The Earth, is in fact, not flat

OK, so how do we know — scientifically — that the Earth is a sphere?

At a very basic level, we can see the Earth’s curvature through satellites that we’ve launched into space. Additionally, through the use of high-powered telescopes, we’ve been able to examine planets both in our solar system and beyond, and all of them are spherical in shape.

There is a very deep, fundamental reason why the Earth is round: the force of gravity depends upon the distance between two interacting objects, and the only three-dimensional object you can make with a single distance is a sphere. We can measure the behavior of gravity in the laboratory with a variety of highly sensitive experiments. Each of these experiments shows that the force of gravity depends only on the mass of the two objects and the distance between them. If, on the other hand, you wanted to form a flattened object using gravity alone, the force of gravity would have to depend upon two, perpendicular distances in two perpendicular directions.

Now, let’s backtrack to the time before satellites and telescopes. Why did people once think that the Earth was flat?

The primary reason that ancient people believed that the Earth was flat was that it looks flat from our vantage point on the ground. Most people throughout history never traveled more than a few miles from their place of birth, so the horizon that they saw was always the same. Moreover, most people were more worried about meeting the necessities of life than they were about the shape of the Earth.

The misconception that the Earth must be flat because it looks flat to us arises simply because the Earth is big. The height of an adult is much less than one millionth of the Earth's radius. In order to see the curvature of the Earth in a single field of view, you would need to be perched above the surface a sizable fraction of that radius, and one millionth wouldn't be considered "sizable.”

What clues changed their thinking?

This state of affairs started to change about 2,500 years ago during the Iron Age, especially with the Greeks. There are two primary reasons that the Greeks knew the Earth was round:

• Lunar eclipses. First, they saw that during a lunar eclipse the shadow of the Earth always had a round profile. This happened regardless of the time of night that the eclipse occurred, the season, or the direction that the shadow crept across the Moon's surface. The only object that casts a circular shadow no matter how you shine a light across it is a sphere. Any other shape would not be able to cast a round shadow under this variety of circumstances.
  
• Star patterns. The second observation is how the pattern of stars changes as you move north and south. If you were to stand at the North Pole, Polaris (the North Star), would be directly overhead. On a flat Earth, Polaris would always be visible — no matter how far away from the North Pole you moved, it would still be above the horizon. However, by the time you reach the equator, Polaris is on the northern horizon, and it disappears entirely once you move into the southern hemisphere. You can't see Polaris from Australia. In fact, the ancient Greeks calculated the circumference of the Earth using this effect and produced an answer that was strikingly close to what we measure today.
  

If you want to prove that the Earth is a sphere, here’s an experiment you can do today:

Quito, Ecuador and Nairobi, Kenya are two cities on the equator. Fly to either and you'll see Polaris on the northern horizon. Cape Town, South Africa and Melbourne, Australia are well south of the equator. Fly to either and you won't see Polaris at all. A few minutes drawing the predictions of the two competing models on a piece of paper is sufficient to exclude one of those models. And, this isn't the only demonstration you can do from the ground.

The rotation question.
People who are receptive to the idea of an unmoving earth often say, "I can't believe I am on a globe spinning through space. Astronomers say I'm going in a dizzying speed of a thousand miles per hour around earth's axis, while it hurtles 70,000 mph in an orbit around the sun and the sun moves around the galaxy at 450,000 mph, and the galaxy is supposedly going somewhere faster than anyone can imagine. I don't feel any such motion, so I can't believe it."

If you were on an large ocean liner or cruise ship sailing at steady speed on a very calm day you could be below decks in your room with the portholes covered and you'd not feel any motion of the ship. No experiment you could do with readily available tools could detect the ship's motion. Maybe you'd feel the vibration of the engines, but you wouldn't be able to tell which direction the ship was moving, or how fast, or if it was moving at all.

But some one may object. Maybe you can't detect straight-line motion, but what about spinning motion? Surely you could tell if you were on a rotating carousel, even if you were blindfolded. Yes, you certainly could, and we can, by similar means, detect the rotation of the earth even if we were in a closed room and could not observe the apparent motion of the stars. If you were on that cruise ship, and it were sailing around a circular path, you could, with suitable instruments, detect that fact even below decks in a windowless room.

The Foucault Pendulum. 

There are many evidences of the earth's rotation. The one most people have heard about is the Foucault Pendulum (Léon Foucault, 1819-1868), a pendulum swinging slowly on a very long wire from a fixed support. As the earth rotates underneath the pendulum, its plane of swing relative to the room it is in slowly precesses at a constant rate, easily observed. Science museums around the world have such pendulums, and some university physics buildings do also. Pendulums at different latitudes have different precession rates, and the data confirms not only the earth's rotation, but the fact that it is round.

Gyroscopes. 

A spinning wheel, mounted so that it can freely turn in any direction with respect to the earth, will maintain spinning about a fixed axis as the earth turns underneath. Its behavior as a function of latitude is clear evidence that the earth is round and that it rotates. It can also tell us the earth's rotation period even if the gyroscope were in a closed room.

Stellar aberration. 

The earth's motion relative to the stars is measurable. James Bradley (1693-1762) demonstrated this, reporting his results in 1728. The earth rotates on its axis and revolves around the sun. Incoming light from stars passing into a telescope has finite speed, so from the time it enters a telescope objective to the time it reaches the eyepiece, the telescope has moved due to the motion of the earth. So the telescope must be aimed slightly forward (in the direction of its motion) to compensate for this. Measurement of this tilt angle confirms what was already known about the motion of the earth. This size of this effect is dependent on latitude, so it also provides evidence that the earth is round.

Some flat-earthers try to explain this as an atmospheric refraction effect. They conveniently overlook the fact that stellar aberration is biased in direction, because of the earth's rotation. Atmospheric refraction would be seen in all directions.

The Sagnac effect. 

The Sagnac interferometer sends two light beams around a closed path in opposite directions. The two beams then pass through a partially silvered mirror and then are recombined, forming an interference pattern. Rotation of the entire device is revealed by the shift of the interference pattern of the two beams. This effect is the basis of laser gyroscopes now often used to replace mechanical gyroscopes. They also reveal the earth's shape and rotation. The Sagnac interferometer and the mechanical gyroscope work on entirely different principles, but they both reveal the rotation of the earth, and give the same result for its rotation speed.

Atmospheric cyclones and anticyclones,

demonstrate earth's rotation and the direction of that rotation. These are examples of Coriolis effects. The fact that they have opposite sense of rotation in the northern and southern hemispheres is evidence of an antisymmetry about the equator. This would not be the case even on a spinning flat disk earth. Ocean currents have the same sort of symmetry, though these depend on several influences, including winds, and water density. These also support the fact of earth's rotation, but the interpretation of the evidence is complex.*

The Compton tube. 

Physicist Arthur Holly Compton (1892-1962) devised a neat way to demonstrate earth's rotation in 1913. It is called the "Compton tube". It is seldom mentioned in textbooks. It uses a large 1 meter diameter glass torus filled with liquid having small particles in suspension. It is aligned in a plane east-west (parallel to the equator) and allowed to stabilze for a day or more, then it is quickly rotated 180 degrees about its diameter. The suspended particles are observed with a microscope and for a few seconds they rotate with respect to the tube, the motion damping out in about 20 seconds. This motion occurs because the liquid and the particles were initially moving with the tube around the earth's axis. After the tube flips, the liquid and the particles, are now moving in the opposite direction inside it. Their speed can be observed with the microscope. This device not only indicates the direction of North, but also shows which direction the earth rotates. I find only one brief mention of this on the Internet. See: Science World, Compton tube. Compton proposed this as an inexpensive and practical method for demonstrating the earth's rotation in a school laboratory.*

Long distance projectiles

 ..and unguided missiles must be launched at a calculated angle and direction if they are to hit their targets. This requires taking the earth's rotation into account in the calculations. They must also use the fact of the earth's surface curvature.*

During a World War I naval engagement near the Falkland Islands British gunners were puzzled why their precisely aimed guns were falling to the left of the German ships. Their guns had been corrected for the earth's rotation, but the correction was for the Northern hemisphere. The Falkland islands happen to be in the Southern hemisphere. So the guns failed to correctly account for the earth's rotation. Who says it doesn't matter whether you accept the rotation of the earth?

Launching earth satellites

 into orbit takes advantage of the earth's rotation. With the exception of satellites intended for polar orbits, they are launched heading eastward. This requires the least fuel. If launched westward they would need to gain as much speed as the earth (in the wrong direction) plus the additional speed to achieve orbital speed. Satellites in low orbit must reach speeds of about 17,500 mph. The launch rocket on the launching pad already has velocity toward the East due to the Earth's rotation. Launches near the equator gain 1,000 mph from Earth's rotation.*

The Doppler effect

 is the shift of frequency of light (or any electromagnetic radiation) due to the relative velocity of source and receiver. If source and receiver are moving toward each other the frequency rises; if they are moving apart the frequency decreases. We now have orbiting earth satellites that can image the earth with Doppler radar, primarily for gathering atmospheric data. This data confirms (a) the round shape of the earth, and (b) its rotation. We even left a cube corner light reflector on the moon that allows us to send a beam of laser light to the moon, which directly reflects back to the earth station that originated it. This can accurately measure the distance to the moon as it changes due to the moon's orbital eccentricity. This can even measure the change in this distance due to the earth's rotation relative to the moon, confirming the round shape of the earth. Similar confirmation of the earth's shape and rotation comes from global positioning systems (GPS).*

Falling bodies. 

A stone dropped down a deep vertical mine shaft is deflected slightly eastward. This is because its velocity at the earth's surface is greater than the velocity of the earth at the bottom of the shaft, which is nearer earth's center. This is true in either hemisphere of earth, but the amount of deflection depends on latitude, being greatest at the equator and reaching zero at the poles. The deflection also happens with a stone dropped from a high tower. This effect is also clear evidence of the sphericity of earth.*

In fact this experiment has a long history. Some early philosophers denied the rotation of the earth. They argued that if it were rotating, a stone dropped from the mast of a ship would not fall to a point on deck directly below, but would fall behind the mast. Experiment proved otherwise, because, as Galileo argued, the stone at the top of the mast was initially moving just as fast as the deck below, and retains that speed as it falls. So he concluded that even on land, if the earth moves, a stone dropped from a high tower falls to a point directly below. So the earth's rotation, he said, doesn't affect motion of objects on earth.

But he was only approximately correct. There's more to this story. Galileo had not considered the fact that on a round rotating earth the top of a high tower actually moves faster than the ground at the base of the tower, due to it being farther from the center of the earth. So the stone at the top is also moving faster than the ground below, and retains this velocity all the way down. So it gets ahead of the ground and falls to the east of the point just below, opposite to the prediction of the flat earthers. The effect is small, because the difference in velocity at top and bottom is so small, but it is measurable with precision instruments. Flat earthers predicted the ball would fall beind the mast if the earth were moving, but it actually falls ahead of the mast. So the flat earther's experiment actually is another evidence for a round, rotating earth.*

Atomic Clocks.

Standards of length and time are now based on the microwave radiation from atomic energy level transitions. Supercooled cesium-133 has been used for this for several decades. Now atoms of strontium-87 and ytterbium-171 offer a threefold increase in the precision in such measurements. Einstein's special relativity theory predicts that clocks suffer time dilation when they are moving, and this has been experimentally conclusively confirmed. Now, with strontium-87 clocks we can even compare the difference in speed between networked clocks located at different places on earth. In 2017 Pacóme Deva of the Paris Observatory and his colleagues did this, using optical fiber links between London and Paris, and one between Paris and Braunschweig, Germany. While this was reported in the press as a test of special relativity, it also serves as a measure of earth's axial rotation and its sphericity, since the cities being compared are at different latitudes, and therefore different distances, from the earth's axis, and have a different velocity due to that fact. [New Scientist, 25 March 2017, p. 11,] *

The round earth.
The ancient Greek philosophers knew the earth was round, and they based that conclusion on hard evidence, observable by anyone, even without telescopes. The shadow of the earth on the moon during a lunar eclipse shows by the progression of changing shape of the shadow's edges that both the earth and the moon are round. This shape progression is the same whatever time of the day or day of the year that the eclipse occurs. That would not be the case for the shadow of a flat disk.

The night sky. 

Early civilizations were quite aware of the starry night sky, with its apparently fixed patterns of stars, leading to those patterns (constellations) being given names. These patterns were reliably constant and unchanging in shape. Ocean navigators knew them well, and used them for determining latitude at sea. As they traveled south, the north star's elevation decreased, and the southern sky constellations, previously not visible, were now above the horizon. Either the entire sky dome had obligingly shifted northward as they sailed south, or they were sailing on a round earth.

The ancient Greeks 

also knew the size of the round earth. Eratosthenes of Syene (276-194 BCE) calculated that by geometric triangulation, using a measured baseline from Alexandria to Syene (now Aswan). His baseline data wasn't very accurate, but his method was sound and his result was within 10% of the modern value. High school physics students sometimes repeat this experiment, using baselines spanning several U.S. states.*

The Internet makes it possible for near-instant communication anywhere on earth. A group of people in various countries could do this experiment. They would observe, at a pre-agreed time, the length of the shadow of a perfectly vertical pole. The data, combined with their geophysical coordinates, would allow a repeat of Eratosthenes measurement of the size of the round earth.*

The equatorial bulge

 is a result of the earth's rotation, which distorts the earth into a slightly oblate ellipsoid of revolution. Isaac Newton (1642-1726/27) gave a mathematical proof of this rotation effect in his Principia (1687). He showed that a rotating self-gravitating near-spherical elastic body would increase its radius at the equator and decrease its radius at the poles. This is a very small effect, but is measurable by precise surveying techniques along with measurements of stellar positions at various latitudes. These measurements, along with Euclidean geometry, determine the variation in the length of one degree of latitude as a function of latitude. During the 17th and 18th centuries scientific expeditions were sent to Northern countries to measure the "polar flattening", and the effect was confirmed. Today GPS systems further confirm it in both hemispheres, and the fact that the effect is nearly symmetric about the equator not only confirms the fact that the earth is round and that it is rotating about its polar axis, but also confirms the correctness of Newton's gravitational law.*

Seismology 

gives further evidence. Earthquakes propagate waves through the body of the earth. The direction and arrival times of these disturbances at surface monitoring stations allow us not only to pinpoint the location of the earthquake epicenter, but also to learn the composition of the inner earth itself, and how the inner shells refract seismic waves. All of these consistently support the spherical shape of the earth and its inner shells. No flat earther has ever even tried to account for this massive body of data by using a flat earth model.*

Radio transmission.

Radio waves refract and reflect from ionized layers of earth's atmosphere. This has been observed and utilized in radio communications, and is well known to ham radio enthusiasts. This refection of radio waves allows radio signals to go far around the curvature of the earth.

Very high frequency radio waves have the smallest reflection. Lower frequency waves reflect strongly, at heights that depend on the frequency. Measusrement of the intensity and direction of waves at the receiving antenna confirms the curvature of earth and atmosphere, as well as the height of the reflecting layers. The interpretation is complex, just as for the seismic waves of the previous section.

Gravity.

Another argument heard from flat earthers is this: "If a round earth were really spinning at 1000 mph surely everything on it would fly off into space like mud from a spinning wagon wheel."That invalid conclusion ignores the force due to the earth's gravity, which not only keeps us on the surface of the earth, but also holds the earth together. Without gravity the earth would never have formed in the first place.

Even in Newton's time critics dismissed gravity as an "occult force", for they couldn't get their minds around the notion that bodies could exert forces on other bodies without touching them, and without anything material between them.

Flat earthers usually dismiss and deny the existence of gravitational force. They say, "Things fall because they are heavy. What else do we need to know?" Scientists want to know more.

• Isaac Newton (1642–1726/7) showed that his inverse square law of gravitational force demonstrated what kept the moon in its orbit and how the planets orbit the sun.
  
• Observations of planetary motions are fully consistent to very high precision with their mutual gravitational forces (Newton's formula), with the constancy of the vacuum speed of light, and with Euclidean geometry.
  
• Henry Cavendish (1731-1810) demonstrated that a gravitational attraction existed even between objects in the laboratory. He measured it using two lead balls suspended by delicate torsion pendulums, confirming the inverse square law of gravity within the confines of a laboratory room. Even today high school students repeat this experiment, measuring the force between two pails of sand, or two concrete blocks, suspended by strong Mylar tape. Commercial versions of Cavendish's apparatus, small enough for a table top are commercially available.*
  
• Earth satellites orbiting the earth demonstrate the correctness of Newton's gravitational law. At a 200 mile altitude the gravitational force is about 10% less than at the earth's surface. The periods of these orbits agree with calculations from Newton's gravitational law.*
  

Another argument heard from flat earthers is this: "If a round earth were really spinning at 1000 mph surely everything on it would fly off into space like mud from a spinning wagon wheel."That invalid conclusion ignores the force due to the earth's gravity, which not only keeps us on the surface of the earth, but also holds the earth together. Without gravity the earth would never have formed in the first place.

Even in Newton's time critics dismissed gravity as an "occult force", for they couldn't get their minds around the notion that bodies could exert forces on other bodies without touching them, and without anything material between them.

Flat earthers usually dismiss and deny the existence of gravitational force. They say, "Things fall because they are heavy. What else do we need to know?" Scientists want to know more.

• Isaac Newton (1642–1726/7) showed that his inverse square law of gravitational force demonstrated what kept the moon in its orbit and how the planets orbit the sun.
  
• Observations of planetary motions are fully consistent to very high precision with their mutual gravitational forces (Newton's formula), with the constancy of the vacuum speed of light, and with Euclidean geometry.
  
• Henry Cavendish (1731-1810) demonstrated that a gravitational attraction existed even between objects in the laboratory. He measured it using two lead balls suspended by delicate torsion pendulums, confirming the inverse square law of gravity within the confines of a laboratory room. Even today high school students repeat this experiment, measuring the force between two pails of sand, or two concrete blocks, suspended by strong Mylar tape. Commercial versions of Cavendish's apparatus, small enough for a table top are commercially available.*
  
• Earth satellites orbiting the earth demonstrate the correctness of Newton's gravitational law. At a 200 mile altitude the gravitational force is about 10% less than at the earth's surface. The periods of these orbits agree with calculations from Newton's gravitational law.*
  

Light Refractions

Flat earth believers dismiss gravity. But in order to account for what they consider the "apparent" motion of sun, moon and stars they must invoke a contrived hypothesis that light paths bend and light changes its speed in a very complicated manner that even they have not expressed in mathematical and physical detail. Flat earthers have no explanation for the cause of this mysterious effect.They must assume some equally mysterious cause for their assumed severe distortion of distances on earth's surface, varying from N to S pole. Their justification for these is, "That's just is the way it is." Gravity deniers called Newton's gravity an "occult" force. This is clearly a double standard, for whatever might cause their light path bending is also an "occult" force.

Bending of the path of light is called refraction. It occurs when the light path traverses materials of different optical density. There is one refraction effect that argues for the round earth and against the flat earth. The atmospheric blanket of the round earth is curved, and therefore refracts light in a way well understood from laboratory experiments. The atmosphere acts as a huge lens wrapped around the earth. This allows us to see a bit "around" the curve of the earth. The cause of this refraction is the decrease of atmospheric density with height. It also distorts the shape of the rising and setting sun and moon (flattening them), an effect easily observable by anyone. This causes the sun to rise earlier and set later than would be expected if there were no atmosphere. Stars and other celestial objects that would otherwise be "over the horizon" can be seen. This is why, above the Arctic Circle one can see the sun move all the way around the sky near the summer solstice, remaining above the horizon all the way around the sky. It is also the reason that in some cases a total lunar eclipse at sunset or sunrise can be seen when both the sun and moon are above the horizon. Astronomers, even amateur ones, must be aware of, and compensate for, this effect when observing stars near the horizon. This is not a mysterious effect, but one due to atmospheric refraction, and the physics of it is well understood. It is quite different from the sort of severe light bending that flat earthers must assume to support their earth model.

This atmospheric refraction is constant and ever-present. It is not to be confused with the localized and temporary optical phenomena due to temperature inversions near the ground. See the discussions of atmospheric refraction and looming, sinking towering and similar refraction phenomena in the Wikipedia. These are confined to situations where the light path is very close to the earth's surface, usually, but not always, over water.

The horizon question.

The Horizon Question
Such temperature effects on air near the ground are important because flat-earth believers often challenge conventional geodessy by citing the fact that sometimes, under certain conditions, one can see over an ocean surface farther than Euclidean geometry would predict. They neglect to mention the situation where sometimes one cannot, even on a clear day, see as far as simple geometry would predict. This is called "cherry-picking" data—accepting data that fits your hypothesis and ignoring any that doesn't. The reason for both observations is the temperature gradient of air near the ocean surface—whether the temperature rises with height, of decreases with height. This causes refraction of light, deviating it from a straight line path. Interpretation is difficult, for to predict what you will see, you would need to know the temperature variations all along the line of sight. To my knowledge, this has never been done in that much detail. But the results of this are measurable and undeniable.

Historically, such observations have been used by all sides of the earth shape question. Those who accepted the round earth cited the fact that ships sailing toward the horizon are seen to disappear gradually, from the hull then to the top of their masts. That's the usual case, when there are no temperature gradients along the line of sight. When conditions happen to be just right, a ship that ought to have already disappeared "over the horizon" actually seems to rise above the water (looming), or even rise up above the apparent horizon. Sometimes it will appear upside down, and this is called an illusion, though it is also a refraction phenomena. All these effects can be seen over land as well. Flat earthers ignore those cases. They look only at those cases where local conditions give the result they want.

Then there's an interesting phenomenon not generally well known to most people. The surface of the ocean is of variable curvature, and sometimes over considerable distances, is nearly flat! This is because the surface level is affected by the profile of the ocean floor beneath it—a gravitational effect. Over an undersea mountain, the water level actually rises. Over an undersea hollow or trench the level is depressed into a slight valley. This is confirmed by satellite radar observations. So the ship you are looking at with binoculars may actually be higher than you assumed, sailing up the slope of one of those water surface "humps". Your line of sight may be over a relatively flat ocean surface, or even across a surface "valley". Your magnified view of the ship may, on a calm day, be stable, without the shimmering or distortion that you would expect of a mirage.

For evidence of earth's shape and spin, we look to "cleaner" experiments (described above) that aren't confused by variables difficult to control and measure. We also do many independent and fundamentally different sorts of measurements to see whether they agree with each other. Those experiments have been done.

The validity of the round earth model does not depend on messy, complicated and selective observations such as the horizon observations. We need to look at the abundance of diverse evidences (cited above), and the fact that they all lead to the same conclusion about the earth's round shape, and all agree on its spin (and its direction of spin). These agree on size and spin rate to great precision.

How round is the earth?
Flat earthers today sometimes argue "Scientists say that earth's rotation causes an equatorial bulge, and some even say there are also other bulges, one in the Southern hemisphere, so the earth is really "pear shaped". Yet their pictures of earth from space show a perfectly round earth image. These are contradictory, so they must be faking the evidence carelessly."

The departures from sphericity of the earth are relatively small. For example, the equatorial bulge raises the equator about 22 km, which is only 0.4% of the earth's radius. All other bulges and surface features are much smaller. This is an example of how sensitive our measuring instruments have become—that we can even measure this. Suppose we made a scale model of the earth the size of a bowling ball, with the highest mountains and deepest ocean trenches to scale. This model would be smoother than a regulation bowling ball. And it would appear, to the unaided eye (or to a camera), perfectly round. The atmosphere, on this scale, would be a coat of clear varnish. If you doubt this, do the calculations yourself. Flat earthers hardly ever show any mathematics to support their claims. One wonders why.

Symmetry.
A common feature of many of the above arguments arises from symmetry of physical phenomena on the earth. Most of them provide hard experimental evidence for symmetry around the N-S axis of the earth, and mirror symmetries about the equator. This is what you'd expect with a round, spinning earth, and for phenomena that are gravity dependent. But they make no sense with a flat earth of any kind, and flat earth advocates have no good explanation for that.

The size of the earth
Eratosthenes of Cyrene (276-194 BCE) calculated an estimate of the size of the earth by triangulating the sun with a baseline between Alexandria and Syene. His estimate of the distance between these cities was crude, so his result differed from the modern value by around 10%.

In the 18th century accurate maps were important for commerce, so land surveys used triangulation and surveying instruments to make them. These depended on accurate determination of distances to use as baselines of a triangle, but instruments and methods were better than Eratosthenes had. Surveyor's transits measured angles with great precision. Of course, they assumed that light paths are straight, but as these surveys were over land this was not usually an issue and could be avoided or corrected for. These results clearly indicated that the triangles could not be lying on a flat surface. Over large areas, the curvature of earth's surface could be calculated. This was not surprising, for we already had abundant evidence that the earth was round. Such surveys were even precise enough to show that the earth was not exactly a sphere, but was slightly "flattened" at the poles, bulging at the equator. It is a very small effect, but was confirmed by many such surveys, especially those at higher latitudes.

Does this alone "prove" that the earth is a round ball? No, for if the earth were a hollow shell of the same size with us waling on the inside, the triangulation results would be the same! This is a result of Euclidean geometry. This fact might be welcomed by proponents of the notion that the earth is really a hollow sphere with us on the inside, but they are not well enough versed in mathematics to even understand it. [See the link to "The Hollow Earth" at the end of this document.]

Conclusion.

The evidence for a round, rotating earth is abundant, consistent and undeniable, and much of it can be observed and confirmed by anyone willing to look for it. You don't need a research grant, and needn't be a member of the scientific establishment. The evidence comes from many independent natural phenomena, and the interpretation of that evidence is straightforward enough to be understood even without a degree in physics or astronomy.

Those who deny the round, spinning earth and deny gravity do not know or appreciate the abundance of confirmatory evidence from many independent sources. Though the evidence arises from diverse experiments, it all fits (is consistent with) the laws of physics and our understanding of the geometry of the universe to a precision equal to that of our most sensitive instruments. The deniers generally treat all physics laws as separate and independent. They do not grasp the fact that physics laws are all part of a rigorous, integrated and unified logical/mathematical network. This is the beauty and strength of physics. Deniers have not constructed any system based on the flat earth model that can equal conventional physics in comprehensiveness and predictive power, such that it accounts for all of the evidence. They complain that conventional physics is "too complicated" for their comprehension. But where is their simpler model that can equal conventional physics in its precise and experimetally confirmable results?

Flat earth believers reject all evidences for a round, rotating earth with contrived and often absurd and inconsistent arguments, or dismiss some of them as hoaxes. Their arguments are hand-waving, rationalizations, empty, unsupported assertions and contrived rationalizations. It is usually not worth anyone's time to refute their incredibly lame assertions and arguments by using mathematics and physics because those who accept those arguments do not have sufficient understanding of mathematics and physics. Many even deny the laws of geometry and physics entirely. However, even sophomore physics majors know enough classical physics to find the flat earth arguments laughable, and cannot imagine how anyone could believe them.

It is difficult to believe that flat earthers are serious. One suspects they are just having fun challenging accepted science. It is easy for them, of course, for they assume no obligation to present testable hypotheses, gather precise data, or do the necessary mathematics to make correct interpretations and inferences from that data.

Another possible explanation is that some people really don't care about scientific evidence and solid arguments. They prefer to live in a fantasy world of their own devising, untroubled by any necessity to justify it by scientific inquiry. They love to repeat unverified or invented "facts" and "magical" concepts. To refute every one of their bogus "evidences" would occupy a diligent researcher a lifetime. It would be futile anyway, for the believers would just invent some other fantastic arguments.

None of them ever feel any obligation to square their fantasies with physics and mathematics. None have ever tried to develop a comprehensive alternative physics that would support their claims and account for all experimental evidence from all fields of physical science. They should be obligated to do this, for their claims certainly are not supported by conventional physics. Their pretensions are not science, but science fiction and fantasy.

I must admit that there are also many people who accept the round, rotating earth, yet cannot defend their position, don't know the evidence, and haven't sufficient understanding of physics and mathematics to argue for their position. This document is a small effort to remedy that. Before I wrote this docuement I used to test college students by asking them to write a short essay defending their notion of the shape of the earth, supporting their position with hard evidence and sound argument. Most failed miserably on the first attempt. (These were not science majors.) I encouraged them to consult reliable library references. [I had to give some of them directions how to find the library.] After several attempts one student complained, "I'm frustrated. I don't know why the earth is round. All my teachers said it was, and I believed them. Isn't that enough?" I responded, "No, that is not enough." Education is a failure for many, because they "learn" only by memorizing empty assertions and slogans without understanding the underlying evidence and arguments.

Functionally, the earth is (for most people) indistinguishable from flat.

That's why maps (of the earth) are generally drawn on flat pieces of paper.

The fact that there is the mariana trench and mount everest would mean the world's surface is uneven, meaning it is not smooth, meaning it is not flat.

BOOM mic drop

That's like saying: well distinctionally, the sun rotates around the earth,

That's why maps (of the earth) are generally drawn on flat pieces of paper.

Or maybe it’s just easier to carry around. Most people have believed in a spherical Earth for a millennium. 

Though as a statement of fact, it would be false to say that the Sun actually revolves around the Earth.

I don't think that axioms are meant to be as whimsical as you are  suggesting.

You really only NEED to account for the curvature of the earth (IFF) you're a ballistics technician or a satellite engineer.

MOST people never need to account for this incidental curvature or the earth. 

For example, when drafting blueprints or city planning, you're dealing with what is effectively and practically a flat surface

Most people weren’t city planners. I’ll go as far as to say most people didn’t use maps.

Anyway, many ancient civilisation took into account the rotation of the Earth when planning their monuments. 

Incidental curvature not so much.

Incorrect. It is false to say that the sun revolves around the earth. You could claim, "It is perceived as if the sun revolves around the earth" but to claim that it does is factually incorrect. 

Yeah, so we agree?

Honestly I don't understand the popular outrage against "flat-earthers".

They're much less dangerous than most religious people.

The opinion "the earth is flat" is generally harmless.

I’m allergic to ignorance regardless of ideology. 

My adult sister not believing in dinosaurs comes to mind.

And I know for sure she lacks critical thinking skills. 

Only one axiom is needed, and in order to claim anything else regarding the universe you would have to accept it - which is - the universe is as we observe it. 

That's it.