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History of Mathematics

Engineering Physics
Engineering Mathematics

History of Mathematics
Carl Boyer
735 pages

Open: History of Mathematics
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Numerous histories of mathematics have appeared during this century, many of them in the English language. Some are very recent, such as J. F. Scott's A History ofMathematics; a new entry in the field therefore should have characteristics not already present in the available books. Actually, few of the histories at hand are textbooks, at least not in the American sense of the word, and Scott's History is not one of them. It appeared, therefore, that there was room for a new book—one that would meet more satisfactorily my own preferences and possibly those of others.

The two-volume History of Mathematics by David Eugene Smith was indeed written "for the purpose of supplying teachers and students with a usable textbook on the history of elementary mathematics," but it covers too wide an area on too low a mathematical level for most modern college courses, and it is lacking in problems of varied types. Florian Cajori's History of Mathematics, still is a very helpful reference work; but it is not adapted to classroom use, nor is E. T. Bell's admirable The Development of Mathematics. The most successful and appropriate textbook today appears to be Howard Eves, An Introduction to the History of Mathematics, which I have used with considerable satisfaction in at least a dozen classes since it first appeared in 1953. I have occasionally departed from the arrangement of topics in the book in striving toward a heightened sense of historical mindedness and have supplemented the material by further reference to the contributions of the eighteenth and nineteenth centuries especially by the use of D. J. Struik, A Concise History of Mathematics.

The reader of this book, whether a layman, a student, or a teacher of a course in the history of mathematics, will find that the level of mathematical background that is presupposed is approximately that of a college junior or senior, but the material can be perused profitably also by readers with either stronger or weaker mathematical preparation. Each chapter ends with a set of exercises that are graded roughly into three categories. Essay questions that are intended to indicate the reader's ability to organize and put into his own words the material discussed in the chapter are listed first. Then follow relatively easy exercises that require the proofs of some of the theorems mentioned in the chapter or their application to varied situations. Finally, there are a few starred exercises, which are either more difficult or require
specialized methods that may not be familiar to all students or all readers. The exercises do not in any way form part of the general exposition and can be disregarded by the reader without loss of continuity.

Here and there in the text are references to footnotes, generally bibliographical, and following each chapter there is a list of suggested readings. Included are some references to the vast periodical literature in the field, for it is not too early for students at this level to be introduced to the wealth of material available in good libraries. Smaller college libraries may not be able to provide all of these sources, but it is well for a student to be aware of the larger realms of scholarship beyond the confines of his own campus. There are references also to works in foreign languages, despite the fact that some students, hopefully not many, may be unable to read any of these. Besides providing important additional sources for those who have a reading knowledge of a foreign language, the inclusion of references in other languages may help to break down the linguistic provincialism which, ostrichlike, takes refuge in the mistaken impression that everything worthwhile appeared in, or has been translated into, the English language.

The present work differs from the most successful presently available textbook in a stricter adherence to the chronological arrangement and a stronger emphasis on historical elements. There is always the temptation in a class in history of mathematics to assume that the fundamental purpose of the course is to teach mathematics. A departure from mathematical standards is then a mortal sin, whereas an error in history is venial. I have striven to avoid such an attitude, and the purpose of the book is to present the history of mathematics with fidelity, not only to mathematical structure and exactitude, but also to historical perspective and detail. It would be folly, in a book of this scope, to expect that every date, as well as every decimal point, is correct. It is hoped, however, that such inadvertencies as may survive beyond the stage of page proof will not do violence to the sense of history, broadly understood, or to a sound view of mathematical concepts. It cannot be too strongly emphasized that this single volume in no way purports to present the history of mathematics in its entirety. Such an enterprise would call for the concerted effort of a team, similar to that which produced the fourth volume of Cantor's Vorlesungen uber Geschichte der Mathematik in 1908 and brought the story down to 1799. In a work of modest scope the author must exercise judgment in the selection of the materials to be included, reluctantly restraining the temptation to cite the work of every productive mathematician ; it will be an exceptional reader who will not note here what he regards as unconscionable omissions. In particular, the last chapter attempts merely to point out a few of the salient characteristics of the twentieth century. In the field of the history of mathematics perhaps nothing is more to be desired than that there should appear a latter-day Felix Klein who would complete for our century the type of project Klein essayed for for the nineteenth century, but did not live to finish.

A published work is to some extent like an iceberg, for what is visible constitutes only a small fraction of the whole. No book appears until the author has lavished time on it unstintingly and unless he has received encouragement and support from others too numerous to be named individually.


Chapter I. Primitive Origins 1
1 The concept of number. 2 Early number bases. 3 Number language and the origin of counting. 4 Origin of geometry.

Chapter II. Egypt g
1 Early records. 2 Hieroglyphic notation. 3 Ahmes papyrus. 4 Unit fractions. 5 Arithmetic operations. 6 Algebraic problems. 7 Geometrical problems. 8 A trigonometric ratio. 9 Moscow papyrus. 10 Mathematical weaknesses.

Chapter III. Mesopotamia 26
1 Cuneiform records. 2 Positional numeration. 3 Sexagesimal fractions. 4 Fundamental operations. 5 Algebraic problems. 6 Quadratic equations. 7 Cubic equations. 8 Pythagorean triads. 9 Polygonal areas. 10 Geometry as applied arithmetic. 11 Mathematical weaknesses.

Chapter IV. Ionia and the Pythagoreans 48
1 Greek origins. 2 Thales of Miletus. 3 Pythagoras of Samos. 4 The Pythagorean pentagram. 5 Number mysticism. 6 Arithmetic and cosmology. 7 Figurate numbers. 8 Proportions. 9 Attic numeration. 10 Ionian numeration. 11 Arithmetic and logistic.

Chapter V. The Heroic Age 69
1 Centers of activity. 2 Anaxagoras of Clazomenae. 3 Three famous problems. 4 Quadrature of lunes. 5 Continued proportions. 6 Hippias of Ellis. 7 Philolaus and Archytas of Tarentum. 8 Duplication of the cube. 9 Incommensurability. 10 The golden section. 11 Paradoxes ofZeno. 12 Deductive reasoning. 13 Geometrical algebra. 14 Democritus of Abdera.

Chapter VI. The Age of Plato and Aristotle 91
1 The seven liberal arts. 2 Socrates. 3 Platonic solids. 4 Theodorus of Cyrene. 5 Platonic arithmetic and geometry. 6 Origin of analysis. 7 Eudoxus of Cnidus. 8 Method of exhaustion. 9 Mathematical astronomy. 10 Menaechmus. 11 Duplication of the cube. 12 Dinostratus and the squaring of the circle. 13 Autolycus of Pitane. 14 Aristotle. 15 End of the Hellenic period.

Chapter VII. Euclid of Alexandria 111
1 Author of the Elements. 2 Other works. 3 Purpose ofthe Elements. 4 Definitions and postulates. 5 Scope of Book I. 6 Geometrical algebra. 7 Books III and IV. 8 Theory of proportion. 9 Theory of numbers. 10 Prime and perfect numbers. 11 Incommensurability. 12 Solid geometry. 13 Apocrypha. 14 Influence of the Elements.

Chapter VIII. Archimedes of Syracuse 134
1 The siege of Syracuse. 2 Law of the lever. 3 The hydrostatic principle. 4 The Sand-Reckoner. 5 Measurement of the circle. 6 Angle trisection. 7 Area of a parabolic segment. 8 Volume of a paraboloidal segment. 9 Segment of a sphere. 10 On the Sphere and Cylinder. 11 Book ofLemmas. 12 Semiregular solids and trigonometry. 13 The Method. 14 Volume ofa sphere. 15 Recovery of the Method.

Chapter IX. Apollonius of Perga 157
1 Lost works. 2 Restorations of lost works. 3 The problem of Apollonius. 4 Cycles and epicycles. 5 The Conies. 6 Names of the conic sections. 7 The double-napped cone. 8 Fundamental properties. 9 Conjugate diameters. 10 Tangents and harmonic division. 11 The three-and-four-line locus. 12 Intersecting conies. 13 Maxima and minima, tangents and normals. 14 Similar conies. 15 Foci of conies. 16 Use of coordinates.

Chapter X. Greek Trigonometry and Mensuration 176
1 Early trigonometry. 2 Aristarchus of Samos. 3 Eratosthenes of Cyrene. 4 Hipparchus of Nicaea. 5 Menelaus of Alexandria. 6 Ptolemy's Almagest. 7 The 360 degree circle. 8 Construction of tables. 9 Ptolemaic astronomy. 10 Other works by Ptolemy. 11 Optics and astrology. 12 Heron of Alexandria. 13 Principle of least distance. 14 Decline of Greek mathematics.

Chapter XI. Revival and Decline of Greek Mathematics 196
1 Applied mathematics. 2 Diophantus of Alexandria. 3 Nicomachus of Gerasa. 4 The Arithmetica of Diophantus. 5 Diophantine problems. 6 The place of Diophantus in algebra. 7 Pappus ofAlexandria. 8The Collection. 9 Theorems of Pappus. 10 The Pappus problem. 11 The Treasury of Analysis. 12 The Pappus-Guldin theorems. 13 Proclus of Alexandria. 14 Boethius. 15 End of the Alexandrian period. 16 The Greek Anthology. 17 Byzantine mathematicians of the sixth century.

Chapter XII. China and India 217
1 The oldest documents. 2 The Nine Chapters. 3 Magic squares. 4 Rod numerals. 5 The abacus and decimal fractions. 6 Values of pi. 7 Algebra and Horner's method. 8 Thirteenth-century mathematicians. 9 The arithmetic triangle. 10 Early mathematics in India. 11 The Sulvasutras. 12 The Siddhantas. 13 Aryabhata. 14 Hindu numerals. 15 The symbol for zero. 16 Hindu trigonometry. 17 Hindu multiplication. 18 Long division. 19 Brahmagupta. 20 Brahmagupta's formula. 21 Indeterminate equations. 22 Bhaskara. 23 The Lilavati. 24 Ramanujan.

Chapter XIII. The Arabic Hegemony 249
1 Arabic conquests. 2 The House of Wisdom. 3 Al-jabr. 4 Quadratic equations. 5 The father of algebra. 6 Geometric foundation. 7 Algebraic problems. 8 A problem from Heron. 9 Abd al-Hamid ibn-Turk. 10 Thabit ibn-Qurra. 11 Arabic numerals. 12 Arabic trigonometry. 13 AbuT-Wefa and al- Karkhi. 14 Al-Biruni and Alhazen. 15 Omar Khayyam. 16 The parallel postulate. 17 Nasir Eddin. 18 Al-Kashi.

Chapter XIV. Europe in the Middle Ages 272
1 From Asia to Europe. 2 Byzantine mathematics. 3 The Dark Ages. 4 Alcuin and Gerbert. 5 The century of translation. 6 The spread of Hindu-Arabic numerals. 7 The Liber abaci. 8 The Fibonacci sequence. 9 A solution of a cubic equation. 10 Theory of numbers and geometry. 11 Jordanus Nemorarius. 12 Campanus of Novara. 13 Learning in the thirteenth century. 14 Medieval kinematics. 15 Thomas Bradwardine. 16 Nicole Oresme. 17 The latitude of forms. 18 Infinite series. 19 Decline of medieval learning.

Chapter XV. The Renaissance 297
1 Humanism. 2 Nicholas of Cusa. 3 Regiomontanus. 4 Application of algebra to geometry. 5 A transitional figure. 6 Nicolas Chuquet's Triparty. 7 Luca Pacioli's Summa. 8 Leonardo da Vinci. 9 Germanic algebras. 10 Cardan's Ars magna. 11 Solution of the cubic equation. 12 Ferrari's solution of the quartic equation. 13 Irreducible cubics and complex numbers. 14 Robert Recorde. 15 Nicholas Copernicus. 16 Georg Joachim Rheticus. 17 Pierre de la Ramee. 18 Bombelli's Algebra. 19 Johannes Werner. 20 Theory of perspective. 21 Cartography.

Chapter XVI. Prelude to Modern Mathematics 333
1 Francois Viete. 2 Concept of a parameter. 3 The analytic art. 4 Relations between roots and coefficients. 5 Thomas Harriot and William Oughtred. 6 Horner's method again. 7 Trigonometry and prosthaphaeresis. 8 Trigonometric solution of equations. 9 John Napier. 10 Invention of logarithms. 11 Henry Briggs. 12 Jobst Biirgi. 13 Applied mathematics and decimal fractions. 14 Algebraic notations. 15 Galileo Galilei. 16 Values of pi. 17 Reconstruction of Apollonius' On Tangencies. 18 Infinitesimal analysis 19 Johannes Kepler. 20 Galileo's Two New Sciences. 21 Galileo and the infinite. 22 Bonaver.tura Cavalieri. 23 The spiral and the parabola.

Chapter XVII. The Time of Fermat and Descartes 367
* 1 Leading mathematicians of the time. 2 The Discours de la methode. 3 Invention of analytic geometry. 4 Arithmetization of geometry. 5 Geometrical algebra. 6 Classification of curves. 7 Rectification of curves. 8 Identification of conies. 9 Normals and tangents. 10 Descartes' geometrical concepts. 11 Fermat's loci. 12 Higher-dimensional analytic geometry. 13 Fermat's differentiations. 14 Fermat's integrations. 15 Gregory of St. Vincent. 16 Theory of numbers. 17 Theorems of Fermat. 18 Gilles Persone de Roberval. 19 Evangelista Torricelli. 20 New curves. 21 Girard Desargues. 22 Projective geometry. 23 Blaise Pascal. 24 Probability. 25 The cycloid.

Chapter XVIII. A Transitional Period 404
1 Philippe de Lahire. 2 Georg Mohr. 3 Pietro Mengoli. 4 Frans van Schooten. 5 Jan de Witt. 6 Johann Hudde. 7 Rene Francois de Sluse. 8 The pendulum clock. 9 Involutes and evolutes. 10 John Wallis. 11 On Conic Sections. 12 Arithmetica infinitorum. 13 Christopher Wren. 14 Wallis' formulas. 15 James Gregory. 16 Gregory's series. 17 Nicolaus Mercator and William Brouncker. 18 Barrow's method of tangents.

Chapter XIX. Newton and Leibniz 429
1 Newton's early work. 2 The binomial theorem. 3 Infinite series. 4 The Method of Fluxions. 5 The Principia. 6 Leibniz and the harmonic triangle. 7 The differential triangle and infinite series. 8 The differential calculus. 9 Determinants, notations, and imaginary numbers. 10 The algebra of logic. 1 1 The inverse square law. 12 Theorems on conies. 13 Optics and curves. 14 Polar and other coordinates. 15 Newton's method and Newton's parallelogram. 16 The Arithmetica universalis. 17 Later years.

Chapter XX. The Bernoulli Era 455
1 The Bernoulli family. 2 The logarithmic spiral. 3 Probability and infinite series. 4 L'Hospital's rule. 5 Exponential calculus. 6 Logarithms of negative numbers. 7 Petersburg paradox. 8 Abraham de Moivre. 9 De Moivre's theorem. 10 Roger Cotes. 11 James Stirling. 12 Colin Maclaurin. 13 Taylor's series. 14 The Analyst controversy. 15 Cramer's rule. 16 Tschirnhaus transformations. 17 Solid analytic geometry. 18 Michel Rolle and Pierre Varignon. 19 Mathematics in Italy. 20 The parallel postulate. 21 Divergent series.

Chapter XXI. The Age of Euler 481
1 Life of Euler. 2 Logarithms of negative numbers. 3 Foundation of analysis. 4 Infinite series. 5 Convergent and divergent series. 6 Life of d'Alembert. 7 The Euler identities. 8 D'Alembert and limits. 9 Differential equations. 10 The Clairauts. 11 The Riccatis. 12 Probability. 13 Theory of numbers. 14 Textbooks. 15 Synthetic geometry. 16 Solid analytic geometry. 17 Lambert and the parallel postulate. 18 Bezout and elimination.

Chapter XXII. Mathematicians of the French Revolution 510
1 The age of revolutions. 2 Leading mathematicians. 3 Publications before 1789. 4 Lagrange and determinants. 5 Committee on Weights and Measures. 6 Condorcet on education. 7 Monge as administrator and teacher. 8 Descriptive geometry and analytic geometry. 9 Textbooks. 10 Lacroix on analytic geometry. 11 The Organizer of Victory. 12 Metaphysics of the calculus and geometry. 13 Geometrie de position. 14 Transversals. 15 Legendre's Geometry. 16 Elliptic integrals. 17 Theory of numbers. 18 Theory of functions. 19 Calculus of variations. 20 Lagrange multipliers. 21 Laplace and probability. 22 Celestial mechanics and operators. 23 Political changes.

Chapter XXIII. The Time of Gauss and Cauchy 544
1 Early discoveries by Gauss. 2 Graphical representation of complex numbers. 3 The fundamental theorem of algebra. 4 The algebra of congruences. 5 Reciprocity and frequency of primes. 6 Constructible regular polygons. 7 Astronomy and least squares. 8 Elliptic functions. 9 Abel's life and work. 10 Theory of determinants. 11 Jacobians. 12 Mathematical journals. 13 Complex variables. 14 Foundations of the calculus. 15 Bernhard Bolzano. 16 Tests for convergence. 17 Geometry. 18 Applied mathematics.

Chapter XXIV. The Heroic Age in Geometry 572
1 Theorems of Brianchon and Feuerbach. 2 Inversive geometry. 3 Poncelet's projective geometry. 4 Pliicker's abridged notation. 5 Homogeneous coordinates. 6 Line coordinates and duality. 7 Revival of British mathematics. 8 Cayley's ^-dimensional geometry. 9 Geometry in Germany. 10 Lobachevsky and Ostrogradsky. 11 Non-Euclidean geometry. 12 The Bolyais. 13 Riemannian geometry. 14 Spaces of higher dimension. 15 Klein's Erlanger Programm. 16 Klein's hyperbolic model.

Chapter XXV. The Arithmetization of Analysis 598
1 Fourier series. 2 Analytic number theory. 3 Transcendental numbers. 4 Uneasiness in analysis. 5 The Bolzano-Weierstrass theorem. 6 Definition of real number. 7 Weierstrassian analysis. 8 The Dedekind "cut". 9 The limit concept. 10 Gudermann's influence. 11 Cantor's early life. 12 The "power" of infinite sets. 13 Properties of infinite sets. 14 Transfinite arithmetic. 15 Kronecker's criticism of Cantor's work.

Chapter XXVI. The Rise of Abstract Algebra 620
1 The Golden Age in mathematics. 2 Mathematics at Cambridge. 3 Peacock, the "Euclid of algebra." 4 Hamilton's quaternions. 5 Grassmann and Gibbs. 6 Cayley's matrices. 7 Sylvester's algebra. 8 Invariants of quadratic forms. 9 Boole's analysis of logic. 10 Boolean algebra. 11 De Morgan and the Peirces. 12 The tragic life of Galois. 13 Galois theory. 14 Field theory. 15 Frege's definition of cardinal number. 16 Peano's axioms.

Chapter XXVII. Aspects of the Twentieth Century 649
1 The nature of mathematics. 2 Poincare's theory of functions. 3 Applied mathematics and topology. 4 Hilbert's problems. 5 Godel's theorem. 6 Transcendental numbers. 7 Foundations of geometry. 8 Abstract spaces. 9 The foundations of mathematics. 10 Intuitionism, formalism, and logicism. 11 Measure and integration. 12 Point set topology. 13 Increasing abstraction in algebra. 14 Probability. 15 High-speed computers. 16 Mathematical structure. 17 Bourbaki and the "New Mathematics."