How to Measure the Earth’s Circumference?

I have completed the second chapter of my book. This section tells about different approaches to the issue of estimating the Earth's circumference. You can read the first page where I explain how I picked up the medium for the estimation.

 If the earth is a globe, its circumference can be roughly determined.

The two giants of the ancient world endeavored to carry out this tremendous task. Both of them estimated the angle of elevation of heavenly bodies above the horizon at distant points presumably located on the same meridian in order to reach the measure of one degree. They also assessed the distance between these spots by converting travel days on land (1) or at sea (2) into an accepted standard of length.

The fact that each of these estimations held versions proves that the scholars were not happy with their initial scores and, being confronted with sound criticism, had to improve their performance. The degree of uncertainty only grows if we take into account the inconsistency between ancient and modern units of measurement.

          The modern science rates the equatorial circumference of the earth at 40,075 kilometers where each meridian degree stands for 111.319 kilometers. (3) Medieval academics possessed a wide choice of measurement techniques, processing a variety of values. 

In the Greek-speaking world, a conventional unit of length to estimate distances both on land and at sea was a stadium, whose value would differ from region to region. The problem with this standard lies in its ambiguity: there are no fewer than five different variants and any talk of “precision” of an ancient estimation leaves an uneasy sense of misjudgment.

Table 2: Stadium to meters conversion (4)

          As we can see, the maximum gap between the opposite values can reach 25 percent (157:209=0.75).  The mean or “consensus” equivalent (184.6), which cuts by half the span of error in this ocean of uncertainty, approaches the Attic stadium.  It also allows contrasting various estimations and matches the Roman mile to stadium conversion (1,480 meters: 8 =185 meters), as reported by Strabo: “if one reckons as most people do, eight stadia to the mile” (5).

Accidentally, the modern metrical value of nautical mile-1.85 kilometers-coincides with our assessment. We will apply this amount for our further calculations to appreciate the charms and wonders of mathematical geography.