CART

(0) items

Foundations of Geometry,9780131437005
This item qualifies for
FREE SHIPPING!

FREE SHIPPING OVER $59!

Your order must be $59 or more, you must select US Postal Service Shipping as your shipping preference, and the "Group my items into as few shipments as possible" option when you place your order.

Bulk sales, PO's, Marketplace Items, eBooks, Apparel, and DVDs not included.

Foundations of Geometry

by
Edition:
1st
ISBN13:

9780131437005

ISBN10:
0131437003
Format:
Paperback
Pub. Date:
1/1/2005
Publisher(s):
Prentice Hall
List Price: $86.66

Buy Used Textbook

(Recommended)
Usually Ships in 2-3 Business Days
U9780131437005
$60.66

Rent Textbook

We're Sorry
Sold Out

eTextbook

We're Sorry
Not Available

New Textbook

We're Sorry
Sold Out

More New and Used
from Private Sellers
Starting at $1.14
See Prices

Questions About This Book?

What version or edition is this?
This is the 1st edition with a publication date of 1/1/2005.
What is included with this book?
  • The Used copy of this book is not guaranteed to include any supplemental materials. Typically, only the book itself is included.

Related Products


  • Foundations of Geometry
    Foundations of Geometry




Summary

For sophomore/junior-level courses in Geometry; especially appropriate for students that will go on to teach high-school mathematics.This text comfortably serves as a bridge between lower-level mathematics courses (calculus and linear algebra) and upper-level courses (real analysis and abstract algebra). It fully implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers. Foundations of Geometry particularly teaches good proof-writing skills, emphasizes the historical development of geometry, and addresses certain issues concerning the place of geometry in human culture.

Table of Contents

Preface ix
1 Euclid's Elements 1(16)
1.1 Geometry before Euclid
1(1)
1.2 The logical structure of Euclid's Elements
2(1)
1.3 The historical importance of Euclid's Elements
3(2)
1.4 A look at Book I of the Elements
5(3)
1.5 A critique of Euclid's Elements
8(3)
1.6 Final observations about the Elements
11(6)
2 Axiomatic Systems and Incidence Geometry 17(14)
2.1 Undefined and defined terms
17(1)
2.2 Axioms
18(1)
2.3 Theorems
18(1)
2.4 Models
19(1)
2.5 An example of an axiomatic system
19(6)
2.6 The parallel postulates
25(2)
2.7 Axiomatic systems and the real world
27(4)
3 Theorems, Proofs, and Logic 31(12)
3.1 The place of proof in mathematics
31(1)
3.2 Mathematical language
32(2)
3.3 Stating theorems
34(3)
3.4 Writing proofs
37(2)
3.5 Indirect proof
39(1)
3.6 The theorems of incidence geometry
40(3)
4 Set Notation and the Real Numbers 43(9)
4.1 Some elementary set theory
43(2)
4.2 Properties of the real numbers
45(3)
4.3 Functions
48(1)
4.4 The foundations of mathematics
49(3)
5 The Axioms of Plane Geometry 52(42)
5.1 Systems of axioms for geometry
53(3)
5.2 The undefined terms
56(1)
5.3 Existence and incidence
56(1)
5.4 Distance
57(6)
5.5 Plane separation
63(4)
5.6 Angle measure
67(3)
5.7 Betweenness and the Crossbar Theorem
70(14)
5.8 Side-Angle-Side
84(5)
5.9 The parallel postulates
89(1)
5.10 Models
90(4)
6 Neutral Geometry 94(41)
6.1 Geometry without the parallel postulate
94(1)
6.2 Angle-Side-Angle and its consequences
95(2)
6.3 The Exterior Angle Theorem
97(5)
6.4 Three inequalities for triangles
102(5)
6.5 The Alternate Interior Angles Theorem
107(3)
6.6 The Saccheri-Legendre Theorem
110(3)
6.7 Quadrilaterals
113(3)
6.8 Statements equivalent to the Euclidean Parallel Postulate
116(7)
6.9 Rectangles and defect
123(8)
6.10 The Universal Hyperbolic Theorem
131(4)
7 Euclidean Geometry 135(26)
7.1 Geometry with the parallel postulate
135(2)
7.2 Basic theorems of Euclidean geometry
137(2)
7.3 The Parallel Projection Theorem
139(2)
7.4 Similar triangles
141(2)
7.5 The Pythagorean Theorem
143(2)
7.6 Trigonometry
145(2)
7.7 Exploring the Euclidean geometry of the triangle
147(14)
8 Hyperbolic Geometry 161(33)
8.1 The discovery of hyperbolic geometry
161(2)
8.2 Basic theorems of hyperbolic geometry
163(5)
8.3 Common perpendiculars
168(3)
8.4 Limiting parallel rays and asymptotically parallel lines
171(10)
8.5 Properties of the critical function
181(4)
8.6 The defect of a triangle
185(4)
8.7 Is the real world hyperbolic?
189(5)
9 Area 194(31)
9.1 The Neutral Area Postulate
195(3)
9.2 Area in Euclidean geometry
198(8)
9.3 Dissection theory in neutral geometry
206(7)
9.4 Dissection theory in Euclidean geometry
213(3)
9.5 Area and defect in hyperbolic geometry
216(9)
10 Circles 225(39)
10.1 Basic definitions
226(1)
10.2 Circles and lines
227(4)
10.3 Circles and triangles
231(7)
10.4 Circles in Euclidean geometry
238(6)
10.5 Circular continuity
244(3)
10.6 Circumference and area of Euclidean circles
247(8)
10.7 Exploring Euclidean circles
255(9)
11 Constructions 264(20)
11.1 Compass and straightedge constructions
265(2)
11.2 Neutral constructions
267(3)
11.3 Euclidean constructions
270(2)
11.4 Construction of regular polygons
272(4)
11.5 Area constructions
276(3)
11.6 Three impossible constructions
279(5)
12 Transformations 284(43)
12.1 The transformational perspective
285(1)
12.2 Properties of isometrics
286(6)
12.3 Rotations, translations, and glide reflections
292(8)
12.4 Classification of Euclidean motions
300(3)
12.5 Classification of hyperbolic motions
303(1)
12.6 A transformational approach to the foundations
304(6)
12.7 Euclidean inversions in circles
310(17)
13 Models 327(18)
13.1 The significance of models for hyperbolic geometry
327(2)
13.2 The Cartesian model for Euclidean geometry
329(2)
13.3 The Poincaré disk model for hyperbolic geometry
331(5)
13.4 Other models for hyperbolic geometry
336(5)
13.5 Models for elliptic geometry
341(4)
14 Polygonal Models and the Geometry of Space 345(47)
14.1 Curved surfaces
346(11)
14.2 Approximate models for the hyperbolic plane
357(6)
14.3 Geometric surfaces
363(6)
14.4 The geometry of the universe
369(6)
14.5 Conclusion
375(1)
14.6 Further study
375(9)
14.7 Templates
384(8)
APPENDICES
A Euclid's Book I
392(6)
A.1 Definitions
392(2)
A.2 Postulates
394(1)
A.3 Common Notions
394(1)
A.4 Propositions
394(4)
B Other Axiom Systems
398(10)
B.1 Hilbert's axioms
398(2)
B.2 Birkhoff's axioms
400(1)
B.3 MacLane's axioms
401(1)
B.4 SMSG axioms
402(3)
B.5 UCSMP axioms
405(3)
C The Postulates Used in this Book
408(3)
C.1 The undefined terms
408(1)
C.2 The postulates of neutral geometry
408(1)
C.3 The parallel postulates
409(1)
C.4 The area postulates
409(1)
C.5 The reflection postulate
410(1)
C.6 Logical relationships
410(1)
D The van Hiele Model
411(1)
E Hints for Selected Exercises
412(9)
Bibliography 421(4)
Index 425


Please wait while the item is added to your cart...