100 Great Problems of Elementary Mathematics (eBook)

Heinrich Dorrie (Autor)

eBook Download: EPUB

2013
416 Seiten
Dover Publications (Verlag)
978-0-486-31847-9 (ISBN)

Infos

Problems that beset Archimedes, Newton, Euler, Cauchy, Gauss, Monge, Steiner, and other great mathematical minds. Features squaring the circle, pi, and similar problems. No advanced math is required. Includes 100 problems with proofs.

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"e;The collection, drawn from arithmetic, algebra, pure and algebraic geometry and astronomy, is extraordinarily interesting and attractive."e; — Mathematical GazetteThis 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' "e;Problema Bovinum,"e; 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 Dörrie's "e;Triumph der Mathematik"e; to the English-language audience for the first time.

Inhalt

ARITHMETICAL PROBLEMS1. Archimedes' Problem Bovinum2. The Weight Problem of Bachet de Méziriac3. Newton's Problem of the Fields and Cows4. Berwick's Problem of the Seven Sevens5. Kirkman's Schoolgirl Problem6. The Bernoulli-Euler Problem of the Misaddressed Letters7. Euler's Problem of Polygon Division8. Lucas' Problem of the Married Couples9. Omar Khayyam's Binomial Expansion10. Cauchy's Mean Theorem11. Bernoulli's Per Sum Problem12. The Euler Number13. Newton's Exponential Series14. Nicolaus Macerator's Logarithmic Series15. Newton's Sine and Cosine Series16. André's Derivation of the Secant and Tangent Series17. Gregory's Arc Tangent Series18. Buffon's Needle Problem19. The Fermat-Euler Prime Number Theorem20. The Fermat Equation21. The Fermat-Gauss Impossibility Theorem22. The Quadratic Reciprocity Law23. Gauss' Fundamental Theorem of Algebra24. Sturm's Problem of the Number of Roots25. Abel's Impossibility Theorem26. The Hermite-Lindemann Transcendence TheoremPLANIMETRIC PROBLEMS27. Euler's Straight Line28. The Feuerbach Circle29. Castillon's Problem30. Malfatti's Problem31. Monge's Problem32. The Tangency Problem of Apollonius33. Mascheroni's Compass Problem34. Steiner's Straight-edge Problem35. The Delian Cube-doubling Problem36. Trisection of an Angle37. The Regular Heptadecagon38. Archimedes' Determination of the Number Pi 39. Fuss' Problem of the Chord-Tangent Quadrilateral40. Annex to a Survey41. Alhazen's Billiard ProblemPROBLEMS CONCERNING CONIC SECTIONS AND CYCLOIDS42. An Ellipse from Conjugate Radii43. An Ellipse in a Parallelogram44. A Parabola from Four Tangents45. A Parabola from Four Points46. A Hyperbola from Four Points47. Van Schooten's Locus Problem48. Cardan's Spur Wheel Problem49. Newton's Ellipse Problem50. The Poncelet-Brianchon Hyperbola Problem51. A Parabola as Envelope52. The Astroid53. Steiner's Three-pointed Hypocycloid54. The Most Nearly Circular Ellipse Circumscribing a Quadrilateral55. The Curvature of Conic Sections56. Archimedes' Squaring of a Parabola57. Squaring a Hyperbola58. Rectification of a Parabola59. Desargue's Homology Theorem (Theorem of Homologous Triangles)60. Steiner's Double Element Construction61. Pascal's Hexagon Theorem62. Brianchon's Hexagram Theorem63. Desargues' Involution Theorem64. A Conic Section from Five Elements65. A Conic Section and a Straight Line66. A Conic Section and a PointSTEREOMETRIC PROBLEMS67. Steiner's Division of Space by Planes68. Euler's Tetrahedron Problem69. The Shortest Distance Between Skew Lines70. The Sphere Circumscribing a Tetrahedron71. The Five Regular Solids72. The Square as an Image of a Quadrilateral73. The Pohlke-Schwartz Theorem74. Gauss' Fundamental Theorem of Axonometry75. Hipparchus' Stereographic Projection76. The Mercator ProjectionNAUTICAL AND ASTRONOMICAL PROBLEMS77. The Problem of the Loxodrome78. Determining the Position of a Ship at Sea79. Gauss' Two-Altitude Problem80. Gauss' Three-Altitude Problem81. The Kepler Equation82. Star Setting83. The Problem of the Sundial84. The Shadow Curve85. Solar and Lunar Eclipses86. Sidereal and Synodic Revolution Periods87. Progressive and Retrograde Motion of the Planets88. Lambert's Comet ProblemEXTREMES89. Steiner's Problem Concerning the Euler Number90. Fagnano's Altitude Base Point Problem91. Fermat's Problem for Torricelli92. Tacking Under a Headwind93. The Honeybee Cell (Problem by Réaumur)94. Regiomontanus' Maximum Problem95. The Maximum Brightness of Venus96. A Comet Inside the Earth's Orbit97. The Problem of the Shortest Twilight98. Steiner's Ellipse Problem99. Steiner's Circle Problem100. Steiner's Sphere ProblemIndex of Names

Produkt-Details

• Erscheint lt. Verlag: 9.4.2013
• Reihe/Serie: Dover Books on Mathematics
• Reihe/Serie: Dover Books on Mathematics
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber ► Natur / Technik ► Naturwissenschaft
• ISBN-10: 0-486-31847-8 / 0486318478
• ISBN-13: 978-0-486-31847-9 / 9780486318479