Euclidean Geometry

• The Greek mathematician Euclid first systematized ordinary plane geometry around 300 BC. Euclid presented postulates of geometry that he considered so basic as to be self-evident -- such as a straight line being the shortest distance between any two points, that any line can be extended indefinitely, that a circle can be drawn with any center and any radius and that all right angles are equal to each other.
  
  Euclid's fifth postulate, also known as the parallel postulate, defines the notion of parallel lines. To put it simply, it states that any two lines will eventually meet unless the angle between them and a third line crossing them equals 180 degrees. In this one case, the two lines will extend indefinitely without meeting.
  
  

Flat Versus Curved

• Because of its relative complexity, the parallel postulate was not accepted as readily as the others, and mathematicians debated for many centuries about whether it could be derived from other rules. Finally, in the 19th century, several mathematicians constructed new geometries in which the parallel postulate did not hold. All of these take place on curved surfaces, and all of them cause geometrical figures such as lines and triangles to behave differently than on a plane.
  
  

A Small Adjustment

• Spherical geometry is a special case of what is called elliptic geometry, in which there are no such things as parallel lines; two lines that start out parallel will always eventually meet. This changes the way that shapes are constructed. For example, in plane geometry, the three angles that make up a triangle always add up to 180 degrees. In spherical geometry, on the other hand, the sum of these angles is more than 180 degrees.
  
  On the Earth's surface, two meridians going north from the equator at 90-degree angles will meet at the North Pole at some angle. This third angle creates a sum greater than 180 degrees.
  
  

Brave New World

• While some aspects of spherical geometry date back nearly to Euclid, the discovery of its mathematical basis yielded several applications. The first, logically enough, was in mapping; knowledge of exactly how shapes behave on a spherical surface allowed for improvements in map projections.
  
  More dramatically, non-Euclidean geometry turned out to be very important in cosmology. Albert Einstein deduced that space is curved, in his theory of general relativity; scientists since have attempted to find the exact way in which it curves. Spherical geometry and other non-Euclidean geometries have been highly useful in this area.