9780198507581

Geometry Ancient and Modern

by
  • ISBN13:

    9780198507581

  • ISBN10:

    0198507585

  • Format: Hardcover
  • Copyright: 2001-06-21
  • Publisher: Oxford University Press

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Summary

This book offers a guided tour of geometry from euclid through to algebraic geometry. It shows how mathematicians use a variety of techniques to tackle problems , and it links geometry to other branches of mathematics. Many problems and examples are included to aid understanding.

Author Biography

John Silvester teachers geometry to first-year mathematics students at King's College London

Table of Contents

History and philosophy
1(11)
Before Euclid
1(1)
Euclid
2(1)
Parallels and non-Euclidean geometry
3(2)
Hilbert and Birkhoff
5(1)
Coordinate Geometry
6(1)
Pythagoras: theorem or axiom?
7(2)
Philosophy of this book
9(3)
Advice to readers
10(2)
Drawings and constructions
12(12)
Equipment
12(1)
Ruler and compasses
13(2)
Bisection, parallels, and subdivision
15(5)
Triangles
20(4)
Answers to exercises
23(1)
Plane geometry
24(28)
Points
24(1)
Lines
24(1)
Vector notation
25(2)
Distances and orthogonality
27(4)
Isometries
31(6)
Angles
37(7)
Preamble
37(3)
Abstract trigonometry
40(2)
Angle as area
42(2)
Polar coordinates and rotations
44(2)
Dot products
46(6)
Distance from point to line
48(1)
Angle bisectors
48(1)
Answers to exercises
49(3)
Triangles and triangle formulae
52(30)
The centroid
53(1)
The orthocentre
54(1)
The sine formula
55(1)
The circumcentre
56(1)
The incentre and excentres
57(3)
The cosine formula
60(5)
Similarities
65(2)
Barycentric coordinates
67(5)
Menelaus and Ceva
72(10)
Answers to exercises
77(5)
Isometries of R2
82(34)
Fixed points and lines
82(2)
Groups of isometries
84(8)
Complex formulae for plane isometries
92(13)
Classification of plane isometries
93(7)
Orientation
100(5)
Finite groups of plane isometries
105(11)
Answers to exercises
110(6)
Isometries of Rn
116(49)
Matrix forms for isometries of R2
116(1)
Matrix forms for isometries of Rn
117(5)
Classification of isometries
122(7)
Permutation groups
129(3)
Polyhedra
132(7)
The Platonic solids
139(6)
Finite groups of isometries of R3
145(20)
Answers to exercises
153(12)
Circles and other conics
165(30)
Proper and improper conics
167(4)
Plane sections of a cone
171(3)
Classification of conics under isometry
174(4)
Circles: angle theorems
178(7)
Circles: rectangular properties
185(5)
Stereographic projection
190(5)
Answers to exercises
192(3)
Beyond isometry
195(27)
Klein's definition of geometry
195(1)
Euclidean geometry: similarities
195(9)
Classification of similarities
196(3)
Classification of conics (again)
199(1)
Centres of similitude and the nine-point circle
199(4)
Dilatations
203(1)
Affine geometry
204(18)
Affine regular polygons
206(3)
Classification of conics (yet again)
209(2)
A selection of affine theorems
211(4)
Collineations
215(3)
Answers to exercises
218(4)
Infinity
222(45)
Mobius transformations
222(22)
Lines and circles
224(2)
Cross-ratios
226(3)
Angles
229(5)
Applications
234(5)
Hyperbolic geometry
239(5)
Projective geometry
244(23)
Homogeneous coordinates
247(2)
Duality
249(2)
Projective transformations and cross-ratios
251(4)
Classification of conics (for the last time)
255(4)
Elliptic geometry
259(2)
Answers to exercises
261(6)
Complex geometry
267(34)
Complex projective geometry
267(4)
Algebraic geometry
271(30)
Intersections of curves
272(4)
Nine associated points
276(4)
Applications
280(6)
Singular and non-singular cubics
286(11)
Answers to exercises
297(4)
Bibliography 301(2)
Notation 303(2)
Index 305

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