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http://m.youtube.com/watch?v=I9a2KK3RqUw

Norman ties down orientation to cyclical rotation. He implies that lineal orientation is subordinate to rotational orientation.
Thus Within a a cyclically ordered  triangle the ordered side AB has to be set = to -BA , and within an ordered parallelogram BC //DA means DA=-AD=-the vector BC.
The order links the ordered side to a parallel vector/ Träger or line segment on that construction line and to its reverse or opposing vector by mirror reflection or half circle rotation.

What about quarter rotation? We find the signed "area"/form under cyclical rotation behaves exactly the same as i the  magnitude for rotation by a quarter turn .

Some wanted to implement this in IT as "eastern logics". It's like Ghandi that truth looks different from different perspectives. But realy it is how they in India can coexist together.

The Stoikeia begins;
A mark is not a part of anything.
And thus a drawn mark is an extent without form
And also a drawn mark is bounded by marks.
And thus a good drawn mark is one out of which dual marks lie upon itself
And thusly a surface upon which light shines is a unit coming to be of extension and form .
Thus a surface upon which light shines is bounded by drawn marks.

A footabl surface upon which light shines is one out of dual marks of which the good drawn marks lie upon itself.
And so a footable surface upon which light shines knee bend is where in foot pacing two drawn marks streching away from one another and not upon a good drawn mark placed together , towards one another the drawn marks meet .


The descriptions are dynamic and active.

As you continue your meditations Alef may your insight prove rewarding and useful. . For the form seen with the eye( eidos) is distinct from the form formed by the hand with clay( plassos) . And Socrates played a game with men of intellect only, and so did Plato.
The concept of being, or ontology for a man imprisoned and shackled is distinct from that which he may form as a free agent.

In a society, culture or family who really is free?

Thus the sages of the east and far east despised no one of humble birth for they saw that greatness was reincarnated and had to be sought out by the wise.

In the west some bad men sought to extinguish this light in greed and madness.
Those who treasured the eidos were gifted to form the plassos by the Musai. Or those who formed the plassos perfected it in the eidos and treasured their own creation, what came to be through their activities. .
Do you accept fact or myth?
The wise say accept both as illusions and enjoy your life.

Thus fractals generate joy for the wise and terror for the uninitiated .

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http://m.youtube.com/watch?v=EiP3Jf5hnAI

Starting with lecture 37 Norman presents his Förderung on trigonometry . Essentially he eliminated the need for tables of trigonometric values.
While Norman appears to not respect circular measures, this is not the case. Circular functions are normally truncated infinite series. The question is : how useful is this computational load? How practical and how accurate?

While the Pythagoreans had no fear of incommensurable magnitudes, nor of approximations they nevertheless highly valued precise or exact results to processes. They preferred to contain approximations between 2 exact results wherever possible.

In book5 Eudoxus opines that  magnitudes in a ratio or logos, meaning a situation of study about which many words may be said or written in comparison or contrast, when honed down to extensive magnitude has 3 cases. The magnitudes are dual or equal to each other by duplication or fitting precisely upon each other or some other process of comparison or production.
If they are not dual then the cases are , in order, the first is greater than, or the first is lesser than the second.

The study of these cases are not limited to static extensions either, for then the dynamic reality in which we live wold be unrelated to our studies, and in fact nothing could be developed, dualed or multiplied!

In his study of magnitudes Eudoxus starts with the drawn mark. A continuously moving extension. Thus continuity is inherent by motion. What creates this mark is an inherent power within either the mark or the observer/ creator / producer or both.  Thus a drawn mark may be marked in segments or parts at will. Lesser drawn marks are parts of greater ones, and greater drawn marks may be apprehended as multiples of lesser ones.
It is the relationships between these partitions of wholes or dualing into multiple forms of wholes that is explored in book5. The most used one is the Logos Analogos relationship.

Here Norman makes use of this basic relationship to depict spatial extensive magnitudes of interest. But he uses advanced ideas from book 7 to establish his laws.
In fact the root of these advanced ideas are in Book 2 of the Stoikeia , where dynamic parallelograms and rotations are normalised by the quarter turn or ortho.
In nook 2 the metrons established in book 1: trilateral quadrilaterals, trigonia and their relationships are used to establish quantification or metrication. We can define counting, and a counter or scalar magnitude in book 2 . That counter is usually a parallelogram or a dual drawn mark. A normalised parallelogram is called a Rectilineal parallelogram.

There are corners in these works but no degree or radian measure. The measure that is used is the chord of a circular arc, and this is compared to the diameter of the circle in which it is formed.

The quadrance and the spread thus relate to squares drawn on these chords and diameters.
However, the spread is specific to lines emanating through the centre of a citcle . The spread relates lines intersecting marks with that circle to the ratio between the square on the perpendicular and the square on the radius, that is it uses the quadrances of these  compared line segments in a circle to define spread.
The chord could have been used but we are used to half angles( corners at the intersection of 2 lines bisected) and these help in 3 dimensions considerably, especially in cases of rotation.

Why did we spend so long calculating tables?
The Sumerians set the trend, and of course they had no uniform view of a normalised reference frame. Despite astrologers fixing the ecliptic and many other quantity standards , the use of tables conducted the mi d away from establishing a normalised reference frame in the spheres of reality. Or rather, the absence of the idea of a vector /Träger did not stand out as a simplification of calculation processes, or a way to present calculations algebraically in relation to extensive magnitudes.

We owe many people a debt for this insight but foremost must be Justus Grassmann who reduced his eras learnings to these simple fundaments called vectors/ Trägeren. This inspired Robert and principally Hermann to publicise these ideas, even in the teeth of Gauss and Riemann !
Sir William Rwan Hamilton deserves mention, but it is the primary educators the Grassmanns who rolled it out where it could grow! In the minds of young students.  

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http://m.youtube.com/watch?v=xiQNkF_svVw
The issue with sin^2 is the indeterminacy of acute and obtuse corners, which is why the chord is clearer. However, the geometry has to be consulted to determine obtuse or acute reference.