6.4: Revisiting Euclid's Postulates
Without much fanfare, we have shown that the geometry \((\mathbb{P}^2, \cal{S})\) satisfies the first four of Euclid's postulates, but fails to satisfy the fifth. This is also the case with hyperbolic geometry \((\mathbb{D}, {\cal H})\text{.}\) Moreover, the elliptic version of the fifth postulate differs from the hyperbolic version. It is the purpose of this section to provide the proper fanfare for these facts.
Recall Euclid's five postulates:
- One can draw a straight line from any point to any point.
- One can produce a finite straight line continuously in a straight line.
- One can describe a circle with any center and radius
- All right angles equal one another.
- Given a line and a point not on the line, there is exactly one line through the point that does not intersect the given line.
That the first postulate is satisfied in \((\mathbb{P}^2, \cal{S})\) is Theorem \(6.2.3\).
The second postulate holds here even though our elliptic space is finite. We can extend line segments indefinitely because the space has no boundary. If we are at a point in the space, and decide to head off in a certain direction along an elliptic straight line, we can walk for as long and far as we want (though we would eventually return to our starting point and continue making laps along the elliptic line).
The third postulate follows by a similar argument. In the previous section we defined a circle about any point in \(\mathbb{P}^2\text{.}\) And since we can walk an arbitrarily long distance from any point, we can describe a circle of any radius about the point.
The fourth postulate follows since Möbius transformations preserve angles and the maps in \(\cal{S}\) are special Möbius transformations.
The fifth postulate fails because any two elliptic lines intersect (Theorem \(6.2.5\)). Thus, given a line and a point not on the line, there is not a single line through the point that does not intersect the given line.
Recall that in our model of hyperbolic geometry, \((\mathbb{D},{\cal H})\text{,}\) we proved that given a line and a point not on the line, there are two lines through the point that do not intersect the given line.
So we have three different, equally valid geometries that share Euclid's first four postulates, but each has its own parallel postulate. Furthermore, on a small scale, the three geometries all behave similarly. A tiny bug living on the surface of a sphere might reasonably suspect Euclid's fifth postulate holds, given his limited perspective. A tiny bug in the hyperbolic plane would reasonably conclude the same. Small triangles have angles adding up nearly to \(180^{\circ}\text{,}\) and small circles have areas and circumferences that are accurately described by the Euclidean formulas \(\pi r^2\) and \(2\pi r\text{.}\) We explore geometry on surfaces in more detail in the next chapter.