Draw a circle radius r and extend its radius to 3r, by marking off r 3 times. This creates a ruler

Bisect the ruler to find the centre for a semi circle radius 1.5 r.

Draw an acute arc on the small circle and extend the side containing the arc to cut the semi circle.

Create a right triangle by these means in the semi circle. Draw parallel lines to trisect the sine length of the right triangle and extend these to the semi circular perimeter. Connect these points on the semi circle to the centre of the small circle. Where they cross the petimeter of the small circle is a good approximation to a third of the given arc.

This uses the angle subtended by an arc at the centre of the semicircle is twice that subtended at the petimeter

This approximation is less accurate the nearer the arc gets to a quarter arc, and better the nearer the arc reduces to nil turn. It is good below a 1/6th turn.

The method can be improved by using more sections, and indeed the method allows us to section an accute arc as many times as we like with increasing accuracy.

The behaviour as the arc approaches a quarter turn and as the sectioning becomes very large is interesting. There is no angle that is small enough to exist between a tangent and its intersection point on a circle! This is a proposition in the Stoikeia book 3. Thus we can not find a section that at the same time as it approaches the orthogonal radius extends outside of the circle without cutting it in 2 places.

The circle thus bounds an infinitely divisible process . The accuracy cannot be expressed in a finite ratio. When a finite ratio is found it is a special quanta and we define it as uniform, but it actually is beyond our ability to empirically determine.

Pi is transcendental precisely because it is beyond our ability to resolve it.

Pi, I and the ratio of the perimeter of a semi circle to its diameter represent the concept of a transcendental ratio, a ratio we can state but not commensurate.

Drawing a arbitrary angle and then trisecting it’s ray segments allows a triangular arallel grid to be drawn. The grid has interesting roper ties. First it houses an arc that has a ratio to a larger arc 1:3 . But the larger arc is 3 times the rotational displacement of the smaller. Thus 3 of the smaller arcs can be topologically transformed to the larger arc.

Finally three of these small arcs nestle in a close region near the large arc within an array of rhomboid forms. These forms are transacted by the large arc. This indicates that by moving the next part of the trisection to this area may give a highly accurat pair of intersection points for trisecting the larger arc. These drawn back to the angle vertex may give a good approximation to trisecting an angle.

The fascination of attempting to proportion an arc by the radial proportions and parallel lines and circle theorems , albeit approximately has led me to attempt a kissing circles design! Using the arc subtending an angle at the permeter theorem I attempt to section an arc into 6 pieces, roughly equal, by by using a kissing circle of r and 3r , and positioning the angle to be trisected at the kissing point. Sectioning g the diameter of the 3r circle into six I use parallel lines to the angle at each segment to section the arc of the 3 r circle. Using the last 2 I subtended an angle at the kissing point of approximately a third of the given angle.

Can you illustrate the first drawing?

The diagrams are something I will try to provide as I learn how to use WordPress.

You may find this link of interest. It shows the circular Gnomon from which I conjecture the rectilineal gnomon was derived.

http://mathworld.wolfram.com/Arbelos.html