Spherical Geometry

The sole exception to this rule is one of the main characteristics of spherical geometry. Two points which are a maximal distance apart, namely half the circumference, are said to be at opposite poles, such as the North and South poles. Every point P has its opposite pole, called its antipode P, obtained by drilling from P straight through the center of the sphere and coming out the other side. “Digging through the earth to China," so to speak.

Non-Euclidean geometry is an example of a paradigm in the history of science.

The major difference between spherical geometry and the other two branches, Euclidean and hyperbolic, is that distances between points on a sphere cannot get arbitrarily large. There is a maximum distance two points can be apart. In our model of the earth this maximal distance is half the circumference of the earth, roughly 12,500 miles. Distance between two points on a great circle is always the shortest distance between them. The distance between Philadelphia and New York is 120 miles, not the 24,880 miles it would take to go around the other side of the world. The triangle inequality holds in spherical geometry.

In flat space Find any two points and draw the STRAIGHT LINE they define in space. At A and B erect perpendicular, straight lines. Mark these angles.

In Spherical Geometry Find any two points A and B on the equator and draw the STRAIGHT LINE they define. At A and B erect perpendicular, straight lines. Mark these angles.

Find the midpoint of AB and call it Q. Construct a perpendicular from AB at Q. Project it until it eventually meets AO and BO. (It must meet them since there are no parallels in this geometry.) Where will they meet?

repeat the construction. Bisect AQ and from that point erect a perpendicular that will pass through O. Bisect QB and from that point erect a perpendicular that will pass through O. By repeating this process indefinitely, we can divide the original interval AB into as many equal sized parts as we like. Perpendiculars raised from each of these points will all pass through the point O. As before, all these perpendiculars will have the same length.

There is also a circle in the figure. While the line AGG'G''G''' is a straight line, it also has the important property of being the circumference of a circle centered on O. Every point on AGG'G''G''' is the same distance from O. That is the defining property of a circle. And what an unusual circle it is. It has radius AO. That radius AO is equal in length to each of the four segments AG, GG', G'G'', G''A that make up the circumference. Radius = AO Circumference = AG + GG' + G'G'' + G''A AO = AG = GG' = G'G'' = G''A That means that the circle AGG'G''G''' has the curious property that Circumference = 4 x Radius Contrast that with the properties familiar to us from circles in Euclidean geometry Circumference = 2π x Radius A longer analysis would tell us that the area of the circle AGG'G''G''' stands in an unexpected relationship with the radius AO. Specifically Area = (8/π) x Radius 2 In Euclidean geometry, the area of a circle relates to its radius by Area = π x Radius 2

Let us return to our starting point. Euclid's achievement appeared unshakeable to the mathematicians and philosophers of the eighteenth century. The great philosopher Immanuel Kant declared Euclid's geometry to be the repository of synthetic, a priori truths, that is propositions that were both about the world but could also be known true prior to any experience of the world. His ingenious means of justifying their privileged status came from his view about how we interact with what is really in the world. In our perceiving of the world, we impose an order and structure on what we perceive; one manifestation of that is geometry. The discovery of new geometries in the nineteenth century showed that we ought not to be so certain that our geometry must be Euclidean. In the early twentieth century Einstein showed that our actual geometry was not Euclidean. So what are we to make of Kant's certainty? Einstein gave this diagnosis in his 1921 essay "Geometry and Experience."