Table Of ContentUndergraduate Texts in Mathematics
Series Editors
S. Axler
K. A. Ribet
Undergraduate Texts in Mathematics
Readings in Mathematics
Series Editors
Sheldon Axler
San Francisco State University
Kenneth Ribet
University of California, Berkeley
Advisory Board
Colin Adams, Williams College
Alejandro Adem, University of British Columbia
Ruth Charney, Brandeis University
Irene M. Gamba, The University of Texas at Austin
Roger E. Howe, Yale University
David Jerison, Massachusetts Institute of Technology
Jeffrey C. Lagarias, University of Michigan
Jill Pipher, Brown University
Fadil Santosa, University of Minnesota
Amie Wilkinson, University of Chicago
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
year undergraduate mathematics students at North American universities. These
texts strive to provide students and teachers with new perspectives and novel
approaches.Thebooksincludemotivationthatguidesthereadertoanappreciation
ofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthat
illustrate key concepts as well as exercises that strengthen understanding.
Forfurthervolumes:
http://www.springer.com/series/666
and
http://www.springer.com/series/4672
Alexander Ostermann Gerhard Wanner
•
Geometry by Its History
123
Alexander Ostermann Gerhard Wanner
Department of Mathematics Section deMathématiques
Universityof Innsbruck Université de Genève
Innsbruck Genève
Austria Switzerland
ISSN 0172-6056
ISBN 978-3-642-29162-3 e-ISBN978-3-642-29163-0
DOI 10.1007/978-3-642-29163-0
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012936154
MathematicsSubjectClassification(2010):01-01,51-01,51-03
!Springer-VerlagBerlinHeidelberg2012
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In Memoriam
ALFRED FRO¨LICHER
(1927–2010)
Preface
ἀγεωμέτρητος μηδεὶς εἰσίτω
(literally:“non-geometersdon’tenter.”Inscriptionattheentrance
toPlato’s Academy387 B.C. (perhapsonlyalegend), andon the
frontispiece of Copernicus’ De revolutionibus 1543 A.D.)
Geometry,sonamedatleastsincePlato’stimes,istheoldestbranchofmath-
ematics. It contains many beautiful results, elegant ideas and surprising con-
nections, to which many great thinkers have contributed through the cen-
turies.Amongthese areThales,Pythagoras,Euclid,Apollonius,Archimedes,
Ptolemy, Pappus, the Arabs, Regiomontanus, Copernicus, Vi`ete, Kepler,
Descartes, Newton, the Bernoullis, Euler, Monge, Poncelet and Steiner. In
this book, we study geometry from the texts of these masters as closely as
we judge useful and roughly in historic order, in accordance with the famous
maxim of Niels Henrik Abel, “I learned from the masters and not from the
pupils”. This explains “by Its History” in the title, without aiming at a com-
plete history of the subject.1
Geometryaroseinthedawnofscienceandlaterbecameoneoftheseptem
artes liberales as partof the quadrivium. We startfrom the beginning of geo-
metry,motivatedbypracticalproblemsofmeasurement(μετρέω),thenfollow
its development into a rigorous abstract science by the Greek philosophers,
untiltherichperiodwithmoreandmoresophisticatedproblemsandmethods
inthelaterGreekandArabperiod.Thesecondpartofthebookthendescribes
thevictoryofthemethodsofalgebraandlinearalgebra,thegrowingaudacity
indealing withinfinite processesandwe seehowallotherbranchesofscience
grew out of geometry: algebra, calculus, mechanics (in particular celestial
mechanics).Sowehopethatthebooknotonlyconstitutesagoodintroduction
to the study of higher geometry, but also to the study of other branches of
science, especially for those students who intend to become teachers.
However,duetotherapidsuccessofalgebraicmethods,syntheticgeometry
slowly lost its place in university education, a development already deplored
byNewton,morethanthreecenturiesago(firstsentenceofhisTreatise of the
Methods of Series and Fluxions, 1671, p.33) ...
“Observing that the majority of geometers, with an almost complete
neglect of the ancients’ synthetical method, now for the most part apply
1Michel Chasles’ Aperc¸u historique has 800 pages and weighs fivepounds.
vii
viii Preface
themselves to the cultivation of analysis [...] I found it not amiss, for
the satisfaction of learners, to draw up the following short tract ...”
... a development which accelerated during the twentieth century (Coxeter,
Introduction to Geometry, 1961, p.iv):
“For the last thirty or forty years, most Americans have somehow lost
interest in geometry. The present book constitutes an attempt to revi-
talize this sadly neglected subject.”
What is not honoured at the university also disappears, a generation later,
fromthehigh-school.WequotefromA.Connes(NewsletteroftheEMS,March
2008, p.32):
“We must absolutely train very young people to do mathematical ex-
ercises, in particular geometry exercises—this is very good training.
I find it awful when I see that, in school, kids are taught recipes, just
recipes, and aren’t encouraged to think. When I was at school, I re-
member that we were given problems of [...] geometry. We went to a lot
of trouble to solve them. It wasn’t baby geometry. [...] It’s a shame we
don’t do it anymore.”
We have made all efforts to produce an interesting and enjoyable book,
intended mainly for students of science (at the beginning) and teachers
(throughout their career), by including many illustrations, figures, exer-
cises and references to the literature, so that we suggest with Copernicus’
De revolutionibus
Igitur eme, lege, fruere [Therefore buy, read and enjoy].
Acknowledgements. It is our duty—and pleasure—to thank many peo-
ple for their help, beginning with Ernst and Martin Hairer, the assistants
P.Henry,W.Pietsch,A.Musitelli,M.Baillif,C.Extermannandthe librarians
A.-S.Crippa, B.Dudez and T.Dubois at the University of Geneva. A long
discussion with J.-C.Pont, expert in the history of mathematics and science,
helped to reshape the first two chapters. Our colleagues and friends J.Cash,
B.Gisin,H.HerdlingerandC.Lubichreadtheentirebookandgavenumerous
suggestions. Further we thank our wives Barbara and Myriam.
Our special thanks, however, go to J.Steinig, who read the entire book
four times (!!) and suggested thousands of corrections and grammatical im-
provements. He pointed out errors and sloppy arguments, and supplied us
with many references and better proofs. And if ever “eme, lege, fruere” is
really justified, it is also his merit.
Innsbruck and Geneva, Alexander Ostermann
December 2011 Gerhard Wanner
Contents
Part I Classical Geometry
1 Thales and Pythagoras .................................... 3
1.1 Thales’ Theorem ........................................ 3
1.2 Similar Figures.......................................... 5
1.3 Properties of Angles ..................................... 7
1.4 The Regular Pentagon ................................... 9
1.5 The Computation of Areas ............................... 11
1.6 A Remarkable Babylonian Document ...................... 13
1.7 The PythagoreanTheorem ............................... 14
1.8 Three Famous Problems of Greek Geometry ................ 19
1.9 Exercises ............................................... 21
2 The Elements of Euclid.................................... 27
2.1 Book I ................................................. 29
2.2 Book III. Properties of Circles and Angles .................. 39
2.3 Books V and VI. Real Numbers and Thales’ Theorem........ 42
2.4 Books VII and IX. Number Theory ........................ 43
2.5 Book XI. Spatial Geometry and Solids ..................... 45
2.6 Book XII. Areas and Volumes of Circles, Pyramids, Cones
and Spheres ............................................ 48
2.7 Epilogue ............................................... 52
2.8 Exercises ............................................... 54
3 Conic Sections............................................. 61
3.1 The Parabola ........................................... 62
3.2 The Ellipse ............................................. 64
3.3 The Hyperbola.......................................... 70
3.4 The Area of a Parabola .................................. 72
3.5 Exercises ............................................... 73
ix
x Contents
4 Further Results in Euclidean Geometry.................... 79
4.1 The Conchoid of Nicomedes, the Trisection of an Angle ...... 79
4.2 The Archimedean Spiral.................................. 81
4.3 The Four Classical Centres of the Triangle.................. 82
4.4 The Theorems of Menelaus and Ceva ...................... 87
4.5 The Theorems of Apollonius–Pappus–Stewart............... 89
4.6 The Euler Line and the Nine-Point Circle................... 91
4.7 Excircles and the Nagel Point ............................. 93
4.8 Miquel’s Theorems ...................................... 94
4.9 Steiner’s Circle Theorems................................. 98
4.10 Morley’s Theorem .......................................104
4.11 Exercises ...............................................106
5 Trigonometry..............................................113
5.1 Ptolemy and the Chord Function ..........................113
5.2 Regiomontanus and Euler’s Trigonometric Functions .........116
5.3 Arbitrary Triangles ......................................119
5.4 Trigonometric Solution of Malfatti’s Problem ...............121
5.5 The Stereographic Projection .............................124
5.6 The Spherical Trigonometry of Right-Angled Triangles .......127
5.7 The Spherical Trigonometry of General Triangles ............133
5.8 The Area of a Spherical Triangle ..........................139
5.9 Trigonometric Formulas for the Conics .....................140
5.10 The Great Discoveries of Kepler and Newton................141
5.11 Exercises ...............................................150
Part II Analytic Geometry
6 Descartes’ Geometry ......................................159
6.1 The Principles of Descartes’ Geometry .....................159
6.2 The Regular Heptagon and Enneagon......................162
6.3 The Trisection of an Angle and Cubic Equations ............164
6.4 Regular Polygons in the Unit Circle........................166
6.5 Van Roomen’s Famous Challenge..........................168
6.6 A Geometric Theorem of Fermat ..........................170
6.7 Heron’s Formula for the Area of a Triangle .................171
6.8 The Euler–BrahmaguptaFormula for a Cyclic Quadrilateral ..174
6.9 The Cramer–CastillonProblem ...........................175
6.10 Exercises ...............................................178
7 Cartesian Coordinates .....................................185
7.1 Equations of Lines and Circles ............................185
7.2 The Problem of Pappus ..................................188
7.3 Conic Sections: Poles, Polars and Tangents .................189
