
Gelten is an interesting concept in German. As much as I can grasp from Grassmann it is the potency or power of an idea or thing that establishes its validity or reality. Through power a thing exists or becomes..these concepts it seems are ultimately Socratic Platonic, but Hegel is the paramount modern philosopher of them!
Schliermacher is putatively said to be a major influence on Grassmann, but I think this cleric was a greater influence on Justus Grassmann and was a follower if not a student of Hegel.

In any case it is Hegel Grassmann mentions not Schliermacher.

Thus geltenden I took to mean a potable  quality , in the sense that it was none zero and increasing. On further reflection I used the Newtonian idea of assignability. This comes from Newton's consideration in the method of Fluxions, where he diminishes a quantity( note , not a magnitude!) to its last assignable value. This represents, in the method of exhaustion our inability to go on with the process.

So this assignable value reminded me of the tables of values for the sine and cosines etc. within these tables are all numbers from +/-infinity to +/-1 to 0 . They nestle there in their rank and file containing a secret : the Indian decimal polynomial system!

These tables are quantitative up to exhaustion. Thus +/- infinity is written as ad infinitum, and understood not to be a number but an indefatigable process! Thus Grassmann introduces numbers in their ranks as quantitie in his geometrical constructions. The partial nesting of the trig line segments and the ordinary line segments he realised gave a metric or a quantitative evaluation to all of his geometric musings by an overlapping ratio.

In the nature of his development of his initial and fundamental " laws or rules" the preeminent symbols of which are
AB=–BA
and
AB+BC=AC

Where A,B,C can represent or label, or be handles for: points, line segments,plane segments,tetrahedral spaces, complex crystal structures etc Grassmann was granted the extra insight  that 2 sorts of line segments are noted in the formal structures. The algebras for both are inter communicant, but nevertheless different. I have called them ordinary and trig line segments

He first came upon the notion of trig line segments as he reworked, reprocessed ( re-edited and redacted) the Analytical Mechanics of Lagrange. It became clear that these types of line segments, we're associated with swivelling or pivoting in the plane and in space. This swivel was first presented as a perpendicular projection, but later it was generalised by Grassmann as a type of swinging arm ( trig line segment) and producted with the exponential function as an operator. It is these trig line segments and the exponential operator that form the most general Algebra he could conceive. It is a quotient algebra,where the quotient represents the exponential operator.

The exponential operator is designated as such because it is in fact an infinite or ad finitum process..thus we cannot consider it to be a potable value in the trig tables, but it is clearly a part and mentioned in the trig tables. Thus immediatel we realise that the trig tables relate to this operator as particular and assigned values. However, the tables canbe extended at either end by this exponential process, and they canbe interpolated, again by this ad infinitum process.

In fact, within the Indian number system, the numbers themselves represent a polynomial in exponential terms, the base of which is 10.


Thus Grassmann identifies the exterior product by the swinging line segment swinging apart to get an assignable value, and the interior product by the swinging arms swinging together to get an assignable value.

Ordinary line segments have a fixed magnitude that does not therefore assume or presuppose any rotation or dynamic change in the line segment. The Algebra for that situation is clear and well defined in Grassmanns earlier work.

However for all dynamic situations the line segments need to be trig line segments. These represent fundamentally rotating space. Thus in Grassmanns Algrbra the fundamental dynamic is rotation and extension.thus the fundamental spatial orm is a vortex in the GrassmannQuotient Algebras!

The exterior product is based on fixed line segments, but they can represent dynamics by the use of scale factors. Thus these dynamics are based on transformations or extensions. For a very great many engineers and mathematicians this is sufficient to describe all kinds of dynamics even circular. But for the geometers from Newton and from Euoxus before him, the ultimate description had to be based on the sphere, and the sphere in its dynamic form is the gateway to all vortices!

When Descartes utilised the paradigm of vortices to explain orbits, it was taken seriously until Newton demonstrated a formula and a failure. The formula is now called a gravitational law; the failure was his inability to fathom the mathematics of fluid dynamics using his principles.!

Despite knowing the exponential base value, and logarithms and the binomial series and Fluxions, and infinitum processes, and the most sublime philosophy of Quantity, he could not see how to combine it all. This fell to Sir Roger Cofes as a happy and tragic moment in a worthy career! He established the logarithmic relationship
ix= ln( cosx+isinx)
Both De Moivre and Newton were on the wane in their powers of analysis, but Cotes was up and rising., when he was struck dead. The loss was palpable to Newton who remarked " Had he lived we would have learned something!"

It was about 80 years later when Euler put it all together in the Cotes Euler form
e^ix= cosx +isinx
and about another 60 years for Hamilton o come up with his Quaternions and the quaternionic form of this formula
 e^q = cosM+ LsinM
Where q= a+LM
Where a and M are " real" and L is the imaginary part of a quaternion., M being the magnitude or modulus of the quaternion.

In the meantime Grassmann had done his general analysis of how to do and learn and conceive of mathematics based on simple line segments. It was the trig line segments which in their dynamism furnished the link to the geometry and construction of imaginaries as arcs of rotation( in the plane at least). this is not to say that Hamilton did not realise this, but rather that Hamilton recognised Grassmanns formulation and expression of the idea as better and more general than his own. Not least because it avoided reams of calculations and keeping track of sign, but also because it could be generalised to arbitrary parameters( ie not just 4) .

The inner product is vitally necessary to make this quotient algebra work meaningfully.

Commentary.
In the Grassmann " melded " notstion( zusammenhängend) which he arrived at by am intense process of reworking and editing guided by the principles of the sum and the product of line segments as constructions.
Thus points A,B instruct a straight line segment. The line segment scaled constructs a ray . 2 rays construct an angle when summed, unless they are parallel, then they add or subtract quantity.( nb! not magnitude!)according to a magnitude called direction and a magnitude called orientation .
The magnitude of orientation is given or presumed. The magnitude of direction is given and fixed by convention.

What a line is thus becomes a moot point. Later we will find it can only really be defined by rotation about a given point! However starting with 3 points they form a line if the product BC extended= the product –BA extended indefinitely . Thus qAB + qBC =0  which Möbius used to define his line in his Barycentric calculus .
Of course AB = –BA gives AB + BA =0  but this is clearly a line segment or a single ray. This formula defines direction of a ray but not orientation . The orientation is assumed given.

The properties of points as generators of straight lines begs the question how? The simplest construction is by a mechanically straightened edge. But this too begs he question : how is a straight edge made? The surprising newer is by dual points constructed inside a crcle using a compass pair!

Ordinary line segments are Inthe mixed or melded form cAB where c is the quantity by some measuring scheme( metric)  and AB is the line segment generated with the direction A to B.

In contra distinction a trig line segment exists in a relationship. At first it seems to BR by perpendicular projection
Thus for for cAB and bAC we can write ccosøAC and bcosøAB. These are trig line segments in the line segments AB and AC.

Now I said that it seems like these are perpendicular projections, but in fact Grasdmann introduces them as a swinging arm protected onto the horizontal position. Thus they represent a dynamically rotating ordinary line segment or an ordinary  line segment that changes its orientation relative to a perpendicular to a projection line segment or if you prefer a start position for the swinging ordinary line segment.

These trig line segments thus have a particular role: when considering geometry in dynamic relative rotation to its initial conditions these forms of line segments are fundamental.

Now the Algebra while being inter communicant with the ordinary algebra for line segments has a remarkably different product rule due to the cosine circular function and in part the hyperbolic cosine function. This can be constructibly demonstrated using the rhombus construction in figure 1

The processing of the labels is subtle.
First is used to identify a corner angle, not to measure it. So how do we measure it? ā is used to symbolise a measurement in radians. So now we can say = ã radians.

But since is only labelling a corner we can replace it by another label. In this case it is the product ab where a and b are trig line segments( because we are dealing with relative rotations)

ab is thus a parallelogram product, in which case it is an inner product because these are trig line segments, or it is an intersection point for the 2 trig line segments. The appropriate interpretation is the intersection point. The reason being that the inner product becomes 0 at /2. The intersection point not only labels the point where the angle is located but so the trig line segments that form the angle. What it does not do is quantify or measure tha angle. That has to be done by the appropriate radian measure.

Now we can establish the operator as a quotient that multiplies, pro icing the Algebra with a new type of notation to perform multiplications. This notation is a logarithm of the exponential. Thus all multiplication can be done by addition of Logs! In one fell swoop Grassmanns notation allows quotient multiplications to be done by logarithmic addition, in addition the quotients are clearly the corners of the geometric inner product. Thus we have a geometrical representation of quotient multiplication which is clearly angular addition. These angles are corners of inner product parallelograms so now rotations can now be represented by logarithms which add like angles, are measured in radians and which look like inner product parallelograms!

The quotient notation or labelling can then be used to transform trig line segment quotients into exponential operators recording and manipulating rotations.

As you can see, no normal to the plane is required , but the line segments do have to be trig line segments to function properly.

The beauty of trig line segments is that if you fix their relative orientations they can be treated as ordinary line segments.

Commentary

E^iø = coshø+ i*sinhø

This analog suggests itself or reveals itself from Grassmanns argument  for a construction of the imaginary magnitudes. Using the E instead of e is to remind one that the hyperbolic functions are to be used , and thexprct hyperbole not circles

You cannot serve God and Mammon!

This ecclesiastical dictum, put into the mouth of Jesus, shaped the course of Prussian mathematics and ultimately it's groundbreaking impact on Western thought!

The debate arose as a religious difference. The philosophical differences had been simmering for centuries. Galileo and his Memesis  Garibaldi, Descartes and Leibniz, to name but a few. But in Prussia , the political and national turmoil brought about by several humiliating defeats by the French led to the debate about reform. The chief philosophical argument for reform was that of Humboldt. He recognised that if Prussia was to survive it needed self actualising individuals who would think creatively for themselves, for their futures and ultimately for the future of Prussia! The evidence seemed to be demonstrable in the burgeoning uccess of the French and their imperial conquests. New ideas, new technologies and new politics were all evident in the French " revolution" .

In order not to be engulfed by these revolutionary passions Prussia had to reform, and Humboldt focused on education. In the meantime the ecclesiastical powers were fighting to maintain their control over education. Their enemy was not the emperor of Prussia, but the burgeoning industrial concerns in the rest of the world. This was Mammon!

The growing success of technologies made certain groups and individuals very wealthy. It enabled them to demand certain things of their workforce and certain protections of their intellectual and industrial properties. Rural Prussia , the power base of the clerics was increasingly made to compete with these demands and repeatedly failed to deliver the social and economic goods! God it seemed was on the side of the industrialists!

The theosophical debate was whether God actually intervened in the world, and if so how? On the one hand clerics were preaching faith in a god who shapes history especially the Holy Roman Empire now seated in Prussia, while industrialists were pointing out that those who held a mechanical philosophy were receiving more of the earths bounty. The debate was very divisive with the rtionalists( those that believed in gods revlatIon through Jesus) accusing all others of being irrational. The debaters broke into schisms and dome even came to blows and pitch battles.

In the midst of this, a new philosophical breed of clerics struggled to provide sneers. Schliermacher, Hrgel, Kant all were engaged in the debate, while Humboldts followers eventually pressed the advantage and got the imperial seal on a sketchy programme of educational reform.

It was meant to involve all the society in the debate and it had to, because there was no 5 year plan, just Humboldts ideas! Everybody was executed to bring their expertise to the tak, and all levels of society in education were expected to mingle to discuss how to bring it about. The natural meeting places were the universities as such, nd all teachers were called to meetings, iscusdions, debates on the way forward, intermingling with the  academics and the philosophers.

It was at such meetings that a young Justus Grasmann first heard the call and dedicated his life to educational reform in Sczeczin. This is how the Grassmann came to be involved in the topic of the Ausdehnungslehre.

In the meantime Kant had managed to broker a peace between the warring factions about God and mammon, now represented as man's ingenuity, on the bais of ome fine philosophical argumentation, and the work of Newton. Newton was widely regarded in Prussia as an apostle of God, bringing god' s truth about the laws of nature. Thus, despite Newtons unconventional beliefs, he was used to join together believers in a truce. All would wait and see how god would unfold the universe according To Newton's laws. If so then Gid cited and it was described by mathematics!

This is for me a great fallacy, but even in Kant's brokered peace there were dissenters. Mathematics was not rational and thus revealed and handed down by god; instead it was constructible, and arrived at by the hands and minds of men!

The debate became: was Mathemtics discourive( revealed in iscusdions with god and others) or constructed?
It turned out that Justus believed his tutor that it was constructed and set out to prove it in Geometry! Really it was to establish it in arithmetic, but he could not found multiplication logically either in Algebra or arithmetic or even logic itself! His resort was to geometry or confusion!

His problem was not his analytical ability, or his rigid formalism, it was Aristotle ! Immediately it was Legendre's Euclidean Geometry, which was a concoction of parts of Euclid's Stoikeia and Legendres own pragmatic approach informed by his engineering and Mechsnical background. In addition he was schooled in the Aristotelian system developed by Aristotles Lyceum and copied by the Islamic scholars. Thus few realised the difference between Playos Academy and Aristotles Lyceum. They were deemed to both be Platonic. But in fact The Academy was Pythagorean! Aristole rejected several key Pythgorean notions particularly in regard to Arithmoi! Thus Aristotle's teachings were Platonic, but no longer faithfully Pythagorean.

The view of the Stoikeia, when it was made public was that it was a Greek geometrical text after the fashion of Aristotle? Nothing could have Ben Rutherford from its purpose!

The philosophical introduction which the Stoikeia constitutes is markedly different from he Aritotelism introduction to rhetoric, grammar and logic! In addition, the Stoikeia is Muse lead, requiring the student to meditate and seek guidance by the muses. Aritotle replaced the muses by Peripatetics! These were teachers who walked up and town in front of a class! This was unheard of before! Students gathered round a seated teacher hoping to eavesdrop on his conversation with his chosen students!

Thus the philosophy of the Stoikeia does not sit well with the Aristotelian pedagogic or logical approach!. Grassmanns attempts to " fix" geometry was misguided by Legendre, and many others before him. The case of the 5th postulate is the most famous embarrassment of the era. It culminated in so called nonEuclidean Geometry which turned out to be a version of ancient and well used and researched spherical geometry!

Gauss was appalled and embarrassed and hot Riemann to throw out a challenge to engineers and physicists! Essentially it was a request for help to get Maths back on track, dealing with real world probems based on a real well founded Grometry or set of geometrical notions..

The solution fell to Hermann Grassmann. It turned out that though it was innovative and entirely novel thinking on his part, he had managed to recapture Pythagorean thinking and thought process.. His approach involved tackling the philosophical Kantian issues first , using to him a clearer version of Hegelian dialectic.min so doing he created a work, part time, and in between very demanding periods of his life , which fundamentally shook the philosophy of Peano, A N Whitehead, Hamilton, and a number of others internationally. However it did not seem to have affected Gauss!

We see that to this day Mammon won the argument. Mathematics had to deliver technological benefits! They were called for by Riemann and thus Gauss. They were worked on by the Grassmanns, and Now they underpin all our modern mathematically nd theoretical physics.

This is why Hermann was at pains to point out all his proofs were constructible!

Commentary.
I have reviewed the translation of the section on the imaginary magnitudes, in the light of Grasssmann knowing and studying the work of Euler on the circular functions.

In this regard the whole passage represents hiscanlysis nd reworking of Euler's wok in tems of the sum and product of line segments, however it is crucial that the line segments are trig line segments. In this section he discusses the cos line segment snd the sine line segment.. Given these 2 line segments he is able to use an snakogy with the sinh nd the Cish circular functions, by imagining the sin as the sinh and the cs as the cosh. He then derives Eulers formulae from the line segment sums , taking care to rotate the swinging are by  90° to get the sine analogy correct.

However his argument or presentation relies on the identity e^ix= cosx + isinx which is derived from the infinite series expansions by Euler. That being given one can demonstrate theat
Coshix = cosx
And sinhix = isinx
Thus e is relateable directly by the imaginary arcs to e^i(a)

The fact of the matter is that neither sinh or cosh are close to sin and cos in tabled values, so such an equation is not sustained for "real" values. However for these imaginary arc values they mysteriously work!

What Grassmann is showing is that geometrically these processes make sense. In particular they make sense as rotation.

One new analogue in Grassmanns promoted idea is that i not only rotates the plane , and every line segment in it, it also necessarily rotates every circular arc segment concentric to the centre of rotation. .

Rotating arc segments takes us into the concept of phases and phase angle!

A phase essentially is a notion of a position or status in a cycle . Thus  i changes the phase of a rotation by pi/2.

The arc logarithms is another new concept. An arc in a sector marked out by two radii can be defined as the logarithm of the quotient of these 2 lines. We could say that degrees are the logarithms of the 1° sector, with the 2 radii forming the intersecting lines of the quotient operator. While we tended to just measure Grassman realised that a product could be formed that rotated the swinging arm or arc round the disc. The addition of angles was now able to be viewed as logarithms of a proportioning process, ie a geometric mean process.

The exact nature of this process is discussed usually under the heading of the roots of unity. These come about as a result of the Cotes DeMoivre  theorems. In this sense a degree is the 36o th root of unity
, while i is the 4 th root of unity.
As Grassmann observed the Menge or crowd of all these different logarithmic scales or logarithmic schemes is infinite in magnitude! Because we are so poorly educated we do not currently know these things!

When I was developing the polynomials of rotations thread I had only intuition, and brute force to guide me. I made many mistakes, based on false assumptions and did not realise how to get from one firmly fixed plane of rotation to another. It is heartening to realise tha neither did Grassmann at this stage, and Hamilton had sent some 10 years blundering into his solution! The importance of parallel or concentric motions came to me later. Eventually I was able to construct the Newtonian triples, something Hamilon was not able to achieve satisfactorily. This all by flashes of insight into this analysis here!

Commentary
It seems plausible that Grassmann in the section on the development of the inner product into the quotient algebra, introduces a a third type of line segment: the hyperbolic trig functions line segment!

Not only does this extend the generality of the trig line segments it suggests a future development of the line segment types into the general function line segment.

Such a ,one segment would essentially be n ordinary line segment with a function as a factor in a product with the quantity of a line segment. In one development I can imagine using functions bases on the trig tables, in another line of development I can think of new tables being developed by using the richt angled triangle to measur off points along a general curve..

I did this for Theodorus spiral and developed a function called sint= øsinø. Plotted against cos it gives a spiral .

The possibilities are endless.

Commentary
While not earth shattering it is important to realise that the line segment AB fir example is a representation of a family of parallel line segments throughout space.
Thus when Grassmanns quotient operator swings a line segment into another orientation it swings all the family into that orientation. This is the meaning of his statement about every line segment in the planar case. However, this makes a point not unique in space, but rather a representative of a family of points!

To specify a unique poit we have to actually uniquely identify it. Even if I uniquely identify it that is only with regard to a uniquely identified reference frame! Thus every aspect of our mathematics is relative and subjective. The only invariants are formulaes expressing invariant relationships   As everywhere in space. This is a big assumption, and one that needs to be empirically justified.a formal proof is just an illusion in this case?

Commentary
Further insights are addressed in the following thread.
http://www.fractalforums.com/index.php?action=post;topic=1218.15;num_replies=29

Commentary
I often disagree with other mathematicians on fundamentals. In this case I feel that rotation is a fundamental transformation that then defines all other. Consequently it action has a big job to do!

When I first coined the term Spaciometry in the fractal Maths foundation thread I recognised the difficulty and thus defined rotation as an experience: it is an experience of the boundary of a quantity or form.

A magnitude is an experience of sensory excitation, so I would not define rotation as an experience of an experience foundation ally. We get a better apprehension when we define foundational concepts in as many sensory modalities as pertinent, and in every case quantity or quanta invokes bounded ness and boundaries. Such concepts are necessarily subjective, but they are basic and have easy referrents so can be communicated usually accurately.

Rotation as an experience is thus a magnitude which can be further defined adjectively. So part rotation means we see only part or feel only part or receive expchoes from only part of a forms boundary. Total rotation means we experience all of its boundary.

One of the realisations that makes the definition so tricky is the reliance on the circle or sphere. It is such a fascinating form that no wonder we model rotation on it, but it is in fact an ideal created by pragmatism. I have meditated greatly on Laz Plaths trochoid app which applies circles to the notion of arbitrary rotation, and as far as curves go trochoids or roullettes are a more satisfactory notion of rotation!

The complexity of rotation is well illustrated in any of his trochoidal presentations.

The notion of a point, a pragmatic seemeioon often is left out of the notion of rotation being considered as the centre. However experience of material points, reflection and translation indicates that we cannot remove points from the transformation. Points necessarily itate also.

Rotating about a point is generally replaced by rotating about an axis. And it is this axial concept that removes the centre from consideration. The work of Grassmann on Schwenkunglehre while not complete allows one to see rotation in terms of cyclically traversing a boundary . Whatever the path of thst boundary it is clear that to end up where you started from is a rotation! The rotation of a point as it traverses a boundary is transmitted to the tangent concept. The line segment concepts of Grassman and the ancient Greeks enable us to sum the rotations of points by summing line segments in the form AB + BC =AC , an absolutely fundamental formula for the physical world!

Thus the concepts of rotation do require this infinite geometric series approach, which can then be simplified to logarithmic addition. It is just not usual to have this pointed out in such a clear geometrical form!

As Grassmann states" whoever searches for these adopted concepts is unlikely to ever find them ! "  . He believed he had been given an insight that was from God or his Muse!

Ausdehnungslehre 1844 continued Ausdehnungslehre Vorrede

---

Yet I am to me, regardless of all trouble related on the form , still conscious of  the great unfulfillment of it
True I have the complete Thing worked through multiple times in different forms, sometimes in Euclidean  form Expositions and propositions for best possible rigour, another time in a melded development form with best possible overview guides, at other times both with one another interleaved, in relation to which the given overviews precede  and then let the Euclidean form follow on.

Truly I have been of it completely aware, that by the repeated  processing around it many an improved form would step from here , that is of a bit, more rigorous, of a  bit, clearer guided overview given. But from the pervading Conviction, that I could  hope for no   full satisfaction  , and the plain fact,  the truth compared to it, i must the presentation still always only leave wanting, I myself concluded with the form that seemed best to me at the time, with the form which steps forward  arising from here.

A special reason  for hindsight leniency also thereby i hoped to find, that to me the time for processing these was by virtue of my appointed duties only most extremely  and meagerly and piece wise apportioned, and also to me my post did not present the opportunity the lively Freshness to extract,through memoranda out of the field of this expertise, or also of  related content,  which like a life giving breath,must blow  through the whole work, even if it should seem like a lifegiving arm stretched out to the organic body of wisdoms.
Yet even if also a professional duty, which such memoranda out of  the field of this expertise should become my singular work, as my  wish goal and deep yearning stand to me before my eyes,yet still I believe the processing of this expertise will not until the achievement  of  the goal be allowed to be postponed,  markedly since I can hope through the processing of this volume  by myself the way to the goal to be able to ride.

Stettin 28 June 1844

Commentary
I only just realised that when I was in a British school in Germany we studied the history of these times! Not only the Napoleonic wars but more cogently the  period of Bismarcks rearmament of Prussia and his industrial reforms! Of course we never studied the intellectual and cultural history of the times at the secondary level.

http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Laplace.html

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lagrange.html

The French Renaissance in Science and. Mathematics came through ome dark and troubled times in Rurope, mostly relaying to the Spanish kings seeking to regain control from the Islamic empire. But the Islamic empire was the one bright light in the gloom and dismal darkness of plague ridden Western Europe.

These are interesting times, with many connections. Grassmann was in a time when despite his provincial status great opportunity for advancement by merit existed.  Gauss, Möbius and Euler were out on the Ruropean limb, as was Newton, but from east to west Europe similar changes were taking place with communication becoming faster and more reliable, and printing more democratic!

Good news. I have now gained access to the article :
Der Ort der Hamilton'schen Qaternionen in der Ausdehnungslehre.

I will start a thread to translate that after this one!

I do not propose to translate the whole Ausdehnunglhre here on this forum, because I do not think that is necessary, or completely relevant.
I chose the Vorrede for several reasons, the main one being it is a fundamental introduction to Grassmanns thinking, and to his interpretation or promotion of complex or imaginary arithmetics. Similarly the short article on Quaternions will be instructive.

I will of course continue to " durchbearbeit" the Ausdehnunslehren in my blog site
 Jehovajah.wordpress.com.

As you know I love this forum and have had most of my major insights while posting here. Also, I have said it before: this forum is the most important forum on fractal imaging on the web in my opinion.

Long may that continue.

Commentary
 Having seemingly started out in a bold fashion, declaring great clarity and inight in what some might view as arrogance, Grassmann finishes with a tale of Woes, anxieties and dissatisfactions!

Last few sections have been discussing his struggle to find time, peace and muse to properly process and edit the work he was publishing. Even up to the last minute he was making modifications, worried whether he was rigorous enough, whether his overviews were good enough. But in addition he admits to be struggling with known imperfections in his work.

As a publisher and editor he wants his book to sell. It must then appeal to a target audience. It seems he was undecided who to edit and format the text for? In addition to that the applications of the method were so numerous and deep that he felt hopelessly inadequate. But he was a fighter. This first volume for. All its shortcomings and compromised decisions, compromised by lack of quality time and opportunity, he hoped in hindsight to have demonstrated its truth and importance, and thus to find a way to progress to the completion of his plan

Although he has no direct appeal for help, indirectly he begs for it

What follows gives him nearly 15 years to perfect his system almost singlehandedly, and the help in the processing, editing and presentation of his work by his brother Robert. For at least 2 years they laboured to redact the 1844 texts and the additional texts Grassmann had set apart for the second volume,

He had 17 years to perfect his system , because no one in Prussia was interested enough to buy his book.eventully he burned most of his first print run!

Commentary

http://youtube.com/watch?v=2t61M-3jniQ

This video relates how we think and frame "law", and draw theories and apply them

http://youtube.com/watch?v=ZYodF7ikj5g

This is a remarkable insight into the nature of geometry of the line and the circle! I have taken and recommended Normans course on universal hyperbolic geometry, but this is the best description of duality I have ever seen him give. To me it represents the deepest insight into nature of geometry, and as yet I cannot trace its equal in Grassmanns original work.

The explanation of bilinearity is at its clearest here, but his thinking I would distinguish as bilinearity!

Notice how he highlights the circular radial values of 1,0,-1.  These values pervade the Clifford Algebra of today, but I am sure that Hermann used only 0 to characterise his products, deriving 1 or -1 as particular constraints when designing applicable algebras. I will know more when I translate the article on Hamiltons Quaternions..

Perpendicularity is a key concept but you have to let it go into the general dual form Norman reveals if you want to understand how your subjective mind processes space! Things which are locally perpendicular rvertheless seem to meet at a distance beyond our horizons or on our horizons. We know this is not anywhere near infinite space, yet our reasoning makes little discernible distinction between this vanishing point on the horizon and some mystical point at " infinity ".  In fact the vanishing point behaves like any conceptual point at infinity.

We have to acknowledge our limitations even at the level of subjective processing. We cannot really imagine " nothing" neither can we imagine " infinity". All we imagine is discernment disappearing outside of the sphere of our view. Beyond that, as they say " there be monsters!". However Aristotle decided to populate it with mythical entities as inversions of our own selves. Religions populate it with good and bad entities . As an adult human being you are free to choose!

Commentary.

I have reshaped the final translation as is my wont! Over time I will probably redact and review the whole translation in this thread but for now I have done what seems the best to me at this time.

Precisely the spirit of Hemanns closing words of his Vorrede.

The translation into English would have been more difficult if I were attempting to create a literary masterpiece. Instead I was struggling to understand the mind of a particular man in his times. The form of language, the allusions to biblical and cultural references( Michaelangelo's : breathing life into Adam) and his hubris tempered by trying to advance himself in academia by playing the game, all come through, along with his determination never to give up!

Much of the language, in its convoluted form is very hypnogogic. Thus his wish to be free to carry out his desires burns through the page in this last part, and is a direct appeal to his intended audience for patronage! Unfortunately in his time it fell on deaf ears. Unlike some of the mathmatical greats, LaPlace Leibniz, La Grange, Euler, Steiner( also from Stettin) it never fell right for him. In every sense he had to work to achieve everything he gained in his life, while carrying one of the most precious insights god has given to man, if you will!

Still, from his humble mustard seed the great tree of Modern theoretical and quantum Physics has undoubtedly grown.

It is unfortunate that it has been bent out of shape by the intransigencies of the human heart, but it was ever thus!

(http://upload.wikimedia.org/wikipedia/commons/6/64/Creaci%C3%B3n_de_Ad%C3%A1n_(Miguel_%C3%81ngel).jpg)

Group theoretic approach that influenced Justus Grassmann.
http://youtube.com/watch?v=VSB8jisn9xI

Commentary

The more I translate about Grassmann the more I understand the unique confluence of historical streams meet up in and around him.
https://openlibrary.org/books/OL24340512M/William_Gilbert_of_Colchester_physician_of_London

It is clear that in the history of European mathematics Hermann lived at a crucial epoch, but also crucial in the history of the physical sciences, and the history of actions and nations, of states. But it is publishing that makes the vital link between it all, for Stettin far from being a cultural backwater in the Prussian mire was a minor cultural and print setting community. Situated on the Rhone valley, not far from Leipzig a major print setting town, Stettin enjoyed all the advantages of a dormitory centre to a major european publishing centre.

Thus not only was Grassmann away from academia sufficiently to be independent minded, but also he had access to the major titles of works of worldwide importance. He took full advantage of this nexus to new and old publications and technological advances in printing and became very well read, but independent in the direction of his studies, often taking direction from his father and his brother Robert as well as his uncle who actually brought him up.

De Magnete by Gilbert was published in about 4 editions European wide, one edition of which was published by a printer in Stettin!

This is a good explanation of how Grassmanns Ausdehnunglehre assisted the mathmatical thinking of physicists.

https://www.youtube.com/watch?v=n27_QneBkqE

There are several ways in which this presentation is logically sounder than the Stanford University  one.
The concept of time has always been problematic. Lagrange dismissed time as an observable quantity or magnitude! He stated:" there is only velocity!"

We could say the velocity is made from more primitive concepts of distance displacement and time, but we would be misleading ourselves because we cannot have displacement without time displacement. It is the ratio of these 2 displacements we perceive as speed.

This presentation makes it clear that these perceptionsl primitives are as such relative , and that the fundamental relativity of our thinking, the arbitrary " one" as Hermann put it has to be included as a choosable measurement standard  or Metron in our mathematical descriptions.

Just found on Mactutor a partial translation of the 1844 Vorrede

http://www-history.mcs.st-andrews.ac.uk/Extras/Grassmann_1844.html

While this translation is " smoother" it is less informative in crucial ways. It speaks more of the translators mindset than it does of hermanns. However this is an inevitable consequence of translation !

However it has helped me to see my translations, though awkward are not that bad!  ;D

Anyone else is more than welcome to post their own in this thread. In fact I would welcome it.

Revisiting the translations of the Einleitung with my newly acquired adjectival skills has proved beneficial. So I am aiming to work through these translations again in that light.

I would encourage any and everybody who feels able to submit a translation to this thread as per the outlined heading recommendations.

The headings clearly identify which page in the Vorrede you are translating and whether it is a commentary or a translation. That is it.

While translation into a language other than English is welcomed it would perhaps best be located in a sister thread in the appropriate language section.

So come on you fractallers, try your hand at a bit of Grass, man! Lol! :dink:

WoW
You ,JEHOVAJAH,just was made a series of texts and scientifical movie series.
Unfortunethly that those movies, wasn't broadcasted on DISCOVERY SCIENCE TV or SCIENCE TV or other similar TV channel.
Can you request the TV channel "Discovery" to broadcasting your movies from your topics: "Light", "The Theory of Stretchy Things", "Twistor" etc.
Would be cool, and you would can become famous.
 :peacock: :wow: :thumbsup1: :smileysmileys:

 :D

Thanks hgjf2. I do not want to become famous but I see no reason why that would not be a good and interesting project!

I would very much like you to get the ball rolling , that is to make initial contacts if you wish. I am maxed out right now with everything else, and do not want to distract from my programme unduly.

Some relevant background on Möbius
http://youtu.be/-ECOwPNkoys
http://www.youtube.com/watch?v=-ECOwPNkoys

Loving this !

http://m.youtube.com/76tNP31_CQk
http://www.youtube.com/watch?v=76tNP31_CQk

Hallo Jehovajah,

when I first read your thread "The Theory of Stretchy Things" I was wondering what are you talking about.
I started right at the beginning. When I read your first post I had the impression that some important backgroud information was missing.
The mirracle was revealed when I read kram's thread on geometric algebra.

So in your first post of this thread some background inforamtion like the following would be helpfull for the uninitiated reader would be help full:

Hermann Günter Graßmann(1809-1877)

Hermann Graßmann developed a new branch of mathematics in his book "Die Ausdehnungslehre" from 1844. In this book he introduced the outer or exterior product. In modern notation written as Bivector or a wedge b. Graßmann published his book in the same year as Hamilton anounced the discovery of the quaternions. But did not receive the same fame as Hamilton during his lifetime. Perhaps he was not taken seriously by his contempories because he was only a high school teacher.

@hermann
Happy new Year in 2017.
Thank you for your continued support and posts in these threads .
I loved the walk by the river in the "Fractal" snow!!.

It has been a tough year for me to be able to continue working on these threads, but it is my aim to do so, until I die or finish the translations .

It is not that Hermanns work has not been translated either, because Kannenberg has done a wonderful job. Rather it is that Hermanns and Justus were High school teachers! This material was meant to be taught to elementary children up to Gymnasium level!

So why was it said to be so obscure?

My translations are an attempt to get behind that obscurantism.

When I tried to translate my poor German seemed a hindrance, but gradually I realised it was essential to revealing where other translators had gone too far in interpreting.

I found and tried to keep the analogicall statements or metaphorical descriptions of the ideas both Justus and Hermanns used to be both graspable and elegant!

So the exterior product is the" outward stepping" product, where the line segments jostle past each other in an outward stepping direction!  Like 2 feet that step or spread the legs further and further apart at each step!

The interior product was in contrast the colliding together product, where the 2 line segments got closer and closer until they met or collided, one of the line segments being perpendicularly dropped onto the other.

This kind of wordy description is anathema to most " serious" mathematicians! That is why it was rejected as unsuited to its audience!
In 1862 Robert Grassmann redacted Hermanns work into the more acceptable mathematical jargon of his day, and rescued Hermanns from Obscurity.

It's a great story, and does require some background, which is why I am focussing on Justus booklet now in the V9 thread .

When I finished the work in this thread I knew I had to go on to do the Einleitung or induction, and then on to the first chapter. But I also felt that Hermanns was relying on his Fathers unique contribution to the understanding of Geometry and Mathematics. So I started to focus on that work, once I got my hands on it.

I am surprised by how much of its content is actually standard mathematical education at primary level, but with the Grassmann twist!
For example: who knew that crystallography could be enhanced by these ideas? Or who thought that combinatorics would be structured by these notions?
Certainly not  Justus until he struggled through his intuitions!

And it is that dedication to intuitive exploration and adjustment using the methigpdologies of the Pythagorean school as adduced from intense study of Euclids extant works, including The Stoikeia, that sparkles through the Ausdehnungslehre of 1844

In addition to Hus Fathers influence, Hegel was a major influence on the Young Hermanns, even if he rejected the usual way Hegel was taught in his time .

So this little book, the Ausdehnungslehre, that took up so much of his spare time is a real gold nugget for many reasons .
That fact is attested to by so many famous physicists and Mathematicians who owe their intuitive insights to the work of the Grassmanns.

I propose to communicate in a brief form some applications of Grassmann's theory which it seems unlikely that I shall find time to set forth at proper length, though I have waited long for it. Until recently I was unacquainted with the Ausdehnungslehre, and knew only so much of it as is contained in the author's geometrical papers in Crelle's Journal and in Hankel's Lectures on Complex Numbers. I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science.

It is amusing that the quote above represents the religious ideal of humility in an academic setting.

Thanks for the quote Hermann and it is a fitting  end quote for this thread .

It shows how the work of elementary educators is profoundly important to succeeding generations. Many adult academics strive to make important lasting contributions to their higher academic fields and are directed away from the foundations. Both Justus nd Hermann were serious about establishing the foundations of Prussian mathematics and physics in the reform of education ordered by the Prussian emperor.

They had nationalistic goals to make Prussia a leading scientific and industrial technological force to be reckoned ith. They drew heavily on the best in their day and spoofed this understanding the next generation, aiming to make them proud examples of Prussias self actualising dream.

They we're staunch supporters of the Emperor and against the attempted revolution brought on by the collapse of the economy after the defeats by France particularly in 1806. So their work was revolutionary in the zeitgeist of the time. But amazingly it was by returning to Pythagorean principles as expressed in Euclids work that the major revolution was effected! The Stoikeia revealed as a discourse in basic natural philosophy of dynamics and quantification of dynamic systems.
Klein attempted to academically harness the transformational dynamics and in so doing obscured the work of the Grassmanns to academics of his day, except those who could see its intuitive value.

Gauss and Ruler dominated the minds og academics who could not be seen to be placing the work of elementary teachers over the complex abstractions of their professors.

The move toward abstraction thus obscured the simplest observations of Justus. Hermann in attempting to be academically recognised tried to present these ideas abstractly, but failed to identify his audience correctly, like Justus, who aimed at elementary school teachers. It took Robert to redact the work in 1861 in a form suited to Acadmics.
Nevertheless it is the raw power of the 1844 version that resonates through its imperfections.