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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!

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!

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.

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

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

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
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?