This week we take a look at chapter 9, "Non-Euclidean Geometry and Nonorientable Surfaces." This chapter contains information that I could have been taught in 9th grade geometry but never was. In fact, I thought much of what was discussed in the latter part of the chapter (which included 3-D non-euclidean geometry and higher dimensions) could easily have been incorporated into higher level geometry in high school.
I found particularly interesting the section that dealt with the development of non-euclidean geometry and the relationship between reason and geometry. Having read "Critique of Pure Resaon" and absolutely hated it, I found it particularly amusing that Immanuel Kant and his followers were in the thick of this. I never realized that geometry had played such an integral part in the philosophical problems of the times. What interests me is whether or not geometry was at the forefront of the reason movement or was it merely a minor, insignificant point that merely added creedence to the basic principles that came out during the Age of Enlightenment? Also, did Kant himself actually write anything on the subject of geometry or was this merely implied by his writings on other subjects? It seems like a simple and logical connection, but I just wonder whether or not it truly was that simple.
Also, concerning Karl Friedrich Gauss, why have I never heard of him before? In geometry we hear about Euclid and devote the entire year to learning euclidean geometry, but why isn't Gauss and his non-euclidean geometry ever discussed even if it is only mentioned?