## Geometry and Algebra in Ancient CivilizationsOriginally, my intention was to write a "History of Algebra", in two or three volumes. In preparing the first volume I saw that in ancient civiliza tions geometry and algebra cannot well be separated: more and more sec tions on ancient geometry were added. Hence the new title of the book: "Geometry and Algebra in Ancient Civilizations". A subsequent volume on the history of modem algebra is in preparation. It will deal mainly with field theory, Galois theory and theory of groups. I want to express my deeply felt gratitude to all those who helped me in shaping this volume. In particular, I want to thank Donald Blackmore Wagner (Berkeley) who put at my disposal his English translation of the most interesting parts of the Chinese "Nine Chapters of the Art of Arith metic" and of Liu Hui's commentary to this classic, and also Jacques Se siano (Geneva), who kindly allowed me to use his translation of the re cently discovered Arabic text of four books of Diophantos not extant in Greek. Warm thanks are also due to Wyllis Bandler (Colchester, England) who read my English text very carefully and suggested several improve ments, and to Annemarie Fellmann (Frankfurt) and Erwin Neuenschwan der (Zurich) who helped me in correcting the proof sheets. Miss Fellmann also typed the manuscript and drew the figures. I also want to thank the editorial staff and production department of Springer-Verlag for their nice cooperation. |

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### Sadržaj

1 | |

Pythagoras and the Ox | 14 |

Euclids Proof | 29 |

Areas of Plane Figures | 38 |

The Broken Bamboo | 53 |

The Historical Development | 66 |

The Application of Areas | 80 |

The Sum of a Geometrical Progression | 94 |

The Motion of the Apogees and Nodes | 127 |

Archimedes Upper and Lower Limits for ws | 141 |

Pythagorean Triples | 154 |

Part B Babylonian Egyptian and Early Greek Problems | 158 |

Circling the Square as a Ritual Problem | 172 |

Apollonios Rapid Method | 185 |

The Volume of a Pyramid | 200 |

Liu Hui and Euclid | 202 |

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### Uobičajeni izrazi i fraze

ancient apogee Apollonios applied approximation Archimedes arithmetic Aryabhata Aryabhatiya astronomical Babylonian texts Bhaskara Brahmagupta Cairo papyrus calculated ch'ih Chinese Nine Chapters Chinese text chord circle segment circumference commentary computed correct cube cubits diagonal diameter difference Diophantos divided divisor Egypt Egyptian epicycle equal Euclid Euclid's Elements Euclidean algorithm formula Geometric Algebra geometrical constructions given numbers Greek height Heron Hindu hypotenuse identity inscribed integer Kaliyuga line segments linear Diophantine equations linear equations Liu Hui longitudes mathematical mathematician means megalithic Megalithic Yard method of solution multiplied Neugebauer numbers of revolution obtains Omar Khayyám pair Pell’s equation planets polygon problem Proklos proof proportion proposition Pythagorean triangles Pythagorean triples quadratic equations radius ratio rectangle remainder result right-angled triangle rule says scribe sexagesimal side similar Sines solve square root straight line subtract Sulvasutras text BM Theorem of Pythagoras theory tion tradition translation treatise Type unknown volume