The trigonometric convention for measuring angles and computing the sine and cosine and tangent Ratios has the circle divided into 4 quadrants. The convention makes the principal direction on the horizontal orientation to the page. In 3d that page swivels as easily as any universal joint!

The principal direction is then always to the right of the circles centre , on this horizontal diameter.

However this then divides the diameter into two rays, each starting at the centre and directed contra to each other. These are termed ray segments and are as such radii of the circle. This old term reflects the fixed value of the length of the diameter and thus the radius or displacement from the circles centre.

Euclid defined, or possibly Apollonius redefined this property of a circle in terms of seemeia, or indicators. These " points" lay on the intersection of every diameter with the circle form, such that every point of intersection was the same line segment( length) from a unique point called the centre.

It is hardly ever pointed out that this is a mechanical description of a drawn form. As Newton observed, Mechanicl principles lie at the base of geometry. That is to say empirical and metrical concerns precede and define the fundamentals of Geometry prior to any reasoning or ratioing.

For example the 4 quadrants are empirically determined; the 6 sectors are empirically discovered and determined, various figures are mechanically constructed.

It is because these constructions are so basic and repeatable that they are universally held to be true. The " perfection of them is hardly noticeable. However, who has not had the slightly disconcerting experience of segmenting a circular arc into 6 by the Radius displacement only to find it does not quite fit?

That is when you make the 6th segment mark it does not land on the hole made by the compass point!

As a young geometer it is easy to believe that your fumbling attempts are the problem. More skill , practice and care should eliminate the discrepancy, but it does not. Instead what we do is perfect the result. After all it is a pragmatic thing to do. However I find it highly suspicious that the perimeter length of the circle is a little over

**6** 1/4 radii. The curved arc naturally should be longer than the chord on which it sits, but only 1/24 of a radius longer seems too small!

We accept it because we have no other measure by which to compare. In fact the issue is neurological. Our sensors are set to enhance certain signals for definition and contrast. So the boundary of a form is enhanced to clarify it. The area or volume bounded by a perimeter is also enhanced, but when we use a Metron the surprising differences in the counts between similar figures is just one of those empirical spatial behaviours that defies simplistic pattern making and reveals the flaw in pure reason not grounded in empirical data.

This is how it is. This is what our fabulous unconscious processing delivers up to us to reason with; to compare and contrast and distinguish by symbol, name or some internal experience of memory.

It is the " perfecting" of our memories that blinds our inquiries. Those who have insights have not obscured the flaws or ignored our process of pragmatic perfectioning.

Our conventional system of assigning Values to the ratios of line segments in a right angled triangle has been augmented over the centuries. The mechanically derived tables were initially partial and limited to a few indexing arcs.. These arcs associated with the appropriate corners became the concept of a corner angle..

The perimeter of a circle was divided in the Sumerian times by the Akkadians, into 360 parts. Each part was a day in an annual cycle of seasonal observations of the stars, moons and planets. Thus 360 was empirically derived, by marking sighting lines on a great circle.

During the Napoleonic years the Ecole in Paris decided to make the quadrant part of the metrical system. It was divided into a thousand sectors! For each sector the sine value was calculated. Needless to say it took years of calculation to complete, but was never introduced into general use! Copies of these greate tomes are in the library in the French Acadrmy.

Thus the angle, the arc and the circle are indissolubly linked. Even if you, like Norman Wildberger consider the current angle measure flawed you cannot escape the relationship between the right angled triangle and the circle.Thales theorem only asks you to accept what is empirically demonstrable: all orthogonal angles are equal. This statement can never be deduced, it must be induced from the circle properties and empirical measurement.

Before directed line segments were introduced, in particular contra line segments were distinguished by these circle measurements. Thus a diameter was evaluated as 2 radii, but the radii were rays contra from the centre. They were said to be opposite or 180° to each other. Imposing a – sign onto this ancient angle or arc measure system of defining direction was bound to lead to confusion!

In the first quadrant all ratios were positive(+). This meant that the principle orientation had the direction drawn to the right of centre, or vertically above the centre for the relative vertical orientation.

The relative vertical orientation was introduced precisely because of the right angled triangle.. Once the principal orientation is fixed the relative vertical is also fixed.

However the quadrant is limited to a quarter arc. Pragmatically this was not a problem because you only needed a quarter arc to define the other quadrants! Similarly if we defined a sector of a 1/6th arc then we would have to apply it in each sector. This means we are pragmatically forced to repeat values in each sector , and thus confuse ourselves as to relative orientation!

To overcome this 2 quadrants are used to define the trig ratios. 2 scales were written on every protractor. It was crucial to state which scale one was using.

Eventually the mathematical board decreed that 4 quadrants be used, that contra rays be used and that the ratio signs be distributed in the quadrants by the CAST mnemonic. It also decreed that the conventional principal rotation from the principal orientation was counterclockwise!

As usual this was ignored at the elementary level. So we still refer to the interior angles of closed forms and the exterior angles of the same forms., or we defy the board by rotating the figure or paper into the " legal" orientation and then measuring!

These games we play have serious consequences. We become confused and use incorrect values and our calculations then do not match the mechanical reality! Which do we believe? Unfortunately most are taught to believe the calculation!

However is the decreed method practical? For example how do you measure an angle between 2 lines when both are not horizontal? That angle is the Difference between 2 decreed angles, and that forces us to accept that angle measures are fundamentally differences between decreed angles..

Using that notion our sine table values have to guide us to the correct ratio values for angle differences up to 2

radians.

The overlaying of the negative(–) contra sign onto the older rotated contra orientation is still a fundamental confusion in the notation. But mathematicians seem to love these quirks!

If I start with 2 contiguous rays, some call these parallel and touching, I always want to call them collinear, but perhaps I might just use collineal; then giving one the conventional principal rotation means it's relative orientation changes anti clockwise but it's principal direction remains positive (+). By this I mean moving away from the centre the arrow I attach in the initial position is fixed in that line. By this means I can perceive how my experience of its direction gradually changes from one direction on the principal orientation line to the contra direction. To see this clearly I have to do a vertical projection from a fixed point in the rotating ray.

Now I associate a decreed rotation with a projected direction on the principal orientation. Note that the ray projects a direction onto the principal orientation not just a length.

So an angle difference strictly projects a ray difference or a direction difference onto the principal orientation.

The simple minded approach often ignores these subtleties. To be fair, they arise only due to the need to be consistent, but that need arises because confusion reigns else!

Why do we make it so confusing? Or rather allow it to be so ambiguous? This ambiguity is confused with flexibility, or rather as we all know it allows a fudge factor to adjust our confused thinking to empirical evidence?,

These issues were all considered deeply by the Grassmanns in an attempt to promote rigour in these kind of discussions. Justus in particular showed a dogged determinism not to give in to the logical inconsistencies, but it is so hard to eliminate them in confused thought. Hermann by his system of notating everything at least made them obvious, even if he then was forced to define them away!