With my recently developed highly skeptical world view, I have been greatly enjoying the revival of discussions on the shape of the Earth for a number of reasons:

  1.  It turns out that in reality, every living person, whatever they believe, relies upon the assumption that the surface of a large, undisturbed body of water will lie flat and horizontal whenever they do any practical work.  This is especially true of engineers, such as myself, even on large scale projects covering tens of kilometres such as mines.
  2. When one is challenged to make ones own observations on this subject, one finds that ones preconceived ideas have a huge effect on the way we interpret what we see.
  3. The technical difficulty is not that high, while the heat of the internet discussion is very high.
  4. The use of this topic to provide a dividing line between “normal” people and any person that has a more skeptical view of the way that one is governed.

As my personal investigations have unfolded, the various objections raised has prompted me to organise the investigation into topics:

a)  The default assumption for every practical purpose is that the Earth is flat.

b) The basic geometrical considerations of the spherical model

c)  The effects of atmospheric refraction in modifying the sightings made

d)  How surveyors deal with curvature and refraction

e) The consideration of focal lengths, vanishing points and magnification

f)  The “transmissivity” of the atmosphere, i.e. the measure of how far light reflected from an object wil travel in air before being completely blocked from reaching the observer and in particular, the Fresnel zone.

In this page, I will cover my current thinking on these items.


In Reality, EVERYONE Assumes The Earth Is Both Flat and Unmoving

The reader will probably be quite shocked to realise that, despite what they may or may not believe, they personally will always assume that the earth is both flat and unmoving for any practical task they undertake.  Firstly, lets define our terms.

Clearly, the land forms of the earth are not flat in their entirety.  Moreover, the oceans also rise and fall with the tide.  However, the surface of large, undisturbed bodies of water, such as lakes are assumed to lie flat and horizontal.  More importantly, it is assumed that if a pipeline or channel is used to connect two lakes, then the water will flow from “higher” lake to the “lower” lake or remain still if the surface of the lakes is at the same level.  The relative height of these lakes, and all other topographical features of surveyed land, is measured by comparing its height to that of “sea level”.  “Sea Level” is a concept that was originally based on careful measurements of tide movements, with the average height of the sea at any one point becoming the “zero datum”.  The height of a geographic point as measured by typical land surveys is given as a “Reduced Level (RL)”, that is, the height of that point compared to an imaginary “sea level” directly, vertically below it.

It turns out, that for every practical purpose, EVERYONE single one of us ASSUMES that points of equal RL lie on a completely flat, horizontal plane.  There are NO exceptions.  When we use a spirit level, when we dig a drain, build a house, a road, a rail track, a skysraper or a mine, we ALWAYS assume that all points of equal RL lie on a flat, horizontal plane.  What is astonishing is that this assumption has never been the cause of construction errors.  No person, be they drain digger or architect or engineer has EVER had to make a correction to this assumption to allow for the curve in the earth that they BELIEVE is the true shape of the earth.

Even more significantly, all aeronautical engineering and navigation is also based on the assumption that the earth is both flat and unmoving.   The use of critical gyroscopic instruments assumes that all points of equal RL lie on a flat horizontal plane and that, furthermore, that the only movement that needs to be taken into account for navigation purposes is that of the plane.

Similarly, no person is ever required to make a correction for the rotation of the earth in calculating either the route or the flying time when flying from, say, New York to Los Angeles and then back again.


Basic Geometrical Model of the Spherical Earth

I came across a simple way to calculate the expected distance to the horizon from any observer making an observation to the sea level horizon on a spherical planet.  I neglected to reference the information, so i hope that the originator of this method gets his due recognition somehow.


The method used is easily picked up from this diagram.  If an observer places his eyes (or other surveying equipment)  a height “h” above sea level, then the distance to the horizon of calm sea level, “X”, can be easily calculated using the arithmetic shown in the diagram.

curvaturediagram4The model of the Earth that is considered by convention to be the most accurate is that it is a PLANET which is roughly spherical but slightly bulging in the middle or perhaps slightly pear shaped ball with an average radius of 6,371 kilometres.

In the case, this calculation simplifies to the very close approximation X =3.57 x SQRT(h), where X is in km and h is in metres.   For example, an observer 9 metres above sea level should find that curve of the earth flips over at 3.57 x 3 = 10.7 km away.

So let us plug these values into the equation and see what we should expect with respect to the distance between the ocean horizon point and an observer at a height h above sea level on the planet Earth..


Atmospheric Refraction

In certain circumstances, the path of light travelling from an object to the observers eye or telescope can deviate from a straight line as a result of atmospheric refraction.   The most significant variables in calculating the net affect of refraction are air density, water vapour content and temperature differential with height.  In the case of surveying observations made more or less horizontally, the temperature differential is orders of magnitude more important than that of density and vapour differentials.

Rarely articulated is the knowledge that  refraction can result in 3 quite different outcomes… light curves upward, light goes straight and light bends down, depending on the values of the temperature gradient (dT/dh):

  1. Light will curve upwards when the colder ground (or sea) absorbs heat from the atmosphere, causing the atmospheric temperature to increase with distance from the ground and therefore dT/dh > 0.
  2. Light will curve downward when the warmer ground emits heat into the atmosphere, resulting in a negative temperature gradient, i.e. dT/dh < 0.
  3. Light will travel in straight lines  no heat transfer takes place (dT/dh = 0) or on windy days when turbulence mixes the atmosphere sufficiently to eliminate a temperature gradient.

When refraction for surveying is discussed, the general observation is made that the  atmosphere refracts the horizontal line of sight downward, making the level rod reading smaller and that he typical effect of refraction is equal to about 14% of the effect of earth curvature.  Moreover, it is also said that the effect increases the closer to the ground the light ray get and errors in the region of 5mm/km have resulted.

In fact, it is well understood that when dT/dh > 0, the effect is curve the light ray upwards away from the modelled curve of the earth, which should have the apparent effect of giving the earth a smaller radius and a tighter curve.  Equally, there are many occasions when dT/dh =0, particularly on windy days when turbulent mixing of the atmosphere eliminates the temperature gradient and this circumstance is not limited briefly to evening and morning.

How  Surveyors Account for The Curve of the Earth and Refraction

In everyday practical life, surveyors involved in civil and mining projects undertake “Plane Surveys”.  A Plane Survey is based on the assumption that area being surveyed is built upon a plane surface starting at “sea level”.  The determination of sea level will be taken up later in this note and this level is the “zero point” for what surveyor call the Reduced Level or RL.  That is to say, the “RL” of sea level is assumed to be zero.

Plane Surveys are said to be perfectly valid for distances under 200km.  Empirically, this assumption has been repeatedly demonstrated to be PERFECTLY valid in the case of  an uncountable number of surveys  undertaken over hundreds of years.

While this may seem unremarkable, one should note that the deviation of from the straight and narrow of the putative curve of the earth over a linear distance of 200km is 3,200m, a difference in elevation that is difficult to miss on the basis of reconciling the overlapping of plane surveys.

Clearly, one would expect that these plane surveys that require high levels of precision with respect to RL would need to take into account the curvature of the Earth  over relatively small differences.  For example, in surveying a railway track, a linear distance of 2km requires a correction of 300mm, an amount which cannot be ignored when the gradient of the track is critical.

No doubt the reader will be somewhat surprised to learn that standard techniques employed by professional surveyors simply avoid the need to take curvature/refraction into account.  The need to correct for errors due to curvature and refraction is eliminated by using a combination of:

  1. Proper field procedures (taking shorter shots and balancing shots) can practically reduce errors
  2. Wherever possible, staff readings should be kept at least 0.5 m above the ground,
  3. Using short observation distances (25 m) equalized for backsight and foresight
  4. Simultaneous Reciprocal Trigonometrical Heighting
  5. Observations made at each station at exactly the same time, cancels the effects of curvature and refraction

HowSurveyorsAccountfor CaR

The above image is a copy of a slide taken, under fair use rules,  from  http://facstaff.cbu.edu/~gmcginni/classes/CE%20115%20Field%20Measurments/PowerPoint%20Presentations/Leveling%20Theory.pdf

Focal Length and Perspective an Overview

In making optical observations, one can make of use simple rules which apply to the interpretation of what is being observed.

The first is that when humans focus on any particular object, then our eye is fixed on a particular distance.  The net effect of this is that objects which are at different distances are likely to be indistinct and are likely to be unobserved psychologically.

How Far Can Light Travel in Air

One of the stranger comments that I have heard in recent times about the shape of the earth is that “if the earth were flat, then we would be able to see China from California..”

The problem with this argument is that the atmospheric gases and particulates absorb and eventually block the path of light.

The different gases which make up the first layer of the atmosphere above sea level consist of N2 (78%), O2 (21%), CO2, H2O, CO, SO2, etc. Each of these gases absorbs light at different parts of the spectrum and, as a result, various combinations of conditions result in more or less absorbtion.  For observations at sea level, the components which vary the most is carbon dioxide and gaseous water. If you are interested in some more detail, this is an ok source:


H2O is most variable in the atmosphere.
CO2 varies seasonally.

Generally speaking, this means that light reflected off an object in China could not reach the shores of California, even if the Earth was indeed flat.  Nonetheless, some spectacularly long observations have been made, for example:

What’s the record for visibility without help from the silhouetting effect? I think that might belong to the report of the expedition led by Korzenewsky (1923), who reported seeing snow-capped peaks of a mountain range 750 km away. Conditions were perfect: the lower atmosphere was in shadow at sunset; the peaks were quite high (4650 meters, or over 15,000 feet); they were covered with white snow, increasing their visibility; and there must also have been considerable looming to bring these distant features above the observers’ horizon. As the observation was made on June 1, near the peak of superior-mirage season, the looming is not improbable, though the amount required is hard to believe. The observers themselves were in the deserts of Turkestan [now southeastern Kazakhstan] at a height of nearly a kilometer, where the dryness of the air favored extreme clarity, and looking across a broad, sandy depression. And, of course, much of the air path was in thinner air well above ground level, because of the mountains’ height.”


Testing the Model

To check the spherical model of the world I live in, I went to my local beach at North Shore in Victoria, Australia with a pair of binoculars as the sun was setting.  I stood at water level and looked east with the sun  setting behind me in the west.   I could easily see the coastline of the Bellarine Peninsula at water line up to Portarlington and easily see the channel marker at that point.


As the attached image from Google Earth shows, this distance is well over 20 km.  According to the spherical model, I needed to be around 40 metres or 130′ in the air before I would be able to make the observation that I did.  I have made similar observations using a telescope between Indented Head and Port Melbourne.  These observations would have required that I be 130 metres or 400′ in the air.

The second aspect of this discussion is the calculation of the declination of the horizon line from the horizontal.  The angle “a”,  shown in the diagram, can be calculated from as a = acos(X/(R+h).  The “declination” of the horizon is 90-a.   For a commercial aircraft travelling at 11,500 m above sea level, the distance to the horizon should be 383km and the declination to the horizon should be a clearly observeable 3.4o from the horizontal.  As any person that flies in a plane will observe, once alerted to the possibility, the horizon remains firmly at eye level.

The conclusion here is simply that the Earth is NOT a near spherical planet with a radius of 6,371 km.

This is really only the start of a remarkable situation.  In my very long career as a Mining Engineer, i have often either conducted or made use of plane surveying.  I have never experienced the need to make use of the spherical model in any of the survey work that I have undertaken or used to establish mining reserves, haul roads, rail lines, conveyor belts or any other item, even though the distance often involve tens of kilometres.  A small survey of my engineering colleagues reveals a similar outcome.  In fact, it turns out that there is NO case of any civil or mining engineer applying corrections for sphericity when building a rail line, canal, tunnel, skyscraper, aqueduct or pipeline…. no matter if these projects extend over hundreds of kilometres.

In other words, for the practical world in which we live, there is NO actual application of spherical (i.e. “geodetic”) modelling in any engineering projects of any kind, whatsoever.

By the way, every person can make the simple observations that I have made… and I urge you to do so.


A Discussion of “UN Flag” Flat Earth Model

One of the models of the Earth that has been proposed is that it is flat and looks like the UN Flag.  https://en.wikipedia.org/wiki/Portal:United_Nations#/media/File:Flag_of_the_United_Nations.svg

Until quite recently, the routes chosen to fly between continents in the southern latitudes were consistent with this map.  Consider the following examples:

a)  To fly from Melbourne to Nairobi, practically all flights flew first to Bangkok or some other nearby city, and then over India into Nairobi… a straight line on the UNFlag model.

b)  In flying from Perth to Johannesburg, the Qantas flights would stop at the Cocos-Keeling Islands… another straight line on the UNFlag model.

c)  There were no direct flights from Australia or New Zealand to South America

d)  There were no flights over Antarctica

Within the past few years however, a number of changes have taken place, namely:

e)  Qantas now flies directly from Sydney to Santiago and return.  My sister-in-law has taken this return flight twice.   Air New Zealand also states that it has flights from Auckland to Santiago

f)  South African Airways now travel directly from Perth to Johannesburg

However, there are still no flights over the Antarctic.

Google Earth has no useable photographic information inside the the S85 latitude circle.


15 thoughts on “Sphericity

  1. sasilik

    There is no arguments for you equation. Its for calculating distance to the horizon if earth were perfect sphere and there was no refraction. You can use it and get good enough results. Problem is with your observations and arbitrary conclusions you draw from them. You just see something, claim that its channel marker and claim that you should not see it from where you are. You don’t take account your elevation, you don’t verify that what you see is really what you claim it to be, you don’t take account the height of the object you see and you don’t take account of the air refraction. You should look at the earth curvature calculators like
    which are going to give you quite good estimation how much something is hidden when you observe something. If you really understand math then please, make same kind of calculator for flat earth which gives you estimation how much you see some object from certain distance. Otherwise there is no evidence that earth isn’t round or that the earth may be flat.


    1. anounceofsaltperday Post author

      Hi saslik, thanks for taking the time to comment. I clicked on both links but only the second one was active. Personally, I think the explanation that I have given of the curvature calculation is easier to comprehend than that provided in the second link.

      With regard to refraction, my view is that this argument is inverted and that it is refraction that adds to the illusion that the surface of large bodies of water are curved. The key consideration in refraction is the thermal gradient dT/dh, which is largely eliminated during windy days when the thermal gradient is eliminated by air turbulence. It is under these conditions that more distant “anomalous” observations are frequently made.

      My intention in this blog was simply to highlight my own observation and provide the necessary information to allow other people to make their own observations.

      Here is a simple observation that you can make if you are fortunate enough to be able to sit at a beach facing west. Go and sit at the waters edge and watch the sun set. The horizon line will just over 2.5 km or 1.5 miles away. As that sun sets, do you observe that to occur only 1.5 miles from where you are sitting? That certainly has not been my experience.


      1. sasilik

        Both links work for me without problems. Maybe first one its blocked for you for unknown reasons. Refraction is also always a factor and it does not create illusion that surface is curved. It enables you see little farther and raises farther objects a little higher than they would be without refraction.
        I don’t also quite understand your question about sunset. I observe sun setting and going behind horizon line. Horizon line is about 2.5 km away. Sun is not. So, sunset does not occur 2.5 km away but sun is going behind horizon line which is 2.5 km away.


      2. markko

        Refraction is most of time one way. Weird cases are possible but usually its is like I described. It raises objects behind horizon higher. And what about these pictures? They show situations which are impossible on flat earth.
        I also want to comment about your blog intent being “highlight your observation and provide necessary information”. It really does not seem that way. You provide equation for calculating the distance to the horizon. To the horizon! Not to the objects behind the horizon or how much these objects are obstructed by the horizon. And then you are going to make observations where you ignore all other pieces of information which are crucial to understanding why you see what you see. Observer height from the sea level. How tall is observed object. How much refraction you may have. And then you state that distance to the horizon is that and you should be at some height to see the things which are behind the horizon. Not mentioning how much of the objects bottom you see or don’t see. You are misrepresenting information and drawing false conclusion from it because you look at the object behind horizon and in same time using distance to the object as distance to the horizon. Its not right. If you see the object behind the horizon then that does not mean that the horizon must be at same distance as the object is. You can’t use the equation for calculating distance to the horizon to explain why you see object behind the horizon. This doesn’t work that way.


      3. anounceofsaltperday Post author

        Hi Markko, thanks for taking the time to comment. In my observation I clearly state that I am standing at the water line and that I can see channel markers in the water at Port Arlington. There is no need to consider the height of items “behind the curve” because the channel markers can clearly be seen at the point of entry into the water. Similarly the beach front along the Bellarine Peninsula. What always surprises me is that people will defend a model of spherical shape that they have never personally applied. Have you personally ever applied the curve calculation or have you, like everyone else, assumed that surface of large bodies of water are flat and horizontal if otherwise undisturbed?


  2. markko

    There definitely is need for taking account height of items behind the curve. Your visual observation “I just see things” does not count for anything when you don’t mention how high object is and how much of it is obscured. If you claim that you see these markers at distance of 20 km from top to bottom entirely and clearly then there is need for photographic evidence because its quite extraordinary claim. For now its just your claim that you see something that you think are channel markers but there is no evidence that they actually are and that you see them entirely from top to bottom. I’ve seen it before. People claiming to see something and when afterwards there is some more research then it comes out that they really saw something else.
    And the curve calculation have been applied many times. For example https://www.youtube.com/watch?v=MoK2BKj7QYk . There is also link to flickr image where the calculation data is shown. What surprises me is that people don’t even bother do more research and limit their work only very basics and then misinterpret all observations that come afterwards. And ignore all observations that go against their initial thought. You have done your cursory observation which really consists of – I went to the beach and I saw something – and you conclude from that that earth may be flat and ignore every other observation which are not possible on flat earth.


    1. anounceofsaltperday Post author

      I don’t understand your complaint with my curve calculation… are you saying it is incorrect? Can you be specific? With regard to the video you link, let’s review the data given at 4’38” showing the tower at a distance of 47.9km. From the observer at 2.1m, the distance to the horizon is 5.17km. The distance from the Turning Turso to that same horizon point is, therefore, 47.8-5.17=42.73km. By the curve calculation, we expect that 143.3 metres of the tower to be obscured in the observation and that only 46.7m of the tower should be visible. Clearly and unambiguously, there is considerably more than 120 metres of the tower in view. This observation therefor clearly disproves the idea that the earth is a curve of 6,371km. If you are going to reply, could you also advise where you have personally used the curve calculation in your life when making decisions about digging drains etc?


    2. anounceofsaltperday Post author

      just confirming that I am replying to sasilik as well as markko.

      OK, so we are not arguing about my arithmetic we are arguing about refraction. Your view is that refraction can only go one way. I have put in my blog the reasons why you are incorrect with respect to dT/dh and the relativity of ground and air temperatures, wind turbulence and other factors. Your argument regarding refraction is also, forgive the pun, circular. If the refraction is occurring in the manner you describe then that effect will EQUALLY be the reason why a foreshortened Turning Torso is observed if the sea between them is flat and horizontal. Clearly, the observations need to be taken under circumstances where refraction is minimal and the distance is sufficiently short that it can be ignored. An example of EXACTLY the circumstances that I have described are provided in this video. The lakes surface is well below freezing point and refraction can be eliminated as a contributing factor. https://www.youtube.com/watch?time_continue=9&v=bh1KN5gbPvU

      How is the refractive condition of the air established anyway? If the earth is assumed to be curved, then the number chosen will be adjusted to make the assumed curvature correct… exactly as you have done in the calculations that you make.

      Please provide an instance where you have been required to make use of curve calculations in your personal or professional life? I have never been required to do so and I am a Mining Engineer that would reasonably expect to need to make use of such calculations. During the time that I was doing private light aircraft pilot training we were also required to assume the earth was both flat AND unmoving.


  3. sasilik

    Problem is that you don’t have curve calculation and you don’t even understand that. You have equation for the distance to the horizon. And only that. And then you observe objects which are quite farther behind the horizon and you use this like these objects were right at the horizon line. You really don’t understand what you are doing wrong do you?
    With regard to the video I mentioned that also that there is flickr image where there are shown all data. If you can’t see it on video description then there it is – https://www.flickr.com/photos/138443523@N08/25238470304/
    There you can see also how much refraction plays part in this.
    I also don’t get your fixation that I personally must do this kind of calculation in real life situation. Especially when digging drains in my backyard or doing some small local work. I have referred you to person who has done it and data matches round earth, not flat.
    I also don’t get your fixation only with this one specific type of observation which you do quite superficially and ignore all other observations. Like how sun does not change its angular size from sunrise to sunset. How can sun set. How sun can illuminate bottoms of the clouds or how it does not illuminate clouds about 2 km high but illuminates clouds which are at 5 or more km high at same place above your head. How can’t you see same stars everywhere. What makes things disappear behind horizon. How can you calculate the obstruction of tall object by horizon line on flat earth. Why stars rotate on sky in different directions on both hemisphere. Why things don’t fall straight down when you let them drop down at high place. Why does not ship wake converge to the point at see when it should do it on flat plane.And there are more.
    Its kind of useless to obsess over one thing which you are doing wrong and draw conclusions from there when all other observations just don’t fit anywhere in this model.


      1. sasilik

        Sasilik and markko are same. Sasilik is my alias. I don’t know why there is sometimes markko and sometimes sasilik. As I said, look at the data in flickr image. Notions about all other observations what you just ignore are still standing. And I try one more time to explain what you are doing wrong.
        You have equation for calculating distance to the horizon. That is X. Then you observe the object at the distance D=X+x1. Distance to the horizon stays same for you but for some mysterious reasons you assume that to see the object at distance D you must have at the height where there horizon line would be at D. It sin’t so. You can see objects which are beyond horizon line and you can’t use the distance to the horizon to calculat ewhat you see behind the horizon and what you don’t. Also the refraction plays quite much par for objects behind the horizon as seen in data for Turning Torso observation which you again just ignore.


      2. anounceofsaltperday Post author

        With regard to refraction, what you call “normal” occurs when the ground is warmer than the air. In both my observation and the Turning Torso observation, the water is COOLER than the air, and this reverses the refraction effect (dT/dH). This is reason why any of the tower is obscured. In both cases, the observations DISPROVE the claim that the earth is a sphere with a radius of approximately 6,371km.


      3. sasilik

        Your claim about refraction effect being reversed is wrong and baseless. You have no recorded information about temperatures above the water in both places.
        In regard with the video there is no 120 meters tower visible in point F where the distance is 47.9 km and observer height is 2.1 m. I don’t get from where you got this number. All data and calculations are in the https://www.flickr.com/photos/138443523@N08/25238470304/ and they match quite closely the data of round earth with radius of 6371 km.
        Your number of the tower being obscured on your drawing is number without taking account refraction and it is on the same table on flickr and there is also a number when refraction is taken account. It just shows that you must take account the refraction and can’t rely just on geometry.


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