The Formula for the Drop from Flat Earth to Round Earth

The point of this article is to introduce people to the “drop” formula used to compare between the Flat Earth model and the Round Earth model. This is one of the examples where people should be armed with mathematical knowledge to investigate the truth or otherwise of certain controversies, or conspiracies no less. I am not telling people what to think, as with the hordes of propagandists nowadays. I am teaching people how to think. I am giving people some tools investigate the truth for themselves. We learn mathematics not for the sake of exams, or doing silly homework exercises, but to empower ourselves to seek truth for ourselves, independent of other people’s propaganda.

Let us cut to the chase. Say you start at a point A on the surface of the Earth and you move in a straight line a distance x away, you would end up at a point B if the Earth is flat, but at another point C if the Earth is round. [See the diagram, not drawn to scale.] Suppose you shine a laser beam straight from A to B, and you sail off from A, if you notice that after travelling a distance x, the boat has “dropped” a distance d (or equivalently, the laser beam appears to have risen by a height of d), then this would be lend evidence to the thesis that the Earth is round. If not, then we have some alternate conclusion. Of course we are assuming that the laser beam is powerful enough and there is no refraction to distort the results.

OK, so what is the formula? And how to we derive it?

Suppose you think this formula is way too complicated, or maybe your calculator’s square root button is kaput. We can derive an approximate formula using Newton’s Binomial Expansion. Remember that Issac Newton did not have any electronic calculators in his day, and he would have worked things out by hand, if he were given this problem to solve.

Example

The radius of the earth is (said to be) 6 371 km. As an example of calculation, for a distance of 5 km, say, we put in x = 5 and R = 6371 to get
d = 5² ÷ (2 × 6371) = 0.001 962 (kilometres).
This means a drop of 1.962 metres.

The drop grows approximately proportional to the square of the distance. So if you double the distance, the drop will be four times. If you triple the distance, the drop will be nine times. Now you are empowered with some knowledge of mathematics. Now you can check the calculations of other people trying to sway you one way or the other. You can call their bluff (if need be). Can you FEEL THE POWER OF MATHEMATICS ??? Can you taste the liberating assurance that you can figure things out yourself?