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Author Topic: Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre  (Read 15906 times)
Description: Grassmann Mathematische Annalen (1877) Volume: 12, page 375-386
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« Reply #360 on: April 28, 2017, 04:19:01 AM »


H. GRASSMAN 376 die mittlere jener drei Gruppen bilden. 1ch will diese Art der Multi plication die mittlere nennen, und zwar hauptsächlich deshalb, weil sie, wie sich sogleich zeigen wird, zwischen den beiden Hauptarten der Multiplication, die ich die ,,äussere" und die ,,innere genannt habe, die Mittelstufe bildet. Die äussere Multiplication hat nämlich Zu Bedingungsgleichungen die zwei Gruppen (2) und (3) und die innere die zwei Gruppen (1) und (2). Ich habe das aussere Product zweier Strecken a und b mit [ab], das innere Product derselben mit [a l b] bezeichnet und werde in dieser Abhandlung unter ab (ohne scharfe Klammern) stets das mittlere Product der Strecken a und b verstehen. Dann ergiebt sich sogleich, dass das mittlere Product ab zweier Strecken sich darstellen lässt in der Form 

(4) ab  = [a|b] + u'[ab]


 wo  und u' constant und zunächst willkührlicli, jedoch nicht null sind. Aus den Bedingungsgleichungen (2) ergiebt sich, dass es für das mittlere Product zweier Strecken 

n(n–1)/2 + 1 
 von  einander unab- hängige Einheitsproducte giebt, von denen eins (etwa e1^2) dem  inneren Producte [a|b], die andern(e1e2,e1e3,e2e3, u. s. w.) dem äusseren Producte [a b] zu Grunde liegen. Im Raume, wo die Anzahl der von einander unabhängigen Strecken drei beträgt, also n = 3 ist, ist also die Zahl der Einheitsproducte, auf die die mittlere Multiplication zurückführt, gleich vier. Die Bedingungsgleichungen der mittleren Multiplication werden dann 
e3e2 = -e2e3, e1e3  = -e3e1, e2e1 = -e1e2
(b) e1^2 = e2^2 = e3^2

 Aber das wesentlich Eigenthümliche der mittleren Multiplication im Raume als einem Gebiete dritter Stufe ist, dass die Anzahl der von einander unabhängigen Einheitsproducte in (a) gleich der Anzahl der Einheiten ist, und man daher jene auf diese zurückführen kann. So bleiben also dann die Einheiten des Productes wenn man noch die in (b) zu Grunde liegende Zahleinheit hinzunimmt, dieselben wie die ursprünglichen. Diese einfache Beziehung verschwindet bei den Ge- bieten höherer Stufe, so dass die mittlere Multiplication in der Aus dehnungslehre welche Gebiete beliebiger Stufe behandelt, keine ein fache Bedeutung behält. Ich beschränke mich daher auf den Raum und nehme an, dass die drei zu Grunde gelegten Einheiten drei gleich lange zu einander senkrechte Strecken sind, deren Länge 1 beträgt. Nun habe ich in der Ausdehnungslehre (&2 50 , 51) nach gewiesen, dass die Bedingungsgleichungen der äusseren Multiplication noch bestehen bleiben, wenn man statt der ursprünglichen Einheiten beliebige andere einführt, und (&2 330 ff.), dass, wenn e1, e2, e3 einen


H. GRASSMAN 376 form the middle of these three groups. This kind of multiplication is called the middle one, chiefly because, as will be shown, it is the intermediate stage between the two main types of multiplication, which I have called the "external" and the "inner" (2) and (3), and the inner two groups (1) and (2), respectively.I have the external product of two segments a and b with [ab], the internal product Is denoted by [alb], and will always be the Intermediate product of the segments a and b in this paper (without sharp brackets), and it is immediately evident that the intermediate product can be represented in the form from two lines

(4) ab =  [a | b] + u '[ab]


 Where  and u 'are constant and initially arbitrary, but not zero. From the conditional equations (2) it follows that for the intermediate product of two line segments

n (n-1) / 2 + 1
 Which are independent of one another, of which one is the basis of the inner product [a | b], the others (e1e2, e1e3, e2e3, etc.) to the external product [a b]. In the space where the number of independent segments is three, ie n = 3, the number of unit products to which the intermediate multiplication is derived is equal to four. The conditional equations of the intermediate multiplication are then
(a) e3e2 = -e2e3, e1e3 = -e3e1, e2e1 = -e1e2
(b) e1 ^ 2 = e2 ^ 2 = e3 ^ 2

 But the essentially peculiar characteristic of the middle multiplication in space as a third-order domain is that the number of mutually independent unit products in (a) is equal to the number of units, and therefore can be reduced to these. Thus, therefore, the units of the product remain the same as the original one, if we add the number of units on the basis of (b). This simple relation disappears in the higher-order areas, so that the Intermediate multiplication in the Extensive Magnitude  theory, which treats any region of any stage, does not remain simple. I therefore limit myself to the space, and suppose that the three basic units are three equally long line segments perpendicular to each other, the length of which is 1. Now I have shown in the theory of Extensive Magnitudes (& 2 50, 51) that the conditional equations of the external multiplication still remain if we introduce arbitrary others instead of the original units, and (e1,e2,e3 one



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« Reply #361 on: April 29, 2017, 12:35:12 PM »

Is this the invention of the geometric product by Hermann Grassmann?

Hermann
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« Reply #362 on: May 01, 2017, 11:39:17 AM »

Is this the invention of the geometric product by Hermann Grassmann?

Hermann
Actually Hermann, this is the invention of Justus Grassmann who clearly cites his sources!

The geometric product has taken on a different connotation over time, but it has it's roots in area calculations.

What is remarkable is that we as mathematicians lost sight of the Stoikeia. Here the most general analytical and synthetical processes are described, discoursed and taught. It is an algebra of lines and planes not of alpha numerical symbols. We never grasped that until Justus Grassmann, Abel and a few other group and Ring theorists went back to nature!
Natural Philosophers contested the prevailing view that theology and thus Euclid were of divine rational origin. Instead they argued divine rational powers were ambient in the environment around man and it was his task to analyse and construct useful models from these god ordained powers. Thus our rational and intellectual thought had to be carefully constructed. It was not divinely revealed to but a few, but rather accessible to all thinking animates.

In that view the area formulae are suddenly dusted off as incredible results of human ingenuity. They are the geometric products! And much more, the ratios within the right triangle were viewed as a divine ordinance. Centuries were spent counting the parts , and thus calculating these ratios. They are the basis of any practical notion of rational numbers which may reveal an infinite process beyond our ability .
Later, arrogant men claimed godlike powers to compute thes infinite ratios in principle! It was and is a delusion! One which Cantor spent many years in an asylum trying to come to terms with!!

By going back, avoiding the assumptions of the French geometers like LeGendre, Justus was able to recover the original dynamic nature of the Stoikeia and thus the dynamic nature of these geometric products.

It was Hermann however who set it out with Robert in a format we call lineal Algebra ( better lineal) which demonstrated what a unifying concept was being proposed by the Greeks revealed again by his father Justus, and by dint of his own unflagging work on a part time basis.

Once it was recognised by his academic peers, however, it took on many forms, variations and misinterpretations, the most noteable of which was the Gibbs,Heaviside interpretation of a vector algebra.
Hamilton and the Grassmanns really thought through an algebraic way of representing lines by symbols , even if the mathesis of the imaginary numbers they both held was radically different.

Hermanns mathesis was based on Eulers geometrical identification of i with the quarter arc   But in fact it was not the arc length but the trigonometric ratio that Euler identified i with, and through that with the process of the exponentail series expansion, something Cotes had done with the logarithmic expansion one 7 decades earlier.

It is these ratios of geometric magnitudes that are the basis of the processes of the geometric products and thus the geometric algebra .

What is clear is that a Vector / Träger was not a new idea, but an old idea highlighted and put to a new use . Whereas Justus takes time to explain, Hamilton obscures it in academic hubris and mathematical symbols .
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« Reply #363 on: May 01, 2017, 02:32:00 PM »

What is clear is that a Vector / Träger was not a new idea, but an old idea highlighted and put to a new use . Whereas Justus takes time to explain, Hamilton obscures it in academic hubris and mathematical symbols .

I read somewhere, that the name 'Vector' was first used by Hamilton and from there went into Vector Analysis by Gibb's and so found its way into physics and engineering.
Hermann Grassman also talks about directed length in the 'Vorrede' of the 'Ausdehnungs Lehre'. May be he was not the first who used a directed length, but it is possible, that he reenvented the Vector?
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« Reply #364 on: May 01, 2017, 03:07:11 PM »

Some how I have Problems, correcting typos in the last post, so I'll try a second post

I read somewhere, that the name 'Vektor' was first used by Hamilton and from there went into Vector Analysis by Gibb's and so found its way into physics and engineering.
Hermann Grassman also talks about direkted length in the 'Vorrede' of the 'Ausdehnungslehre'. May be he was not the first who used a directed length, but it is possible, that he reenvented the Vector?
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« Reply #365 on: May 01, 2017, 06:08:10 PM »

Some how I have Problems, correcting typos in the last post, so I'll try a second post...

It's that dang "apostrophe problem" again!

This forum is on it's last legs and will be replaced with a more modern one soon.

It's running on a old version of php and the apostrophe character is forbidden since it can be used for "php injection"  {a malicious way to hack a forum}.

You can always go to the subject line of your post in the editor when making a post or editing one, and remove or replace the apostrophe.

Then you can edit as usual.

Note that this happens with private messages too.

Fear not - our updated forum is in the works and is in the "pre-alpha" testing stage.

Alpha testing will start soon with invitations sent to our more active members, followed by a public Beta test, and then a Grand Opening!

Sorry to divert from your topic.

Carry on   afro
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« Reply #366 on: May 01, 2017, 06:42:17 PM »

Thank You for the Information Sockratease.
Also when I have removed all the apostrophes I still have a problem to save the changes.

Hermann
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« Reply #367 on: May 01, 2017, 10:32:23 PM »

Thank You for the Information Sockratease.
Also when I have removed all the apostrophes I still have a problem to save the changes.

Hermann

That's odd.

I don't see any apostrophes removed from post titles though.

Did you remove the one in the place showed in this picture?



Maybe edit that part first, then edit the post??

Me confused.  It worked the other three times this happened to members!

Oh well, either way it will no longer be a concern in just a few months  afro
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« Reply #368 on: May 02, 2017, 01:35:48 AM »

Thanks Socratease .
Yes, it works .
@Hermann. The modern use of the word vector was coined by Hamilton. He took it from the Latin vehore, meaning to carry . What we see here is Justus using the German for carrier Träger to denote construction lines. But he goes further; these construction lines impart orientation and the facility to carry a magnitude in either direction. This was written prior to 1829, before Hamilton published his paper on a science of Pure Time, and his work on Quaternions in 1841 . Thus the word is Hamiltons, but the idea of a line carrying a magnitude and direction is older. I do not claim Justus coined it, but that he explained it clearly. Hermann in fact comments on this word coined by Hamilton as unnecessary word smithing!
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« Reply #369 on: May 02, 2017, 01:58:44 AM »

@Hermann
The form that Hermann calls the Middler or intermediate product is presented as the geometric product of two vectors in Clifford algebras. It is usually strangely presented referring to Two main products usually no one has heard of . Here Herman presents the form of the Middler product and then restricts it to 3 dimensions because it was too complex to resolve in higher dimensions. The point is when this product form is so restricted it becomes possible to carefully derive the Quaternion results.

When Clifford read this paper in particular he followed it precisely for the Quaternions and the biquaternion algebras. So this was his source . But in truth this type of product is discussed in the Einleitung / Induction of the Ausdehnungslehre.

That Clifford algebras take this model to design higher dimensional products is thus understandabke. What is not so easy to grasp is their presentation of this form. It is not the general product, but the mould for constructing a product in any dimension. It is a synthetical form, not a product per se. It is a guide in constructing a product.
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« Reply #370 on: May 05, 2017, 01:57:39 PM »

Hallo Sockratease,

thanks for the information!
Without the apostrophe the correction mode now works!

Hermann
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« Reply #371 on: June 08, 2017, 01:15:20 PM »

The concept of a vector is a quantity with magnitude and direction . Well so ome say . But in fact both Hamilton and Justus Grassmann used an old geometrical practice: using construction lines in developing an accurate sketch .
A construction line gives the essential path or direction, orientation and locus of a form. These lines then " carry" the exact. Magnitude or rather quantity of magnitude desired for that path( direction, orientation, locus).

Restricting the description of position to orthogonal axes or orthogonal " vectors" of unit quantity, allows direction and orientation o be given by a "vector sum" of these orthogonal vectors . This in Hamiltons mind was given in the form  ai + bJ + ck. this he called a vector but Hetmann calls it a directed line segment.

The fourth quantity Hamilton thought of as a number on the calculation axis. Hermann considered it to be the inner product of the defining line segments for a Quaternion . This inner product is a calculation of the dropping of a perpendicular between the two defining line segments. It turns out to be a measure of the magnitude in the direction given by the wedge product of the 2 defining line segments .
Reducing it still further to unit line segments corresponds to directions in space determined by spherical coordinates and magnitudes determined by spherical radius.

Quaternions despite their complicated products are magnitudes of direction!

The products of these magnitudes of direction give us in addition a magnitude of rotation to a new magnitude of direction .
As Hermann said : the Grassmann presentation is a lot easier to grasp .
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« Reply #372 on: July 19, 2017, 10:22:22 AM »

Xxx I finished up some work in the thread "the operator I is more complex than that" regarding circular arc vectors. It became apparent that the i notation is a unit diameter quarter circle .2i can be summed by the unit radius quarter circle . This unit radius circle is often called the unit circle  because it gives an area of /Pi square units. However a unit diameter gives us a perimeter of Pi which can be pragmatically measured, whereas the area can not.

The unit diameter is a line segment in a plane for i,j,and k circular arcs we choose orthogonal planes xy,xz,yz. We choose them in this order to maintain an anticlockwise arc measure. . Thus the diameter along x axis is rotated anticlockwise to give diameter along y axis.. We can see this by placing a clock in the xy plane with 12 along the x axis in the positive direction. . To ensure the z axis is also consistently an antilockwise rotation we rotate the clock into the xz plane and see that it is an anticlocise rotation towards the up positive direction. Then we can rotate the x axis in a measured anti clockwise direction, and construct the xz plane. As well as the yz plane. . Now we can swing the clock in a measured anti clockwise manner into the yz plane and confirm that our constructing rotation is indeed consistently anti clockwise.

All this is done by the observer sitting in one position moving the clock face. If the observer moves or rotates then we quickly become unsure of what is or was clockwise!
If we agree these orthogonal axes then we can drop the confusing use of clockwise and anticlockwise instead any 3 or More sequenced points A, B,C,.. In a plane defined by ABC   determine a rotation by their order ABC. No matter where the observer is situated ABC is the direction of rotation amongst the points. Thus ACB is the opposite rotation .

This convention avoids much confusion .

Rotations are thus defined in a plane and not by a centre in the plane. The centre is in fact constructed from the rotation.
If ABO defines a plane and BCO an orthogonal plane and CDO a mutually orthogonal plane to the first 2, then O can be set as the centre of a sphere and if A,B,C,D lie in the surface of that sphere we can define several rotations in space by 3 sequenced points.

The use of i,j k to reference orthogonal rotations in orthogonal planes has been obscured, even Hestenes work Iin the geometric lgebra has not penetrated to primary school level.

Now I particularly identify these familiar notations with quarter circle arc vectors. the preceding explanation should be read with that concept in mind.

When it comes to Quaternions then we should identify the x,y,z as unfortunate references for numbers and scalars. I would perhaps write s + xi + yj + zk to depict the scalar(s) the x axis count of circular arc vector sums, the y axis and the z axis count of the same.
X,y,z can then be vectors along the axes , while s is just a number . In practice it determines the size of the unit diameter in relative terms ans is a logarithmic number.

Hermann definitely defines this part of the Quaternion as a quantity that is some multiple of1 a unit number not a unit line segment.

Because the planes are mutually orthogonal we cn use the count of circular arcs just like an orthogonal set of axes, in which case the rotations disappear and a point vector is defined in space scaled by s. but we can follow the rules of summing circular quarter arcs to determine an alternative point. Based not on a cuboid but a type of paralellepiped .
Multiplication transforms this point onto a sphere of increasing diameter, thus the track is generally trochoidal in space.


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