V9

[jehovajah]

<a href="http://www.youtube.com/v/KJEJUGz1ReE&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/KJEJUGz1ReE&rel=1&fs=1&hd=1</a>
http://m.youtube.com/watch?v=KJEJUGz1ReE
Tesla was using different cyclus or modular arithmetics to explore frequency ratios for his AC research . However Grassmann used the cyclus to design a 3 dimensional product for line segments capturing rotational motions .
Hamilton designed a quaternion product using i which has a cyclus of 4 .Grassmann got 4 by designing a product that had a rational product and a non commutative product which nevertheless could be congruent to the sign switched commutated  form

[jehovajah]

This is an experimental post and will be edited
I am trying out Dragon Dictation! I have to find a way to progress my meditative translation

This is a google translation of the photographed page of Justus page 6

00 52 are developed. The task arising from this can be regarded as the chief task of that part of the geometrical theory of combination with which we are concerned here. Three-position to given points Outgoing radii, if not lying in one plane, determine a physical angle, or a spherical triangle. If we assign to the angles which they form under themselves (the sides of the spherical triangle), and to the ratios themselves, we pass through the one, as through the other, from the domain of the unmixed geometrical doctrine of cambitation, Which is independent of all magnitudes, and every difference possible in this respect leads to a series of derivative forms. We shall here begin with the simplest of all conceivable cases, but we shall find that in certain respects he comprehends all the rest in himself. This is the case in which three lines, as the 3 dimensions of the space, are perpendicular to each other, and equal in magnitude. We shall call the regular system of figures by means of the crystallographs. It is now easy to overlook the fact that, under the given conditions, the initially arising bodies of the cubes will be. I consider it the most distributive, the position of the lines which determine the shape, according to the assumed stand-

I have yet to read the German myself and produce my meditative translation, but this is a better starting point for continuing this thread and for others to contribute their own translations.

Please contribute if you can xxx

My recent meditative working on the text

The personal exercise hereout springing to mind can be tracked as the most important exercise of the indicated part of the geometrical combinatorial doctrine, with which we occupy ourselves here.
Three radii, their layout according to the given items of a points outgoing radii, even if they do not lie in one plane, are concording to a bodily spatial corner or a spherical triangle.
One gives concording magnitudes to the corners they construct under thmselves( the sides of the spherical triangle ) and to the radii concordingly outwardly holding magnitudes, thus one steps hereout through one, how one steps through the other; out from the field of the unmixed geometrical combinatorial doctrine, which should be independent of all magnitudes, all outwardly held magnitudes
And in this relationship each of possible differing qualities guides onto an array of " away leading" shapes!

We would begin here at the nearby with the simplest of all thinkable cases, to find rather, coming to be that it begrips already in itself onto conscious mental attitude all remaining entities .

This case specifically is the one by considering which the three lines, as concording to the three dimensions of space, are going perpendicularly onto one another

[jehovajah]

http://m.youtube.com/watch?v=qhbuKbxJsk8
Quite a lot of detail in this video, but what underlies it all is rotational motion discretised with straight lines emphasised.

These are the line figures Justus begins to analyse in his work I am translating here

[jehovajah]

Kunde refers to those who are subscribers to or patrons of a certain interest group or activity . Thus Justus is writing to those communities that love Mathematical and Natural items or topics of interest and are willing to pay to maintain that subject interest and its development.
It seems not many were that interested in Justus ideas, except his publisher and his sons

The Grassmanns were a great publishing family. They lived near to one of the great publishing houses in all of Europe, and Robert even started and maintained his own publishing business.

As primary educators they were expected to prepare and publish their own teaching material for their students, and so not only were they very aware of the need to I Struct a printer very carefully, but also of the necessity of providing accessible material for a diverse range of abilities.

Here Justus prepares an article or a tract to attract subscribers. He hoped to publish in one or more of the well known magazines with reliable subscription bases but was unsuccessful in getting his material accepted for print!

Nevertheless he printed the material off himself for use with his students and any interested parties.
His sons were educated in the schools he had responsibility for, and in a manner akin to the Pestslozzi style.
He also collaborated briefly with Jakov Steiner, the European greatest synthetic geometer since Newton! To synthetic geometers the magnitudes they worked with we're dynamic elemental entities, not as we find today algebraic symbols dealing more with quantity of extension rather than the extensive magnitude itself.

Thus naturally a line segment is a living dynamic entity not captured by a number, but by an endless dynamic line whose intension is as dynamic as its extension . Continuity like curvature is an irreducible un analysable quality or property of magnitude. The creative point is a dynamic conception of position . It's dynamic creations the magnitudes of lines surfaces and space itself are mysteries to which we may assign one creation as a measure of all the others by an inductive method or system . Numbers as precise symbols in their own right do not measure or count. We consciously utilise metrons to count and measure and announce the results as counts of a specified Metron. We call the Metron a standard unit in the SI system, and recognise it as a fundamental di- mension, that is a split in the measuring process of a magnitude , usually split by orientation, but also by sensory perceptibility. Thus mass, length, time are sensorily split by kinaesthetic, visual, and motion/ change perceptions.

It is the combinatorial aspects of synthesis that so intrigued Justus, and this is why precise Numbering is so secondary to the overall method of description or depiction.

Norman J Wildberger has made a stab at making mathematic dependent on numerals, but shows time and again the problems of doing so!

The old wisdom is to use metrons of which the Arithmoi are the most noteable creation of the mind. These metrons are not numbers but rather extensible magnitudes arranged into patterns called mosaics. It was what the Pythagorean school was most noted for, and something Atistotle failed to understand, preferring a rhetorical basis to interacting with the cosmos.

The particular mosaic Justus is intrigued by is one constructed principally from lines!

These mosaics are dealt with in depth by Eudoxus in books 5 and 6 of the Stoikeia . The modern mathematician is unaware of the difference between a line segment and a length! The methods of Eudoxus are taught by using length and then horrors using numerals! These methode do not rely on length or numerals but on counting!

Thus Logoi are different relationships of counting comparable things of which the most commonly known is the Analogos method! In book 5 Eudoxus defines several more methods of counting comparables ( homogenous by their dynamic generation sic "the creative point") and in each case the synthesis of the comparables is set out positionaly. Thus they may be alongside each other, underneath each other as sides of any dynamic form , relative to some reference line or locus or point or any combination of those.( for example the legs of a point on the circle perimeter that stand on the diameter )

Many interesting properties of form were found by these synthetic comparisons, the most well known being Pythagoras theorem .
We see why algebra is definitely symbolic arithmetic, because the use of letters or marks to denote extensible forms is a natural tool in the synthetic geometers tool box. However, the system Justus and others pioneered was robust enough to replace symbols by other symbols as long as it was done in a conistent and coherent manner.
DesCartes method, though influential was not consistently worked out. Wallis later inspired a consisten set of orthogonal lines called axes, and researchers like Abel, Steiner and the Grassmanns popularised this consistent and coherent system by demonstrating its wide utility in classifying forms.

The risebofbthe Cartesian coordinate system to denote position only took a long time , and much was lost by adopting it as  superior fom of geometry ! It made the vector( Träger) a mystery instead of a natural extensive magnitude.

Everything that Hermann needed to construct ( synthesise) his LINEAL algebra was and is worked out in this booklet by Justus Grassmann. What Hermann did and to a less well known extent Robert , was show the applicability of this method to physics in particular beyond the classification of crystal forms, and to a general geometry of any number of dimensions, defined by the system or method.

[jehovajah]

Another experimental post.
I have restarted working on the translation of Justus work with failing eyesight . However modern accessibility applications mean I can tackle it with hope of a readable outcome .

I have previously studied two methods of interacting with the text. Dragon speak allows me to dictate my meditative thoughts on a passage or a word, while Google translate supplies me with a reasonable guide translation of the original photographed old Prussian German .
I will post both as I work through some  250+ pages of detailed thinking and teaching by Justus

D0102 gleich gross angenommen werden. Das hieraus enstehende System von Gestalten, wollen wir mit den Krystallographen das regelmässige nennen.
Es ist nun leicht zu übersehen, dass unter den gegebenen Bedingungen der zunächst entstehende Körper der Würfel sein wird. Ich halte es für das vortheilhafteste, die Lage der Linien, welche die Gestalt bestimmen, nach dem angenommenern Stand-puncte des eigenen Körpers zu beschreiben zu  benennen. Diese Bezeichnungsart setzt uns jedesmal selbstt in die Mitte der zu beschreibenden Gestalt
D0102 have the same size. We shall now call the regular system of forms by means of the crystallographers. It is now easy to overlook the fact that the present conditions will be the bodies of the cubes. I consider it most advantageous to describe the position of the lines which determine the shape, according to the assumed stand-point of the central body. This type of expression always puts us in the middle of the figure to be described,

It is clear that proof reading of the OCR is a necessary part of this process !

[jehovajah]

Today's translation

also in dieselbe Lage, in welcher sich der Astronom in Beziehung auf den Himmel befindet, oder in Rücksicht auf das Planetensystem auf den heliocen- trischen Standpunkt, Die Krystallographen sind ge wohnt, von Aussen zu beschreiben, welches ich für sen weniger vortheilhaft halte. Ihre Ansicht verhält sich zu der unsrigen, wie die eines künstlichen Himmels- globus zur Ansicht des. Himmels unmittelbar oder in jener am Rücksicht, wie das Ptolomäische Weltsystem zum Copernikanischen. Da wir es hier allein mit der Gestalt zu thun haben so kann es gleichgültig sein ob sie die Grösse eines Sandkorns hat, oder an dem unermesslichen Gewölbe des Him mels beschrieben ist. Das unmessbar Grosse in der Astronomie kann mit dem unmessbar Kleinen in der Krystallographie (etwa Hauy's integrirendem Molecul), füglich, unter einer Form betrachtet werden. Beides gelangt zur Anschauung, indem wir ihm end liche Abmessungen geben, und wir werden finden, dass die Kugel für beide Arten der Entwickelung mit gleichem Vortheile angewandt werden kann. Von den 3 auf einander senkrechten Linien sei nun die eine vertical, OU (Fig. 1.), die beiden an- dern horizontal, und zwar die eine nach rechts und links, RL, die andere nach vorwärts und rückwärts, VH, alle drei durch denselben Punct M gehend. Durchschneiden wir nun jede in gleicher Entfernung vom gemeinschaftlichen Durchschnituspuncte M aus mit einer darauf senkrechten Ebene, so erhalten wir den Würfel. Man stelle sich in die Mitte seines Zimmers, das Gesicht gegen die eine Wand gekehrt, und denke sich durch diesen Standpunct die beschrie- benen Linien gezogeh, so ist klar, dass, abgesehn von etwanigen Unregelmässigkeiten, die Wände sammt

That is, in the position in which the astronomer is related to heaven, or in consideration of the planetary system, on the heliocentric standpoint, The crystallographs are habited, from outside, which I consider to be less advantageous. Their view is analogous to ours, like that of an artificial heavenly globe to the view of the heaven, either directly or in the latter, as the Ptolomian world system to the Copernican. Since we have only to deal with the form here, it can be indifferent whether it has the size of a sand-grain, or is described on the immense vault of heaven. The immeasurable greatness in astronomy can be regarded with the immeasurable little in crystallography (such as Hauy's integrating Molecul), feasible, under a form. Both attains to the point of view, by giving it final dimensions, and we shall find that the sphere can be applied with the same advantage for both kinds of development. Of the three lines which are perpendicular to each other, let the one vertical, the other two horizontal, one to the right and the left, the other to the forward and backward, to the other, to the three Through the same point M. If we now cut each at a uniform distance from the common intersection point M with a plane perpendicular to it, we obtain the cube. Let us stand in the middle of his room, his face turned against the one wall, and if this line of sight see the lines described, it is clear that, apart from some irregularities, the walls are mingled

[jehovajah]

Today's translation


Decke und Fussboden jene Ebenen, so weit sie sich gegenseitig begrenzen vorstellen können. Man möge dieses immer nahe Versinnlichungsmittel nicht vornehm. verwerfen; es erleichtert die Orientirung ungemein, und kann nicht leicht zu einer falschen Ansicht führen. Jede Linie zerfällt nun vom Durchschnittspunkte aus in zwei Theile, welche ich, in sofern sie zur Construc tion der Ebenen dienen, und dieselben tragen, "radii con- structores", oder tragende Strahlen, auch schlechthin Träger nenne. Die Richtungen, nach welchen wir die Träger des Würfels angenommen haben, heissen mir, die Hauptrichtungen, und es werden hier, wie in jedem andern Systeme, je zwei, die zu derselben Linie gehören, mit demselben Buchstaben bezeichnet, von denen aber der eine den Accent erhält. Auf diese Weise sind in Fig. 1 die Träger der Würfelflächen bezeichnet: MO= b, MU=b', MR=c, ML=c', MV=d, MH=d'. Ueberhaupt sollen diejenigen Träger, welche den Combinationen ursprünglich zum Grunde liegen, und ihre Elemente geben, Elementarträger genannt werden. Man kann sich nun vorstellen, die Ebenen lägen anfangs alle durch den Mittelpunkt M, so dass also je zwei zusammenfielen, und fingen nun an sich auf ihren Trägern gleichmässig fortzubewegen, so dass in jedem beliebigen Moment alle Träger von M aus gleich gross waren, so wird man immer einen fort und fort wachsenden Würfel haben, und diese Construction kann in einem beliebigen Puncte ge- hemmt werden, um einen Würfel von bestimmter Grösse zur Anschauung zu bringen. Die Gestalten hören dadurch gewissermassen auf, starre und unver- änderliche Producte zu sein, und erscheinen, wenig-

Ceiling and floor, those levels, as far as they can mutually limit each other. One should not take this near-natural remedy. discard; It greatly facilitates the orientation, and can not easily lead to a false view. Each line now divides from the average point into two parts, which, in so far as they are used for the construction of the planes, are called radii constructores, or bearing rays, also simply bearers. The directions according to which we have assumed the carriers of the cube are called the principal directions, and here, as in any other system, two each belonging to the same line are designated by the same letter, Accent. In this way, the carriers of the cube surfaces are designated in FIG. 1: MO = b, MU = b ', MR = c, M L = c', MV = d, MH = d '. In general, those carriers which originally form the basis of the combinations, and their elements, are called elementary carriers. It may be imagined that the planes lie at first all through the center M, so that two each have collapsed, and they began to move on their carriers, so that at any given moment all the carriers of M were of the same magnitude We shall always have an ever-increasing cube, and this construction can be inhibited at any point in order to render a cube of a certain size. The structures thus cease to be rigid and unalterable products, and appear to be little-


To be
Processed

[jehovajah]

Commentary

The details that Justus establishes here are of fundamental importance.
Many may know about Felix Klein , the Erlangen Proeject and the notion of "Dynamic Geometry" or transformational geometry, but not realise that this was a reactive movement to the way geometry was taught as a static perfection!
In fact , geometry as a term is a misnomer if applied to the Stoikeia or to the Pythagoreans

Panta Rhei, and it took a revolution in France to free the academic mind from centuries of encrusted misinformation, and to give modern, young fresh minds the chance to directly engage with the Pythagorean Text.
Of these, the most profound thinkers and educators were the Grassmanns and the Natural Philosophical school, who attempted to start again from first principles.

Thus the construction of form within a spherical overjacket was of ancient origin. And the notion of a vector( Träger) that is a radius of construction an ancient practice! That Hamilton called it a Vector and Justus a Träger attests to the uniformity of this practice amongst geometers.

And so now I see that a vector carries the final line segment of a form when it is precisely constructed using a compass and ruler or straight edge method.

The line of construction is and always has been for the purpose of orientation and drawing direction of a line segment. The magnitude marked off on this construction line is called the length of the line segment.
Ths a construction line carries the magnitude of the line segment, it imparts to the line segment it's orientation, and allows the constructor to decide upon a direction of carry!

Justus says it is a " bad idea" to call it just a vector/ Träger!  However lazy as we are we just do it anyway!!

The other point to be made is that construction arcs are Also to be called vectors/Träger!

Now pay attention to the Dynamism of Justus construction and conception! Thus continuity is encapsulated as a dynamic motion, not a static property of a line segment at all. Continuity is imparted to a line segment by the vector/ Träger too!

The conception of dynamic Strecken leaps out of the page when you read the very first Words of the Ausdehnungslehre 1844

[jehovajah]

Today's translation

wenigstens ihrer Grösse nach, in einer unendlichen Ent- wickelung begriffen, gleichsam fliessend, und sind nur für einen Moment dieser Entwickelung zur be- stimmten Anschauung gebracht. Durch diese vor stellung tritt es klarer vor dem Geist hin, dass nicht das endliche Product, sondern der Act der Produc tion es ist, welcher in dieser Entwickelung das Haupt- moment ausmacht Die Träger geben uns also nur die Richtungen und Verhältnisse gewisser Bewegungen an. Zwischen diesen soll nun combinirt werden. Diese Combina tion ist eine Combination der Bewegungen, muss also nach der reinen Bewegungslehre bestimmt und beurteilt werden. Ich nenne deshalb diesen Tieil der geometrischen Combinationslehre die pho ro nomische Combinationslehre Eine Conbi nation in diesem Sinne ist also von ganz anderer Art und Beschaffenheit, als die von welcher vorhin die Rede war und bei welcher man auf das Raum- liche sah, was den verbundenen Elementen gemein war, oder doch durch sie bestimmt wurde. Hier ist es die Richtung und relative Grösse einer Bewegung, welche aus zwei oder mehr andern zusammengesetzt ist. Zwei oder mehr Bewegungen lassen sich aber zusammensetzen oder combiniren, wenn man die eine einem Punct auf einer Linie, die andere der Linie selbst beilegt. Kommt noch eine dritte Bewe- gung hinzu, so kann man diese der Fläche, in wel cher die beiden erstern vor sich gehen, zuschreiben. Man könnte die zweifach zusammopgesetzte Bewe- gung eben so als die Bewegung oder den Weg des Durchschnittspunctes zweier sich bewegenden Linien, die dreifache als den Weg des Durchschnittspunctes dreier sich bewegenden Ebenen ansehn. Wir be-

At least in size, in an infinite development, are, as it were, flowing, and have been brought to a definite view only for a moment of this development. By this representation, it is clearer in the mind that the finite product is not the final product, but the act of production, which is the main moment in this development. The carriers, therefore, give us only the directions and relations of certain movements. We shall now combine them. This combination is a combination of the movements and must therefore be determined and judged according to the pure theory of motion. I therefore call this part of the geometrical theory of combination the phoronomic theory of combination. A combination in this sense is, therefore, quite different in nature and composition from that which was mentioned earlier, and in which one looked at the space, Elements, or was determined by them. Here it is the direction and relative magnitude of a movement composed of two or more others. Two or more movements, however, can be composed or combined even if the one movement laying by  a point on one line, the other movement laying by the line itself. If a third motion is added, this can be ascribed to the surface in which the two former occur. The two-fold aggregate motion may be regarded as the motion or the path of the intersection point of two moving lines, which are threefold as the path of the intersection point of three moving planes. We are

To be processed further

[jehovajah]

Commentary

Note that Justus, in combining movements, does not refer to a resultant line between the two movement lines or vectors or construction lines , instead he describes a process of laying one movement to a point in its line, and the other movement to its line . .

While this is not truly clear it is clarified by the reference to the intersection point. One movement is laid by one line up to the intersection point with the other line against which the other movement is laid. .

Later this is clarified as placing the beginning and end points of Strecken or line segments in a specific pattern to depict the movement. This is the idea of vector sums!  There is no vector resultant mentioned here, nor in Hermanns opening discussion of his first apprehension of summing line segments.

The phoronomic reference is arcane to most, but stereoscope and Opthamology are derived from this once vibrant field of study. The topic is how the eyes focus and track the objects and motions in the field of vision , and how 3 dimensional vision arises. . Thus a lot of these construction line demands and conventions are imported from that doctrine. Into Justus thinking here.

[jehovajah]

Today's translation

halten die erste dieser beiden Ansichten als die übliche und darum geläufigere bei.. Dass die Lehre von der zusammengesetzten Bewegung ganz der rei- nen Mathematik angehöre, unterliegt, sobald man dabei von den bewegenden Kräften ganzlich absieht, keinem Zweifel, wiewohl die reine Bewegungslehre bis jetzt in dem Lehrbüchern derselben keine Auf- nahme gefunden hat. Für unsern Zweck genügt das, was Fischer im 4ten Capitel seines Lehrbuchs der mechanischen Naturlehre (3te Auflage, Berlin 1826) über die Theorie der gleichformigen Bewe gung, von der hier allein die Rede sein wird, vor- getragen hat. Man kann die wenigen Lehrsätze, auf welche es dabei ankommt, auch aus jedem Lehrbuehe der Mechanik nehmen. Die, von denen hier Anwen- dung gemacht werden soll, sollen spälerhin auch noch zusammengestellt werden. Als Lehnsäze führen wir hier nur an, dass, wenn die einfachen Bewegungen gleichformig sind, die zusammengesetzte bei zweien nach der Diagonale des Parallelo gramms, bei dreien nach der Transversale des Parallele pipedums so nenne ich die gerade Linie von einer Ecke des Parallelepipedums in die gegenüberstehende) geht welches darch Richlung und Verhältniss der einfachen Bewegungen be- stimmt ist. Die Combinationen der Träger lassen sich nun, wie andere Combinationen mit und ohne Wieder holung, und nach verschiedenen Anordnungen auf- stellen, wobei nur zu bemerken ist, dass eine Com- bination zwischen den gleichnamigen Trägern, wie b und b' niemals slattfinden darf, weil diese entge- gengesetzte Bewegungen bezeichnen, welche sich entweder ganz oder theilweise aufheben, und die

Hold that the first of these two views is the usual one, and therefore more common. The fact that the doctrine of the composite movement belongs entirely to the pure mathematics is, no doubt, as far as the moving forces are concerned Now in the textbooks of which he has not found a copy. For our purpose, what Fischer, in the fourth chapter of his textbook of mechanics of nature (3rd edition, Berlin, 1826), presented the theory of uniform motion, of which we shall be concerned here. One can take the few doctrines, which are important, from every doctrine of mechanics. The ones which are to be applied here are also to be compiled. As lemmings, let us assume here that, if the simple movements are uniform, the two are parallel to the diagonal of the parallelogram, and three to the transverse of the parallel pipedum, I call the straight line from a corner of the parallelepipedal to the opposite ) Is the determination and the relation of the simple motions. As with other combinations with and without repetition, the combinations of the carriers can be set up according to different arrangements, with the exception of the fact that a combination between the carriers of the same name, such as b and b ' Because these represent opposing movements, which either cancel wholly or partly;

To be processed

[jehovajah]

Today's translation


Natur entgegengesetzter Grossen haben. Die Auf- gabe, die Complexionen der Elemente bcd b'c'd' zu entwickeln, gehört also in die Classe derjenigen, bei welchen die Verbindung gewisser Elemente ver- boten ist.

Vorläufige Construction der vollkommen be- stimmten Gestalten.

Wir wollen nun, gleichsam zur Probe, und um den Leser nicht durch Vorbereitungen, deren Zweck er noch nicht übersehen kann, zu ermüden, aus den gegebenen Elementen die Complexionen ohne Wie derholung entwickeln, und die durch sie bestimmten Gestalten construiren,— damit wenigstens an einem Beispiele erhelle, wohin diese Combinationen führen, —und sodann den Faden da, wo wir ihn fallen liessen, wieder aufnehmen.

Die Combinationen aus den gegebenen Elemen- ten sind nun folgende:

Unionen b. b'. c. c'. d. d'.

Binionen bc, bc', bd. bd'
b'c.b'c'.b'd.b'd'.
 cd. cd'. c'd. c'd'.  

Ternionen: bcd. bcd' bc'd. bc'd'. b'cd. b'cd'. b'c'd. b'c'd'.

Eine höhere Classe kann es nicht geben, da die gleichnamigen Elemente nicht verbunden werden dürfen.

Nature of the opposite size. The task of developing the complexions of the elements bcd b'c'd 'thus belongs to the class of those in which the connection of certain elements is forbidden.

Provisional construction of perfectly defined figures. Let us, as it were, to the rehearsal, and not to weary the reader by means of preparations whose object he can not overlook, to develop the complexes without repetition from the given elements, and to construct the figures determined by them Give an example of where these combinations lead, and then resume the thread where we let it fall.

The combinations of the given elements are now as follows:

Uni onen  B '. C. C '. D '.

Bin ions bc, bc', bd. bd'
b'c. b'c'. b'd. b'd'.
 cd. cd'. c'd. c'd'.
Ternionen: bcd. bcd'. bc'd. bc'd'. b'cd. b'cd'. b'c'd. b'c'd'.

A higher class can not exist, since the elements of the same name can not be connected.


To be processed

[jehovajah]

Commentary
Note Justus points to the diagonal and the "transversal" as associate determinations or resultants of simple movement sums. We take this for granted , as a doctrine from Mechanics. He sets Fischers work as a textbook. This is a direct reference to Newtons Astrological principles! Newton established this association in this book. However this is too advanced for primary school teachers and hard to get, so Fischer is recommended.

The sources of vector/ Träger concepts are mundanely revealed by this reference.

We see, then how closely Hermann followed his Fathers ideas and presentation. But Hermann made one crucial change and observation and that was with regard to 3 line segments in the plane. It revealed to him that the main planar shape for study was the parallelogram and it's 3d equivalents,Parallelepipeds  not the cube!

On d you make that determination, then 2 products / multiplications become fundamental: the spreading apat product, that is the sides of the parallelogram , which really evaluate when the sides are orthogonal ; and the colliding together product , that is a dropped perpendicular onto one side from the tip of the other side, which evaluates according to the trigonometric evaluation of the perpendicular and the receiving side. The outcome is the variation of the " outer sides" product by a trig ratio. This cane to be called the " inner line segment"  product, or just the inner product. Both evaluate the so called area of the figure, but , and it is a big but, the answer is not important! What is important is the ratios of accessible figures and forms. So the area of a parallelogram can be expressed as multiples of a unit parallelogramm( a rhombus) which then makes the count invariant irrespective of the angle between the vectors/ Trägers. The inner product then enable a comparison relative to squares and cubes not the count but the shapes!!

Freed from standardised evaluations, the geometer can manipulate and synthesis the pure forms that are appropriate to the circumstances.

What about angles? They are not necessary. Trigonometry is about ratios in right nglrd triangles, boy angles.

So why are angles made to seem so important?
In a student Maths set you have the set squares and the protractor. The protractor reduces the number of set pieces required to construct a figure. Using a ruler, a compass and a protractor a wide range of figures can be constructed. In addition, using trig tables standard values can be evaluated.

The length of a circular arc is NOT problematic, but it is often inaccessible to Astrologers . The arc length can be calculated by using Analogous ratios and trig tables.. But where it is accessible it is most accurately measure by rolling the arc on a flat surface without slipping. This moves the circle centre along the dimeter parallel to the flat surface the required displacement .

Again this is a curved measurement apparently straightened out, but it is an unmoveable axiom, demand that curves can never be straightened!! So this measure is in fact a ratio of an arc to a straight line on a flat surface. The ratio changes for curved and trochoidal surfaces dramatically.

So Justus establishes a method of synthesis that requires counting not measuring , later he deals with the trigonometric ratios that becom necessary only then! It is the last few chapters of his booklet, and is simply traditional higher mathematics related to his method. Hermann, following his fathers set out plan hoped to produce a second volume to express this part of his fathers life long work, but the poor reception of his 1844 work meant he nearly gave up on ever achieving that goal. However his brother Robert, also promoting his fathers work and insight redacted and reprinted Hermanns work in a format that he judged would go own better with its target audience, and he was right, but it was never going to burn the house down while Gauss and Riemann were alive. Once they died then Hermanns work came into its own, and outshone Roberts several publications on the subject!

[jehovajah]

<a href="http://www.youtube.com/v/bMlX28tIqzA&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/bMlX28tIqzA&rel=1&fs=1&hd=1</a>
Norman develops the planar geometry using the dot product. And a defined cross product for the plane.
Norman spends a great deal of time filling in the gaps Herrmann and almost everyone else left . So he defines natural numbers using data structure notation , the concept of anti matter rather than opposing directions or mirror reflections or rotations through Pi radians.  He gets as far as integers but has defined rational numbers in an alternative series.

The theory of numbers he promotes is firmly based on the logoi cases in the Stoikeia.

However he misunderstands the role of numbers and the role of magnitudinally extensive forms in the Stoikeia. He correctly identifies the generality of the form over the expressions using labels and so algebraic forms ( symbolic arithmetic )
It reminds me of the fantastic development of real no.s by Hamilton, using the concept of flowing or propagating time.
It was also brilliant but flawed! The issue always is continuity, and that is a motion concept. Like curvature can not be straightened out, so continuity can not be divorced from  motion even jerky motion.

Any ways Norman is coming back to Justus ideas on geometry and kinematics.

[jehovajah]

http://m.youtube.com/watch?v=x6UofU0QJI4
This series of videos in which Norman explains the spread , relates it, at last to the unit circle.
Of course for any circle an arc on the perimeter is unique and thus any regular gigue that contains that arc as a diagonal will also be unique in area. Thus area can be a measure of arc in a proportional way, related to the square of an arc " length" .

Norman points out that arc length is more naturally related to sector area, but that calculation is more complicated.

We should not let calculative difficulty o scurr what is observation ally simple!

Of course the utility of arc length is questionable. In fact Normans whole premise for rational trigonometry is that arc length is restrictive! This is true because Pi is an incommensurable ratio, and rarely do we need to construct perfect spheres or circles.

However, the deeper mysteries of physical motion are inaccessible to straight lines, even though we can confidently establish metrons with them. In point of fact we can establish excellent metro s only through the use of circuar construction lines! And similarly we can construct better curve approximations to general curves using circular arc elements.

We ca not make what is curved straight so we need to expand our notion to u its of curved extensions as well as straight ones.

We use a fudge factor called Pi to reveal that the algebra is the same even if the metrons are curved.

So the area of a sector is just so e fudge factor multiple of the area of a parallelogram and both are a measure of the contained arc squared.

Squating an arc seems a novel idea, but in fact it is procedurally identical to Squaing any lineal element. There is no unique measure called area. We standardise a process of counting a Metron that " divides" or we place down upon a region in a contiguous manner, very much a ong and dance process!

So whether we use a line or an arc squared as a measure of rotation is a trchnicallity of calculation. The most direct measure is to draw a standard circle and ompare arcs on that circle, but comparing areas volumes etc does provide a more accessible measure in many instances even if it is a none li Ealing method that distorts the numerical results from uniformity.  Evenso it does provide a unique up to sign, measure of rotation , but not of distance travelled by a rotating disc with no slippage. For that you do need the most direct measure which is the arc to diameter ratio..
For a perfectly flat surface the centre of the disc moves along the diameter ib a direct proportion to the arc turned. This is a possible definition of the notion of arc length : the arc: length ration of movement measured at the diameter.

You will note that this length varies with the surface the disc rolls over, in particular that the circle within or upon circles motion produces not only incredible patterns but incredible arc lengths.