Spherical Geometry And Navigating The Globe

Geometry derives its meaning from the Greek words geometria and geometrein which mean "measuring the earth". Geography, on the other hand derives its meaning from the Greek words geographia and geographein which mean "describing or writing about the earth". One would expect words so similar in meaning to be similar in concept as well. However, the two fields were separate and distinct until the days of ancient Greece, when Ptolemy (astronomer, mathematician and geographer) made use of geometry in reasoning more about the earth and its shape:

"In Geography one must contemplate the extent of the entire earth, as well as its shape, and its position under the heavens, in order that one may rightly state what are the peculiarities and proportions of the part with which one is dealing... It is the great and exquisite accomplishment of mathematics to show all these things to human intelligence..."


Interestingly enough, it was also Ptolemy and not Christopher Columbus who discovered that the earth was spherical and not flat, and stated his rationale in the Almagest 1300 years before Columbus sailed around the world:


"If the earth were flat from east to west, the stars would rise as soon for westerners as for orientals, which is false. Also, if the earth were flat from north to south and vice versa, the stars which were always visible to anyone would continue to be so wherever he went, which is false. But it seems flat to human sight because it is so extensive."

Like geometry and geography, the worlds of spherical geometry (used in geography) and planar geometry (commonly taught in most geometry courses) are closely related and yet extremely different.

Anybody who has completed high school level geometry (or to some extent, elementary geometry) knows that in Euclidean or planar geometry, two parallel lines never meet, the sum of the three angles of a triangle add up to 180°, and the shortest route to get from one point to another is a straight line. In the world of spherical geometry, two parallel lines on great circles intersect twice, the sum of the three angles of a triangle on the sphere's surface exceed 180° due to positive curvature, and the shortest route to get from one point to another is not a straight line on a map but a line that follows the minor arc of a great circle. Maps provide a way of translating the spherical view of the world to a planar view, by projecting the Earth's topologies and locations to a flattened surface using Hammer, Mercator or cylindrical methods. A consistent and standard representation that minimizes projective distortions is yet to be established.

The discovery of spherical geometry not only changed the history and the face of mathematics and Euclid's geometry, but also changed the way humans viewed and charted the world. Using this new knowledge, explorers and astronomers used the circular path of stars to navigate the earth to discover new lands and reason about the cosmos.

Less than 50%...

[link to www.math.ubc.ca]


Can "Flat Earthers" explain this?