![]() In order to navigate the Earth’s surface, such as on a ship on the ocean, all we need is these two units of measure – the direction in which to travel, which can be given using a compass as an angle measured from due north, and the distance by which to travel, which can be calculated using a ruler on a map together with measurements of the ship’s speed, or, for longer distances, with calculations based on star positions.Įuclid was able to create a powerful way to systematize the knowledge the Greeks developed regarding “the measure of the Earth” – geometry. Replica of a Trireme, the ship used by Greeks such as Alexander the Great (around 334 BC) for both warfare and exploration. Euclid’s system was especially powerful and influential since it organized Greek knowledge regarding geometry – the measure of the Earth’s surface – into a coherent, logical system. These tools were made into a system of geometry by a Greek man named Euclid, who lived around 300 BC. The ancient Greeks, assuming the Earth was flat, developed a powerful set of tools to use to navigate the surface of the Earth, such as sailing across the Mediterranean Sea. It is also possible to determine that the Earth is not flat simply by making measurements directly on its surface, using a property known as holonomy. Both of these observations imply that the Earth is not flat. You can see this directly on the Earth by looking at the Earth’s curved shadow as it crosses the moon, and by watching ships as they approach from the sea – the first part of the ship you see is the top of the mast, and the body of the ship becomes visible as it sails closer, which is a result of the fact that the ship is in fact sailing around the curved Earth, not on top of a flat Earth. ![]() So how did we find out that the Earth is round, without flying to the moon? This literal view of the flat Earth is also promulgated in the Old Testament of the Bible, and other historical sources. This has both literal and symbolic connotations – the man is both literally sticking his head beyond the edge of the Earth, and also, symbolically, attempting to gaze beyond the Earth’s surface into the unknown. In this 1888 French depiction of medieval cosmology, a man stretches his head beyond the Earth’s firmament, attempting to gaze into the heavens. Here is a symbolical depiction of the edge of the world: Flammarion’s 1888 engraving depicting the edge of the Earth, from tumblr: knowgnosis Many of the people of the ancient world are said to have feared travelling too far in a ship, due to their belief that they would eventually reach the edge of the world, and fall off. The moon is also a sphere (approximately), but from someone on its surface, it appears to be flat. Here’s a NASA image of the Earth as seen from the moon during an Apollo mission in 1969. From our vantage point as relatively small creatures living on the surface of the planet, it appears that we are living on a more-or-less flat surface, like the top of a table, rather than a curved object that’s very close to a sphere in shape. Good point – the Earth appears to be flat overall, with irregularities such as mountains and valleys. ![]() In elliptic geometry the lines "curve toward" each other and eventually intersect.But it doesn’t appear to be flat – what about all the mountains and valleys? In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular these lines are often called ultraparallels. In Euclidean geometry the lines remain at a constant distance from each other, and are known as parallels. (See the entries on hyperbolic geometry and elliptic geometry for more information.)Īnother way to describe the differences between these geometries is as follows:Ĭonsider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any pair of lines intersect. Euclid's 5th postulate is equivalent to Playfair's Postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry.
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