http://m.youtube.com/watch?v=-wokVaaGFgAIn this remarkable series Norman is at last able to present the general beauty of his rational trigonometry in the context of lineal algebra.

It has profound implications for teaching so called mathematics.

There is one misconception he holds on to and that is angles. He misconceives what an angle is, or rather he deals with the general nd pervasive misconception of the term " angle" .

The arc length Ø is dealt with both by the Pythagorean school, Sir Robert Coates, Euler and indeed Newton and DeMoivre. It is a corespondence between the diameter( radius) of a disc and how far its centre moves along the diameter as it rolls on a flat surface. It is a ratio between non homogenous magnitudes where the eternal curve is compared irrevocably to a Rectilineal line by a process of point to point matching. The point on the curve is called the tangent point the Rectilineal line through this point and only through this singleton point is called the tangent line.

In terms of infinite processes this is an unimaginable definition! In pragmatic terms we accept it from a diagram as existing as drawn. There is no logical connection between the drawing nd the statement of definition. The drawing is always perisos( approximate) the definition artios( perfectly fitting)

So the arc as a magnitude is quantified by sub arcs, but the usefulness of these sub arcs depends on a correspondence or ratio( logos) that can be developed in an analogos( proportional) way.

Thus what Norman makes explicit is that ratio and proportion theory Eudoxus sets out in books 5 and 6 of Euclids Stoikeia.

While Norman rightly points out the infinite nature of certain ratios when reduced to a single standard form, it is the attitude that asserts we can ignore pragmatics that is questionable, and indeed misleading.

Eulers and Cotes theorem involve the cosine and sine Ratios, not infinite expressions of the same distinguished not as polynomials, or even power series , but as complete functions!

But here Norman demonstrates how the doctrine of extensive magnitudes leads to algebraic simplicity,uniformity and applicability to multiple dimensions AS directions.

It is the notion of orientations as dimensions that makes lineal algebra so powerful . In addition the direction of travel in a given orientation is fundamental to a specific conception of Dimendion in space. Dimension in general relies upon metrons by which quantification processes are established. So we have dimensions of space, mass and time, and dimensionless quantities like angle which are ratios. When I draw an arc length, I identify a dimensionless quantity called annngle or a corner. It is dimensionless precisely because it is encoded by a ratio, and recorded in ratio tables of approximate "Values" or results of a division process.

The tables have always been exact(artios) or approximate( perisos) results of a difference process called division. The power series expressions of this difference process or division provide a systematic way to find these differing values, and should not be used to replace the underlying ratios.

In the modern world of driving nd the McIntyre world of gears arc length is important, but for construction the sine tables have been specifically calculated to help engineers pragmatically. It is only through the discoveries of men like Ōrsted, Ampère Faraday and Boscovich that the long held belief in the vorticular motion of natural powers has established the applicability of the sine ratios to these invisible yet powerfully manifested phenomena.

As part of this approach to mathematical description it is imperative to understand that circular arcs are legitimate extensible magnitudes that describe rotational dynamics in an algebraic way set out in the Ausdehnungslehre