**Reader Post | By Justin Case**

The place we call earth can be experienced in many very different ways. If you have a mind sensing 2D, maybe a sphere can look flat! And you can come up with incomplete “reasons” why it must be so. When I was a secondary school student, I learned Geometry, Trigonometry, and then Spherical Geometry. I had to expand my awareness beyond understanding mathematical diagrams about 2-dimensional math while looking at a flat piece of paper to comprehending diagrams of 3-dimensional math while still looking at a flat piece of paper.

This route, precisely calculated from Miami to Southampton using spherical geometry, and plotted on a flat map, is a curved line. (courtesy of searoutes.com) It is not because the captain wanted to burn more fuel and spend more time at sea. When a route is calculated for an airline pilot or ocean vessel captain between two points on the earth, it is by default, the shortest path. When it is drawn correctly on an accurate flat map, it is not a straight line, because the path is actually along the surface of a sphere. The speed of this ship is based on nautical miles. One nautical mile is the distance between each minute of latitude change on the spherical surface of the earth while traveling perpendicular to the equator. This will get long and involved, but correcting errors in judgement sometimes requires that.

“**Crow’s Nest**” Hundreds of years ago, the relatively primitive sailing ships had a “Crow’s Nest” on the tallest mast. These sailors realized that the higher you were above the surface of the ocean, the further you could see ahead of you, of course subject to atmospheric conditions. Calculations were possible without too much trouble. On the deck of a ship 4 m above the surface of the ocean, an observer can spot a 20-meter-high sail from at most ±25 km. From a 35-meter-high crow’s nest, an observer will be able to spot the same ship from ±40 km away. (Courtesy of flatearth.ws) The women left at home used the same strategy while waiting for the return of their wandering husband. The area at the highest point of the house was called the Widow’s Walk, where she hoped to be able to see the ship’s return a few minutes earlier. These phenomena occur because the surface of the earth is a curve in all directions from any point.

**“Sea Level”** Does this water in a totally filled test tube look “flat?” It actually is 90 degrees from “level” at the edge of the test tube. Water has many strange characteristics.

**The “Dropping Surface” fallacy.** You can draw a circle on a piece of paper and call its radius 3,959 miles, to portray a section through the earth. Then you can come up with a formula for the distance, to scale, that the curve of the circle deviates, or drops from a horizontal tangent line. (The equivalent of extending a line formed by a carpenter’s level at that point on the earth.) The formula is `d = r * (1 - cos a) where r is the radius of the circle. `

In a length of one mile along that horizontal tangent (level) line the “drop” to the circumference of the circle in the drawing is 8 inches. At 100 miles length along that horizontal tangent line the “drop” to the circumference of the circle in the drawing calculates as 1.26 miles. By the time you have reached a length of 3,429 miles on that horizontal tangent line the “drop” to the circumference of the circle in the drawing is 1,979.5 miles; half the radius. The radius line of the circle at that point is 60 degrees from the starting point radius line as shown below.

That mathematical demonstration has just about nothing to do with any real measurements on the actual surface of the earth. Those dimensions have nothing to do with how far a long river has to “drop” for the water to flow. All false “proofs” like this are a combination of over-simplification and misdirection. If we add a new parameter called “sea level,” the difference between a drawn math demonstration and a real earth is revealed. The Mississippi River travels 2,340 miles from its source to its delta in the Gulf of Mexico. It does not have to “drop” 670 miles from an imaginary tangent line to a circle on a drawing to do that. Lake Itasca, generally accepted as the source of the Mississippi, has an elevation of 1,475 feet above sea level. At the other end, it has dropped to sea level. That is what keeps it flowing “downhill.”

If the elevation of the top carpenter tool above is at sea level, then the one 60 degrees around the surface of the earth, is still at sea level. It has not “dropped” 1,979.5 miles. Its elevation has not changed at all. If your brain is stuck in 2D, you might think that means they are on a flat surface. The value of an elevation is determined using barometric pressure as the variable. That value for any elevation is relative to the initial value at sea level at that time, as it varies with localized pressure variations caused by weather systems. The argument about whether gravity exists or not comes later.

The distance along the circumference of the earth to that point in the drawing is 4,146 miles, or 1/6 of the way around the spherical earth, not 3,429 miles along a flat earth. On the actual planet, the tangent line at r1 is level to the surface of the earth as proved by the carpenter level. And at r2, that tangent line is also level to the surface of the earth as proved by the carpenter level. And those two lines defined by the carpenter levels, differ by 60 degrees in the diagram, and on the earth. The carpenter level is always perpendicular to the radius from that point on the surface of the earth to the center of the earth. It doesn’t matter if you call it the center of gravity, or the source of, or even zero point of radial electric field emanations.

**Correcting the flight of an airplane.** To say that an aircraft should arrive at a destination to the west much faster if the earth is a sphere and spinning at 900mph to the East, is another example of both over-simplification and subsequent misdirection. Motion always has to be described **relative** to an observer. A lot of relatives are going to show up now. This false aircraft claim assumes that somehow the point of destination is moving relative to the aircraft at 900mph at time zero. Is that supposed to occur as the aircraft leaves the ground? If that were true, it could just sit motionless 100 feet above the ground and wait for the destination to arrive! But motionless relative to what point of observation?

The planet rotates at a speed of ~900 mph along the equator if observed from a theoretical “fixed point” in space. Of course, it is only “fixed” relative to the axis of rotation of the earth. That point would actually have a very complex motion relative to the sun. As the point of interest moves north or south from the equator, the speed of the surface (parallel to the equator) is reduced as (cos)L where L is the latitude in degrees. At 60 degrees latitude the speed of a point on the surface would be half of 900, or 450mph.

Observed from and relative to that fixed point in space looking at the line of the equator and perpendicular to the axis of rotation, both the airplane and its destination (if both are on the equator) are moving to the east at 900mph.

The difference between speed and velocity needs to be discussed here to maintain scientific accuracy. Velocity has an additional dimension of direction. 900mph of speed to the east is only observed exactly at the line of sight from the observation point to the center of the earth. Because the objects are rotating on a sphere, as the angular distance (a) of an object from that tangent point on the surface increases from 0 to 90, its observed speed along that tangent line decreases as s=(1-sin(a))

The object’s velocity remains constant as it moves with the surface of the earth on the x axis of the equator, and also the additional dimension of the z axis, which moves it further from the observation point. As it is going out of view on its rotation at 90 degrees, its apparent speed to the east reaches zero because its direction of velocity is all on the z axis. I warned it would get involved. For our purposes here, we can ignore that complication and use the constant speed of ~900mph for objects near the line of sight.

To continue, since the aircraft and the destination are both observed to be moving east at 900mph, they have no relative motion. They stay the same distance apart. It doesn’t matter if the aircraft is on the ground or hovering 100 feet up. Another complexity is the atmosphere. Normally the speed of the wind relative to the ground reaches zero, at the ground surface. From that fixed point in space, those atoms and molecules in the air would also be moving east at 900 mph. But relative to the earth, air on the ground, the aircraft, and the destination all have no relative motion.

As the aircraft is flying west toward its destination at 500mph ground speed, from that fixed point of observation in space, it would now appear to be moving east at 400mph, as the destination continues to move east at 900mph. Their distance from each other would be decreasing at the rate of 500 miles each hour, which is the ground speed of the aircraft. As the aircraft slowed for arrival and then taxied to the gate, its motion observed from the fixed point in space would speed up to reach 900mph to the east again. The aircraft then has no speed or motion relative to the destination terminal building.

The velocity of the atmosphere (the wind) relative to the surface of the earth, becomes chaotic as elevation above the surface increases. It quickly decouples from the surface due to the rotation of the spherical earth and atmospheric pressure deviations. This can have a significant effect on the “ground speed” of an aircraft. If the aircraft has an airspeed of 500mph to the west, and the wind is 100mph to the east, the aircraft’s motion relative to the ground is reduced to 400mph. It would take longer to arrive at a destination to the west because of the effect of the wind, and this is often the case.

**The Horizontal Flight Fallacy.** Another similar incorrect claim is that if an aircraft is flying on a true “level” course its distance from the surface of a spherical earth would increase in the same way discussed in the “Dropping Surface Fallacy” above. Using that error, this fallacy says that in the first mile of flight it is actually 8 inches further from the surface, and after 100 miles of “level” flight it would be 1.26 miles further from the surface using the same formula d = r * (1 – cos a). This error occurs by not understanding how an airplane stays in level flight. As explained earlier, elevation above sea level is based on barometric pressure. Now GPS also allows 3-dimensional position calculation by using signals from multiple GPS satellites. But to follow the pressure instrument idea, level flight is defined as flying to maintain a constant atmospheric pressure. This directly correlates with elevation above sea level.

If you think of the line of flight in terms of the drawing of the circle and tangent line, then you could say that the flight path needs to drop 8 inches from that tangent line in the first mile. From there the discussion becomes more complicated. To continue with the circle and tangent line idea, in the first mile, the aircraft was actually flying an average of ~0.014 degrees “below” that tangent line. It does not need to “drop” according to the formula above in the “second” mile. The instrument which determines level flight is now pointing to the center of the earth along a line rotated 0.014 degrees from the original one. There is now a new tangent line which is perpendicular to a new radius 0.014 degrees from the original, and the aircraft is flying another “first” mile.

In practice, the deviation from manually maintained level flight is continuously plus and minus more than 0.014 degrees. After a mile it averages out to being the same distance above sea level, based on the barometric pressure.You can also break the two-dimensional geometry down to smaller increments. After flying 500 feet, the line to the center of the earth, and its perpendicular tangent line has changed by 0.001 degrees. After 50 feet the change is 0.0001 degrees. As you plot this with increased resolution, the tangent line segments become a smooth curve, not a straight line. In a two-dimensional mind, level flight has to be in a perfectly straight line. In a two-dimensional drawing of the three-dimensional world, level flight is a curved line a fixed distance from, and concentric with the center of the earth.

**Orbiting the Earth.** Obviously, you cannot orbit a flat earth. Claiming that everything NASA has done and televised is a lie or a fake movie, to support the idea of a flat earth, is just ridiculous. Staying in orbit around a spherical planet is a mathematical balancing act between gravity, velocity, and elevation. This explanation can use that same circle and tangent line drawing, and relate to the level flight discussion as well.

An orbiting satellite is always falling toward the surface of the earth directly below, under what we will call the force of gravity. But its forward(orbital) velocity is such that in each increment of time, it has traveled far enough along an imaginary tangent line, that the distance to the surface of the spherical earth, the “dropped” distance, has increased by the same amount that gravity caused it to fall. So, it is now the same elevation directly over a new point on the surface of the earth, on a new tangent line, that differs from the first by a small angular increment of one complete orbit. This same interplay of force and speed continues with each increment of time. The result is a circular orbit at a constant elevation, or more complex elliptical orbits with varying elevation and speed are possible.

The energy needed to accomplish a desired orbit also depends on the total weight being launched. This is a situation where the rotational velocity of the earth can be used to reduce the energy needed to obtain the necessary velocity of the spacecraft to achieve orbit. This is why they usually start out in an easterly direction, to get a little extra “push.” That initial extra hundreds of miles an hour is not relative to any point on the surface as it lifts off, but as it enters orbit, it adds to the achieved velocity.

The lower the elevation, the higher velocity needed to maintain orbit. At about 150 miles above the surface of the earth, a velocity of about 17,000 mph is needed to maintain orbit. The shuttle missions were mostly designed to put a lot of weight in orbit, and had orbit altitudes of about 190 miles to 330 miles.

I have a copy of some video taken by a shuttle astronaut on a mission, using his personal video camera. I got it from a 100% dependable source that worked with astronauts at NASA. It shows the process of decelerating from orbit all the way to landing back on earth. It was a continuous video, with no edits. It was very interesting to hear and see the conversations among the crew during this time. Also shown was the “black-out” time during re-entry as the increasing density of the atmosphere and the initial high velocity of the shuttle temporarily surrounded the shuttle with a high temperature plasma. Nobody could watch this video, and still believe NASA is a total hoax, and that we live on a flat earth!

Finally, here is something found on social media. The identity and employer of this person is easily confirmed from the information he provides.

“For those of you that don’t know. This is what I’ve been working on for the last roughly 10 years. In a little more than 3 weeks we will be launching a human rated vehicle back to the moon. I am the lead mechanical and power officer of this flight. We’re responsible for operating all of the electrical and the mechanical systems on the vehicle. Rick and Judd do a great job of describing what all we do in this video. When they say solar arrays. Batteries. Eclipse. Pyrotechnics, Jettison, Main Parachutes, or fbc jettison. Drogue chutes, Postlanding 2 hrs power down. That’s all my team. I get the privilege of being the ascent/entry MPO and I’m on console for the 2 major burns. I also got to do a media day today with all of the other leads where around 60 media members came by and talked to us while we sat on console and asked us detailed questions. That was pretty cool too. Hope you enjoy the video and watch as we launch (hopefully) at the end of the month.”

https://www.youtube.com/watch?v=K5ovFY175zo

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