Introduction:
Today, the groups will work with two physical aids,
(S)A transparent sphere with ruler and right-angle.
(D)Circles with centers and file cards. 

On the Sphere

Question 1.
Prove that the shortest distance between two points
is realized on a great circle passing through them.

Answer. This open ended. I was looking for some good 
ideas. You could have drawn a spherical triangle ABC
and measured that the length of AB is shorter that
BC added to CA (count the hatchmarks on the ruler).

Question 2.
Interpret Points = points and Lines = great circles 
and Parallel= no common Point. 
Check each of Euclid's Postulates, starting with 
E1: Two Points determine a unique Line joining them.  

No, antipodal points count for two Points and are 
joined by infinitely many Lines (the longitudes.)

E5: Through every Point off a Line there is 1 Parallel.  

No. There are no Parallels at all here. Any two great 
circles meet, twice, at antipodes.

E2: A line Segment joining two points can always be 
    extended in both direction.

Yes, just keep walking along the great circle. You'll
never have to stop, but you'll be covering the same
route forever. The Line is not infinitely long, but
it never stops.

Question 3. (deferred) 
Decide what Circle and Right angle is on a sphere.
Check these two postulates, but not very deeply!

E3: A Point (center) and a Segment (radius) determine
    a Circle. 
E4: All Right angles are Equal.

Question 4. Check whether the Angle Sum Theorem holds. 

No. Consider a Rectangled Triangle from the north pole
to the equator, along the equator for 90 degrees and
then back up to the porth pole. This Triangle has 
angle sume of 270 degrees.

Question 5.(defer!) Check whether SAS holds?

Question 6.(defer!) Does Pythagoras hold?

Next, you can repair the problem with antipodal points
in these three ways. (Only two for today.)

Interpret Points = antipodes, Lines= great circles

Question 7.
Show that this reestablishes the uniqueness in E1. 
What doesn't this reinterpretation fix? Is E5 true?

Now all those great circles joining antipodes are 
just a long Lines all going through a single Point.
But through two Points that are not antipodal, there
is a unique plane through them and through the center.
Therefore there is a unique Line joining them. 

Next Points= points in the southern hemisphere
excluding the equator. Lines = great arcs. 

Question 8. Now is E5 true? why.

Finally, it's true again. Here's why. Given an Line
and a Point there is exactly one great circle through
the Point and which meets the great circle through 
the original point exactly on the equator. But we
have eliminated the equator. So these two great arcs
don't meet, and so Playfair is true again.

(Defer) Finally, add the points on half of the 
equator and study this model of the projective plane.

Notice that in the third example, central pojection
sets up a 1-1 correspondence between the lower 
hemisphere and the plane tangent to the south pole.
Think of looking with one eye from the center of the
sphere, through the hemisphere, at stuff in the 
base plane. 

So E5 *had* to be true, since it holds in the 
Euclidean plane!