Synopses & Reviews
Problems that beset Archimedes, Newton, Euler, Cauchy, Gauss, Monge and other greats, ready to challenge today's would-be problem solvers. Among them: How is a sundial constructed? How can you calculate the logarithm of a given number without the use of logarithm table? No advanced math is required. Includes 100 problems with proofs.
Synopsis
"The collection, drawn from arithmetic, algebra, pure and algebraic geometry and astronomy, is extraordinarily interesting and attractive." -- Mathematical Gazette
This uncommonly interesting volume covers 100 of the most famous historical problems of elementary mathematics. Not only does the book bear witness to the extraordinary ingenuity of some of the greatest mathematical minds of history -- Archimedes, Isaac Newton, Leonhard Euler, Augustin Cauchy, Pierre Fermat, Carl Friedrich Gauss, Gaspard Monge, Jakob Steiner, and many others -- but it provides rare insight and inspiration to any reader, from high school math student to professional mathematician. This is indeed an unusual and uniquely valuable book.
The one hundred problems are presented in six categories: 26 arithmetical problems, 15 planimetric problems, 25 classic problems concerning conic sections and cycloids, 10 stereometric problems, 12 nautical and astronomical problems, and 12 maxima and minima problems. In addition to defining the problems and giving full solutions and proofs, the author recounts their origins and history and discusses personalities associated with them. Often he gives not the original solution, but one or two simpler or more interesting demonstrations. In only two or three instances does the solution assume anything more than a knowledge of theorems of elementary mathematics; hence, this is a book with an extremely wide appeal.
Some of the most celebrated and intriguing items are: Archimedes' "Problema Bovinum," Euler's problem of polygon division, Omar Khayyam's binomial expansion, the Euler number, Newton's exponential series, the sine and cosine series, Mercator's logarithmic series, the Fermat-Euler prime number theorem, the Feuerbach circle, the tangency problem of Apollonius, Archimedes' determination of pi, Pascal's hexagon theorem, Desargues' involution theorem, the five regular solids, the Mercator projection, the Kepler equation, determination of the position of a ship at sea, Lambert's comet problem, and Steiner's ellipse, circle, and sphere problems.
This translation, prepared especially for Dover by David Antin, brings Dorrie's "Triumph der Mathematik" to the English-language audience for the first time.
Table of Contents
ARITHMETICAL PROBLEMS
1. Archimedes' Problem Bovinum
2. The Weight Problem of Bachet de Méziriac
3. Newton's Problem of the Fields and Cows
4. Berwick's Problem of the Seven Sevens
5. Kirkman's Schoolgirl Problem
6. The Bernoulli-Euler Problem of the Misaddressed Letters
7. Euler's Problem of Polygon Division
8. Lucas' Problem of the Married Couples
9. Omar Khayyam's Binomial Expansion
10. Cauchy's Mean Theorem
11. Bernoulli's Per Sum Problem
12. The Euler Number
13. Newton's Exponential Series
14. Nicolaus Macerator's Logarithmic Series
15. Newton's Sine and Cosine Series
16. André's Derivation of the Secant and Tangent Series
17. Gregory's Arc Tangent Series
18. Buffon's Needle Problem
19. The Fermat-Euler Prime Number Theorem
20. The Fermat Equation
21. The Fermat-Gauss Impossibility Theorem
22. The Quadratic Reciprocity Law
23. Gauss' Fundamental Theorem of Algebra
24. Sturm's Problem of the Number of Roots
25. Abel's Impossibility Theorem
26. The Hermite-Lindemann Transcendence Theorem
PLANIMETRIC PROBLEMS
27. Euler's Straight Line
28. The Feuerbach Circle
29. Castillon's Problem
30. Malfatti's Problem
31. Monge's Problem
32. The Tangency Problem of Apollonius
33. Mascheroni's Compass Problem
34. Steiner's Straight-edge Problem
35. The Delian Cube-doubling Problem
36. Trisection of an Angle
37. The Regular Heptadecagon
38. Archimedes' Determination of the Number Pi
39. Fuss' Problem of the Chord-Tangent Quadrilateral
40. Annex to a Survey
41. Alhazen's Billiard Problem
PROBLEMS CONCERNING CONIC SECTIONS AND CYCLOIDS
42. An Ellipse from Conjugate Radii
43. An Ellipse in a Parallelogram
44. A Parabola from Four Tangents
45. A Parabola from Four Points
46. A Hyperbola from Four Points
47. Van Schooten's Locus Problem
48. Cardan's Spur Wheel Problem
49. Newton's Ellipse Problem
50. The Poncelet-Brianchon Hyperbola Problem
51. A Parabola as Envelope
52. The Astroid
53. Steiner's Three-pointed Hypocycloid
54. The Most Nearly Circular Ellipse Circumscribing a Quadrilateral
55. The Curvature of Conic Sections
56. Archimedes' Squaring of a Parabola
57. Squaring a Hyperbola
58. Rectification of a Parabola
59. Desargue's Homology Theorem (Theorem of Homologous Triangles)
60. Steiner's Double Element Construction
61. Pascal's Hexagon Theorem
62. Brianchon's Hexagram Theorem
63. Desargues' Involution Theorem
64. A Conic Section from Five Elements
65. A Conic Section and a Straight Line
66. A Conic Section and a Point
STEREOMETRIC PROBLEMS
67. Steiner's Division of Space by Planes
68. Euler's Tetrahedron Problem
69. The Shortest Distance Between Skew Lines
70. The Sphere Circumscribing a Tetrahedron
71. The Five Regular Solids
72. The Square as an Image of a Quadrilateral
73. The Pohlke-Schwartz Theorem
74. Gauss' Fundamental Theorem of Axonometry
75. Hipparchus' Stereographic Projection
76. The Mercator Projection
NAUTICAL AND ASTRONOMICAL PROBLEMS
77. The Problem of the Loxodrome
78. Determining the Position of a Ship at Sea
79. Gauss' Two-Altitude Problem
80. Gauss' Three-Altitude Problem
81. The Kepler Equation
82. Star Setting
83. The Problem of the Sundial
84. The Shadow Curve
85. Solar and Lunar Eclipses
86. Sidereal and Synodic Revolution Periods
87. Progressive and Retrograde Motion of the Planets
88. Lambert's Comet Problem
EXTREMES
89. Steiner's Problem Concerning the Euler Number
90. Fagnano's Altitude Base Point Problem
91. Fermat's Problem for Torricelli
92. Tacking Under a Headwind
93. The Honeybee Cell (Problem by Réaumur)
94. Regiomontanus' Maximum Problem
95. The Maximum Brightness of Venus
96. A Comet Inside the Earth's Orbit
97. The Problem of the Shortest Twilight
98. Steiner's Ellipse Problem
99. Steiner's Circle Problem
100. Steiner's Sphere Problem
Index of Names