Simple approximate methods for dividing circles and arcs
These methods are based on bisecting an angle, bisecting a line, and equilateral triangles.
We can demonstrate how to divide a circle into quarters by bisecting a line. By this method we can find the chord that quarters circular arc
We can demonstrate that we can divide the circle into 6 equal parts by using equilateral triangles. By this we can find the chord that divides the circular arc into sixths (1/ 6)
Rotating these two chords onto a diameter, the same diameter, enables us to compare the two chords.
The chord that divides the circular arc into five equal arcs lies between these two chords on the diameter. We can now find this by trial and error, and using Euclid’s algorithm or method of division. We will know by this process that dividing the circular arc equally does not divide the diameter into equal parts.
To trisect semicircle. If we didn’t know that the radius trisects a semicircle, we can proceed in the following manner.
Construct the semicircle, and then construct a semi circle on the same centre which has three times the diameter. Bisect the largest semicircle. This gives us the cord that divides the semicircle into two. Mark off the diameter of the smaller circle on the damage of the larger circle and also Mark off this chord. We know that the chord which trisects a larger semicircle lies between these two chords. We can find it by trial and error and Euclid’s method .
In the first description we have one method of trial and error. The interpolation of the correct chord is done on the diameter.
In the second description we have two methods of getting the correct chord : one is by using the Arc and the other is by using the chord. Using both together will give us a faster convergence.
This method is worked in the arc and the diameter of the larger circle using estimates and corrections from the smaller circle. The estimates improve as the curvature difference becomes less noticeable in the arc projections.
So in the case of the semicircle being divided into three, we use a diameter of the semicircle as the chord estimate in the larger circle. We Mark this off on the diameter of the large circle and then step it off around the circle. It will be too short ( because the curvatures are different) . We then take the shortfall in the larger circle and project it back down on to tthe small circle to give us a one third estimate of the arc length in the larger circle, And the chord that marks it off.
We can use this chord in two ways; on the diameter to extend the first estimate of the chord or on the circle to mark off an estimated third arc length on the circle, enabling us to then draw a revised chord on the circle. Rotating the chord down onto the diameter we can compare the two new chord estimates. The larger one will be used to mark off the new trisection. first. If it is too long then the shorter one will be used.
Again the shortfall will be projected down onto the smaller circle in order to obtain a estimate of the third of the shortfall arc and the associated chord . We know by this process the bounds between which the desired chord lies as marked out on the diameter of the larger circle. We quickly obtain an accurate approximation if not an accurate result.
In this case we check the method because the result should be the radius of the larger circle.
The method is applicable to any number of divisions and any general arc.