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That just means that the curvature is everywhere the same.

The surface of a sphere has constant curvature. Of a sphere, such as is our own earth, to very good approximation. The geometry of 5 NONE proves to be very familiar it is just the geometry that is natural to the surface The New Geometry of 5 NONE is Spherical Geometry We now risk a new danger: we may just overlook the fact that we are really dealing with new and different geometries. It is a trick that makes visualizing the geometries of 5 NONE and 5 MORE easy. We will see in this chapter how this arises. Recognizing that fact makes it easy to visualize these new geometries and one rapidly develops a sense of the sorts of results that will be demonstrable in them. Naturally in surfaces of constant curvature. The geometry of 5 NONE and the geometry of the other postulate 5 MORE turn out to be the geometries that arise The surprising thing is that this is not so. You would be forgiven for thinking that the new geometry of 5 NONE is a very peculiar and unfamiliar geometry and that there is no easy way to comprehend it as a whole. So the sum of its angles is three right angles (and not the two right angles dictated by Euclid's geometry). The triangle OGG', for example, has three angles, each of one right angle. Each of its quadrants are triangles with odd properties. Its cirumference is both a circle and a straight line at the same time. Its circumference is only 4 times is radius (and not the 2π times its radius dictated by Euclid's geometry). We constructed a circle with center O and circumference G, G', G'', G'''. The outcome was a laborious construction of circles and triangles with some quite peculiar properties. We drew lines and found points only as allowed by the various postulates. In the last chapter, we explored the geometry induced by the postulate 5 NONE by means of the traditional construction techniques of geometry familiar to Euclid. The New Geometry of 5 NONE is Spherical GeometryĮuclid's Postulates and Some Non-Euclidean Alternatives.