# In ancient times, how did people conclude that the shape of Earth is a sphere?

This is more of a philosophical question, but I want a mathematical explanation. During ancient times, it was well accepted that the surface of Earth was spherical. People first observed this when they saw that starting from a point going to any direction they ultimately reach the same point, but by this logic how did they conclude that or even guess that this might be a sphere? Why did nobody guess that, say, a torus might also be possible? As far as I know, at those time there were lots of mathematicians. If they were convinced by this logic then they must have had some logical explanation, and I do not think that for them just staying on the surface would at all be possible to guess the structure of a surface.

One of my logical thoughts for this is the following: If I take a telescope and stand on a field and look up, my vision will never get limited by the surface of Earth. Mathematically, if I draw a line perpendicular to the surface at a point, then that line will never intersect the surface. But doing the same experiment at each point on Earth is not very convincing. Can anybody please provide me a better logical explanation of how people of that time concluded that the shape of Earth is a sphere?

• Look up this link. It talks in detail about what would a toroidal earth would be like. Maybe you'll find an answer in there. And it's a really interesting thing to read anyway.
– G-man
Commented Mar 10, 2015 at 18:43
• Eratosthenes measured the angle of the sun on the same day in two different geographic locations and got data that was not far off.
– rschwieb
Commented Mar 10, 2015 at 18:43
• Wikipedia has a pretty long article about the history of the concept of a spherical Earth.
– Rahul
Commented Mar 10, 2015 at 18:43
• "People first observed this when they saw that starting from a point going to any direction they ultimately reach the same point." I would be surprised if they had actually done that. Did they cross the polar regions or re-discover America? Commented Apr 30, 2015 at 19:54
• I think its a mistake to think that the determination of the shape of the earth was mathematical in the way we think it now; qualitative arguments are possible: for example, once it's been grasped that the earth was some object hanging in space and one observes that the two substantial objects in the sky that look solid are the moon and the sun, both of which are round; then a possible induction would be that that too must be the shape of the earth. Commented May 21, 2015 at 18:55

The ancients understood that a lunar eclipse is caused when the earth gets between the sun and the moon. They saw that the shadow the earth casts on the moon is round. From that it wasn't too far of a leap for them to conclude the earth is a sphere.

• I think it's particularly important for answers that are common historical folklore to include references. Commented Apr 18, 2015 at 19:08

Eratosthenes of Cyrene did an experiment that confirmed that the Earth was roughly spherical, and estimated its circumference around 200 BC.

Note that the lunar eclipse observation mentioned by Gregory Grant suggests that the Earth is round in 2-dimensions (an Earth shaped like a flat disc is consistent with this observation), but does not provide evidence regarding the Earth's 3-dimensional shape.

• Actually, Eratosthenes assumed that the Earth is spherical to make his estimate. Measuring inclination of the Sun on solstice at just two points doesn't really confirm anything about the overall shape. But I agree that round shadow on the moon is equally consistent with a cone, a cylinder, etc. Commented Apr 29, 2015 at 23:26
• One lunar-eclipse shadow shows that the Earth is round in 2D. Several which occur at differing points in the sky relative the observer show that the Earth presents multiple round aspects to the sun. Collect enough and any conclusion other than a spherical shape begins to get very strained, indeed. Commented Jun 2, 2015 at 14:58
• @dmckee This idea can be upgraded to a proof: mathoverflow.net/a/39131/61785 Commented Jul 27, 2019 at 19:36

Aristotle provided a number of arguments for the sphericity of the world in his writings. For example, (and this one has already been mentioned but I don't believe with any attribution to Aristotle) in On the Heavens, he noted that during a lunar eclipse the shadow of the earth is circular in shape.

Also in On the Heavens, Aristotle observed that as one moves to the south, new constellations become visible. Furthermore, a relatively short trip to the north also reveals new stars. So, in addition to concluding that the earth is spherical, he also ascertained that it cannot be of great size.

In ancient times, how did people conclude that the shape of Earth is a sphere?

I believe also that many theorized that if the world was flat then a ship appearing on the horizon would not do it bit by bit but would appear all at once.

The sailors at least knew the earth was not flat.

• I didnt ask about "FLAT" Commented Apr 28, 2015 at 18:25

I believe the answer to your question lies mostly in Greek's cosmic philosophy of the way the universe should be. They saw spheres as one of the most symmetrical simplistic shapes, and if our earth, which for them was at the center of the cosmos, would be the shape of anything it would be a sphere. Of course, the calculations they used and observations they made lined up with a sphere, so it seemed that it was the most reasonable thing for earth to be. Now, many philosophers of science might argue that the only reason one would choose some model over another, which calculate the same and take account of all phenomena, is mostly due to simplicity. Not because one necessarily represents reality more so than the other. Hence, I'm sure a creative philosopher could argue that your torus world view can't be shown to be wrong if you find clever ways to account for all the various phenomena.

Note: I'm not even sure that the the ancients had the mathematical concept of a torus. I'm probably entirely wrong about this. The greeks were fascinating.

• The ancients did indeed have the mathematical concept of a torus, which they called an anchor-ring. Archytas used a torus with inner radius zero to solve the problem of two mean proportionals (or the duplication of the cube) as early as the first part of the 4th century bce. Commented Apr 9, 2015 at 8:47
• Same comment as above: could you please include references for your answer? Commented Apr 29, 2015 at 15:50
• @VicAche Is that question being addressed to me? This is just knowledge I've accumulated from being a double major in math and philosophy. Commented Apr 29, 2015 at 19:04
• @Valentino I happen to "know" the same things but it doesn't make it an acceptable answer here. You definitely should try looking for references Commented Apr 29, 2015 at 19:26
• @VicAche Honestly, I would rather not I have much more important things to do. Also, when this question was asked it was not originally asked on this specific stackexchange. Rather, it was asked on the math stackexchanged, but one does not need text references when answering there. Good day. Commented Apr 29, 2015 at 21:05

In this following link I've found some mathematical argument, and some of those are really interesting to solve our this doubt. Interested people are welcome. https://math.stackexchange.com/questions/854380/how-to-distinguish-walking-on-a-sphere-or-on-a-torus

• The Foucault pendulum was used to determine the Earth's revolution and topology, reference The Pendulum by Baker and Blackburn. A comparison of pendulum behavior on the sphere and torus in Avron, Joseph E., Daniel Osadchy, and Ruedi Seiler. "A topological look at the quantum Hall effect." Physics Today 56.8 (2003): 38-42. The ancient Egyptian architects had sophisticated uses for the pendulum and encoded Earth measures and precession astronomy in the pyramids, reference Peter Tompkins, Christopher Hills, Robert Gilbert and others in the long answer below. Commented Feb 26, 2016 at 19:25

Eratosthenes (276-194 BC) is credited as "the first" person to have measured the circumference of the Earth in our modern history books, but a serious study of the circumstances shows that Eratosthenes did not actually measure anything, and that he simply copied an earlier method of the Egyptians. Eratosthenes was a Greco-Egyptian director of the Library of Alexandria at the time, and he had access to the ancient records of science collected during the Alexandrian era which contained not only Egyptian but Mesopotamian and Persian knowledge of a variety of kinds. Among these documents was one from which Eratosthenes copied a method of deriving the Earth’s circumference, but the method itself (as we will see) shows that the actual experiment and record of this in Egypt occurred as early as 3750 BC, making “the first” to actually measure a portion of the Earth’s circumference and calculating were most probably Predynastic Egyptians living up to 3500 years before Eratosthenes’ time.

History of Cartography "Around 350 BC Aristotle put forward six arguments to prove that the Earth was spherical and from that time on scholars generally accepted that indeed it was a sphere."

How Aristotle Proved the Earth is Spherical (explains the lunar eclipse observation too)

Aristotle's Astronomy by Thomas Fowler: "To prove that the earth is a sphere"

To prove that the earth is a sphere, he produced the argument that all earthly substances move towards the center, and thus would eventually have to form a sphere. He also used evidence based on observation. If the earth were not spherical, lunar eclipses would not show segments with a curved outline. Furthermore, when one travels northward or southward, one does not see the same stars at night, nor do they occupy the same positions in the sky. (De Caelo, Book II, chapter 14) That the celestial bodies must also be spherical in shape, can be determined by observation. In the case of the stars, Aristotle argued that they would have to be spherical, as this shape, which is the most perfect, allows them to retain their positions. (De Caelo, Book II, chapter 11) ... In Science before Socrates, Daniel Graham argues against the prevalent belief that the Presocratic philosophers did not produce any empirical science and that the first major Greek science, astronomy, did not develop until at least the time of Plato. Instead, Graham proposes that the advances made by Presocratic philosophers in the study of astronomy deserve to be considered as scientific contributions.

...

Virtually all philosophers came to accept Anaxagoras' theory of lunar light and eclipses. Aristotle endorsed Anaxagoras' theory of eclipses as a paradigm of scientific explanation. Anaxagoras' theories launched a geometrical approach to astronomy and were accepted as foundational principles by all mathematical astronomers from Aristarchus to Ptolemy to Copernicus and Galileo-and to the present day.

... Also D.R. Dicks in Early Greek Astronomy to Aristotle, says speculation about the spherical earth can be dated to at least the 6th century BC.

Supporting historical references found on the History Wiki: Spherical Earth

The earliest evidence for a spherical Earth came from an ancient Phoenician expedition for ancient Egypt. The Egyptian pharaoh Necho II, during his reign from 610 BCE to 595 BCE, employed Phoenician sailors to circumnavigate around the entire African continent, then known as "Libya". In The Histories (written c. 431 BCE - 425 BCE), Herodotus described how the Phoenicians reported the sun being observed shining from the north. ... With this expedition, the Phoenicians and Egyptians were thus the first to discover evidence of the Earth being curved and therefore spherical.

The ancient Egyptians ahead of Eratosthenes

Eratosthenes gets the credit for the first earth measure of any accuracy. The Egyptians had the same knowledge some 2000 years earlier. The height of the Great Pyramid x 43200 = Earth Polar Radius The base perimeter of the Great Pyramid x 43200 = Earth Circumference.

.

The Egyptian Heritage in the Ancient Measurements of the Earth, by Gyula Priskin, Published in Göttinger Miszellen, 208 (2006), pp. 75-88.

Abstract: The comparison of an Egyptian text recording the north-south extent of the country with the descriptions of Eratosthenes’ and Posidonius’ experiments to measure the earth’s circumference reveals that early Hellenic mathematical geography borrowed much information from Egyptian science. A further analysis of all the figures circulated for the length of the meridian in antiquity reinforces the case that Hellenic geographers were greatly influenced by Egyptian ideas when they formed their opinions on the size of the earth.

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Egyptian Estimates of the Size and Shape of the Earth From Stecchini's The Pyramids of Egypt.

The Acropolis Width and Ancient Geodesy Nicholas Kollerstrom

“Was the Earth measured in remote antiquity?” This was the stirring question with which Berriman opened his book, Ancient Metrology. To be sure, the question had earlier been tackled in Nicholson’s Men and Measures, as to whether a knowledge of Earth’s dimension had afforded the original basis for units of measure. Here we inquire, specifically, as to whether the ancient Greek units of measure were related to the circumference of the Earth. This hypothesis tends to be related to the notion that a global, maritime civilization had once existed in prehistory. Our inquiry is therefore in some degree related to the thesis propounded by Francis Bacon, in his New Atlantis: “You shall understand (that which you will scarce think credible) that about three thousand years ago, or somewhat more, the navigation of the world (especially for remote voyages) was greater than at this day.” Could there have been a civilization of prehistory which vanished, but left behind its geodetically-defined units? ....

The position here advocated could be described as Newtonian, insofar as Isaac Newton believed, that one function of ancient temples was that of expressing the proportions of the world ....

From his study of the Hermetic texts of the Egyptians, Newton stated in The System of the World what some controversial authors are trying to say today:

It was the ancient opinion of not a few, in the earliest ages of philosophy, that the fixed stars stood immoveable in the highest parts of the world; that, under the fixed stars the planets were carried about the sun; that the earth, as one of the planets, described an annual course about the sun, while by a diurnal motion it was in the mean time revolved about its own axis;

As Steven Weinberg says in his latest book, To Explain the World: The Discovery of Modern Science, (the Dendera Zodiac is shown on the book cover) "At several points in this book I suggest that, as great as is the progress that has been made in the methods of science, we may today be repeating some of the errors of the past." Indeed, it seems as though some forgotten history is repeating itself in the controversy over early knowledge of precession, the spherical earth and what this means for modern science. For instance, in The Zodiac of Paris by Jed Z. Buchwald & Diane Greco Josefowicz "The Dendera zodiac--an ancient bas-relief temple ceiling adorned with mysterious symbols of the stars and planets--was first discovered by the French during Napoleon's campaign in Egypt, and quickly provoked a controversy between scientists and theologians."

This is a fascinating study of how politics, science, and religion intersected in the heated debates over the meanings of the hieroglyphics on a pair of stones brought from Egypt to Paris in 1821. At the heart of the tale is the question of how we know the past. It has the excitement of a real-life archeology mystery combined with a clash between science and theology that has great resonance for today. --Walter Isaacson, author of Einstein: His Life and Universe

See, Buchwald, J.Z. "Egyptian Stars under Paris Skies." Engineering and Science 66.4 (2003): 20-31. Notice of long standing errors on the Dendera Zodiac English Wiki talk page: "It might be that the Dendera Zodiac is academic hotstuff that not many academics dare touching, ... said Rursus." ...and to be compared with the Dendera Zodiac Google French translation:

Jean-Baptiste Biot, astronomer and Egyptologist, meanwhile, looks in detail at the Egyptian circular zodiac.... and shows that the celestial arrangement goes well beyond the Roman era. ... According to Sylvie Cauville, Egyptologist, author of numerous books including The Eye of Re, the founding charter of the Dendera temple is part of the ancient writings of the library of Cheops.

The Temple of Hathor at Dendera:

The temple complex, as it stands today, was built on the site of an older temple, and is a replica of the original. The present building was first initiated by Ptolemy III, with numerous additions by subsequent Ptolemaic and Roman rulers. The inscription on the present temple states that the original building was erected in the far pre-dynastic times, by the followers of Heru (Horus).

Michael Rice, in Egypt's Legacy, "the precession is fundamental to an understanding of what powered the development of Egypt" (p.10).

Update comment: One item in reference to the torus is the discovery of earth's magnetospheric lines having a dodecahedral vortex configuration. So from the etheric the earth is indeed something of a torus. Author John Michell's ancient metrology discusses the blueprint of megalithic monuments as being related to what he called the New Jerusalem or Cosmological Circle, which has the form of a dodecahedron in three dimensions!

Another update to connect some of the references: The Cosmological Circle is also known today as Plato's Wheel, drawn from his Timaeus. Ernest Pecci shows how the scalene triangles within Plato's Wheel are fundamental to the geometry of the Great Pyramid and that the builders knew the precise circumference of the planet. See The Sacred Geometry of the Great Pyramid: From the Drawing Board of Its Architects by Ernest F. Pecci.

More conservative references show the "Pythagoreans were the first ... to say the earth is round." - George Sarton (noted founder of the academic discipline of the History of Science) Ancient Science Through the Golden Age of Greece. Pythagoras was an Egyptian initiate and some of his knowledge was directly attributed to the Egyptians. Both Kepler and Newton were also privy to some of the secret oral tradition. Kepler acknowledged his debt to the Egyptians. Newton went to great lengths in a paper trying to figure out the Sacred Cubit.

Another "source": Ancient Wisdom Discovery that the Earth is spherical -

The first (official) measurement of the radius of the earth was made by Erasthenes (b. 275 B.C.), who was the head of the great library of Alexandria. He was born in Cyrene, now Libya. It seems likely that the ancient Egyptians, much before Egypt's conquest by Alexander the great, had already grasped the idea of a spherical Earth, and it was from them that this doctrine was adopted by Pythagoras, who, as we know, spent many years of study in Egypt.

The Egyptian Pyramid and the Archytas Doubling of the Cube by Pierre Beaudry

... the Great Pyramid included in its very construction frame the idea of doubling the cube. ... Now, as you all know, the geometric problem that Archytas had to resolve was formulated as follows: {find two mean proportionals between two extremes in the ratio of two to one}. In order to find those two mean proportionals, the Archytas construction for the doubling of the cube required a {cone, a torus, and a cylinder}. ... However, this Greek discovery was based on the more ancient Egyptian discovery as its {necessary predecessor}.

Thus, the Egyptian doubling of the cube is simply a derivative of two astrophysical observations that had to be made at the site of the Great Pyramid in order to establish its architectural design. Those two conic projections, from the North Pole and from the Ecliptic, generate the frame-shadow of the Great Pyramid whose triangular meridian angle, PAM, shows that the two proportional segments, AM and AP, respectively represent the sides of two cubes whose volumes are in the ratio of 2/1. So, it becomes clear that this {is} where the Archytas construction took its origins. ... Ironically, this Great Pyramid triangular frame-shadow of 90°, 52°, and 38° degrees, with its harmonically conjugated segments, AB, AM, and AP, not only reflects the power of successively doubling the cube, but also reflects the golden section, the Great Pyramid paradox of squaring the circle, and the 256 series behind the well tempered musical system.

The Constructive Geometry of Pythagorean Sphaerics: Part I by Pierre Beaudry

Pythagoras had established his Astronomy on the original accomplishments of the Egyptians, who, themselves had received their legacy from Atlas, the original Trans-Atlanticist founder of ancient Astronomy, and the first inventor of the celestial sphere, which, according to Jean Sylvain Bailly, can be dated at about 4,000 BC.

The discovery of Precession Astronomy can be traced back, in an architectural documented form, to the construction of the Step Pyramid of Zoser at Saqqara (circa 3400 B.C.).... Between the years 495 and 491 B.C.,... Khnum-Ab-R'a, who was chief minister of works in Egypt, had left an inscription on a public monument of the valley of Wadi Hammamat, which put on record his 24 architect predecessors, leading back to Imhotep .

...

For more on the Torus and Sphere of the Earth, with animations and illustrations:

Thales Theorem and the Archytas Model by Pierre Beaudry

Prehistoric Egyptian Geodesy: "Giza (Heliopolis) to the Equator = 1/12th the circumference of the earth. (360°/12 = 30° 00’)."

There is one point about ancient Egypt which stands out above all others, an insight critical not only to our understanding of Egypt, but also to our overall understanding of history. The ancient Egyptians observed, and to an important degree understood, the precession of the equinoxes. This point is really a subsidiary correlate to the realization that at least circa 2500 BC the Egyptians knew the size of the earth very precisely. Precise geodetic knowledge is contingent upon precise astronomical observations, and both taken together imply an advanced understanding of geometry, as well as precession. It follows that the ancient Greeks should be taken at their word when they claim that their knowledge is of great antiquity and was derived from Egyptian sources. Indeed it is nothing if not bizarre that modern scholars of the Greek world should go to great lengths to dismiss such claims on the part of the authors of the primary texts themselves, to instead rely on the advice of modern Egyptologists that the ancient Egyptians had no such knowledge.

....

After reviewing the opinions and work of the best of the Egyptian astronomical tradition: Sir Issac Newton, Sir John Herschel, and Sir Norman Lockyer, Neugebauer & Parker, Livio Catullo Stecchini, Robert Bauval, and even Schwaller de Lubicz, and finally, visiting the key sites myself, I believe the situation we are faced with is one in which it can be demonstrated that c2500 BC someone designed and oversaw the construction of an object, the Great Pyramid, which encoded exceedingly accurate geodetic information along with profound geometric insight and subtly. While it is disorienting to recognize that so early in the chronology of human civilization there stands such a discontinuous alpha point, it exists, and attempting to dismiss its implications is no substitute for grappling honestly with them.

_

The Culture of Astronomy by Thomas Karl Dietrich: "The Great Pyramid at Giza demonstrates that it is a scale model of the Earth in the proportion of 43,200 Great Pyramids equal to one Earth." Also, from Cosmo Myth, Thomas Karl Dietrich discovered

Substantiating evidence that Eratosthenes did not discover the size of the Earth, but only read about it in the Library of Alexandria (Culture of Astronomy). John Michell already discusses this fallacy in some detail in his famous The New View over Atlantis (p.135-6). Thomas G. Brophy in The Origin Map (p.110-11) actually states that Eratosthenes used data from the latitude and longitude of the ancient Nabta Playa which was also used in the Piri Re'is Map from Istanbul noticed by Charles Hapgood in Maps of the Ancient Sea Kings.

And ... though a mathematical explanation was asked for, Richard Grossinger says in The Night Sky: The Science and Anthropology of the Stars and Planets- "History changes everything, and we must struggle to regain primary experience, whether visionary or scientific." (p.100) The shape of the earth was commonly known in the ancient practice of shamanism (including the Egyptians). Also, E. C. Krupp, Skywatchers, Shamans & Kings: Astronomy and the Archaeology of Power.

OK ... a little math :-) ... The approximate diameter of the earth in miles: 7920 = 8 x 9 x 10 x 11. These numbers are encoded in the geometry of the Cosmological Circle. Michael Schneider, Constructing the Cosmological Circle:

The Cosmological Circle is a geometric diagram that has appeared in the arts, crafts, architecture, religion and literature of cultures around the world, and is associated with their golden ages. Because it's the visual representation of the harmony naturally inherent in the structure of the numbers 1 through 12, it encodes the ideal patterns and proportions toward which nature's forms strive. ... Its dimensions are described as the Heavenly City described in The Book of Revelation, seen in the plan of the Buddhist Brobodur temple, Stonehenge, Glastonbury, and described as Plato's ideal city Magnesia.

Also, 7920 = 55 x 144. Notice the Fibonacci numbers and that 144/55 is approximately equal to the Golden Number squared. Half the value of the the Golden Number squared is equal to the midradius of the dodecahedron. Schneider says the geometry is just a mask for the Number Canon. Now, the question is how did they discover the Number Canon?

The Number Canon removes the geometric mask and looks into the numbers at the heart of the Cosmological Circle. The way the twelve numbers combine and organize into an all-encompassing whole was the model for most of the ancient world as a microcosmic representation of the harmonious universe. It was used as a standard for defining relationships among weights, measures, music and the proportions of sacred art and architecture.

According to WolframAlpha, ... this answer requested by AnubhaV ... is mathematically correct.

Supporting references -- Thoth: Architect of the Universe, by Ralph Ellis, in a review by Elliot Malach he says:

The author ties the measurements and mathematics of the pyramids, Stonehenge and Avebury with the myth of Thoth, who educated mankind in math and the mysteries of the heavens, leaving repositories of knowledge throughout the Earth. Those repositories may not be "inside" these megalithic structures, but instead the fundamental mathematics encoded in the architecture of these structures themselves.

From Wikipedia Thoth:

The Egyptians credited him as the author of all works of science, religion, philosophy, and magic. The Greeks further declared him the inventor of astronomy, astrology, the science of numbers, mathematics, geometry, land surveying, medicine, botany, theology, civilized government, the alphabet, reading, writing, and oratory. They further claimed he was the true author of every work of every branch of knowledge, human and divine.

• Can you summarize in more detail what you found in the references? Commented Apr 10, 2015 at 22:38
• You can edit that in to your answer. Commented Apr 12, 2015 at 17:43
• Most of these references, if not all, are controversial and highly speculative. I'm far from being an expert in the field, but I don't think any of these books meet the criteria implicitly required in this site. Commented Apr 14, 2015 at 1:00
• I disagree, the question isn't controversial or speculative. The worst case scenario would be that there isn't a conclusive answer, which would still be a very valid answer. Also, which particular field are you talking about? In the amazon review of one of these books we read "Shows how the censorship of nonofficial Egyptology as well as new archaeological discoveries continued under Antiquities Minister Zahi Hawass. " So I assume we aren't talking about "official" egyptology. In my opinion this is nothing more than pseudoscience. Commented Apr 14, 2015 at 22:00
• I invite you to defend your position in meta.hsm.stackexchange.com/questions/216/…, where I've raised my concerns about your sources and the pseudoscience problem in general. B.D. Josephson (physics nobel laureate) researches paranormal things nowadays, should we believe everything he believes? Commented Apr 15, 2015 at 18:28