In my 2019 book, Falling Flat: A Refutation of Flat Earth Claims, I evaluated many of the claims that flat-earthers make. With so many flat-earth claims and with more of them arising since publication of my book, I didn’t cover all the flat-earth claims out there. That is why I continue to update my study of the flat-earth movement with new web articles and blog posts. Some people may think it is a waste of time to discuss the flat-earth movement. However, the flat-earth movement presents a danger to the church. Self-professed biblical flat-earthers apply a flawed hermeneutic to Scripture. Their hyper literal approach to poetic and prophetic books wrings unfounded meanings from those passages. Flat-earthers must create in their minds a vast satanic conspiracy about the earth’s shape to support belief that the earth is flat. This often results in a caustic attitude that damages couples, families, and churches. Many flat-earthers end up leaving their churches and not assembling with fellow believers as we are commanded to do. Therefore, it is important that I continue researching the flat-earth movement and report on new developments in that movement.
Flat-earthers must create in their minds a vast satanic conspiracy about the earth’s shape to support belief that the earth is flat.
One of the flat-earther arguments that I have not examined before is claims that there are many government documents admitting the earth is flat. I thank Michael Toon for compiling a list of 44 government documents that some flat-earthers claim admit that the earth is flat (hereafter, I will simply refer to this as “the list”). Here I will examine most of the documents on that list. If these documents truly admit that the earth is flat (and that is a big if), then those admissions could be evidence that the earth indeed is flat and not a globe. After all, wouldn’t government agencies, particularly scientific agencies, be in possession of knowledge of the earth’s true shape? On the other hand, if those government agencies were part of the cabal that promulgates the lie that the earth is a globe, why would those same government agencies admit that the earth is flat? Flat-earthers incessantly insist that governments lie, so why do flat-earthers change course and accept the word of a government agency that consistently lies? This is particularly true of NASA, the source of the first government document that I will discuss.
Probably the most common government document that flat-earthers cite as admitting the earth is flat is a report titled “Derivation and Definition of a Linear Aircraft Model” produced by three researchers at NASA’s Ames Research Center in California. Sure enough, one does not have to read very far (the first sentence of the text) in this report to see that this paper derived a linear model for an aircraft “flying over a flat, nonrotating earth.” For most flat-earthers, that is all they need to see. They reason, “Why would NASA go to the trouble to derive a model of aircraft flying over a flat, nonrotating earth, if the earth is spherical and rotating?” This is typical of the way flat-earthers make their arguments—they focus on one or two words and ignore the context and the other words around the few words they focus on. Here is the first sentence in this report:
This report documents the derivation and definition of a linear aircraft model for a rigid aircraft of constant mass flying over a flat, nonrotating earth.
Flat-earthers focus on the description of the earth as being flat and nonrotating and ignore the description of the aircraft as being rigid and having constant mass. Aircraft are not rigid. When seated over the wing of an airliner, I often watch the wingtip rise a few feet as we take off. Materials flex when forces are applied to them. In the case of the wings of aircraft, the lift generated is applied to the wings, and the wings flex upward in response to lift. The control surfaces of aircraft also move, departing from the assumption that aircraft are rigid. Hence, aircraft are not rigid. Nor do aircraft have constant mass. As the flight proceeds, stored fuel is consumed, reducing the mass of the aircraft considerably.
So, why would three researchers derive the equations that govern the flight of aircraft when all four of their starting assumptions are false? When solving problems in physics and engineering, it is important to identify and state all the assumptions one must make in solving the problems. Many of these assumptions are simplifying assumptions. That is, they are assumptions that are not necessarily true but are of little consequence. For instance, in my teaching of physics, I showed students how to solve mechanics problems. We usually ignored the effects of friction and wind resistance. If the speeds of objects under consideration were low and the surfaces were relatively smooth, then wind resistance and friction did not greatly affect the outcome. Inclusion of the more realistic situation, such as friction and wind resistance, would have made the problems far more difficult to solve. In many cases, the effects of other factors, such as wind resistance and friction, could be added as corrective factors to the solution. We also often assumed that objects we studied in mechanics were point objects, even though they weren’t. We needed to consider the physical extent of objects only when considering rotational aspects of their motion. Even then, we could add rotational motion as a correction to the initial solution treating objects as point masses.
Knowing that simplifying assumptions are often made in problem solving in physics and engineering, let us examine the four assumptions in the NASA report. While aircraft flex some while in flight, the amount of flex is relatively small, and the flexure does not seriously affect the flight of the plane. Hence, flexure can be ignored. A change in the mass of an aircraft causes a change in weight and hence the amount of lift required. While fuel consumption is significant over a long period of time, moment by moment, the amount of mass lost due to fuel consumption is so small that it does not affect flight. If one wished to account for mass loss over a long flight, one need only solve the equations of flight over sections of the flight with a slightly modified mass in each section.
Flat-earthers misunderstand what is meant by “flat earth” in this document. It is not a denial that the earth is spherical. In these sorts of studies, the simplifying assumption of a flat earth can have two possible meanings. One meaning of the flat-earth assumption is to ignore local topography, overlooking hills and valleys, as if all positions on the ground have the same elevation. In some of the documents I examine here, that is the intended meaning. However, in this paper, the flat-earth approximation probably refers to the second meaning. The second meaning of the flat-earth approximation is to consider limited range in horizontal distance. In geometry, we say that locally a sphere appears flat. That is, if the horizontal distance is considerably less than the curvature of the sphere, then the surface of a sphere can be modeled in the plane tangent to the sphere at the location under consideration with little loss of accuracy. In this paper, aircraft are modeled in flight moment by moment. Hence there is no need to consider the earth to be spherical. Just as one could model the loss of mass by fuel consumption as a series of small increments, the spherical earth can be modeled by considering a series of small increments of flight.
As for a nonrotating earth, this means that the Coriolis effect has little impact on flight. Indeed, this is true. The Coriolis effect is very feeble—the Coriolis effect becomes significant only over great distances on the earth, and then only if there is no controlled flight. Pilots or autopilots continually tweak the controls of aircraft, swamping the tiny Coriolis effect. Again, the model of flight in this paper is for moment by moment, not long flights in their entirety. Consequently, this NASA document does not contain an admission that the earth is flat and nonrotating.
The paper that preceded this one describes the FORTRAN program that they used. This paper is titled “User’s Manual for LINEAR, a FORTRAN, Program to Derive Linear Aircraft Models.” Page 15 lists the simplifying assumptions, including,
A rigid aircraft flying in a stationary atmosphere over a flat nonrotating earth.
This adds the simplifying assumption of no wind, something that obviously is not true. Just as topography and a rotating earth can be added to the more general solution, so can wind. An updated version of this program with similar language was published in 1988.
There are many more papers about flight from NASA facilities that use virtually the same language as the above paper when elucidating their simplifying assumptions, including a flat, nonrotating earth. The meaning in each case is the same. They do not constitute admissions that the earth is flat and nonrotating. One example is the 1991 report “An Aircraft Model for the AIAA Controls Design Challenge.”
Page 11 lists the simplifying assumptions:
The nonlinear equations of motion used in this model are general six-degree-of-freedom equations representing the flight dynamics of a rigid aircraft flying in a stationary atmosphere over a flat, nonrotating Earth.
The control surfaces of aircraft must move, and there is flex in the wings, so aircraft are not strictly rigid. A stationary atmosphere means that there is no wind. The earth being flat refers to the earth having not topography (hills and valleys). The nonrotating earth means that the Coriolis effect was ignored. None of these assumptions are strictly true, but considering them would not have significantly changed the results.
An even more detailed list of simplifying assumptions is found on page 6 of a 1978 NASA report with the title “Investigation of Aircraft Landing in Variable Wind Fields.” They are:
As before, none of these simplifying assumptions are strictly true, but their effects of these factors are minimal and, hence, can be ignored for simplicity.
Here is another NASA paper on the flight of aircraft that mentions “a flat nonrotating earth.” This one is titled “General Equations of Motion for a Damaged Asymmetric Aircraft” and was presented at a 2007 conference by two researchers from NASA’s Langley Research Center in Virginia. The purpose of this paper was to develop a model of flight of an aircraft that is damaged and loses mass so that its center of mass is shifted. Obviously, since the aircraft loses mass, the assumption of constant mass as in the previous paper is not made. However, the other assumptions, rigid aircraft and a flat nonrotating earth were made. These assumptions have the same meanings as in the earlier paper, so this document hardly is an unguarded admission that the earth is flat and does not rotate.
Another NASA paper that flat-earthers sometimes mention is “Predicted Performance of a Thrust-Enhanced SR-71 Aircraft with an External Payload,” authored by Timothy R. Connors of NASA’s Dryden Flight Research Center in California in 1997. The purpose of this paper was to evaluate the feasibility of modifying the engines of the SR-71 for use in carrying scientific payloads aboard. Page 8 of this report listed four assumptions made to simplify the modeling of this project. While the earth being flat is not mentioned (at the altitude the SR-71 flew, the earth’s topography did not matter), a nonrotating earth was mentioned. However, the other three assumptions were that the SR-71 was a point mass, no turbulence, and no side forces. With a length of more than 100 feet and a wingspan of more than 50 feet, the SR-71 hardly was a point mass. Furthermore, there is no place in the atmosphere with no turbulence. The assumption of zero side forces probably refers to crosswinds, which isn’t true either. The assumption of a nonrotating earth probably has the same reasoning as in the previous studies—the Coriolis effect is so small as to be ignored.
A 2002 NASA report considered the “Stability and Control Estimation Flight Test Results for the SR-71 Aircraft with Externally Mounted Experiments.” Sure enough, on page 11 the phrase “flat, nonrotating earth” appears. Here is that phrase in context:
These equations assume a rigid vehicle and a flat, nonrotating Earth. The time rate of change of mass and inertia is assumed negligible. The SR-71 configurations studied herein, like most aircraft, are basically symmetric about the vertical-centerline plane. This symmetry is used, along with small angle approximations, to separate the equations of motion into two largely independent sets describing the longitudinal and lateral-directional motions of the aircraft.
Once again, no aircraft, including the SR-71, is rigid. The statement, “The time rate of change of mass and inertia is assumed negligible,” is phrased in such a way as to indicate that the authors know constant mass and inertia are not strictly true, but that loss of mass and inertia are not significant enough to include. The terminology that the aircraft is “basically symmetric about the vertical-centerline plane” is a tacit admission that the aircraft is not strictly symmetrical but is symmetrical enough to justify the assumption of symmetry. The report admits that the authors used “small angle approximations” [emphasis added]. Anyone who has used the small angle approximation knows that it is just that, an approximation, but that if the angle is small, the small angle approximation does not deviate significantly from using the sine or tangent of the angle (but it sure makes the math much simpler). With all this, we are supposed to believe that this report contains some damning admission that the earth is flat and nonrotating?
A similar report, this time sponsored by the Federal Aviation Administration, is the 2015 report, “Derivation of a Point-Mass Aircraft Model Used for Fast-Time Simulation.” Again, the term “flat, non-rotating Earth” is found in the first sentence under the subheading “2.1 Reference Frames.” However, this term has the same meaning as in the previous papers. The first sentence of the report stated that the paper derived equations of motion “for a point-mass aircraft.” Indeed, that term even appears in the title of the paper. Aircraft obviously are not point masses, so why does this paper treat aircraft as if they are? For the purposes of this paper, aircraft can be treated as point masses, much as the earth can be treated as being flat and nonrotating, even though the earth is rotating and has topography so that it is not flat.
A 1988 NASA report was titled “Flight Testing a V/STOL Aircraft to Identify a Full-Envelope Aerodynamic Model.” Once again, the phrase “flat, nonrotating Earth” appears on pages 4–5:
For aircraft problems, the state and measurement models together represent the kinematics of a rigid body for describing motion over a flat, nonrotating Earth.
Note, once again, the simplifying assumption that the aircraft is rigid. As in all such papers, the assumption of a “flat, nonrotating Earth” is a simplifying assumption, not an admission that the earth is flat and stationary.
Here is a 2006 paper titled “Singular Arc Time-Optimal Climb Trajectory of Aircraft in a Two-Dimensional Wind Field” authored by a staff member at NASA’s Ames Research Center.
In our minimum time-to-climb problem, the aircraft is modeled as a point mass and the flight trajectory is strictly confined in a vertical plane on a non-rotating, flat earth. The change in mass of the aircraft is neglected and the engine thrust vector is assumed to point in the direction of the aircraft velocity vector. In addition, the aircraft is assumed to fly in an atmospheric wind field comprising of both horizontal and vertical components that are altitude-dependent. The horizontal wind component normally comprises a longitudinal and lateral component. We assume that the aircraft motion is symmetric so that the lateral wind component is not included.
Notice the many assumptions that are not strictly true:
The justification for ignoring these complicating factors is based upon the realization that they are very small compared to the overall effect under consideration. Furthermore, the effects of the complicating factors could be added to the solution later. In similar manner, this paper does not say that the earth is globally flat and not rotating. Rather, the paper indicates that the effects of topography and the Coriolis effect are not significant in this study.
Here is a 1972 document from NASA’s Langley Research Center titled “Determination of Angles of Attack and Sideslip from Radar Data and Roll-Stabilized Platform.” One of the simplifying assumptions, once again, is “a flat, nonrotating earth.” Within the paper, this is recognized as a limitation of the applicability of the work, for four times in the paper it states, “The method is limited, however, to application where a flat, nonrotating earth may be assumed.” Hence, rather than being an admission that the earth is flat and nonrotating, it is implied that the earth is not flat and is rotating.
Articles about flight simulations made the list as well. An engineering dissertation titled “A Discussion of Methods of Real-Time Airplane Flight Simulation” made the list too. A search for the word “flat” yields this result (page 10):
Spherical coordinates can be inconvenient for two reasons. First, the geocentric latitude, φʹ, is not exactly the same as the geographic latitude used in navigation. This is because the Earth is actually an oblate spheroid, slightly flattened at the poles. And second, radius from the Earth’s center is an unwieldy coordinate. [emphasis added]
Notice that the word here is “flattened,” not “flat.” This statement could not be clearer—this source explicitly states that the earth is spherical, not flat. The word “flat” appears only four times in this dissertation, on pages 11 and 14. Here is the context of all four of those appearances (page 11):
Flat-Earth Coordinates. In many flight simulators, global navigation is not important. For example, the range of flight could be limited to a small area, or the simulator might not care about the airplane’s location. In such cases, it is appropriate to model the Earth as a plane half-space rather than an oblate spheroid. [emphasis in the original]
Again, this could not be clearer—this is not an admission the earth is flat, but rather it stipulates the conditions in which the spherical earth could be treated as flat.
Again, this could not be clearer—this is not an admission the earth is flat, but rather it stipulates the conditions in which the spherical earth could be treated as flat. How did flat-earthers come to think this was otherwise? They must have searched for the word “flat” but did not read what the text actually said. By the way, as a doctoral dissertation, this hardly is a government document.
The academic journal article “Design and Implementation of Flight Visual Simulation System” also made the list. A search revealed that this article does not contain the word “flat.” Perhaps what triggered inclusion of this article on the list is its use of the word “level” (flat-earthers typically think that flat and level mean the same thing). The word level is found on the third page, where the assumptions of the flight simulation are listed:
As with modeling real aircraft, this model for flight simulation assumes the aircraft is a rigid body. More importantly, the paper explicitly acknowledges that the earth is spherical and rotates, but for the purpose of the model, those factors are ignored. Once again, a paper that allegedly supports the notion of a flat, stationary earth under minimal scrutiny says exactly the opposite.
Another flight simulation paper is “Simulator AERO Model Implementation” by a NASA Ames researcher. As is typical of such studies, the simplifying assumptions are stated. On page 10,
The above three equations, plus the three translational equations comprise the equations of motion for the rigid body aircraft.
As I’ve stated many times, there are no rigid aircraft. The relevant simplifying statement is also on page 10:
For the flat, nonrotating earth considered here, any fixed frame of reference can be employed as an inertial frame. [emphasis added]
The wording “considered here” makes it clear that a flat, nonrotating earth is a simplifying assumption.
Another flight simulation source listed is a 1995 paper “Flight Simulation Software at NASA Dryden Flight Research Center.” Note the context of the three times that the word “flat” appears in this document. The first is on page 2:
This structure, with both flat- and oblate-Earth versions, has successfully supported more than 50 different aircraft.
Clearly, this simulation operates in two modes, one in which the earth is treated as flat and the other in which the earth is treated as its true shape, an oblate spheroid. Why the two modes? As I’ve discussed many times already, some applications can use the much simpler flat-earth approximation, but this simpler approach is not warranted in some applications. As page 8 explains:
In most cases, flat-Earth six-degree-of-freedom equations of motion are used. Oblate-Earth equations of motion were developed for the space shuttle simulation and later used in the NASP and follow-on simulation studies. The flat- and oblate-Earth equations of motion use a hybrid axis system that allows forces and moments to be added in the axis systems for which they are commonly computed, thus reducing axis transformations.
Because of its much greater range, the space shuttle could not be simulated in the flat-earth mode.
Another example is the undated NASA Technical Memorandum titled “A Mathematical Model of the CH-53 Helicopter.”
On page 17, the simplifying assumptions are given:
The helicopter equations of motion are given in body axes with respect to a flat, nonrotating Earth. The helicopter is considered a rigid body with mass symmetry about the xh - zh plane. The effects due to the engine angular momentum are neglected.
Again, none of these simplifying assumptions are true, but the effects of the factors ignored are negligible compared to the factors that are considered.
Speaking of helicopters, a 1979 NASA paper was titled “Development and Validation of a Piloted Simulation of a Helicopter and External Sling Load.” Here are the simplifying assumptions from page 33:
The equations of motion for both the helicopter and the external sling load are developed in body axes with respect to a flat, nonrotating Earth. It is assumed for convenience that each body is rigid and that the xh – zh plane and the xl – zl plane are planes of mass symmetry and that gyroscopic effects of engines are negligible.
None of the other assumptions are strictly true, but we are supposed to believe that the assumption of a flat earth is strictly true? And in context, what does the term “flat earth” mean? On page 44, in the section discussing the simulation software, this is said about two airports used in the simulation:
The 7.3-m by 18.3-m terrain model board of the VLDS includes two airports and surrounding terrain, one at 750/1 scale and the other at 1500/1 scale, and is shown in figure 22. There are a total of five paved runways, from 0.6 km to 3.5 km in length. A helipad is located on the 750/1 airport and is shown in figure 23. It consists of a Maltese cross with a 45-m by 45-m border. The terrain is generally flat, and provision is made for variable visibility, variable cloud-base heights, and day, dusk, and night scenes. [emphasis added]
It is obvious that the meaning of “flat earth” refers to the terrain around the airport, not the overall shape of the earth. Lest anyone think that this is a simulated airport, it is a real airport that was entered into the simulation.
Another NASA document that flat-earthers mention is a 1961 report titled “Calculation of Wind Compensation for Launching of Unguided Rockets,” where the term “a flat nonrotating earth” appears in the simplifying assumptions on page 7. There are three key terms in the title of this paper: wind, launching, and guided. The paper made it very clear that it was concerned with winds in and below the jet stream, limiting the scope of this study to altitudes of less than eight miles. The scope of this study does not include placing payloads into orbit or even great range or height within the atmosphere. That is what launch refers to. Rockets that go very far generally are guided. If one wished to consider these other possibilities, then they could be added to the conclusions of this study. With great range or height, one would have to consider the earth’s spherical shape and the earth’s rotation.
Another example used by flat-earthers is a 1971 paper supported by NASA titled “A Method for Reducing the Sensitivity of Optimal Nonlinear Systems to Parameter Uncertainty.” This study was of a general nature, with an example of the principles derived beginning on page 12 of the report. The statement of the problem began:
The example problem is a fixed-time problem in which it is required to determine the thrust-attitude program of a single-stage rocket vehicle starting from rest and going to specified terminal conditions of altitude and vertical velocity which will maximize the final horizontal velocity. The idealizing assumptions made are the following:
- A point-mass vehicle
- A flat, nonrotating earth
- A constant-gravity field, g = 9.8 m/sec2 (32.2 ft/sec2)
- Constant thrust and mass-loss rate
- A nonlifting body in a nonvarying atmosphere with a constant drag parameter KD = ½ρCDS, where S is the frontal surface area.
Notice again that these are idealizing assumptions. The rocket is not a point mass. Nor is the gravitational field constant. Nor is the thrust and mass-loss rate of a real rocket constant. Nor does a body move through the air with constant drag. But are we supposed to believe that, unlike the other assumptions, the earth truly is flat and nonrotating? Of course not. All five assumptions are made to greatly simplify the calculations. Once the problem is solved, one is free to modify the solution by explicitly including these more complicated situations.
A 1993 report from NASA’s Langley Research Center titled “Approximate Optimal Guidance for the Advanced Launch System” is about the launch of payloads into low-earth orbit using a two stage rocket. Its inclusion on the list probably was triggered by a search for “flat earth” that resulted in three instances (page 14, and twice on page 25) as simplifications for part of the modeling. The fact that the report is about launching satellites into orbit probably escaped the flat-earther who found this document. If he had bothered to read the document (or just did a little more searching), he would have found six mentions of the earth being spherical (pages iii, 1, 3, 14, 16, and 146). Admittedly, each instance also included “nonrotating” which obviously was a simplifying assumption. Clearly, this is not a document that admits the earth is flat.
A 1965 NASA paper prepared by Georgia Tech scientists was titled “Atmospheric Oscillations,” which was a study of previous work on upper atmospheric winds. In describing the previous work, this paper said (page 10),
A model frequently used is that of a flat, nonrotating earth. The temperature is assumed either to be constant, to increase or decrease monotonically with altitude, or to be stratified. Gravity is usually considered to be constant. Density and pressure are usually considered to vary exponentially with altitude.
Again, flat-earthers always read the “flat, nonrotating earth” part while ignoring the other assumptions.
Again, flat-earthers always read the “flat, nonrotating earth” part while ignoring the other assumptions. The possible temperature profiles can be generally met in the real world, but none of them are strictly true. Nor is gravity constant, and density and pressure only approximately vary exponentially with altitude. Similarly, while the earth can be considered flat locally, the earth is not flat globally. Furthermore, the earth’s rotation can be ignored in many applications, even though the earth is rotating. Indeed, the next paragraph summed this up nicely:
The most one can profitably simplify the problem is to consider an isothermal atmosphere, plane level surfaces, and a nonrotating earth. This case has been handled by Eckart (1960), Lamb (1932), and Hines (l960). The simplification is not valid for small effects, but general, large effects may be described and discussed.
In other words, when considering small effects, the simplifying assumptions do not work because the magnitude of their influences is comparable to the magnitude of the effects under consideration, but when one considers large effects, the magnitude of the complicating factors are small enough that those complicating factors can be ignored.
A 2000 Army Research Laboratory report titled “Modeling of Atmospheric Effects” made the list. What probably flagged this document was the use of the word “flat” four times (two of those are in the list of key words at the beginning). Here is the first occurrence in the document itself (page 31):
Thus, this model can only be considered valid for flat terrain, small spatial domains, and short time intervals.
Notice that this says, “flat terrain,” not that the earth is flat. Also notice that the model is limited to small spatial domains. That means localized areas. The surface of a large sphere is flat locally. That is why on the next page this sentence appears:
This model works well over a flat-earth and a nonturbulent atmosphere.
Once again, the mere use of the word “flat” was construed to be an endorsement of the notion the earth is flat.
A paper delivered at the 2007 39th Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting titled “Studies on Instabilities in Long-Baseline Two-Way Satellite Time and Frequency Transfer (TWSTFT) Including a Troposphere Delay Model” made the list, probably because of the single use of the word “flat” (page 216):
We tried three different mapping functions. Two are rough approaches, namely a simple plane troposphere (assuming a flat Earth) and the straight “line of sight” through the spherical troposphere shell. While these two functions result in too large or too small values, respectively, we use for the path length computation the mapping function as reported by Niell.
Notice that the assumption of a flat earth is a rough approach. Furthermore, in another rough approach, the researchers considered a straight line through the spherical shell of the troposphere. Rather than this paper being an admission of the earth being flat, it is an admission that the earth is spherical.
A 2001 Army Research Laboratory report is titled “Propagation of Electromagnetic Fields Over Flat Earth.” Flat-earthers obviously saw “flat earth” in the title of this paper but didn’t read much, if any, of the article. The article investigated the propagation of high-power microwaves over the earth. Microwaves are sent and received by line of sight. Since microwaves are electromagnetic phenomena, electromagnetic waves interact with the surface of the earth. The purpose of this paper was to investigate that. A search of this document reveals that the word “flat” appears 14 times. The occurrence of that word that is most revealing of its meaning in this document is in the introduction (page 1):
It is assumed that the transmitting antenna and the target (or receiver) are located above, but near the surface of a flat idealized earth (constant permittivity, ε, and conductivity, σ) ground.
The use of the words “assumed” and “idealized” indicates that this sentence states the simplifying assumptions. Flat-earthers skip over the word “idealized” and see “flat earth.” The parenthetical insertion of constant permittivity and conductivity are part of the clarification of the idealized condition considered. But notice that the next word is “ground.” The term “earth ground” appears several times in this paper. That is a strange way to express the earth (meant in the sense of the realm where we live) being flat rather than a sphere, if that was the intent. The meaning of earth here is to refer to the soil. Since microwave communications are line of sight, they are of limited range on the earth, so assuming that the earth is flat rather than a globe would be a justified simplifying assumption. But that is not the meaning here—the meaning is to ignore hills, trees, and other obstructions. If those factors were figured in, then one would need detailed information about the local topography, which would prohibit solving the problem considered in the general case.
A 2000 Army Research Laboratory report titled “Path-Loss Measurements in a Forested Environment at VHF” made the list. This interesting research project studied the effect of foliage in attenuating radio frequencies, with the hope of developing technology for detecting enemy presence in forests. They conducted measurements in both woods and cleared land at various frequencies and the heights of the transmitter and receiver. The first mention of “flat earth” is on page 2:
We made multipath measurements to provide confidence in the data and to get an idea of how well our measurements of the clearing represented an ideal flat earth. [emphasis added]
Notice that “flat earth” is modified with “ideal,” indicating that this is an idealized, rather than actual, situation. Furthermore, the next sentence reads,
We measured the path loss at a range of 410 m with the receive antenna 2.7 m above the ground, varying the transmitter height from 6 to 21 m in 1-m steps. [emphasis added]
It is difficult to ignore the fact that these experiments were carried out over 410 meters. Using the eight inches per mile squared approximation that flat-earthers are so fond of using, I find that over 410 meters there is a half-inch drop. Diffraction of radio waves exceeds this, so assuming the earth is flat is a very reasonable simplifying assumption. The maximum distance of any test run in this study was 4,700 meters, a little less than three miles. The amount of curvature over this distance is negligible.
Another Army Research Laboratory paper is a 2003 report titled “An Energy Budget Model to Calculate the Low Atmosphere Profiles of Effective Sound Speed at Night.” The Army is interested in this subject to develop methods of finding and identifying potential targets that can be detected by the sound they produce. I suppose that flat-earthers found this document by searching for the term “flat earth,” for the word “flat” appears three times in this document. The first appearance is on page 9:
To briefly examine short range acoustic attenuation at night, we use the low atmosphere profiles of wind speed, temperature, and relative humidity (shown before) as input to a flat earth, nonturbulent acoustic propagation model called the Windows (version) Scanning Fast Field Program (WSCAFFIP). [emphasis added]
Sure enough, there is that term “flat earth.” Once again, flat-earthers lifted these two words out, not seeing the context. Notice that flat earth is one of the properties of the model of acoustic propagation considered. Also notice that the model does not include turbulence. Even on a very calm night, there is some turbulence. However, the effect of turbulence was judged to be so small as to not warrant inclusion in the model, which would have greatly increased the complexity of the model. But what does flat earth mean in this document? The second appearance of that word is on page 12, in the section that discussed comparison of measurements to the model predictions:
The test site was chosen (in part) because of its relatively flat terrain (average slopes were 0.5 degree). [emphasis added]
Hence, flat earth in this paper refers to flat terrain, not the earth in its entirety being flat rather than spherical. The third and final time that the word flat is found is on page 14:
As an example, Figure 11 shows results from the CN2 over-land optical turbulence model (42,43), derived from two levels (above ground) of conventional micro-meteorological information as input (i.e., wind speed, temperature, and humidity retrieved on farmland, over flat barren soil, under mostly clear skies). [emphasis added]
Again, this indicates the meaning of flat earth in this paper is to refer to lack of topography.
Another Army Research Laboratory paper is the 2009 report titled “Computationally Efficient Algorithms for Estimating the Angle of Arrival of Helicopters Using Acoustic Arrays.” The purpose of this paper was to evaluate methods for tracking helicopters by the sound they emit. The term “flat earth” appears four times in this document, the first being on page 9:
Figure 6 illustrates a simple model for multipath, which is a based upon the signal having a single bounce on a flat Earth with propagation that is described by ray tracing for signals in the far-field. [emphasis added]
As before, notice that this is in the context of a model, and all models have simplifying assumptions. The model treated the bouncing of sound off the earth. Since a bounce is very localized, and a spherical earth is locally flat, the meaning of flat in this context refers to flat topography, not the entire earth being flat rather than a sphere. But let’s assume for a moment that the flat-earthers are right, that this paper intends a flat earth in the sense that flat-earthers take it. What did this paper conclude? Here is the second mention of flat earth from page 22:
The estimated phases of the reflection coefficients have a dependency on range. This may be caused by a violation of the assumption of the flat Earth model. [emphasis added]
This appears on page 23:
These results are poor at all times and indicate that the model used to describe the propagation of the signal is not adequate. The assumptions of straight-line propagation, constant reflection coefficient, or reflection off a flat Earth may not be valid. [emphasis added]
And this appears on page 27:
The algorithm assumed multipath could be modeled with a single bounce, a constant reflection coefficient, straight line propagation, a flat Earth, and incident angles that were not near grazing. This algorithm did not work well. [emphasis added]
The conclusion is very clear—the model did not work well, meaning that the model was disproved. Therefore, if this document refers to the earth being flat rather than spherical, then this paper proved that the earth is not flat.
A 2010 Army Research Laboratory report titled “Adding Liquid Payloads Effects to the 6-DOF Trajectory of Spinning Projectiles” considered artillery rounds with liquid payloads. The only mention of flat earth in this paper is on page 1:
A 6-DOF rigid projectile model is employed to predict the dynamics of a projectile in flight. These equations assume a flat Earth.
As in every example, assumption of flat earth is within a model. This model is for artillery rounds, which have relatively short range. On a local scale, the spherical earth is flat. However, the intended meaning of flat earth here is to ignore local topography, such as hills and valleys.
A follow-up paper titled “Adding Liquid Payloads Effects to the 6-DOF Trajectory of Spinning Projectiles” by the same author was published the same month. The same simplifying assumption of flat topology appears on page 1.
The 1999 Marine Corps manual Tactics, Techniques, and Procedures for the Field Artillery Manual Cannon Gunnery also made the list. The only time the word “flat” appears is on page 10-17, where it reads,
For an MPI registration, the orienting point should be in a relatively flat (level) area to eliminate the need to replot the MBL.
This obviously doesn’t refer to the earth’s shape, so it is doubtful that this is what triggered the inclusion on the list. What likely brought the attention of flat-earthers is the statement of the standard conditions of using field artillery (page 4 of the pdf linked above), which includes “no rotation of the earth.” However, the standard conditions include specified air temperature and air density, no wind, and the gun, target, and the meteorological datum plane being in the same altitude. Realistically, none of those conditions are ever met. Furthermore, the rotation of the earth is explicitly mentioned 10 times in the manual as corrections to the standard conditions (pages 3-15, 3-18 [twice], 5-11, 7-19 [twice], 11-50, Glossary 3, Glossary 4, and Glossary 9). In fact, the two mentions of earth’s rotation on page 7-19 describe the use of Table H on the next page to make corrections to range for the earth’s rotation and Table I on the following page for correcting the azimuth for the Coriolis effect. Hence, this government document that supposedly supports a stationary, flat earth does no such thing.
A manual related to the above is the Marine Corps’ TTP for the Field Artillery Cannon Gunnery. It is not clear why this source made the list, for the term “flat earth” appears nowhere in it, nor is there any mention of the earth’s rotation. The term “flat terrain” appears twice (page 5-16 and page 8-1), as does the term “flat areas” (both on page F-2). Even flat-earthers likely would recognize that these could not be construed to mean the earth is not spherical. The flat-earther who decided this was a government document that admitted the earth is flat merely searched for the word “flat” without examining the context.
A similar Army artillery manual Field Artillery Manual Cannon Gunnery made the list too, though that document mentions “flat earth” nowhere. Regarding the earth’s rotation, the language of the Army manual is identical to the Marine Corps manual in that it lists the same standard conditions even in the same format (page 11-1). It even contains the same information for correcting for the earth’s rotation. The Army manual includes the same Table H (page 7-20) and Table I (page 7-21) as the Marine Corps manual does. The Army manual goes into more detail, explaining the need for correction for the Coriolis effect on pages 7-18 and 7-19. Again, rather that this source supporting a flat, stationary earth, this source supports a spherical, rotating earth.
A related manual Field Artillery Gunnery made the list, though it doesn’t use the term “flat earth” or mention the earth’s rotation. However, this manual twice endorses the spherical earth when it defined the range of an artillery round (pages 2-23 and 2-28):
Range – The distance measured on the surface of a sphere concentric with the earth from the muzzle to a target at the level point.
So, rather than supporting the notion the earth is flat and stationary, this source reflects the conventional cosmology.
The firing tables for the above artillery manuals were produced and reported in the Aberdeen Proving Grounds 1967 publication “The Production of Firing Tables for Cannon Artillery.” Since this publication does not mention flat earth, it is a mystery why it is included in the list. This publication went into detail of how the tables were generated, including the correction for the Coriolis effect. The tables H and I of the above manuals are found on pages 101–102 in this publication.
A 2017 report sponsored by several government agencies titled “Telemetry Standards” was included in the list. Sure enough, “flat-earth” appears three times in this document. Let’s look at the first occurrence on page F-2:
At the outset, it should be noted that the curvature of the earth complicates the trigonometry for computing elevation, azimuth, and bearing angles. For example, the elevation angle computed for the path from an AMT ground station to a flight test aircraft 320 km away operating at an altitude of 30,000 feet will be close to zero degrees due to the curvature of the earth. Using a flat-earth approximation, the angle would be computed to be approximately 4 degrees, thus suggesting incorrectly that interference from terrestrial sources would not be received in the main beam of the AMT ground station.
Notice that this document explicitly refers to the earth being spherical, suggesting that assuming that the simplifying assumption of a flat earth can be used as an approximation for some situations, but not for others. The text continues on the next page:
The equations used in the representative examples below assume a spherical earth, as evidenced by the inclusion of the value for the radius of the earth in km (e.g., 6358 km). The flat-earth approximation is obtained by letting the value of the earth’s radius go to infinity.
Again, it could not be clearer that this document assumes the earth is spherical, not flat. The other two occurrences of “flat-earth” in this report support this conclusion.
A 2002 Army Research Laboratory report titled “Automatic Target Acquisition of the DEMO III Program” was included in the list. The word “flat” appears twice. The second occurrence refers to a panel on a device being flat. Here is the other occurrence from page 1:
In other scenarios, only the range to the center of the field-of-view and the depression angle is known so that a flat earth approximation provides the best estimate.
The text is clear that assuming the earth is flat is merely a simplifying approximation on a spherical earth. Again, this document does not state that the earth is flat.
A 2010 Army Research Laboratory report titled “Beacon Position and Attitude Navigation Aided by a Magnetometer” also made the list. This study investigated a method of accurately determining the position and motion of a projectile. The single use of the term “flat earth” occurs on page 3:
This section summarizes and notates three kinds of coordinate systems. The first is the Earth-fixed coordinate system, which is fixed to the Earth with a flat Earth assumption. Denote X, Y, and Z as the unit vectors pointing in the directions of the X, Y, and Z axes, respectively. Without loss of generality, the X, Y, and Z axes point to forward, right, and down, respectively. The second is the body-fixed coordinate system, with three unit vectors Xb, Yb, and Zb pointing to the Xb, Yb, and Zb axes, respectively. The Xb axis is along the object’s symmetric axis, referred to as the spin axis. The other two axes are perpendicular to the spin axis and each other. The Earth-fixed coordinate system and the body-fixed coordinate system are shown in figure 1. [emphasis added]
Notice that this work employs a flat earth assumption in one set of coordinates. What does this mean? It doesn’t mean that the earth is flat. Rather, it is describing a set of coordinates with respect to a point on the earth’s surface. From that reference point, one can define altitude and azimuth, as illustrated in Figure 1. Altitude and azimuth are expressed with respect to the horizon plane, which is tangent to the surface of the spherical earth at the location of the reference point. It is this tangent plane that is meant by the flat earth assumption. In the figure, the altitude angle is θ, and the azimuth angle is ψ. This is the standard way to describe the position of an object above the surface of the earth. Hence, this is not an admission that the earth is flat.
A 1955 Air Force study titled “Review of Sound Propagation in the Lower Atmosphere” made the list. The lone mention of flat earth is on page 1:
In most of the topics to be discussed the problem is to describe the sound field in a region of atmosphere above a flat earth.
Is this an admission that the earth is flat? Hardly. Here is a portion of the abstract (page iii):
The application is to the prediction of sound fields due to aircraft (in flight or on the ground), especially, at distances up to a few miles from the aircraft sound sources.
How far is “a few miles”? Statements elsewhere make it clear that the distances considered were much less than 100 miles. Under that restriction, the difference between a spherical earth and flat earth are minimal, but the mathematics is much simpler with the flat-earth approximation. There were other simplifying assumptions made, such as constant humidity and temperature, no particles of fog or smoke. These are idealized conditions that are never met in reality but come close enough to produce good results.
A 2003 Army Research Laboratory document titled “User Manual for the Microsoft Window Edition of the Scanning Fast-Field Program (WSCAFFIP) Version 3.0” was included in the list. The assumption of the flat-earth simplification was tacitly made on page 4 when the wave equation (equation 2.8) was transformed from Cartesian coordinates to cylindrical coordinates with azimuthal symmetry (equation 2.9). This choice of coordinates makes sense only with plane-parallel symmetry (flat earth, flat atmosphere) centered on the source or the receiver. Figure 2 in the publication shows the plane-parallel symmetry. The sample case of Appendix A in the document had computations of up to 20 km, a distance over which deviation of the spherical earth from a flat earth are minimal. The abstract on the final page stated,
This model works over a flat earth and non-turbulent atmosphere. Even with these restrictions, the model performs very well for many scenarios.
What were the restrictions? The simplifying assumptions of a flat earth and an atmosphere with no turbulence. Why would treating the earth as flat be a restriction if the earth truly were flat?
A 2001 Army Research Laboratory report titled “Scale-Insensitive Detection Algorithm for FLIR Imagery.” What probably triggered the inclusion of this paper in the list was the one occurrence of the word flat in the introduction (page 1):
For example, in some scenarios, it is assumed that the range is known to within one meter from a laser range finder or a digital map. In other scenarios, only the range to the center of the field of view and the depression angle is known, so that a flat-earth approximation provides the best estimate. [emphasis added]
It clearly states that the assumption of a flat earth is an approximation. An approximation to what? A spherical earth, because the earth appears flat locally, and over relatively short distances (a few miles), there is little difference between assuming the earth is spherical or flat. What distance is involved in this report? The FLIR referred to in the title is the Forward Looking Infrared imaging system used for target acquisition developed for the RAH-66 Comanche helicopter, an aircraft that never went into production (two prototypes were built). The maximum range of the armaments on the Comanche was 11 kilometers, less than seven miles.
Among the articles that flat-earthers claim are government admissions that the earth is flat, there is one odd one that uses the word “firmament.” Flat-earthers generally think the firmament mentioned nine times in Genesis 1 refers to a dome over the flat earth. Hence, if a government agency admits that there is a firmament (dome), then that is tantamount to admitting the earth is flat, at least in the estimation of flat-earthers. The document in question is an abstract from a Soviet scientific journal. The CIA regularly translated abstracts and articles from Soviet scientific journals, and the abstract in question was one authored by P. N. Boyko and V. M. Kazachevskiy in 1961. The translation was released in 2000. The translated title of the article was “Photographic Observations of the Spectral Intensity of the Firmament.” You can see the translated abstract on pages 21–22 here. This is a provocative title, and flat-earthers seized upon it, interpreting it as an admission by the Soviet government that there is a hard dome over a flat earth. What is odd about this? This is not an easy abstract to find. One must wonder how flat-earthers ever found it. If you bothered to find the abstract via the link I provided, scroll to pages 17–18. There is an abstract from the previous issue of the same journal by the same authors with a similar title: “Observations of Sky Spectral Brightness by the Photographic Method.” Why does one abstract have the word “firmament,” while the other abstract has the word “sky”? Notice that the translator of the earlier abstract was V. Golikov, while the later abstract was translated by G. Livshits. I suspect that the same Russian word was used in the titles of the two abstracts, but that one translator chose to put it into English as “sky,” while the other chose the word “firmament.” Most English dictionaries recognize “sky” as the meaning of “firmament,” although calling the sky the firmament is a bit archaic. There were several other related papers authored by Boyko and to a lesser extent Kazachevskiy. They all discussed atmospheric transparency and scattering, which directly relate to the two papers in question. This example hardly amounts to an admission that there is a dome over a flat earth. Furthermore, this abstract is from a scientific journal rather than a government agency, though one might argue that under communist governments there is not much difference.
In many instances, it appears that flat-earthers simply searched for the words “flat” or “flat earth” and hastily concluded that the documents involved endorsed the notion the earth is flat.
I have examined more than 40 government documents claimed by some flat-earthers to be admissions that the earth is flat. In none of them did I find any such admission. Rather, even when the term “flat earth” was found in these documents, careful examination with understanding of how physics and engineering methodology lays out simplifying (and technically false) assumptions revealed that the mere use of the term “flat earth” does not constitute a belief that the earth is indeed flat. In many instances, it appears that flat-earthers simply searched for the words “flat” or “flat earth” and hastily concluded that the documents involved endorsed the notion the earth is flat. In some cases, the sources cited as evidence of flat earth had statements that flatly contradicted that conclusion. Therefore, none of the claimed documents indicate the earth is flat.