Soon after the publication of my"Ontwakende W etenschap"the need for an English translation was felt. We were very glad to find a translator fully familiar with the English and Dutch languages and with mathematical terminol· ogy. The publisher, Noordhoff, had the splendid idea to ask H. G. Beyen, professor of archeology, for his help in choosing a nice set of illustrations. It was a difficult task. The illustrations had to be both instructive and attractive, and they had t~ illustrate the history of science as well as the general background of ancient civilization. The publisher encouraged us to find better and still better illustrations, and he ordered photographs from all over the world, with never failing energy and enthusiasm. Mr. Beyen's highly instructive subscripts will help the reader to see the inter· relation between way of living, art, and science of the ancient world. Thanks are due to many correspondents, who have suggested additions and pointed out errors. Sections on Astrolabes and Stereographte Projection and on Archimedes' construction of the heptagon have been added. The sections on Perspective and on the Anaphorai of Hypsicles have been enlarged. In the second English edition I have incorporated an important discovery of P. Huber, which sheds new light upon the role of geometry In Babylonian algebra (see p. 73). The section on Heron's Metrics (see p. 277) was written anew, follOWing a suggestion of E. M. Bruins. Zurich. 1961 B. L.

Inhalt

I. The Egyptians.- Chronological Summary.- The Egyptians as the "inventors" of geometry.- The Rhind papyrus.- For whom was the Rhind papyrus written?.- The class of royal scribes.- The technique of calculation.- Multiplication.- Division.- Natural fractions and unit fractions.- Calculation with natural fractions.- Further relations between fractions.- Duplication of unit fractions.- Division once more.- The (2: n) table.- The red auxiliaries.- Complementation of a fraction to 1.- "Aha-calculations".- Applied calculations.- The development of the computing technique.- Hypothesis of an advanced science.- The geometry of the Egyptians.- Inclination of oblique planes.- Areas.- Area of the hemisphere.- Volumes.- What could the Greeks learn from the Egyptians?.- II. Number systems, digits and the art of computing.- The sexagesimal system.- How did the sexagesimal system originate?.- Oldest Sumerian period (before 3000 B.C.).- Later Sumerian period (about 2000 B.C.).- Sumerian technique of computation.- Table of 7 and of 16,40.- Normal table of inverses.- Squares, square roots and cube roots.- The Greek notation for numbers.- Counting boards and counting pebbles.- Calculation with fractions.- Sexagesimal fractions.- Hindu numerals.- Number systems; Kharosti and Brahmi.- The invention of the positional system.- The date of the invention.- Poetic numbers.- Aryabhata and his syllable-numbers.- Where does the zero come from?.- The triumphal procession of the Hindu numerals.- The abacus of Gerbert.- III. Babylonian mathematics.- Chronological summary.- Babylonian algebra.- First example (MKT I, p. 113).- Interpretation.- Second example (MKT I, p. 280).- Third example (MKT I, p. 323).- Fourth example (MKT I, p. 154).- Fifth example (MKT III, p. 8, no. 14).- Quadratic equations (MKT III, p. 6).- Sixth example (MKT III, p. 9, no. 18).- Seventh example (MKT I. p. 485).- Eighth example (MKT I, p. 204).- Geometrical proofs of algebraic formulas?.- Ninth example (MKT I, p. 342).- A lesson-text (MKT II, p. 39).- Babylonian geometry.- Volumes and areas.- Frustra of cones and of pyramids (MKT, pp. 176 and 178).- The "Theorem of Pythagoras" (MKT II, p. 53).- Babylonian theory of numbers.- Progressions (MKT I, p. 99).- Plimpton 322: Right triangles with rational sides.- Applied mathematics.- Summary.- Greek Mathematics.- IV. The age of Thales and Pythagoras.- Chronological summary.- Hellas and the Orient.- Thales of Milete.- Prediction of a solar eclipse.- The geometry of Thales.- From Thales to Euclid.- Pythagoras of Samos.- The travels of Pythagoras.- Pythagoras and the theory of harmony.- Pythagoras and the theory of numbers.- Perfect numbers.- Amicable numbers.- Figurate numbers.- Pythagoras and geometry.- The astronomy of the Pythagoreans.- Summary.- The tunnel on Samos.- Antique measuring instruments.- V. The golden age.- Hippasus.- The Mathemata of the Pythagoreans.- The theory of numbers.- The theory of the even and the odd.- Proportions of numbers.- The solution of systems of equations of the first degree.- Geometry.- "Geometric Algebra".- Why the geometric formulation?.- Lateral and diagonal numbers.- Anaxagoras of Clazomenae.- Democritus of Abdera.- Oenopides of Chios.- Squaring the circle.- Antiphon.- Hippocrates of Chios.- Solid geometry in the fifth century, and Perspective.- Democritus.- Cone and pyramid.- Plato on solid geometry.- The duplication of the cube.- Theodorus of Cyrene.- Theodorus and Theaetetus.- Theodorus on higher curves and on mixtures.- Hippias and his Quadratrix.- The main lines of development.- VI. The century of Plato.- Archytas of Taras.- The duplication of the cube.- The style of Archytas.- Book VIII of the Elements.- The Mathemata in the Epinomis.- The duplication of the cube.- According to Menaechmus.- Theaetetus.- Analysis of Book X of the Elements.- The theory of the regular polyhedra.- The theory of proportions in Theaetetus.- Eudoxus of Cnidos.- Eudoxus as an astronomer.- The exhaustion method.- The theory of proportions.- Theaetetus and Eudoxus.- Menaechmus.- Dinostratus.- Autolycus of Pitane.- On the rotating sphere.- On the rising and setting of stars.- Euclid.- The "Elements".- The "Data".- On the division of figures.- Lost geometrical writings.- Euclid's work on applied mathematics.- VII. The Alexandrian Era (330-200 B.C.).- Aristarchus of Samos.- Archimedes' measurement of the circle.- Tables for the lengths of chords.- Archimedes.- Stories about Archimedes.- Archimedes as an astronomer.- The works of Archimedes.- The "Method".- The quadrature of the parabola.- On sphere and cylinder I.- On sphere and cylinder II.- On spirals.- On conoids and spheroids.- The notion of integral in Archimedes.- The book of Lemmas.- The construction of the regular heptagon.- The other works of Archimedes.- Eratosthenes of Cyrene.- Life.- Chronography and measurement of a degree.- Duplication of the cube.- Theory of numbers.- Medieties.- Nicomedes.- The trisection of the angle.- The duplication of the cube in Nicomedes.- Apollonius of Perga.- The theory of the epicycle and of the excenter.- Conica.- The conic sections before Apollonius.- The ellipse as a section of a cone according to Archimedes.- How were the symptoms derived originally?.- A question and an answer.- The derivation of the symptoms according to Apollonius.- Conjugate diameters and conjugate hyperbolas.- Tangent lines.- The equation referred to the center.- The two-tangents theorem and the transformation to new axes.- Cones of revolution through a given conic.- The second book.- The third book.- Loci involving 3 or 4 straight lines.- The fifth book.- The sixth, seventh and eighth books.- Further works of Apollonius.- VIII. The decay of Greek mathematics.- External causes of decay.- The inner causes of decay.- 1. The difficulty of geometric algebra.- 2. The difficulty of the written tradition.- The commentaries of Pappus of Alexandria.- The epigones of the great mathematicians.- 1. Diocles.- The cissoid.- 2. Zenodorus.- Isoperimetric figures.- 3. Hypsicles.- The fourteenth book of the Elements.- Anaphora.- History of trigonometry.- Plane trigonometry.- Spherical trigonometry.- Menelaus.- Transversal proposition.- Heron of Alexandria.- Metrics.- Diophantus of Alexandria.- Arithmetica.- Diophantine equations.- The precursors of Diophantus.- Connection with Babylonian and Arabic algebra.- The algebraic symbolism.- From Book II.- From Book III.- From Book IV.- From Book V.- From Book VI.- Pappus of Alexandria.- A por…