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Commentary on §15

I must direct the reader to the induction as indeed Hermann constantly does. The labels are not what they might seem to anyone partially versed in the Mathematical arts. In particular the diagrams reinforce the direct line interpretation, but the treatment is equally applicable to circular arc segments and the endless-like circle to which they relate.

In particular do not regard the = sign as equating, but rather as likening. Many of the labels are not only assigned as representations, according to a general principle of representation, or representational theory, but also have a symbolic status as a likeness..


Hermann draws a distinction between this constrained useage here of the labels to the freer use we may adopt after §99. Hamiltonnalo in his theory of couples took a logically sustainable path to support the way we use algebraic symbols andcarithmetic ones, that is quite freely and often very loosely specified.  This freedom is not safe if it's basis is not understood or comprehended. In addition the free use of analystical knitting will lead to negative assignments per se and that has to be justified in the context of line segments and elements of line segments.

Try to establish that [] = (–[]) to grasp the procedural or logical difficulty that we step over without thought.

The difficulty is not in space or our apprehension of a real extensive magnitude, but in our rules of label manipulation.

When the problem or construction task is "visible" we can move spatial objects relative to each other deftly and correctly, but when it is "invisible ", and not possible for us to reorient the construction we can imagine what the result would be. However, sometimes , even quite often, we reverse the spatial orientations of the result, because we overlooked the effect of changing relative positions, or worse still changing labelling. Here both Hermann andvHamilton draw attention to the constraints an "Expert" must consider to get the correct result.

We are taught to be very blasé about such concordances, but when you build a house or send a manned spaceship to the moon and back, reversing the resultant of a calculation or specific solution has disastrous consequences.

Commentary on §16

One must not overlook how Hermann initially constructs his elemental structures, rooted on the prior exegesis in the induction , the beginning element and the end element.

He does so by interrupting a line segment creating element. This gives us a general sectioning of a line segment by a interruption anywhere within a perceived line segment.

Having now the end element he can extend the line segment endlessly by additional end elements. Thus he establishes the 3 element structure in which addition occurs as pairs, recursively.

Later this structure or format is repeated in the plane, and again in three dimensions. All extensions are continuous. In higher stages they may be generally contiguous. Arc segments may also be described in the same way.

Normans treatment of affine and projective points is an instance of this kind of structuring.
  <a href="http://www.youtube.com/v/H6Px1xsPnRQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/H6Px1xsPnRQ&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/H6Px1xsPnRQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/H6Px1xsPnRQ&rel=1&fs=1&hd=1</a>

Ausdehnungslehre 1844
Section 1
The Extending Magnitude

Chapter 1
Addition and Subtraction  of the simple extensive/ extending Magnitudes of the former Step/ rank/ Stage or the Line Segments

§16.  I now take to my side to scrutinise, in order to reach around to the knittings of differing artformed line segments  , to nearby  me, 2 differing artformed fundamental varyings
And let  the first fundamental varying ( or the running into against set entity of it ) whimsically continue forth an element,
And then the thus varied Element in the second manner of varying likecasely  whimsically stride forth
Thusly I come to be capable therethrough to create whole out of an Element an unending like crowd of new elements
And the totality of the thusly creatable whole Elements I name a System of second Step.

Further I take a third fundamental varying to my side for scrutiny,
which varying does not  once again guide out Beginning elements   from that second step System to an element of this System of second step  
And which varying I besign of its behalf  as independent from both those first ones, and let one whimsical element of that system of second step this third varying( or the running in against set entity of it)  whimsically  continue forth,

thusly comes to be the totality of thusly createable elements in order to build a representation of a system of third step
And there is no restriction set to this manner of creating whole concording to the label  
thusly I  come to be capable upon this method to stride forth to Systems of whimsically higher steps .
Hereby is it important to hold firmly
That all upon this manner created whole elements, are not permitting to become apprehended  as by othermeans already given entities  , rather as  "springing out of a source like" created  whole entities
And that they therehere all,
sofar as they are created whole "springing out of a source like" through differing varyings ,
also seem outwardly as differing, their label concording  .

Thereagainst, is once again clear
That
Concording to the entity the elements one time are created,
they appear off from there  as givens,
and their differing quality or  identical quality can not by another  come to be  to decide upon, as if one goes back  onto  the original creating whole event.

Before I now go over to our personal task, specifically to the knitting of differing varying manners, want I the manifesting  through the Geometrical trackings to come to aid  .
It is specifically clear
That the system of second step to the plane inter-communicates,
and the plane therethrough created comes to be thought of
That all points of a direct line concording to a new,  not enholden in it  direction ( or concording to its running in against entity)  themselves kinematically move forth , whereby then plainly the totality of the thusly creatable points builds a representation of unending- like plane .

It appears thuslywith the plane as a totality from parallel lines which all a given Direct entity intersect

And it is outwardly seeable
That, there these parallel lines do not cut themselves,
and also the "springing out of a source like"   direct entities not yet a second time are hit, all upon that manner created points from one another are Differing ,  and thuslywith the analogy a completed entity is.

Thusly so one reaches to the  complete un end-like space, as to the System of third step, if one the points of the plane concording to a new, not in the plane lying  direction ( or its running into against set entity) kinematically move forth

And further can the geometry not stride forth,
  while the abstract expertise knows no boundary!




Footnote
• how sometimes in the doctrine of space, all points are already originally given through the prior set out Space .

Randys combinatorics for V9

This morning I wrote a meditation on this  § and it was not good enough so Shunya swallowed it back!
Maybe it will be time soon to write it in the mean time bits and pieces are being put forward here.

http://m.youtube.com /JnA8GUtXpXY
<a href="http://www.youtube.com/v/JnA8GUtXpXY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/JnA8GUtXpXY&rel=1&fs=1&hd=1</a>

Combinatorics is not a " mathematical" subject. It is how I, we construct models of space and how space constructs these models in me or us.

We have a demonstration of how combinatorics progresses from the arithmoi to the Geometry!

http://m.youtube.com/eqrQjupV28k
<a href="http://www.youtube.com/v/eqrQjupV28k&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/eqrQjupV28k&rel=1&fs=1&hd=1</a>

We cannot really progress much further without acknowledging the simple crowd doctrine Hermann and adjusts use. There are end-like crowds and unend-like crowds, thus we have bounded and unbounded, or unbound able crowds.

These crowds are crowds of any object Or entity.

Now let us consider the geometrical instance of these crowds. In that case the creating element may usefully be identified as a creating point, that is, an entity that creates points!
Thus first varying is a whimsically chosen orientation and the " point" moves only in that orientation obeying the variation constraints. Such a combination of orientation and motion is called a directed motion or a Direction. . The creating point under this direction creates a Line or Line segment.

The line segment is an end-like  crowd of points, but it has no Finite tally! Thus a finite crowd is perhaps a misleading descriptor of it.

In addition let us also consider the motion that creates an arc segment or a circle. In this case  the varying constricts the creating point to a fixed measure from some central point . That point is mysterious until we develop the second step system.


In the second step system a second varying causes any created point from the first step that obeys its command to move . This creates in fact an entity that consists of directed line segments or circular arc segments that are parallel ( or concentric ) and non intersecting. Or these lines and arcs may also be unend-like . No matter which, they consist of an un" countable" crowd of created points.

Now it is clear that the running into against set direction or motion constraint on the circular arc will produce similar results.. Thus the system of the second step is a finely ruled surface called a plane, consisting in either parallel lines or concentric circles that intersect the system of the first step.

If the system of the first step was a circular arc, then the pattern becomes more intricate. The parallel lines no longer occur. Directed lines that intersect at 1 pont called the centre radiate through the arc or circle;

 or  arcsegments or circles of the same constraint as the first pass through this point called the centre;

or arcs from one central point (a point which is the centre for these arcs) as concentric circles of expanding size intersect this circle in 2 points or just one, of which there are 2 such points called points of tangency;
 
Or indeed this may be the case for every point created in the original arc or circle.

Such a surface is indeed a " plane" surface and clearly has more complex potential than the first type of surface. But they too are an uncountable crowd of creating elements.

From these types of crowds of created elements consisting of so ordered and arranged point creating elements we may proceed to ever more complex but finely ruled lines and / or arcs in the system of the third step.  Such a system usually called space is thus neither empty nor in fact simple, but rather full of every potentiality of motion.

Now Hermann draws an oft misquoted distinction. " geometry" can go no higher than step 3, but his expertise knows no bounds!

Geometry, my friends is not space!

It is clear from the outset that these systems are interpretable by Geometry, but in his day Geometry was restricted to 3 orthogonal axes. Lagrange in fact felt that this was the nature of space also despite pursuing these notions as generalised coordinates. But then everybody believed space was Euclidean, the new nonEuclidean spaces were fictional, and Newton was a Divinely inspired Messenger from god. His laws were immutable because they were the Laws of God and Nature!

For a theologian this is not too far a conclusion to leap to. However it confuses empirical data with "Rational" opinion. In those days Rational meant inspired by the word(Logos) of God. Today scientists and atheists have commandeered the word to attack any religious or superstitious belief system! What poetic justice is that?! Such systems are Irrational while the Rational system remains curiously undefined! For to say it is the "scientific method " is to fall into the greatest state of gullibility ever created!

Any professional scientist will soon disabuse you of the fairy tale nature of the so called scientific method!

So it is then, that some would have you believe that space is an empty nothing, that it consists of only 3 possible " dimensions" and it is predominantly ruled by straight lines. Finally that all forces or impulses act only in straight lines!

If you have a moment read both Örsted and Ampères accounts of their empirical research findings!

V9 will be so important to the understanding in this thread that I urge you to read that thread alongside this one.