The Geometric Viewpoint: A Survey of Geometries

Errata: Section 1.1, Problem 8

"overheard" should be "overhead."

Section 1.2 Remarks after Problem 4 Insert "a power of two or" in the second sentence after "is". Change 65,536 to 65,537. Also, there are more Fermat primes known, including the largest current one (1998) of k =303,088. Section 1.3, Answer to Problem 1(c)

Three points and three lines.

Section 1.4 Example 4 Change "Euclid's first four postulates" to "all five of Euclid's postulates." Add after that sentence "(Note that Playfair's Axiom does not hold.)" Problem 7 c) The answer in the Instructor's Manual should be "No. Consider P = (0,0), Q = (2,1), R = (2,2) and S = (1,2)." Section 1.6 Problem 10 b) Add at the end: The family of planes must together completely cover both solids. Chapter 2, Project 7

The formula in (e) should have a + in front of the square root.

Section 3.1 The paragraph from page 101 to 102 needs to be rewritten. There is no evidence that Riemann knew about hyperbolic geometry at the time of his 1854 talk. In the paragraph on the pseudosphere (page 105) omit the sentence starting with "However." End the next sentence after "portions" and add two new sentences: "It is possible to extend either portion, but the surface is no longer 'analytic.' In particular, it becomes increasingly wrinkled." Problem 4 c) Insert "exactly" between "have" and "one." part f) Add at the end: "We can think of two vertical lines as meeting at 'infinity,' outside of the model, and so they are also sensed parallels to each other." Section 3.2 page 109 Replace the first line with "...either UA or PA. Now UA is below l and l intersects the line PA already at T, giving a contradiction either way. Hence m is the..." Section 3.3 page 114 In the second line of the proof of Theorem 3.3.3, the final symbols should be for the line DE, rather than the line segment DE. Section 3.4, Figure 3.28 The letters G and D should be shifted to the right so that G is on the perpendicular and D is on the line through A and B. Section 3.5 Second paragraph on page 125 Insert the remark "For SAS to hold we need to assume that a side is the shortest part of a geodesic." Chapter 4 biographies of Klein (page 143) and Lie (page 155) friends" to "became close friends for many years."

Change "became lifelong

Section 4.2 Theorem 4.2.5. In the proof the order of composition is wrong both times. The subscripts should be in decreasing order, not increasing order (2,1 and 3,2,1). Theorem 4.2.6 page 142 line 2. Change the P to A'. Section 4.4, Example 5 "Figure 3.23" should read "Figure 4.23." Section 6.3, Problem 9(a)

An extra condition is a + c + f = 0.

Section 6.5, Answer to Problem 4 The matrix should have b rather than -b in both places. In the last two equations omit the minus signs after =. Section 6.6, Problem 8(a)

The interior of the hyperbola is x^2 - y^2 > 1.

Chapter 7, page 265 In the biography of Euler, omit the reputed quote because it was in reference to another mathematician, Joseph Louis Lagrange. Chapter 7, Project 8

The last three parts should be (d), (e), and (f).

Appendix B, page 293 Replace Axiom II-2 with Hilbert's latter amendment of this axiom: If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D.