T-Posts and a Body of Water: Measuring Cross Bars to Show the Shape of the Earth

Since water finds its level, bodies of water such as lakes and canals are a good way to show the spherical nature of our earth as these types of bodies of water do not have uneven terrain at the surface level. This post is about an experiment that you can try yourself with a telescope, wood posts, nails and maybe a boat and about how water, frozen or liquid forms a bulge. This bulge is most observable when it’s directly in front of you, not when it’s at the horizon, meaning side to side. This bulge is also most observable when we view it in terms of miles. I borrowed the experiment from the book

Elements of Physical or Natural Philosophy, 1827, by Dr. Neil Arnott. The chapter I found the experiment in is titled The Earth a Globe or Ball, page 774.

I recommend also reading the Prefatory Notice starting on page iii. It’s a short biography of Dr. Neil Arnott’s life, 15th May 1788 to 2nd March 1874 and it goes over his accomplishments. What I like about him is that he carries his scientific theories into practices that are repeatable.

The Experiment
PART 1: CIRCULAR CURVED SURFACE
Steps:

1) Find a straight canal or frozen lake surface about 4 miles long.

2) Place posts every ½ mile in a straight line with a cross bar at the top in the shape of a T. The T-shaped posts should raise 10 feet above the surface of the water.

3) Stand on a boat or ice in front of the first post such that all the other posts are hidden behind it. In this way you will be assured that all the posts are lined up in a straight line.

4) Using a telescope, sight the differences in height of the cross bars on each post as the posts get farther away from you. Remember, you will not be able to see any vertical post except the first, as the others are hiding behind it, but you will be able to see the cross bars on the posts.

5) What you will find is that on the second post being ½ mile away from the first, the cross bar will have sunk down from the first by 2 inches. On the third post being 1 mile away the cross bar will have sunk down from the first by 8 inches, and the fourth post being 1 ½ miles away will have sunk down from the first by 18 inches, etc.

6) This demonstrates the formula for the curvature of the earth: curvature = (8 inches) x (distance in miles away) squared (c=8d^2 if you will).

Miles away - - -

inches of drop
1/2 - - - - - - - - - 2

1 - - - - - - - - - - 8

1 1/2 - - - - - - - 18

2 - - - - - - - - - - 32

2 1/2 - - - - - - - 50

3 - - - - - - - - - - 72

3 1/2 - - - - - - - 98

4 - - - - - - - - - -128 (If drop is further than 10 feet or 120 inches it will not be demonstrated on post)

The graphic below is an example of cross bar spacing of what one will see with a telescope standing in front of the first post where the posts would be half a mile apart. If you would like to observe more drop you will need a longer post. The table above is in inches and the graphic below is in feet.

One can demonstrate the principle of earth’s curvature on any circular surface. On the following disk of circular foam, 9x1 inches, the edge represents a straight canal or frozen lake surface on earth. The popsicle sticks are 1 inch apart and represent the posts and the tooth picks represent the cross bars. The tooth picks follow the same pattern in drop as the curvature of a sphere falling away from you: 2”, 8”, 18”, 32”, 50”, 72”, 98” and 128”in drop. Although not perfectly calibrated, the popsicle sticks and toothpicks sufficiently demonstrate the principle that distance between cross bars of the T get further and further apart as one moves away on a circular surface due to curvature.

PART 2: FLAT SURFACE

What if we lived on a flat surface? How would the spacing between the toothpicks or T bars be different?

If the earth is a flat surface and the distances of the T posts are still 1/2 mile apart, the drop in the cross bars would appear to be getting closer together. All you would be able to observe is the vanishing point.

One must have the posts at least 1/2 mile or so apart in order to observe the 8 inches X miles^2 drop. If the posts are too close together the drop is not observable.

One can demonstrate what if would look like if the earth’s surface were a flat surface using the flat side of the foam disk. The popsicle sticks are still 1 inch apart at their base. Notice how the cross bars are getting slightly closer together the farther away you observe them.

In addition, on a flat surface one can line up the posts and the T bars to all stand behind each other as in the image below. You can see that there are several toothpicks all lined up as observed by the shadows. This is impossible to do on a curved surface with posts about ½ a mile apart as specified above because the earths curve brings the cross bars out of alignment.

PART 3: COMPARING CIRCULAR TO FLAT SURFACES
How do we know we live on a spherical shaped earth? When one can’t line up the T-Posts so that both the posts and cross bars line up and all hide behind the first T post when placed about ½ mile apart. When measurements are taken in terms of miles, only then do we begin to see the curve of the earth. Ships captains demonstrate this curvature every day as ships sail away from each other when spied through a telescope as ships and their masts sink beyond the bulge or curve of the earth without diminishing much in size at the very predictable rate of 8 inches times distance in miles squared.

*About the quadratic polynomial formula for Curvature of Earth = (8 inches) x (distance in miles away)^2:

This calculation is really only accurate to about 100 miles. The Taylor Expansion series is a better formula to use for overall accuracy past 100 miles.