Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: We will begin with a couple of topics from Chapter 17 (which we only read before).
  
      The feature this week is the story of non-Euclidean Geometry. This is a subject I have taught a few times (it is Math 467 in the TAMU undergraduate curriculum). The treatment in Stillwell is quite superficial (this is a flaw/feature in his book, and perhaps unavoidable as Mathematics has a very long history). One challenge in teaching and discussing non-Euclidean geometry is that there are no good models for it. Studying it in the abstract (e.g. via Saccheri's lines, or more fundamentally his quadrilaterals), one has obviously curved lines that are declared to be straight lines, and it does defy our intuition. Beltrami's model in Section 18.4 has straight lines (segments in the disc) and a reasonable notion of parallel, but distance is odd (how do you compute it?) and angles mean something other than what you see. Many geometers and all modern mathematicians find that Beltrami's conformal models in Section 18.5 to be the right place to work in hyperbolic geometry. Of particular note is the upper half plane model (I also like the two views of projecting to the disc and upper half plane). This makes sense in any dimension (the set of points in Rⁿ with last coordinate positive, where the metric is scaled by the inverse of the last coordinate, generalizing what you see on page 371 for n=2 and then page 372 for n=3.) Still, these models leave something to be desired, in my opinion and from my experience teaching.
  
      You are invited to explore my webpage on the hyperbolic football, as well as the description of constructions with it. My time in Germany and the coronavirus crisis kept me from mailing you information for an activity based on this that I like to do with students and teachers. Last week I was supposed to go to a math teacher's circle in Austin and make hyperbolic footballs. (This has been postponed until the Autumn.)
  
  
• Reading:
  1. Chapters 17 and 18 in Stillwell's book. This is the story of the discovery of non-Euclidean geometry.


• Assignment: Due Monday, 6 April 2020, (HW 12)
  To hand in: We are using E-campus for homework submission.
  1. Please do the exercises in 17.4 about the curvature of the pseudosphere, numbers 17.4.1 and 17.4.2.
  2. Please do exercises in Section 17.6 (there are typos here), numbers 17.7.1 and 17.7.2. Here, excess means in excess of the angle sum of a polygon. Having done that, use the idea of 17.7.2 to deduce that the angle sum of a polygon in the Euclidean plane with n sides is π(n-2). Along the same line of spherical geometry, do problem 18.2.1.
  3. Look up Gauss on the web and write about one mathematical accomplishment of his that is not explained in either book, or only explained in passing. This should be brief; such as a paragraph.