This book is for all lovers ofmathematics. It is an attempt to under stand the nature of mathematics from the point of view of its most important early source. Even if the material covered by Euclid may be considered ele mentary for the most part, the way in which he presents it has set the standard for more than two thousand years. Knowing Euclid's Elements may be ofthe same importance for a mathematician today as knowing Greek architecture is for an architect. Clearly, no con temporary architect will construct a Doric temple, let alone organize a construction site in the way the ancients did. But for the training ofan architect's aesthetic judgment, a knowledge ofthe Greek her itage is indispensable. I agree with Peter Hilton when he says that genuine mathematics constitutesone ofthe finest expressions ofthe human spirit, and I may add that here as in so many other instances, we have learned that language ofexpression from the Greeks. While presenting geometry and arithmetic Euclid teaches us es sential features of mathematics in a much more general sense. He displays the axiomatic foundation of a mathematical theory and its conscious development towards the solution of a specific problem. We see how abstraction works and enforces the strictly deductive presentation ofa theory. We learn what creative definitions are and v VI ----=P:. . :re:. ::::fa=ce how a conceptual grasp leads to toe classification ofthe relevant ob jects.
Author(s): Benno Artmann (auth.)
Edition: 1
Publisher: Springer-Verlag New York
Year: 1999
Language: English
Pages: 349
Tags: Geometry
Front Matter....Pages i-xvi
General Historical Remarks....Pages 1-2
The Contents of the Elements....Pages 3-10
The Origin of Mathematics 1: The Testimony of Eudemus....Pages 11-16
Euclid Book I: Basic Geometry....Pages 17-46
The Origin of Mathematics 2: Parallels and Axioms....Pages 47-50
The Origin of Mathematics 3: Pythagoras of Samos....Pages 51-60
Euclid Book II: The Geometry of Rectangles....Pages 61-71
The Origin of Mathematics 4: Squaring the Circle....Pages 73-78
Euclid Book III: About the Circle....Pages 79-91
The Origin of Mathematics 5: Problems and Theories....Pages 93-95
Euclid Book IV: Regular Polygons....Pages 97-107
The Origin of Mathematics 6: The Birth of Rigor....Pages 109-111
The Origin of Mathematics 7: Polygons After Euclid....Pages 113-120
Euclid Book V: The General Theory of Proportions....Pages 121-134
Euclid Book VI: Similarity Geometry....Pages 135-149
The Origin of Mathematics 8: Be Wise, Generalize....Pages 151-159
Euclid Book VII: Basic Arithmetic....Pages 161-182
The Origin of Mathematics 9: Nicomachus and Diophantus....Pages 183-191
Euclid Book VIII: Numbers in Continued Proportion, the Geometry of Numbers....Pages 193-201
The Origin of Mathematics 10: Tools and Theorems....Pages 203-206
Euclid Book IX: Miscellaneous Topics from Arithmetic....Pages 207-211
The Origin of Mathematics 11: Math Is Beautiful....Pages 213-221
Euclid Book X: Incommensurable Magnitudes....Pages 223-228
The Origin of Mathematics 12: Incommensurability and Irrationality....Pages 229-253
Euclid Book XI: Solid Geometry....Pages 255-265
The Origin of Mathematics 13: The Role of Definitions....Pages 267-269
Euclid Book XII: Volumes by Limits....Pages 271-278
The Origin of Mathematics 14: The Taming of the Infinite....Pages 279-282
Euclid Book XIII: Regular Polyhedra....Pages 283-302
The Origin of Mathematics 15: Symmetry Through the Ages....Pages 303-316
The Origin of Mathematics 16: The Origin of the Elements....Pages 317-320
Back Matter....Pages 321-349
