The shadow cast by a vertical pillar in Alexandria at noon during the summer solstice is found to be 1/8 the height of the pillar. The distance between Alexandria and Syene is 1/8 Earth’s radius. Is there a geometric connection between these two 1-to-8 ratios?

As per the question, the actual height and the shadow length of the vertical pillar is x and x/8 say. By this information we can calculate the length of the line joining the end of the shadow and the top of the building. That’s what we can deduce from such information by the use of pythagoras theorem and by some trigonometric identities. Bus...