Interrogating the 5 Postulates

The 5 Postulates

• Postulate 1. “To draw a straight line from any point to any point.” • Postulate 2. “To produce a finite straight line continuously in a straight line.” • Postulate 3. “To describe a circle with any center and radius.” • Postulate 4. “That all right angles equal one another.” • Postulate 5. “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” • Playfair’s axiom: Given a line, l, and a point, P, not on l, there is exactly one line parallel to l and passing through P.

[url=https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.researchgate.net%2Ffigure%2FThe-great-circle-distance-between-origin-and-destination_fig2_320026251&psig=AOvVaw1-GMEJXc4RYWtOc6cRacP8&ust=1714701916916000&source=images&cd=vfe&opi=89978449&ved=0CBQQjhxqFwoTCMiZk8fw7YUDFQAAAAAdAAAAABAP]The great-circle distance between origin and destination.... | Download Scientific Diagram[/url] — The great-circle distance between origin and destination.... | Download Scientific Diagram

Postulate 1

Postulate 1 holds in spherical geometry but not in the exact way it does in Euclidean geometry. In spherical geometry, lines are defined by geodesics. A geodesic is to a great circle(pictured above) as a line segment is to a line. Therefore, we are Abe to draw a "straight" line from any point to any point.

Postulate 2

While you can extend a geodesic in spherical geometry, it cannot go continuously on forever. This is due to Euclid assuming that lines could always be extended infinitely. In spherical geometry, the longest that a line can be is the length of the great circle which is equal to 2(pi)r.

Postulate 3

Postulate 3 also has no issues. A circle is still described as the set of points a given distance away from one point. You can still draw a circle on a sphere it just may not look exactly like a circle in 2D. The circle can look like it does in 2D if you consider the plane tangent to the center of the circle or think about looking at the circle from directly above it.

Postulate 4

There isn't a whole lot to talk about with spherical and postulate 4 other than that it holds. Just like in Euclidean geometry, all right angles are equal to one another.

Postulate 5

Postulate 5 is an interesting one to discuss. Instead of breaking down his wording, we will talk about Playfair's axiom which is logically equivalent to Euclid's postulate 5. This says that there is one line parallel to another line that passes through a given point. Consider the lines of longitude that are vertical lines on the Earth. Each of these lines is a great circle and as we see on the Earth, all these lines meet at the north and south poles. This is also true for all great circles on any sphere. As Nat showed us in class, all great circles intersect in 2 places on the circle. From this we can conclude that all great circles and further that all geodesics(lines on spheres) intersect and that there are no parallel lines in spherical geometry.