E/NE 2: Let's Not Take Sides

Euclidean and hyperbolic geometry differ in only one respect, namely, the parallel postulate: given a line l and a point P not on l, Euclidean geometry says that there is exactly one line through P which doesn't intersect l, while hyperbolic geometry says that there are more than one. Neutral geometry is the body of results that can be proven in either kind of geometry - i.e., everything that can be proven using the axioms other than the parallel postulate (either version). Among the theorems of neutral geometry are the side-side-side, side-angle-side, and angle-side-angle theorems (conditions under which two triangles can be guaranteed to be congruent), the fact that two circles intersect in at most two points, and the fact that, given our old friends l and P, there is a line through P which is perpendicular to l. There's one more useful result, the Saccheri-Legendre theorem, which will be the subject of this post. (Saccheri was an eighteenth-century geometer who came very close to discovering non-Euclidean geometry, but couldn't quite make the intellectual leap.) This theorem says: The sum of the angles of a triangle is at most 180 degrees. (In Euclidean geometry, it's exactly 180 degrees; in hyperbolic geometry, it's less than 180, and varies depending on the triangle.) I'll put the proof under the cut.
( Saccheri-Legendre )

The argument under the cut is rather involved; if you have trouble following it (even after drawing some pictures), don't hesitate to say so.

Sometimes It's Worth It

Today, in calculus, I began discussing sequences and series. As a teaser, I described an argument that most of my students had probably seen and accepted in high school, and then handed them another argument, superficially parallel, which leads to an obviously false conclusion. I asked, "Why does this argument (pointing to the first one) work, while this one (pointing to the other) doesn't? I'll let you think about that." Then I dismissed the class. One of my students - the same one who caught on to what I was doing on the test a couple of weeks ago - grimaced at this.

Leaving the building, I saw him ahead of me. (The doors are glass; he was already outside.) He glanced back, saw me, and grinned. As I came through the door, he hurried back, pointed at me, and said, "It's the 'infinity minus infinity' case, right?" I said yes, although there was a bit more to it than that. He said, "Great! Now it won't be bugging me all night." He went on his way. As I headed back to my office, I was grinning as broadly as he had been.

Sometimes it's worth it.