The RE isn't a horizontal slope which slopes downwards at 8 inches for every mile. That's just the drop in the first mile. The RE is an arc of a circle which continues to

*bend away* from the observer as distance increases.

Suppose that the earth is a sphere with a

radius of 3,963 miles. If you are at a point P on the earth's surface and move tangent to the surface a distance of 1 mile then you can form a right angled triangle as in the diagram.

Looking over a distance of

*1 mile*, we can use the theorem of Pythagoras:

a

^{2} = 3,963

^{2} + 1

^{2} = 15,705,370

and when we square root that figure we get a = 3,963.000126 miles

Thus your position is 3,963.000126 - 3,963 = 0.000126 miles above the surface of the earth.

0.000126 miles = 12 in * 5,280 ft * 0.000126 mi = 7.98 inches

Hence after one mile the earth drops approximately

**8 inches**.

---

Looking over a distance of

*6 miles*, we can use the theorem of Pythagoras:

a

^{2} = 3,963

^{2} + 6

^{2} = 15,705,405

and when we square root that figure we get a = 3,963.004542 miles

Thus your position is 3,963.004542 - 3,963 = 0.004542 miles above the surface of the earth.

0.004542 miles = 5,280 ft * 0.004542 mi = 23.98 feet

Hence after six miles the earth drops approximately

**24 feet**.