The Theory of Stretchy Thingys

[jehovajah]

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.

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[jehovajah]

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.

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[hermann]

I love the tent constructions of Frei Otto

[jehovajah]

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.

[jehovajah]

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 .

[jehovajah]

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!

[jehovajah]

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.
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[jehovajah]

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.

[jehovajah]

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.

[jehovajah]

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.

[jehovajah]

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,

[jehovajah]

1862 1844 AUSDEHNUNGSLEHRE 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.

[jehovajah]

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.

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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.

[jehovajah]

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! :
 
I have a lot of corrections to make in my exposition, but hey that is the nature of the game!

[jehovajah]

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!