Flat Earth vs. Geometry

As mentioned in the last newsletter, I’ve been asked repeatedly by readers over the years to debunk flat earth-ism.  I’ve not done much, though I briefly touched on it in that edition with respect to lunar eclipses.  I’ve resisted delving into the subject of flat earth.  For one thing, I’ve been teaching serious astronomy for over 25 years, and don’t care to waste time on nonsense.  For another, I don’t care to be the “Shell Answer Man” of whatever fad is currently viral.  As explained in the Introduction to Signs & Seasons:

It seems that astronomy is always surrounded by many controversies. Many Christians have bound up astronomy with astrology, pagan influences, origins, calendar disputes, “The Gospel in the Stars,” geocentrism, and flat-earth theory. But these can become lengthy tangents that draw away from the main lesson.

There’s important work to do in Classical Astronomy, a lot of ground to cover, and too little time on my end.  And flat earth-ism is so far removed from reality for anyone who wishes to truly study the subject of astronomy.  Common observations of the sky readily debunk flat earth tenets.  Also, flat earth-ism is debunked by simple geometry learned by 15 year olds.  The notion of a spherical earth is actually the foundation upon which rests the entire edifice of modern science. 

Flat earth is a deception that appears to be worsening at the present time.  We are living in an era of deception, where the powers-that-be cannot be trusted to tell the truth, from pandemics to our food supply.  Flat earth media resources are abounding, apparently from well-funded sources, including slick video graphics and sensible-sounding explanations.  Like all deception, flat earth preys upon the naive and well-intentioned, and sews the seeds of doubt and distrust rather than objectively approaching the real world.  The rank and file flat earth devotees –including your neighbor or family member or coworker — are innocently caught up in this.  But the main purveyors of this ideology are smooth-talking and silver-tongued.

Does Scripture Teach a Flat Earth?

One common deception is that Scripture teaches flat earth-ism. The Bible includes several verses such as these:

The Lord reigneth, he is clothed with majesty; the Lord is clothed with strength, wherewith he hath girded himself: the world also is stablished, that it cannot be moved.  – Psalm 93:1

Who laid the foundations of the earth, that it should not

be removed for ever. – Psalm 104:5

There are several such verses which connote the stability of the foundations of the Earth.  But there is also this prophecy that contrarily speaks of the shaking and instability of the foundations of the Earth:

And it shall come to pass, that he who fleeth from the noise of the fear shall fall into the pit; and he that cometh up out of the midst of the pit shall be taken in the snare: for the windows from on high are open, and the foundations of the earth do shake.  The earth is utterly broken down, the earth is clean dissolved, the earth is moved exceedingly. – Isaiah 24:18-19

A hyper-literal, out-of-context interpretation of such verses can give opposite teachings. Such verses are in passages intended to glorify the LORD and speak of His power.  These passages are poetic and colorful and are unlike the Genesis narrative that teaches the historical events of the creation story.  But even so, these verses do not specifically suggest the Earth to be flat per se, but merely stationary and unmoving.  The same passages have also been used down through the centuries by geocentrists who taught a spherical Earth set unmovingly at the center of the universe.  But insofar as Scripture might teach about the “foundations of the Earth,” this could just as easily imply the layers of strata in the Earth’s crust, which do not “move” overall compared to the body of the Earth but do occasionally “shake” in places due to earthquakes.

Here’s one verse that might be construed as specifically teaching a flat earth:

The sun also ariseth, and the sun goeth down, and hasteth to

his place where he arose.  – Ecclesiastes 1:5

But if we take this verse to teach a flat earth, we also have accept that Scripture teaches that the sunrise occurs in a “place,” namely the east.  This would suggest that everyone in the world would see the sunrise in the east at the same time, as people once believed in deep antiquity.  But this is blatantly untrue, since we live in a world of time zones.

When our family drove to California this past spring, I did not change the time on the clock in our car but left it set on eastern time.  The Sun went down at the Pacific Ocean at about 10:00 PM according to our car clock, though I knew from my own direct experience that the same car clock recorded sunset at around 7:00 PM back in Ohio only a couple days before, and did so again upon our return.  So clearly, my own direct experience with observing the sunset across time zones refuted a hyper-literal interpretation of Ecclesiastes 1:15.  Either the LORD is a liar, or else the correct context of the verse is to colorfully illustrate a metaphorical lesson, namely that all is vanity under the Sun, and not that the Earth is flat. 

Also, a hyper-literal interpretation of the male pronouns in this verse would imply that the Sun is a “he,” something not supported by the modern science of spectroscopy, which shows the Sun to be mostly composed of very hot hydrogen gas.  At any rate, similar context could be applied to the other verses which are hyper-literally interpreted to suggest a flat earth.

Even if Ecclesiastes 1:5 can be interpreted as teaching a flat earth, the modern flat earth movement does not teach that everyone in the world sees the sunrise and sunset at the same time, contrary to this verse  We’ll get to that later.

Flat Earth and Geometry

The standard understanding of a spherical Earth is that “what you see is what you get” — the appearances of the sky are actually what they appear to be.  A methodical study of the sky leads to the conclusion that we are living upon a globe.  Such a methodical study is conducted by making measurements and using the principles of geometry that have been known and understood since the ancient Greek geometer Euclid.  In fact, the word “geometry” itself means “earth-measurement.”  The application of geometry to astronomy will be the subject of Measuring the Heavens, the sequel to our Signs & Seasons curriculum. 

From a plain and ordinary observation of the sky, it is clear that the Sun rises in the morning, reaches its highest point in the sky at “high noon” (not necessarily the same as “clock noon”) and sets again at the opposite horizon.  The Sun’s daily path across the sky is a slanted circle.  And if one observes these daily paths over the span of the year, one readily sees that these paths are essentially parallel circles.  The Sun rises and sets to the north in the spring and summer, and having a higher elevation at noon, reaching a northern extreme on the summer solstice, the longest day of the year.  Conversely, the Sun rises and sets to the south in the fall and winter, and having a lower elevation at noon, reaching a southern extreme on the winter solstice, the shortest day of the year.  The midpoints between these extremes are the equinoxes, when the Sun rises due east and sets due west, reaching an intermediate elevation at noon.  


It just so happens that this appearance is exactly what we should expect from living on a spherical Earth.  The geometrical appearance of the sunrise and sunset is due to a direct geometrical line of sight to the Sun as it physically appears above our section of the globe at sunrise, and physically passes below our local horizon at sunset.  The Sun’s position in space is relatively fixed.  The daily solar motion is due to the rotation of the Earth, and the seasonal variation from north to south is attributed to the annual revolution of our globe around the Sun.   


However else one cares to interpret these appearances, they just so happen to perfectly appear exactly according to what one could expect to see from the surface of a spinning globe circling the Sun, following simple rules of straight lines according to geometry.  Moreover, the slant of the circular paths of the Sun is at specific angles unique to each location around the world, each oriented at precisely an angle that corresponds to the latitude of that location.  The celestial equator (the path of the Sun on the equinoxes) is slanted at exactly the same angle from vertical according to the latitude of the observer’s location on the globe.  Down through the centuries, this fact has been one of the astronomical methods used by navigators to find their latitude on the surface of the spherical Earth.


Such angle measurements are easily made using the basic mathematical concept of trigonometry.  While not everyone studied trig back in school, the premise is based on forming a right triangle having a certain angle, and then finding that angle from the lengths of the sides of the triangle, using simple division that everyone learns in the 3rd grade.  There are three sides to a right triangle, the opposite side from the angle in question, the adjacent side to this angle, and the hypotenuse, a fancy name for the long diagonal side.  There are established mathematical relationships called sine, cosine and tangent that are numbers resulting from the ratios of these sides, when the length of one side is divided by another side.  An inverse operation (performed using a pocket calculator) produces the angle from the sine, the cosine or the tangent.


Anyone who cares to learn this basic concept of trigonometry can find latitude from measuring the noon elevation of the Sun.  At every location, the solar zenith angle of the Sun’s noon position on the equinoxes is the same as the angle of latitude.  Measuring the Sun’s noon elevation and subtracting from 90 degrees produces this number.  On a ship at sea, this measurement is taken by a sextant, and is used to find the current latitude of the ship.  On the summer solstice, the Sun’s noon elevation is 23.5 degrees higher than on the equinoxes.  And on the winter solstice, the Sun’s noon position is 23.5 degrees lower than on the equinoxes    


While not everyone has mastered the art of angle measurement and finding latitude from the Sun, the equations are very straightforward and provide consistent, reliable results — so reliable that common navigators in creaky wooden ships were able to find their way from Europe all around the world in the 17th, 18th and 19th centuries, before the development of modern electronic systems.  These methods were based on the simple concept of straight lines being observed from the eye to the Sun that formed simple triangles that could be used to derive accurate position data.

These simple calculations work both ways and you can readily calculate the noon elevations of the Sun for the solstices and equinoxes for any latitude in the world.  For our latitude here in Cleveland of 41.5 degrees north, the Sun’s noon elevations are as follows:

Summer Solstice => 90 – 41.5 + 23.5 =  72 degrees;

Equinox => 90 – 41.5 = 48.5 degrees;

Winter Solstice => 90 – 41.5 – 23.5 = 25 degrees.

This article from our site shows the lengths of the noon shadows of the solstices and equinoxes cast by my then-12yo son here in Cleveland.  The height of a person or object and length of the noon shadows respectively form the opposite and adjacent sides of a right triangle.  So the angle of the Sun’s elevation can be found by the inverse tangent of the height divided by the shadow length.  If you measure these lengths from these pictures and crunch those numbers, they’ll match the numbers posted above.  (I know, maybe this sounds hard, but it gets easier with practice! 🙂 ) 

Flat Earth Alternate Reality  

Flat earth-ism teaches that you must disbelieve the evidence of your lying eyes.  According to flat earth-ism, the world is a flat, circular disc 25,000 miles in diameter.  The Sun is a body 32 miles in diameter which floats 3000 miles above the surface of the flat earth, and somehow circles above the flat world each day. 

The appearance of rising and setting is not due to straight lines toward the Sun but somehow due to some sort of “horizon effect.”  The Sun recedes toward the horizon and vanishes from view when it is beyond a certain distance from a given location.  There is no straight line geometry in the flat earth view of the world because the tiny Sun somehow acts like a spotlight and only selectively shines downward onto the area immediately below itself.  The Sun’s rays somehow do not extend outward 360 degrees in all directions, according to the common, everyday principles of straight line geometry, but somehow stop at a certain distance due to this elusive horizon effect.  So nighttime happens when the Sun circles over to the opposite side of the Earth, without physically rising or setting compared to the horizon.


We all observe the rules of perspective in our daily lives, how straight line objects converge to the horizon according to geometry.  As shown in this picture from this site, we all observe that close telephone poles appear larger and telephone poles off in the distance appear smaller, following straight lines to the horizon.  But while the poles appear mimimal at the vanishing point, they still remain above the horiozon, and never actually fall below.  Flat earth-ism contends that the tiny Sun disappears at the vanishing point and never actually sets, but just continually circles out of sight above the flat earth at a constant 3000 miles. 


Flat earth-ism offers convoluted verbal talk-arounds to explain away  the obvious geometrical appearances of sunrise and sunset, without any sort of equations or line diagrams to illustrate this effect.  No explanation is given for why the Sun’s path appears perfectly circular as it crosses the sky, or why the equinoctial sunset forms an angle equal to the latitude of each location on Earth.  However this theory is supposed to work, the flat earth very coincindentally happens to appear precisely the way one a spherical Earth would look.  We’ll unpack some more of these details in a future newsletter, and discuss angular sizes and parallax.

An observer at latitude 40 degrees north is 2760 miles from the equator.  So if we just stick with regular trigonometry, the noon elevation of the Sun on the equinox at latitude 40 degrees north is the inverse tangent of 3000/2760, or 47.4 degrees, which is somewhat close to the actual number.  However, at midnight from this latitude, when the Sun is 9660 miles away on the opposite side of the North Pole, the Sun would be 17.3 degrees above the horizon, which is absurd.  Midnight sun only happens in the Arctic, but according to commonplace 10th grade Euclidean geometry, constant, 24 hour daylight would be the result of living on a flat earth with the Sun continually circling overhead.      


Explanations I’ve read of this alleged horizon effect appear to muddle and twist explanations of atmospheric refraction and horizon distance, which themselves are legitimate.  But there is no science behind the flat earth horizon effect. 

Another aspect of flat earth-ism that remains unexplained is the law of gravity, a key feature of the classical mechanics of Isaac Newton.  Somehow the Sun floats in a circle with a daily motion above the flat earth, and the circle expands and contracts with the seasons, being smallest on the summer solstice when the Sun is closest to the North Pole and largest on the winter solstice, when the Sun is furthest from the North Pole.  And again, this mysterious expansion and conraction perfectly mirrors what one would expect from a spherical globe that rotates while it revolves.


If there really was a flat earth, and if normal straight-line geometry prevailed,  the noon elevations of the Sun over the year would be greatly different than what we could measure from our own backyards.  These elevations would be measured by simple right triangles.

Noon Elevations

Summer Solstice => 69.2 degrees;

Equinox => 47.4 degrees;

Winter Solstice => 34.4 degrees.

Midnight Elevations

Summer Solstice => 20.5 degrees;

Equinox => 47.4 degrees;

Winter Solstice => 14.9 degrees.


Not everyone took math or has an interest in science.  But flat earth-ism preys on the innocence and healthy skepticism of naive, well-intentioned laypeople who have not spent time observing the sky and do not appreciate the legitimate scientific proofs that the world is round.  We live in a world of science and the fruits of science surround us on all sides.  And yet the entire edifice of modern science is based on Newton’s work, which dethroned the ancient geocentric pseudoscience of Aristotle and established an empirical method for studying the natural world to discover how things really work.  Our wondrous technology is the result of three centuries of scientific progress since Newton, ironically including the technology used today to promote flat earth-ism. 

This newsletter was just a brief introduction, hope it was helpful.  I’ll build on these concepts in future newsletters to help explain more reasons why flat earth cannot explain the real world. 

Bookmark the permalink.

Comments are closed.

  • Share on social media