Flat-Earth Theory, Part II:

Problems and Paradoxes of Flat-Earth Theory

It should be abundantly clear that Flat-Earth models have severe problems in reproducing the many characteristics of a nearly spherical Earth. As we have mentioned, when Flat-Earthers are confronted with problems they frequently invoke an ad hoc argument to ‘fix’ the issue. In this section, we review some of the many issues faced by Flat-Earth theory, and the ‘explanations’ presented by the Flat-Earth community.

We found the Web site Debunking Flat Earth Misconceptions to be a comprehensive and invaluable site. They present hundreds of cases where Flat-Earth models fail to explain observable data, and they provide the mainstream explanation for those cases. They present their arguments with impressive graphics that contrast the scientific consensus with Flat-Earth claims.

One of the more prominent Flat-Earthers is Eric Dubay. In 2015, Dubay authored a booklet 200 Proofs Earth is Not a Spinning Ball. This is a strange cornucopia of miscellaneous claims. A number of these rely on appeals to statements in the Bible that seem to refer to a flat Earth. Other claims refer to 19th century books that champion the idea of a flat Earth. Several of these refer to  ‘experiments’ carried out by Flat-Earthers. And others support conspiracy theories maintaining that all photographs showing a spherical Earth are fakes, as well as claims that orbiting satellites around Earth and all space programs (manned or unmanned) are faked, part of a vast worldwide web of conspiracies.

The site Debunking Flat Earth Misconceptions includes hundreds of instances showing the errors in Flat Earth theories. We will summarize just a few of these here, and we make use of their fine graphics. We refer to the “Debunking Flat Earth” Web site for myriad other examples.

II.1: The Southern Hemisphere:

The figure below shows the map of the world according to Flat-Earthers. Some Flat-Earthers found this map, and claimed that it was a secret map depicting a flat Earth. Actually, it is an “Air Map of 1943.” Notations on the map state quite clearly that it is a “North Polar Azimuthal Equidistant Projection,” that is, a projection of the spheroidal Earth onto a plane with the North Pole as its center, and with the lines of latitude as concentric circles about the North Pole.

Fig. II.1: A 2-D North Polar Azimuthal Equidistant Projection map of the world, from 1943.


As is well known, it is impossible to project a sphere onto a plane without distortions, or without cutting the sphere in various places. In this case, the Southern Hemisphere is seriously distorted, and the distortions become progressively larger as one approaches the South Pole.

On a spherical globe, the distance between adjacent lines of longitude increases as one travels from the North Pole to the Equator. The distance between lines of longitude then decreases as one moves from the Equator to the South Pole; at the South Pole the lines of longitude again converge at a point. On this North Pole Azimuthal Equidistant map, the distance between lines of longitude continues to increase until one reaches the South Pole. This means that the distance between points in the Southern Hemisphere appears much greater on a Flat Earth map than on a globe.

On a Flat Earth map, countries in the Southern Hemisphere appear much larger than Northern Hemisphere countries. For example, on the map above South Africa appears to be similar in size to the U.S., while in reality South Africa has less than 15% the surface area of the U.S. Australia appears to be 3 or 4 times as large as the United States while it is actually about 80% as large.

Furthermore, paths on the flat Earth (particularly in the Southern Hemisphere) that lie along a “straight line” are nowhere near a straight line on a globe. This is illustrated in Fig. II.2. In a flat-Earth depiction (L side of Fig. II.2), the three cities Sydney (Australia), Los Angeles (California), and Santiago (Chile) lie nearly in a straight line. This suggests that the duration of flights from Sydney to Santiago (or vice versa) should be the same as the sum of the Sydney — LA flight time plus the LA — Santiago time.

Fig. II.2: Flight path from Sydney to Santiago as shown on Flat-Earth map (L) vs on a globe (R).

However, if we compare those flight times on a globe (the right side of Fig. II.2), we see two things. First, the direct Sydney – Santiago flight time of 12 hours 40 minutes is much less than the sum of the Sydney – LA and LA – Santiago times (13 hr 50 minutes and 10 hr 45 minutes, respectively). Furthermore, the Sydney – Santiago flight route goes nowhere near the U.S.

Flat-Earthers claim that flights between cities in the Southern Hemisphere always make intermediate stops in the Northern Hemisphere. This is because distances between Southern Hemisphere cities get extremely large in the Flat-Earth picture, and because on Flat-Earth maps,  many Southern Hemisphere routes go across the Northern Hemisphere, as with the Sydney – Santiago flights shown in the Flat-Earth map on the left side of Fig. II.2.

Contrary to assertions by Flat-Earthers, there are several direct flights between Southern-Hemisphere cities, as shown in Fig. II.3. By the way, you might be shocked that Flat-Earthers would falsely make claims that can be easily fact-checked. However, Flat-Earthers regularly make such statements. They will flatly (LOL 😊) make statements that are demonstrably false. Presumably this makes an impression on their fellow true believers.

Fig. II.3: Many Flat-Earthers claim there are no non-stop flights between Southern Hemisphere cities. Here are some of those routes.

The solid lines in Fig. II.3 all represent existing non-stop flights originating and terminating in Southern Hemisphere cities. On a Flat Earth, these cities would be so far apart that planes could not make the journey without stopping to refuel.

Continuing our discussion of flights, let’s look at airline flights between antipodal cities, or cities that lie on nearly opposite sides of the globe. Two cities that are nearly antipodal pairs are Auckland, New Zealand and Madrid, Spain (another example is Shanghai, China and Buenos Aires, Argentina). On a spherical Earth, if we fly in a “great circle” route between Auckland and Madrid, the travel distance should be about the same for all routes. Thus, the time should be approximately the same for any flights from Auckland to Madrid with a single stop en route. This is shown on Fig. II.4, where the left-hand side shows routes on a spherical Earth between Madrid and Auckland with intermediate stops in Los Angeles, Buenos Aires and Santiago. We expect the travel times to be nearly equal for all three paths.

Fig. II.4: flight routes between near-antipodal cities Auckland and Madrid.

On the other hand, for a flat Earth the distances traveled would be very different (see the right side of Fig. II.4). Thus the Flat-Earth model predicts flight times to be extremely different for different paths. Here are flight times for six different routes:
• Auckland–Santiago–Madrid: 11h 0m + 12h 40m = 23.67 hours
• Auckland–Buenos Aires–Madrid: 11h 45m + 11h 35m = 23.33 hours
• Auckland–Los Angeles–Madrid: 12h 15m + 11h 5m = 23.33 hours
• Auckland–Tokyo–Madrid: 11h 0m + 14h 10m = 25.17 hours
• Auckland–Shenzen–Madrid: 11h 45m + 14h 0m = 25.75 hours
• Auckland–Perth–Doha–Madrid: 7h 25m + 11h 50m + 7h 15m = 26.5 hours

The flight times are nearly the same, in agreement with the spherical Earth prediction, and completely contradicting the Flat-Earth expectation. The same is true for the Shanghai – Buenos Aires approximate antipodal pair.

II.2: Antarctica

In the Flat-Earth theory that we are reviewing, Antarctica does not exist. At the southernmost latitude, Flat-Earthers allege that there is an ice wall that stretches around the entire outer boundary of Earth’s disc. There are many testable ramifications of this Flat-Earth claim.

First, it would be impossible to fly over the South Pole (since there is no such entity). Flat-Earthers will assert that no one has ever flown over the South Pole, or across Antarctica. They also claim that no one has ever circumnavigated the globe, passing over or across both the North Pole and the South Pole in the process. Unfortunately for them, these statements are false. There have been several trips around the earth that passed over both Poles, some by air and some by land. Fig. II.5 shows a few of these.

Fig. II.5: Routes of groups that have circumnavigated the globe passing over or across both Poles.

Here are a few examples. The precise routes that were followed can be found here, with their description.

“Between November 14-17, 1965, Capt. Fred Lester Austin, Jr. and Harrison Finch took off from Honolulu, Hawaii, circumnavigated the Earth through both the poles, and returned to Honolulu. In 1977, PanAm Flight 50 circumnavigated the Earth through the North and South Pole in order to celebrate PanAm’s 50th anniversary. In 1979, Sir Ranulph Fiennes and Charles R. Burton set out from Greenwich, England to the South Pole, and then headed north to the North Pole and back to Greenwich. The Guinness Book of World Records lists this as the first surface polar circumnavigation. In 1988-1989, Dick Smith circumnavigated the globe passing over both poles using a Twin Otter plane. In 1992, Michael Palin made a documentary for the BBC featuring his travel from the Arctic to the South Pole [this was not strictly a circumnavigation, but Palin started from one pole and ended up at the other]. In 2009, the TAG Transpolar08 flight circumnavigated the Earth through the North and South Pole, at the same time breaking the speed record, with an average speed of 822.8 km/h. On July 9-11 2019, commemorating the Apollo Moon landing, the One More Orbit team broke the polar circumnavigation record using the Gulfstream G650ER aircraft.”

Flat Earthers assert that no one has ever crossed Antarctica since there is no such continent. However, several expeditions have accomplished this feat. These are shown in Fig. II.6 below. Furthermore, many scientific experiments and teams are currently installed in Antarctica.

Fig. II.6: Routes taken by groups that have traversed Antarctica.

Of course, Flat Earthers assert that these people were only fooling themselves, and were in fact traveling in circles. We understand there will be a Flat-Earth cruise in 2020 to see “the ice wall.” We don’t know quite what they have in mind. Certainly, at many points Antarctica appears as a large wall of ice, so sailing up near an ice cliff is hardly “proof” of an ice wall surrounding a flat Earth. By the way, the ship the Flat-Earthers will sail on uses GPS for guidance. GPS is governed by a set of satellites distributed around the (spherical) globe; and GPS assumes the Earth is a globe.

Another thing the Flat-Earthers could test on their Antarctic cruise would be the Midnight Sun. Any location north of the Arctic Circle will experience the Midnight Sun at some point during the winter season, and any point south of the Antarctic Circle will also experience the Midnight Sun during the Antarctic winter. A tell-tale sign is that in the Arctic, the Midnight Sun moves from left to right in the sky, while in the Antarctic the Midnight Sun moves from right to left. The Debunking Flat Earth Misconceptions site has a video showing this behavior. Many Flat-Earthers claim there are no videos of the Midnight Sun in the Antarctic – they are wrong.

II.3: The Stars

Flat-Earthers believe that the stars are located in a half-dome that rotates around the Earth’s disc once per day. The belief in a “celestial sphere” containing the stars pre-dated Copernicus. The attachment of Flat-Earthers to this picture of the stars reminds one of the woodcut published in Camille Flammarion’s 1888 book L’atmosphere Meteorologie Populaire. This shows a man in medieval times who manages to find the point where the celestial sphere intersects the Earth. Poking his head through the celestial sphere, he sees many features on the other side of that sphere, including the gears and machinery that make the Heavens move about Earth.  As a side note, it is widely believed that this is a medieval woodcut, although it first appeared in Flammarion’s book in the late 19th century.  Fig. II.7 was colored by Susanna Magruder.

FIg. II.7: Woodcut from 1888 book by Flammarion.

The axis of the celestial sphere supposedly extends upwards from the North Pole of the Earth. Over approximately a 24-hour period, the celestial sphere will make one complete rotation about Earth. Some Flat-Earthers claim that Polaris, the North Star, is located at the axis of rotation. They assert “The North Star, it never moves.”  In his 1885 book One Hundred Proofs That the Earth is not a Globe, William Carpenter stated that “Since we can (in middle north latitudes) see the North Star … all the year round, it is proof enough for any man in his senses that we have made no motion at all and that the Earth is not a globe.” Thus, true believers claim that the fixed positions of the stars, and the fact that the star Polaris never moves, constitute proof of Flat-Earth Theory.

Unfortunately for them, Polaris does move. First, Polaris is not currently located exactly at the North celestial pole. It is 0.65 degrees from the Pole, and one can measure the motion of Polaris with a decent telescope.  One reason that Polaris appears to be stationary is that it is very far from Earth. Astronomers know that Polaris is 433.8 light-years from Earth. In fact, Polaris is part of a three-star multiplet. Polaris Aa, a yellow supergiant star, is in orbit with a smaller companion star Polaris Ab; and that pair is in orbit with a third star Polaris B.

Fig. II.8: Location and movement of Polaris.

We now know that, due to the precession of the equinoxes (a gravity-induced change in the orientation of Earth’s rotational axis), over time Polaris will be observed to move away from the celestial North Pole. Thus, claims by Flat-Earthers that Polaris is absolutely stationary are completely false.

On this same topic, Samuel Rowbotham asserted that “Throughout thousands of years the same constellations have remained fixed in their same patterns without moving out of position whatsoever. If the Earth were a big ball spinning around a bigger Sun spinning around a bigger galaxy, it is impossible that the constellations would remain so fixed.” In fact, the constellations have not remained fixed. We know that stars are moving relative to Earth, and are moving at varying rates since the stars in a given constellation may be billions of miles apart and moving in different directions. If we study star charts from a few hundred years ago, we can easily detect the motion of various stars. Fig. II.9 compares the current shape of Ursa Major (the Big Dipper) with the configuration shown in John Flamsteed’s 1729 book Atlas Coelestis. The motion of Polaris, caused by Earth’s axial precession, is clearly visible. This is consistent with the mainstream view of cosmology but it contradicts Flat-Earth theory.

Fig. II.9: Shift of stars in the Big Dipper constellation over a 225-year period.

If the Flat-Earth model of Fig. I.6 was correct, with the Earth a flat disc and the stars circling about the North celestial Pole once a day, then everyone on Earth should be able to see all of the stars. In particular, everyone on Earth should be able to see Polaris, the North Star. In fact, people who live more than about 1 degree south of the Equator cannot see Polaris. Furthermore, if you live north of about 25 degrees latitude in the Northern Hemisphere, you cannot see the Southern Cross constellation. This is exactly what one would predict with a spherical Earth, but Flat-Earthers must invent far-fetched and arbitrary excuses why this is so.

Also, in the Southern Hemisphere the stars appear to rotate about the South celestial pole. Unlike the Northern Hemisphere, with the bright star Polaris very near the North celestial pole, there is not a bright star located at the South celestial pole. The South celestial pole is close to the faint star Polaris Australis. People who have traveled to the South Pole indeed see the stars apparently rotating about the South celestial pole (note that Flat-Earthers on a 2020 cruise to the Antarctic will see the stars rotating around the South celestial pole — they will also not see Polaris). This motion is impossible in the Flat-Earth model.

II.4: Lunar Eclipses

Fig. II.10 shows the Flat-Earth conception of the Sun and the Moon. These two bodies are supposed to be the same size, each 3,000 miles above the flat Earth, and they rotate over the flat Earth on opposite sides of circles of changing size. Furthermore, the Sun does not radiate light in all directions, but shines like a searchlight (this is so some areas of Earth can experience sunlight while other areas are in darkness). Flat-Earthers have a difficult time explaining why the Sun rises over the horizon in the morning and sinks below the horizon at night (natural to explain for a spheroidal, rotating Earth).

Fig. II.10: A model with the Sun and Moon orbiting above a flat Earth.

Flat-Earth theory has a hard time explaining eclipses in general, but a lunar eclipse is especially difficult for them. A lunar cclipse occurs when the spherical Earth moves directly between the Sun and the Moon. The Earth blocks all or part of the Sun’s rays from reaching the Moon. A total lunar eclipse occurs with the configuration shown in Fig. II.11. There will be a darker shadow called the umbra, and a less dark region called the penumbra. A total lunar eclipse occurs when the configuration is such that the Moon lies entirely within the umbra. When this occurs, the only light that reaches the Moon is sunlight that is refracted by the Earth’s atmosphere. The green through violet part of visible sunlight gets filtered out in this process, and does not appear on the Moon. Red light is refracted least, and so the Moon appears as blood red during a total lunar eclipse.

Fig. II.11: The configuration of Sun, Earth and Moon for a lunar eclipse, showing the umbra and penumbra.

As the Sun, Earth and Moon move, the Earth’s shadow can be seen moving across the Moon. Since the Earth is a globe, its shadow is an arc of a circle. During some lunar eclipses we can see the shadow of the Earth move across the Moon – yet another proof that the Earth is a globe, and that it moves. Fig. II.12 shows images from a total lunar eclipse in 2019.

Fig. II.12: The Earth’s shadow moving across the Moon from a 2019 lunar eclipse, including the ‘blood red’ Moon at total eclipse.

A lunar eclipse presents major difficulties for Flat-Earthers. Since the Earth is supposed to be a flat stationary disc, there is no way that the Earth can be interposed between the Sun and Moon. Hence Flat-Earthers must invent something to explain why one sees a shadow of a spherical object obscuring the Moon.

For many Flat-Earthers, this mysterious body is an “Antimoon.” This is supposedly a body whose orbit is extremely close to the Sun. It cannot be seen because the Sun is so bright that it ‘hides’ the Antimoon from view. Of course, this is all nonsense. The only purpose for the Antimoon is to provide a lame excuse why the Earth’s shadow appears to cross the Moon.

However, the Flat Earth Society acknowledges that the body obscuring the Moon could be some other object, which they call the Shadow Object rather than the Antimoon. “There is also a possibility that the Shadow Object is a known celestial body which orbits the sun, and which projects its shadow upon the moon; but more study would be needed to track the positions of Mercury, Venus and the sun’s asteroid satellites and correlate them with the equations for the lunar eclipse before any conclusion could be drawn.”

Here, we happen to agree with the Flat Earth Society. The “shadow object” is a known celestial body – it’s the Earth, dummies! But this waffling is what one expects from an organization that is determined to get the wrong answer, or to introduce ad hoc pseudo-explanations devoid of other motivation for every observable that causes them trouble.

II.5: The Earth’s Rotation

In Flat-Earth theory, the Earth is stationary, which means that it does not revolve around the Sun. Flat-Earthers also deny that the Earth rotates once per day. They spend much time heaping scorn on anyone who believes that the Earth moves.

In particular, they claim that it is not possible that the Earth, together with its bodies of water and the lower atmosphere, are all rotating at the same speed. They present a “demonstration” that they claim proves the Earth could not possibly be rotating. Take a tennis ball and soak it with water. Then spin the tennis ball. The water flies off, as shown in Fig. II.13. Flat-Earthers say “See? The water flies off the tennis ball when it spins. This is exactly what would happen to water on Earth if it was spinning!”

Fig. II.13: A Flat-Earth ‘demonstration’ that water would fly off Earth if it was rotating.

Let’s see why the “wet tennis ball” is a terrible analogy. The tennis ball produces a gravitational acceleration inward on the water. There is also, in the reference frame of the spinning tennis ball, an outward acceleration, the centrifugal acceleration due to the spinning of the ball. (From the viewpoint of an observer on the ground, this just represents the effect of the water’s inertia in the absence of a force that will cause it to spin with the ball.) We can easily calculate those accelerations; given a reasonable rotational speed for the tennis ball, the gravitational acceleration inward is 3.32 x 10-9 m/s2 (or 0.00000000332 meters per second squared), and the centrifugal acceleration will be roughly 376 m/s2. The net acceleration is the difference between those two quantities, or approximately 376 m/s2 outward. So naturally the water flies off the tennis ball.

Next, repeat the same calculation for a drop of water at the Earth’s surface. The gravitational acceleration is 9.82 m/s2 inward, while the centrifugal acceleration is 0.03 m/s2 outward. Thus, the net acceleration is 9.79 m/s2 inward. The net force on the drop of water is inward, and it does not fly off the Earth, as summarized in Fig. II.14. The “tennis ball” analogy is an extremely poor one, because the gravitational force from the Earth is a billion times larger than that for the tennis ball, while the outward centrifugal force from the tennis ball is 10,000 times larger than that of the Earth. The gravitational force from the Earth is much greater than for the tennis ball because of the huge difference in the masses of Earth and tennis ball. The centrifugal acceleration depends on the square of the angular speed ω times the radius of the spinning object. For a tennis ball ω will correspond to about 1-20 complete revolutions/second, while for Earth ω corresponds to one complete revolution/day, 100, 000 or more times smaller than for the tennis ball . The huge difference in the square of ω more than compensates for the much larger radius of the Earth. The tennis ball analogy is relevant only for people who reject the concept of gravity!

Of course, Flat-Earthers could discover this by talking to any physicist; the calculation is trivial. The fact that they continue to push this “analogy” shows that they have no interest in answering their rhetorical question.

Fig. II.14. Gravitational and centrifugal accelerations of a drop of water for tennis ball and Earth.

Here is a classic demonstration that the Earth is rotating, and that it makes one full (360o) rotation every day. Take a heavy plumb bob (you can purchase one at a hardware store), and attach it to a long wire. Carefully suspend the plumb bob so that it is hanging straight down. Then give it a push so that it starts swinging straight ahead. Watch this “Foucault Pendulum” for a long time. The direction of motion of the pendulum will change. In the Northern Hemisphere, the pendulum will rotate in the clockwise direction. In the Southern Hemisphere, the pendulum will rotate in a counter-clockwise direction. At the Equator the pendulum does not rotate.

Fig. II.15: A Foucault pendulum rotates in different directions in North and South hemisphere, and at different rates depending on the latitude.

After a certain amount of time, the direction of the pendulum will return to its initial orientation (but not at the Equator). A Foucault pendulum set in motion at the Equator will remain in the same orientation. At the North and South Poles, the pendulum will rotate by 360º in one day (but will rotate in opposite directions). At any other latitude, the pendulum will rotate with angular speed ω = 360º sin φ/day, where φ is the magnitude of the latitude. Thus, when φ = 30º (North or South latitude), it will take 2 days for the Foucault pendulum to make one full revolution, while for φ = 45º it will take 1.41 days to rotate.

The rotation of the Foucault pendulum is a direct demonstration of the rotation of the Earth. Flat-Earthers claim that the “rotation” is not reproducible, that the rotation is random, and that the pendulum will eventually come to rest and have to be re-started. The last of these statements is correct; the combined effects of friction and air resistance will cause the pendulum to slow down. The pendulum needs to be “re-started” at regular intervals. It is also true that great care must be taken to insure that the pendulum begins its trajectory with no initial sideways motion. Unless the initial conditions are very carefully set, the circular motion from the Earth’s rotation will mix with another normal mode to produce elliptical motion.

However, the pendulum does not have to be observed for an exceptionally long time to estimate its period. And the fact that the period increases as one goes from the North Pole to the Equator (where it does not rotate), and that the direction of rotation changes from the Northern to the Southern Hemisphere, is sufficient to demonstrate the qualitative effects of the Earth’s rotation.

However, most of these difficulties can be avoided if one substitutes a gyroscope for the pendulum. With a high-quality gyroscope one can minimize several sources of error. The gyroscope will rotate clockwise in the Northern Hemisphere and counter-clockwise in the Southern Hemisphere. One does not need to observe a complete rotation of the gyroscope, one simply needs to observe the amount of rotation and determine that the angular speed ω is as determined by our formula.

Flat Earther Bob Knodel has achieved notoriety for his “Globebusters” YouTube videos. A scientific debunker who runs the Web site “Flat Earth Lunacy” challenged Knodel to purchase a high-quality ring laser gyroscope, to carry out measurements that would show the Earth’s rotation. A paper by Beverini et al. discussed how one could measure the Earth’s rotation with this instrument.

Apparently Knodel purchased a ring laser gyroscope, with the aim of demonstrating that the Earth does not rotate. According to the Debunking Flat Earth Misconceptions Web site,  Knodel’s measurements indeed demonstrated the Earth was rotating at an angular rate of 15o/hour (one full rotation per day). Furthermore, the gyroscope showed rotation in the “correct” direction (a specific direction in either Hemisphere). It is claimed that Knodel refuses to publicly announce the results of his gyroscope test until he can produce the “correct result” (i.e., a stationary Earth). This shows the lengths to which Flat-Earthers are willing to go to maintain their predetermined conclusions. They have no interest in actually obtaining the correct result, or in minimizing errors in their measurements. Any result, no matter how crappy, that is consistent with their Flat-Earth biases is accepted, while they go to incredible lengths to “explain away” anything that demonstrates the motion of the Earth.

II.6: Size and Distance of the Sun

Flat-Earthers claim that the Sun is only about 3,000 miles above Earth, rather than 93 million miles as determined by astrophysics. Furthermore, the Flat-Earth Sun is only 32 miles in diameter, rather than 864,340 miles in conventional astronomy. Let’s discuss some of the evidence that the Sun is extremely large and very far away.

First, we consider the classic experiment carried out by the Greek scholar Eratosthenes in 205 BCE to estimate the circumference of the Earth. Eratosthenes noted that there was a day when the noonday Sun was directly overhead in Syene (modern-day Aswan, Egypt, at 24º N latitude). Since the Sun’s rays are traveling straight down relative to Earth, a pole in Syene will have no shadow at that time. However, in Alexandria (31º North latitude), poles would produce a shadow at the same hour. Eratosthenes assumed that the Earth was a sphere; he further assumed that the Sun was sufficiently far from Earth that one could approximate the Sun’s rays as reaching Earth on parallel lines. From the measured shadow length in Alexandria and some simple geometry to deduce the angular difference of the Sun’s rays between Syene and Alexandria, Eratosthenes estimated the radius of the Earth as roughly 4360 miles, which is about 10% larger than modern measurements of Earth’s radius.

Fig. II.16: Measurements of the Sun’s rays vs. latitude on a flat Earth (L) vs. a spherical Earth (R).

Flat-Earthers point out that Eratosthenes’ experiment does not prove that the Earth is a sphere. This statement is true – Eratosthenes assumed that the Earth was spherical, and from that assumption he calculated the Earth’s radius. Flat-Earthers state that they can repeat the measurement of Eratosthenes, assuming the Earth is flat. The “Flat-Earth” measurement is shown on the left side of Fig. II.16, in contrast to the “spherical Earth” measurement on the right side. In fact, this is how some Flat-Earthers derive the distance to the Sun; they repeat the Eratosthenes measurement in Flat-Earth geometry and obtain a distance of roughly 3,000 miles.

This statement works for measuring shadows at 2 points in Egypt, Syene and Alexandria. However, we can perform the Eratosthenes experiment at many latitudes, one of which has the Sun directly overhead. If we do this, we can choose angles where Flat-Earth theory predicts shadows that increase in linear fashion with each angle. However, with the spherical-Earth measurement, the shadows will not increase linearly. In fact, if you perform the Eratosthenes measurement with two  post, one of which is directly under the Sun, the calculated distance to the Sun in the flat-Earth picture differs for each distance to the second post. This explains why various Flat-Earth groups calculate different distances for the Sun.

Fig. II.17: the Eratosthenes experiment carried out for posts at different latitudes.

Note that if one of the posts is placed near the North Pole, the resulting shadow for a spherical Earth will be extremely long. This is difficult to reproduce in the flat-Earth picture. This is one of the reasons Flat-Earthers are unable to state the exact distance from Earth to Sun; it varies according to the way they calculate this distance. Flat-Earthers “explain” the difference between the shadow measurements for spherical and flat Earth pictures as due to “refraction” of light (Flat-Earthers invoke “refraction” for many phenomena that cause them trouble). But they do not seem to have a consistent understanding of that term, or methods for accurately calculating refractive effects. In this case, Flat-Earthers argue that the Sun’s rays do not pass in a straight line to the Earth, but they travel in a curved path that allows them to obtain the same results as Eratosthenes.

Finally, consider the size of the Sun. Remember that the Flat-Earth Sun has a diameter of only 32 miles (this is obtained by noting that as viewed from Earth, the Sun subtends an angle of 0.5 degrees). If the Sun was only 3,000 miles away, then one can calculate that its diameter would be about 30 miles. We get a good idea of the relative size of the Sun by observing a transit of the planet Mercury. This occurs when the orbit of Mercury takes it directly between the Earth and the Sun. We then see Mercury moving across the Sun. Fig. II.18 shows a photograph of Mercury (the tiny black dot) passing across the Sun’s surface.

Fig. II.18: Transit of the planet Mercury (the tiny black dot) across the face of the Sun.

Reputable astronomers have measured Mercury’s diameter as about 3,000 miles. Note that Mercury appears as an infinitesimal dot in front of the Sun, with the line showing the motion of Mercury. This demonstrates that the Flat-Earth diameter of the Sun as 32 miles is completely wrong. And how does Mercury get between the Sun and Earth, anyway? In the Flat-Earth picture  the Sun is 3,000 miles away from the Earth and Mercury should be on the celestial sphere, a hemispherical dome 4,000 miles above flat Earth.

We have included only a small number of the many serious issues with Flat-Earth Theory. Anyone interested in a comprehensive debunking of Flat-Earth theory should peruse the extensive discussions posted on Debunking Flat Earth Misconceptions. That should convince any objective observer that Flat-Earth Theory is full of holes and not worth serious consideration – it’s “flat wrong” 😊.

Continued in Part III