Hints and Answers

CHAPTER 1

1.1 i: p any false statement; ii: q any true statement. 1.2 Use contrapositives, e.g.: (ii') If a ~ b, then c ~ d. 1.4 [a] contains all integers 5k + a where k is an integer. There are 5 equivalence classes: [0], [1], [2], [3], [4]. 1.5 FTFTT - FFTFT. 1.6 All points; the empty set.

CHAPTER 2

2.1 AlII-I; only ft and ~ onto. 2.2 2x-l, 2(x-l), 2x'-I, 8(X-l)3, 2(x'-l). 2.3 from, into, of, at, to, under, onto, one-to-one, on. 2.4 A=B={1, 2},D={(I, 1), (2,2), (1, 2)},f(1)? 2.6 TTFFF - TFFTF. 2.7 nn, n!, n!, n!. 2.9 D=z, C= R,f(x) =x. 2.10 D=R,f(x)=-x. 2.12 e.g.: {(x) =x+ 2 if x < 0 and {(x) =x-3 if x ~ O.

CHAPTER 3

3.4 1 from 03; 2 from 01, 3 is 02, 4 through 8 from definition of >; 9 from 7 and 8; 10 from 2, 5, and 8. 3.5 FTTTT - FFFTF. 3.6 {(n) = n+ 1. HINTS AND ANSWERS 495

3.10 Not a or b. 3.13 Historical precedence.

CHAPTER 4

4.2 ,'j'J points of Euclidean plane, Y' Euclidean lines through some fixed point in the Euclidean plane, usual incidence; Euclidean plane; Euclidean line. 4.3 Axiom 1: old P and old Q on unique old line and both off Ix; Pn and Pm on lx' the only line with two old pencils on it; Pn and old Q on old line through Q and parallel to n. Axiom 2/: if old I parallel to old m then both through only PI; if old P on old I and old m, then no other new point on both old I and old m; Pm is only new point on old m and Ix' 4.4 TFTFT - TFTFF. 4.6 Affine plane and Axiom 4: Every line has at most two points on it. 4.7 Y the set of points on a Euclidean sphere, 2' the set of circles on the sphere, usual incidence. 4.8 Let (,':!"3'Y3' '~3) be derived from E in Figure 4.4. Take 1,0, E, P', P as in Figure 4.4. Then P', Pm' m, (in (93, :£'3' ,~) correspond, respectively, to P, Euclidean line in I through 0 and parallel to m, plane through m and 0, I.

CHAPTER 5

5.1 See Figure 4.2. 5.2 (x, y) to (x, y) if x ~ 1 and (x,y) to (x-l,y) if x > 2 determines a collineation onto Ml. 5.4 x= 1; y= -1; y= -2x+ 5; y=x if x ~ 0 and y= Ij2X if x> 0; y=4x- 3 if x ~ 0 and y= 2x-3 if x> 0; 3y= 8x+ 5 if x ~ 0 and 3y= 4x+ 5 if x> O. 5.5 TTFFT - FTTFT. 5.7 (x, y) to (x, y1) gives a collineation from Ml to M9. 5.8 Cardinalityargument. 5.10 Compare with (92, 2'2' ~2) of Section 4.2. 5.18 Use Gauss plane and Exercise 3.22.

CHAPTER 6

6.2 For y= mx+ b, take f( (x, mx+ b)) =x(1 + m2)1/2. 6.3 First sentence of Section 6.3. 496 HINTS AND ANSWERS

6.6 Use Exercise 6.5 on Ml. 6.7 TFTTF - TTFFT. 6.14 Taking P=2 and Q=6 in the Gauss plane, then PQ=4 and ~ PQ=R. 6.15 All except M2.

CHAPTER 7

7.4 Draw the streets and avenues through the two points; consider equation in each of the nine regions in the resulting tick-tack-toe figure. 7.5 Theorem 7.7. 7.7 TFTTF - FTTFF.

CHAPTER 8

~ 8.4 Let Hl and l!..t be such sets. Let A - V - B with l = AB and A E

Hl. Show P E int (VB) implies P ft..Hl but P E H2• So BE H2• Then, -~ Q E int (VA) implies Q ft. Hz but Q E H l • 8.5 Cartesian line through A and B in Figure 8.6 contains A and B but not W. 8.6 TFFTT - TFTFF. 8.8 Same as Cartesian plane. 8.9 A off PQ in Figure 7.5.

8.11 Take M as A in Figure 7.5 with Xl =Yl and x2 = Y2•

CHAPTER 9

9.2 Figures 9.15 and 9.16. 9.4 Figure 9.16. ~ ~ 9.5 All points off both VA and VB. (3,3) is not in inside. 9.6 FTFTF - TTFTT.

CHAPTER 10

10.2 Tetrahedron, octahedron, hexahedron, icosahedron, and do• decahedron. 10.3 FTTTT-TFFFT. HINTS AND ANSWERS 497

CHAPTER 12

12.3 Use Theorem 12.14. 12.5 TTFTF - TTFTF. 12.7 See Section 8.2. 12.9 Yes. 12.15 This is Sylvester's original conjecture. 12.17 False.

CHAPTER 13

~ ~ 13.1 A, E, F on a side of CD; Band D on opposite sides of AC as are B andF. 13.3 Although distance is not Euclidean distance, betweenness for points in the model is the same as Euclidean betweenness. 13.4 Use Sylvester's Theorem. 13.5 TTTTF - FFFFF.

CHAPTER 14

14.4 Xl < x2 < X3 iff V3x I < V3x 2 < V3x3 • 14.5 Lines with nonzero slopes n l and n2 are perpendicular iff n l n2 = -3. 14.6 FTFFF-FTFTF. 14.7 If P= (-1,1), then m'LAVP is Tr13, 2Tr13.

CHAPTER 15

15.1 Veblen's axiom system is categorical, every model being isomorphic to the set of Cartesian points with the conventional be• tweenness relation. However, since Taxicab geometry and Cartesian geometry share the same points and betweenness relation, Veblen's system does not fully describe what is usually called Euclidean geometry.

CHAPTER 16

---+ 16.2 With mLA=mLA' and mLB=mLB', takeD onBC to build /::.ABD as a copy of /::.A'B'C'. 498 HINTS AND ANSWERS

16.5 E.g., let V= (0, 0), A= (0, 5), andB= (4,3). A mirror map for l would take A to B, B to A, and V to V. But VA ¥- VB. 16.6 TFTTF-TTFFT. 16.8 Fold paper so desired points coincide. 16.10 (x, y) to (2x, 2y) for Cartesian plane; (M10, t).

CHAPTER 17

17.3 See Exercise 17.6(gl. 17.4 ASA. 17.6 FTTFF-TTTFT. 17.7 No: (Ml, d, m'). 17.9 Right angles at H in Figure 17.6. 17.13 In Cartesian three-space project distances from plane z= 2y to the x-y-plane.

CHAPTER 18

18.4 See Theorem 18.12. 18.5 FTTTT - TFFFF. 18.10 A-D-E such that 2AD=AE-AC; compare with Theorem 18.16.

CHAPTER 19

19.1 Corollary of Theorem 19.9. 19.3 (x, y) to (2x, 2y); show slope is preserved. 19.4 1; 2 or 6; 6. 19.6 TFTTF - TFTFT. 19.7 From slope argument A (y - y' ) = B (x - x' ); midpoint of (x, y) and (x', y') is on l; solve two equations in two unknowns.

19.12 (x, y) to (alx+ bly+ cl' a2x+ b2 y+ c2 ).

19.13 a l = ±b2 = cos (J and a2 = +bl = sin (J in Answer 19.12.

CHAPTER 20

20.3 Use Triangle Inequality. 20.5 TTFTF - TFFTF. HINTS AND ANSWERS 499

20.9 c> 0 and one (i) greater than, (ii) equal to, (iii) less than sum of other two; (iv): a - b = c= O. 20.12 Use contrapositives of statements in hypothesis.

CHAPTER 21

21.3 Theorem 21.8. 21.4 Not true for the Cartesian plane when Band C on opposite <---> sides of AD. 21.5 TTFFF - FFFTF. ----> 21.6 Assumes congruent segments on BA have congruent projec- --> tions on BC.

CHAPTER 22

22.4 Look at Theorem 23.1 only as a last resort. 22.5 Convex; angles acute or right; opposite sides parallel and con• gruent; obtained by reflecting a Saccheri quadrilateral in lower base. 22.6 FFTFF-FTFFT. 22.7 Theorem 18.17. 22.8 Exercise 21.6. 22.12 Theorem 22.17 and Theorem 18.16. 22.14 Four cases: G=C=K; G =i' C=K; G=K,B-C-G; andG=K, B-G-C. 22.16 See Theorem 33.11.

CHAPTER 23

23.1 For Y implies V use idea of Theorem 23.1. 23.3 Theorem 23.6. 23.4 FTTFT - TFTFT. 23.8 Not under the Hypothesis of the Acute Angle. 23.12 There is a positive number r such that the distance from Cn <---> to AB is greater than r for all n.

CHAPTER 24

24.2 Proposition Y of Theorem 23.7. 24.5 FFTFT - FFFFF. 500 HINTS AND ANSWERS

24.9 In proof of Theorem 24.20, let AD=s and A-F-B such that mLACF=1T/2 to obtain f:::,ACF.

CHAPTER 25

25.1 The rub is in defining area in the first place. 25.4 (must preserve the interior of any angle. 25.5 Possibly no planes.

CHAPTER 26

26.1 See Figure 26.2. For n> 1, ll(do) =1T/(n+ 1). 26.2 Either part of Figure 26.2. 26.3 Consider uBADC and uABCD. 26.5 TFFFF - TFFTF. 26.12 LBAE critical angle for AE, A-E-C, and CD.l AC. 26.18 Exercise 22.16.

CHAPTER 27

27.1 PQ big enough in Figure 26.1 so that 1and mare hyperparallel. 27.2 Figure 23.7. 27.4 Theorem 27.2; p in proof of Theorem 26.3. 27.5 Only g false. 27.10 44000000 - 211 01. 27.11 Nine cases, considering position of c.

CHAPTER 28

28.1 Only Theorem 28.6 is substantially different. 28.2 AAA. 28.3 PmPI= (PmP n) (PnPI)' 28.4 Corollary 28.13. 28.5 TTTTT - FFFFF.

CHAPTER 29

29.1 The perpendicular bisectors of two chords PQ and QR of a hypercircle cannot both be perpendicular to each of two lines. HINTS AND ANSWERS 501

29.2 Corollary 28.13, making a wise choice. ~ 29.4 With 'Y/c= PePd where d ..l AB, you also have a proof that the product of three halfturns is a halfturn in the Euclidean plane. 29.5 Theorem 29.13; P1'Y/M with M the midpoint. 29.6 TFTTF - TTFTF. 29.8 PdPbPcPa = PfPdPaPe = PfPaPdPe = 'Y/Q'Y/p- 29.9 Involutory CTrJprr- 1 fixes point Q iff Q= rrP. 29.20 If rr is odd with center c, then n=c; if rr is even, then rrP= rrPIP for all points P on l.

CHAPTER 30

30.3 rriprr-i and Theorem 29.8. 30.5 1881,1883,1961,1984, MM. 30.6 One is colorblind. 30.8 TTTTF - FF'l'FT. 30.28 See Exercise 30.21.

CHAPTER 31

31.2 Proof of Theorem 31.15. 31.4 Theorem 26.19. 31.5 FFFTT-TTFFF. 31.11 In Exercise 31.10: x=a+b,y=a-b. 31.12 coshx=cosh (x/2+x/2).

CHAPTER 32

32.3 sin II (x + y) cosh (x + y) = 1. 32.4 Use Corollary 32.11 in Corollaries 32.16 and 32.15. 32.5 cos (7T - (J) = -cos (J. 32.6 sinh «x*)/k) = csch (x/k). 32.7 Only (a) and (e) false. 32.12 By Exercise 32.5 or 32.11, less thlln In (1 + V2). 32.16 IfII(r)=7T/3, then 2r=ln3. 32.25 Exercise 32.12 and cosh2 2 < cosh 22. 32.26 H is a multiple of the constant in Corollary 32.14. 32.38 a, b, c integers: no; but if a2 + b2 < c2 < (a + b)2 then ta, tb, tc for some real t. 502 HINTS AND ANSWERS

CHAPTER 33

33.2 See Figure 23.7. 33.3 Btx=Aty. 33.5 Expand by third row. 33.6 TTFFF - TTTTT.

CHAPTER 34

34.3 A" and B" outside Cayley-Klein Model in Figure 34.5. 34.7 Theorem 34.10, Exercise 34.5, Exercise 34.6. 34.8 Corollary of Exercise 34.7. 34.14 E.g., a horopencil and all hyperpencils with center in the horo- pencil. 34.18 Theorem 32.21b with u=a. 34.27 The hypotenuse of an isosceles right triangle with legs of length p is very interesting. 34.28 Exercise 34.27. 34.35 Midpoints cannot be constructed. Notation Index

10 "I,3,3,!, Z, Q, R, C 249 [§JABCD 21 lal 256 [jABCD, 22 Z'\ Q"', R*, Z+, Q', R+ DABCD = DEFGH 25 Ix + yil 261 St::.ABC, SDABCD 30 lub, glb 273 t::.ABC - t::.DEF 61 Ml, M2, ... , M15 281 LJABCD, 66 fJP, 2, d, m int (LJABCD) ~ -4 -~ 67 lllm, PQ 292 AB-CD, 68 "i.,PQ LJABCD - LJPQRS 74 A-B-C 296 H(BC) 77 A-B-C-D -4 --> - --> 339 AB lCD, II m 85 AB,VA 351 P-Q - - 87 AB=CD 375 'Y//, --> 89 int (AB), int (VA) 388 int (P) with P a 96 LAVB convex polygon 98 t::.ABC 402 H(x), AB, int (AB) 145 int (LA VB) 409 IABI 148 int (t::.ABC) , HA(l) 412 int ('6') with '6' a 149 DABCD horocircle 158 mLABC, 414 k LAVB= LCWD 421 S 162 l1. m 435 (AB)'" 192 t::.ABC = t::.DEF 451 oT with T a 197 LABC>LDEF, triangulation LDEF< LABC - -- - 454 oR with R a polygonal 207 AB > CD, CD

AAA, 188,* 199,335 Angle-Base Theorem, 295 Abelian, 21 Angle-Construction Theorem, 159 Absolute, Angle-Segment-Construction Theorem, Bolyai's Theorem, 458 159 Exterior Angle Theorem, 264 Apollonius, 116 four-space, 327 Are, 403 geometry, 48, 128, 239, 324, 327 length, 409, 475 length,299 Archimedes, 29, 115 plane, 242 Archimedes' axiom, 29 Pythagorean Theorem, 443 Area, 450, 456,457,477 three-space, 324 Aristotle's Axiom, 245, 265 trigonometry, 443 ASA, 188, 198 value, 21, 25 ASS, 188, 199 Absolutely perpendicular, 327 Associative, 15, 20 Absurd numbers, 51 Automorphism for a, Acute angle, 161 field, 24 Adjacent, 150 geometry, 218 Affine plane, 37 Axial coordinates, 445 Alexandria, III Axiom, 34 Alternate interior angles, 240 1,66 Angle, 96 2,68 of a biangle, 281 3,133 bisector, 160 4,158 measure function, 158 5,195 of a polygon, 388 6,334 of a quadrilateral, 150 Axiom system, 34 of a triangle, 99 Axis, 445 Angle-Addition Theorem, 159 Angle-Angle Theorem, 337 Base angle, 197

*Page numbers in italics indicate an element of the theory, usually a definition or a theorem. INDEX 505

Base of a biangle, 281 Collineation, 38 Beer mugs, 140 Commutative, 20, 21 Beltrami, 311 Com~nstruction Axiom, 230 coordinates, 447 Complementary angles, 160 Between, 73, 74, 107,402 Complementary segments, 435 Biangle, 281 Complete field, 30 Big Protractor Postulate, 168 Complex Cartesian Incidence Plane, 56 Bijection, 14 Complex number, 10,24 Binary operation, 20 Composition, 15 Birkhoff, 155 Concentric, 226, 354 Bolyai,155,305,307,310,314,489 Concurrent, 67 Bolyai-Lobachevsky plane, 317, 334, 483 Congruent, 223 Bolyai's Theorem for Absolute Geometry, angles, 158 458 segments, 87 BPP, 168 triangles, 192 Brush,318,347 Consistent, 6, 35 Constructions, 479, 480 Calculus, 474 Contrapositive, 2 Cancellation, 22 Convex, Cantor, 27, 28, 31, 75 polygon, 388 Cantor-Dedekind Axiom, 76 polygonal region, 451 Cantor's theorem, 28 quadrilateral, 150 Cardinality, 26 set of points, 90 Carroll, 196 Coordinate, 69, 445 Cartesian, system, 69 line, 75 Corresponding, plane, 50 angles, 240 product, 4 arcs, 413 Categorical, 36 points, 413 Cayley, 60, 312 segments, 413 Cayley-Klein Incidence Plane, 60, 105 cosh, 13,415 Cayley-Klein Model, 283, 444, 449 Critical, Center of a, angle, 296 brush,347 function, 296 circle, 226 value, 296 cycle, 354 Critically parallel, 339 frieze group, 393 Crossbar, 101, 146 glide reflection, 379 Cubic Incidence Plane, 57 horolation, 362 Cut, 31, 240 reflection, 219 Cycle, 318, 354 rotation, 362 Cyclic group, 386 translation, 362, 365 Ceva, 464 Dedekind, 26, 75 Cevian, 465 Dedekind cut, 31 Ceva's Theorem, 468 Defect of a, Chord, 226, 354 convex polygon, 451 Circle, 266, 318 polygonal region, 451, 454 Cleopatra VII, 116 quadrilateral, 261 Clifford parallel, 329 triangle, 261, 262 Closed biangle, 292 triangulation, 451 Closed from a vertex, 292 Degrees, 158 Codomain, 5 Desargues, 464 Collinear, 67 Desargues'Theorem, 140,469 Descartes, 37, 51, 118 506 INDEX

Diagonal,150 of a horocircle, 413 Diameter, 226 of a triangle, 148 Dihedral group, 387 Dilatation, 384 False numbers, 51 Directed distance, 466 Fano, 138 Disjoint, 4 Fano's Axiom, 152 Distance, 68 Fictitious numbers, 25, 51 between horocircles, 413 Field, 23 between lines, 246 of complex numbers, 24 between points, 68 of rationals, 23 directed, 466 of reals, 23 function, 68 of two elements, 25 from point to line, 226 Finite geometry, 42, 138 scale, 319, 414, 422, 449 Finite group, 386 Dodgson, 196 Fix, 220 Domain, 5 Flag, 36 Flat, 326 Edge, 133 Foot, 212 Elements, 26, Ill, 115, 121, 244, 273, Four-Angle Theorem, 161 482 Four-Space, 327 Elliptic plane, 181, 312, 370 Frame, 445 Endpoint, 85 Frieze group, 393, 397 Equiangular, 197 Frieze patterns, 392 Equidistance curve, 318, 356 Fundamental Formula, 319, 429 Equidistant lines, 249 Equilateral, 197 Equivalence class, 7 Galileo,26 Equivalence relation, 5 Gauss, 52, 305, 310 Equivalent, 6 Gauss plane, 52 Generated, 386 biangles, 292 points, 351 Gersonides, 272 rays, 292 quadrilateral, 266, 273 by triangulation, 454 Giordano, 248 Escribed cycle, 442 Giordano's Theorem, 250 Euclid, 26, 29, 115, 121, 464 Glide reflection, 379 Euclidean, Graph,5 constructions, 479, 483 Greek alphabet, 9 field, 30 Greek cross, 388 line, 76 Group, 21 plane, 52, 320, 323 Gudermannian, 441 Euclid's, Common Notions, 123 Halffiat, 325 Definitions, 122 Halfline, 89 Parallel Postulate, 123, 243, 269, 276, Halfplane, 101, 133 284 Incidence Plane, 42, 56 Postulates, 123 Halfturn, 375 Propositions, 125 Helmholtz, 310 Eudoxus, 26, 29, 111 Hilbert, 139, 172, 314 Even isometry, 372 Hilbert's axioms, 172 Exterior, Hinge Axiom, 126 angle of a biangle, 281 Hjelmslev, 139 angle of a triangle, 197 Hjelmslev's Theorem, 384 of an angle, 148 Horocircle, 318, 354 of a circle, 226 Horolation, 362 INDEX 507

Horoparallel, 318, 339 Interior ray, 104,281 Horopencil, 318, 347 Involution, 220, 371 Hoiiel,310 Irrational, 26 HPP, 334, 335 Isometries, 216 Hypatia, 118 of the elleptic plane, 370 Hyperbolic, of the Euclidean plane, 384, 392, 397 Ceva's Theorem, 468 of the hyperbolic plane, 382,392,397 Desargues' Theorem, 470 Isometric models, 35, 449 functions, 415 Isomorphism, 24, 35 geometry, 42, 282, 284,312, 325 Isosceles biangle, 292 Law of Cosines, 432 Isosceles triangle, 197 Law of Sines, 432 Menelaus' Theorem, 467 k,414 Pappus' Theorem, 473 Klein, 60, 312 Parallel Postulate, 317,334,335 plane, 42, 317, 332 Lambert, 305, 310 Pythagorean Theorem, 319,428 quadrilateral, 256 trig functions, 13, 415 trig, 435 Hypercircle, 318, 354 Larger, 197 Hyperparallel, 318, 341 Least upper bound, 30 Hyperpencil, 318, 347 Leg, 212,249 Hypotenuse, 212 Legendre,274, 281 Hypotenuse-Leg Theorem, 213 Length, 87, 409 Hypothesis of the, Leonardo da Vinci, 318, 392 Acute Angle, 255 Leonardo's Theorem, 392 Obtuse Angle, 255, 264, 312 Library at Alexandria, 113, 117, 119 Right Angle, 255, 269, 278 Line, 67, 84, 122 Line of symmetry, 226 Identity mapping, 16 Linear pair, 97 In, 144 Line-Circle Theorem, 232 Incidence, 5 Line-Separation Theorem, 90 Axiom, 66 Line-Triangle Theorem, 149 plane, 36, 55 Lobachevsky, 305, 310, 314 Incident, 67 coordinates, 445 Independent, 6, 35 Longer, 207, 409 Inequalities, 207 Lower base, 249 Infinite set, 26 Injection, 11, 12 Mapping, 10 Injective, 11 Mathemata, 121 Inside, 104, 148 Measure, 158 Intersect, 4, 67 Menelaus, 117,464 Interior angle, 240 points, 465 Interior of, Theorem, 467 an angle, 100, 145 Midpoint, 89 an arc, 403 Theorem, 89 a biangle, 281 MIRROR, 185, 221 a circle, 226 Mirror Axiom, 185, 221 a horocircle, 412 Missing-Quadrant Incidence Plane, 42, 56 a polygon, 388 Missing-Strip Incidence Plane, 56, 103 a polygonal region, 451 Model, 35, 55 a ray, 89 Modulus, 25 a segment, 89 Moulton, 58 a triangle, 148 Incidence Plane, 57, 91, 140, 163,209 508 INDEX

Mukhopadhyaya's pentagon, 443 Playfair's Parallel Postulate, 243, 269, Museum, 113 277, 339 Poincare, 59 n-gon, 388 Halfplane, 285 Negation, 19 Halfplane Incidence Plane, 59 Non-Archimedian, Incidence Plane, 59 field, 29 Model, 284, 358 geometry, 138 Point, 67 Non-Euclidean geometry, 313 of symmetry, 386 Obtuse angle, 161 Pointwise, 220 Odd isometry, 372 Polar coordinates, 445 Off,67 Polygon, 388 Omar Khayyam, 248, 271 Polygonal Inequality, 209 Omar Khayyam's Theorem, 257 Polygonal region, 451 On, 67, 86, 144 Pons Asinorum, 196, 197, 207 On opposite sides, 135 Postulate system, 34 One-to-one, 11 Proclus, 118, 138, 200, 244, 246 correspondence, 14 Projective plane, 40 Onto, 11 Protractor Postulate, 155, 158, 166 Opposite, PSP, 101, 131, 13~ 136 angle, 150, 197 Ptolemy, 11 7, 244 halfplane, 135 Pythagorean field, 30 ray, 89 Pythagorean Theorem, 129,321,428 side, 135, 150, 197 vertex, 150 Quadrant Incidence Plane, 42, 56 Order, Quadrature of the circle, 482, 490 for a field, 28 Quadrilateral, 149, 150 of a group, 386 Quantifier, 3 on a line, 73 Outer end, 226 Radius of a, Outside, 148 circle, 226 horocircle, 403 Pappus, 117, 137, 196,464 hypercircle, 371 Pappus' Theorem, 140,473 Range, 11 Parallel, 6, 36, 67, 249, 323, 327 Ratchet, 389 pencil, 7, 347 Rational, 10, 23, 26 postulates, 42 Cartesian Incidence Plane, 55 Pasch, 74, 138, 140, 314 Cartesian plane, 160, 230 PASCH, 131, 135. 136 Ray, 85 Pasch's Postulate, 103,131,135,136 Ray-Coordinatization Theorem, 87 Passes through, 67, 86 Ray-interior, 104, 148 Peano, 138, 140, 180, 314 Real Cartesian Incidence Plane, 37, 50, Peano's Postulate, 136 55 Pencil, 318, 347 Real number, 26 Permutation, 16 Real Projective Plane, 40 Perpendicular, 162, 325, 326, 327, 328 Rectangle, 249 bisector, 206 Reflection, 219, 224 Philo, 200, 205 Reflexive law, 5 Piecewise congruence, 129, 454 Regular polygon, 388 Pieri, 139, 175, 314 Relation, 5 Pieri's postulates, 175 Remote interior angle, 197,281 Plane-Separation Postulate, 101,131, Riemann, 61, 310 133, 136 Incidence Plane, 61, 312 INDEX 509

Right angle, 161 Squaring the circle, 482, 490 Right triangle, 212 SSS, 188, 199,200 Rotation, 362 Standard notation, 425 Ruler, Star-between, 78 and compass, 479 • Star triangulation, 451 Placement Theorem, 70 Straightedge Axiom, 66 Postulate, 68 Subtended, 403 Subtriangulation, 451 8,421 Superposition, 126, 188, 295 SAA, 188,199 Supplementary, 160 Saccheri, 155, 248, 255, 302, 310 Surjection, 11, 12 quadrilateral, 249 Surjective, 11 Saccheri's Theorem, 264, 275 Sylvester's Theorem, 137 Same cardinality, 26 Symmetric law, 5 SAS, 185, 186,195,221 Symmetry, 386 Scalene, 197 Schweikart, 306, 310 Tables, chairs, and beer mugs, 140 Secant, 226 Tangent, 226, 403 Segment, 85 tanh, 13,415 Segment-Addition Theorem, 88 Taurinus, 308, 310 Segment-Construction Theorem, 87 Taxicab Geometry, 77, 162, 195 Segment-interior, 104 Thales, 111 Segment-Subtraction Theorem, 88 Theorem of, 263, 277 Seven arts, 121 Theory of parallels, 244 Shade, 102, 148 Three-space, 324 Shorter, 207 Transitive law, 5 Side of, Translation, 362 an angle, 96 Transversal, 240 a biangle, 281 Triangle, 98 a line, 100, 135 Inequality, 69, 208 a polygon, 388 Theorem, 234 a polygonal region, 451 Triangular region, 451 a quadrilateral, 150 Triangulation, 451 a ray, 135 Trigonometry, 421 a segment, 135 Two-Circle Theorem, 235, 239 a triangle, 98 Side-AngIe-Side Theorem, 125, 165, 185, Undefined terms, 66 186, 195, 221 Upper base, 249 Similar, 273, 319 angle, 249 sinh, 13,415 Skew, 323, 327 Varignon quadrilateral, 346 Smaller, 197 Vertex, 85,96,98, 150,281,388,451 Snowflake curve, 409, 418 Vertical angles, 97 Sostratos, 113 Space Incidence Plane, 56, 104 VVachter, 306, 310 Sphere Incidence Plane, 60 VVallis, 273 Square, 256 VVeird plane, 107 Undergraduate Texts in Mathematics

(conrinuedfrom page iii

James: Topological and Unifonn PeressinilSuIlivanlUhl: The Mathematics Spaces. of Nonlinear Programming. Jiinich: Linear Algebra. PrenowitzlJantosciak: Join Geometries. Jiinich: Topology. Priestley: Calculus: An Historical Kemeny/Snell: Finite Markov Chains. Approach. Kinsey: Topology of Surfaces. ProtterlMorrey: A First Course in Real Klambauer: Aspects of Calculus. Analysis. Second edition. Lang: A First Course in Calculus. Fifth ProtterlMorrey: Intennediate Calculus. edition. Second edition. Lang: Calculus of Several Variables. Roman: An Introduction to Coding and Third edition. Infonnation Theory. Lang: Introduction to Linear Algebra. Ross: Elementary Analysis: The Theory Second edition. of Calculus. Lang: Linear Algebra. Third edition. Samuel: Projective Geometry. Lang: Undergraduate Algebra. Second Readings in Mathematics. edition. ScharlauiOpolka: From Fennat to Lang: Undergraduate Analysis. Minkowski. LaxlBursteinlLax: Calculus with Sethuraman: Rings, Fields, and Vector Applications and Computing. Spaces: An Approach to Geometric Volume 1. Constructability. LeCuyer: College Mathematics with Sigler: Algebra. APL. Silvermanffate: Rational Points on LidllPilz: Applied Abstract Algebra. Elliptic Curves. Second edition. Simmonds: A Brief on Tensor Analysis. Macki-Strauss: Introduction to Optimal Second edition. Control Theory. Singer: Geometry: Plane and Fancy. Malitz: Introduction to Mathematical SingerlThorpe: Lecture Notes on Logic. Elementary Topology and MarsdenIWeinstein: Calculus I, II, III. Geometry. Second edition. Smith: Linear Algebra. Second edition. Martin: The Foundations of Geometry Smith: Primer of Modem Analysis. and the Non-Euclidean Plane. Second edition. Martin: Geometric Constructions. StantonlWhite: Constructive Martin: Transfonnation Geometry: An Combinatorics. Introduction to Symmetry. Stillwell: Elements of Algebra: MillmanlParker: Geometry: A Metric Geometry, Numbers, Equations. Approach with Models. Second Stillwell: Mathematics and Its History. edition. Stillwell: Numbers and Geometry. Moschovakis: Notes on Set Theory. Readings in Mathematics. Owen: A First Course in the Strayer: Linear Programming and Its Mathematical Foundations of Applications. Thennodynamics. Thorpe: Elementary Topics in Palka: An Introduction to Complex Differential Geometry. Function Theory. Toth: Glimpses of Algebra and Pedrick: A First Course in Analysis. Geometry. Undergraduate Texts in Mathematics

Troutman: Variational Calculus and WhyburnlDuda: Dynamic Topology. Optimal Control. Second edition. Wilson: Much Ado About Calculus. Valenza: Linear Algebra: An Introduction to Abstract Mathematics.