Geometry Ancient and Modern

by Silvester, John R.

• ISBN13: 9780198507581
• ISBN10: 0198507585
• Format: Hardcover
• Copyright: 2001-06-21
• Publisher: Oxford University Press

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