About Mathematics Commentary
Chapter 10. Alternative Geometries
What this chapter is aboutThis chapter is designed to give students some new ideas about geometry: about its history, about its foundations,
and about the benefits it provides us. Except for the first section, it will be entirely new to students. (Even in the 1980s
when solid geometry was taught in high school and analytic geometry was taught in college, the content would have been new to them.)
In it two different geometries are introduced: spherical geometry, which is the geometry that applies to our measurement on the
Earth’s surface, and hyperbolic geometry as a third, more abstract alternative.
I note here that this is my favorite chapter and I believe that it is the best in the book. How sad then that it comes last.
That it does so is due to the insistence of two early reviewers for the Mathematical Association of America. They claimed that students’
hatred of mathematics made its original early placement wrong-headed. I finally went along so here it resides. Note: I cannot
resist adding some comments about that review experience, an experience that some readers will find comparable to their own. The four
MAA reviewers clearly had no communication among themselves for a number of their comments were in direct, even exaggerated contradiction:
this is great vs. you cannot do this. In the end the book was rejected, rightly I believe, because at the time it was dependent on
panels in Texas Instrument programmable calculators and the company did not provide the support the program needed. Happily, that
problem has been solved by our Siena College programmers and our current publisher, Linus Books, is serving us well.
Hopefully some of you who teach with this text will revise their schedule so that this chapter is taught early in their course.
This is easy to do as it is essentially independent of all the other chapters. And I invite feedback about their experiences from
those who do teach it.
Section Notes and Activities:Section 10.1 about triangle area formulas is a kind of red herring. It can serve as a quick review of school geometry for these are formulas
from Euclidean geometry, the geometry that is taught in our schools, and it should be noted, that serves us very well in many
applications. But the point of the introductory problem about the farmer’s property is that it is not solved by any of those
formulas.
For this reason, I suggest that you spend little time on this section other than to have your students see that formulas for
all possible situations have been included and that none of them serve the farmer’s problem.
Section 10.2 about Euclid’s Elements reminds students about the school geometry that almost all of them studied. Before this section is taught,
consider assigning exercises (10.2.2) and (10.2.3) to individual students to present to the class.
Section 10.3 focuses on Euclid’s Postulate 5 and the theoretical problems that arose from it. This is brought into focus by the Playfair
Postulate. We are left with three choices and each of those choices leads to a different geometry. Of course, one of those leads to
Euclidean geometry, our “standard geometry”.
Section 10.4 is about spherical geometry, the Playfair Postulate choice of no lines parallel to a given line through a given point outside that
line. The section text leads to the simple formula of (10.4.9) and the solution of the farmer’s problem through the use of
Panel 10.4.1.
The hardest part of this development is the justification of the formula for the area of a spherical triangle. If you wish to
develop this formula with care, I urge you to obtain a large spherical ball on which you can mark a spherical triangle together with
the lunes made by extending the rays of the triangle angles. This same ball or a similar one can also be used to show how great
circle distances are the shortest between two points.
Section 10.5 and 10.6 depart from the foundations line of development, to which the text will return in the final Section 10.7.
Section 10.5 is about locations on Earth as designated by latitude and longitude. The key to this section is the difference between watch time
and sun time and this is best observed when the sun is at its maxim height, thus at noon.
Today that problem is no longer a concern because our technology easily bypasses it; we get location fixes through extremely
accurate distance and direction measurements by satellites. Until the late 20th century, however, this determination was made by
taking noon sun-sights with a sextant.
Section 10.6 about the distance between two Earth locations, provides a formula and associated panel that gives approximate values.
Section 10.7 on hyperbolic geometry addresses the third alternative to the Playfair Postulate: more than one line parallel to a line through
a given exterior point.
I urge you to place the idea of modifying Euclid into a larger context by discussing and comparing the situation with other
times when accepted thinking was rejected: (1) Einstein’s relativity vs. Newton’s physics; (2) Copernicus’s and Galileo’s rejection
of the Earth as the center of the universe; (3) Darwin’s evolution upsetting religious beliefs; and (4) Wegener’s plate tectonics
vs. a fixed Earth. You might want to refer to Wikipedia’s entry “Paradigm Shifts” before discussing this with them.