A History of Mathematical Notations

 
 
Dover Publications (Verlag)
  • 1. Auflage
  • |
  • erschienen am 16. Dezember 2014
  • |
  • 864 Seiten
 
E-Book | ePUB mit Adobe DRM | Systemvoraussetzungen
978-0-486-16116-7 (ISBN)
 

This classic study notes the first appearance of a mathematical symbol and its origin, the competition it encountered, its spread among writers in different countries, its rise to popularity, its eventual decline or ultimate survival. The author's coverage of obsolete notations — and what we can learn from them — is as comprehensive as those which have survived and still enjoy favor. Originally published in 1929 in a two-volume edition, this monumental work is presented here in one volume.

  • Englisch
  • New York
  • |
  • USA
  • Höhe: 216 mm
  • |
  • Breite: 137 mm
  • 58,38 MB
  • 862 gr
978-0-486-16116-7 (9780486161167)
0486161161 (0486161161)
weitere Ausgaben werden ermittelt
Florian Cajori
  • Cover
  • Title Page
  • Copyright Page
  • Preface
  • Table of Contents
  • Illustrations
  • Volume I Notations in Elementary Mathematics
  • I. Introduction
  • II. Numeral Symbols and Combinations of Symbols
  • Babylonians
  • Egyptians
  • Phoenicians and Syrians
  • Hebrews
  • Greeks
  • Early Arabs
  • Romans
  • Peruvian and North American Knot Records
  • Aztecs
  • Maya
  • Chinese and Japanese
  • Hindu-Arabic Numerals
  • Introduction
  • Principle of Local Value
  • Forms of Numerals
  • Freak Forms
  • Negative Numerals
  • Grouping of Digits in Numeration
  • The Spanish Calderón
  • The Portuguese Cifrão
  • Relative Size of Numerals in Tables
  • Fanciful Hypotheses on the Origin of Numeral Forms
  • A Sporadic Artificial System
  • General Remarks
  • Opinion of Laplace
  • III. Symbols in Arithmetic and Algebra (Elementary Part)
  • A. Groups of Symbols Used by Individual Writers
  • Greeks-Diophantus, Third Century a.d.
  • Hindu-Brahmagupta, Seventh Century
  • Hindu-The Bakhshali Manuscript
  • Hindu-Bhaskara, Twelfth Century
  • Arabic-al-Khowârizmî, Ninth Century
  • Arabic-al-Karkhî, Eleventh Century
  • Byzantine-Michael Psellus, Eleventh Century
  • Arabic-Ibn Albanna, Thirteenth Century
  • Chinese-Chu Shih-Chieh, Fourteenth Century
  • Byzantine-Maximus Planudes, Fourteenth Century
  • Italian-Leonardo of Pisa, Thirteenth Century
  • French-Nicole Oresme, Fourteenth Century
  • Arabic-al-Qalasâdî, Fifteenth Century
  • German-Regiomontanus, Fifteenth Century
  • Italian-Earliest Printed Arithmetic, 1478
  • French-Nicolas Chuquet, 1484
  • French-Estienne de la Roche, 1520
  • Italian-Pietro Borgi, 1484, 1488
  • Italian-Luca Pacioli, 1494, 1523
  • Italian-F. Ghaligai, 1521, 1548, 1552
  • Italian-H. Cardan, 1532, 1545, 1570
  • Italian-Nicolo Tartaglia, 1506-60
  • Italian-Rafaele Bombelli, 1572
  • German-Johann Widman, 1489, 1526
  • Austrian-Grammateus, 1518, 1535
  • German-Christoff Rudolff, 1525
  • Dutch-Gielis van der Hoecke, 1537
  • German-Michael Stifel, 1544, 1545, 1553
  • German-Nicolaus Copernicus, 1566
  • German-Johann Scheubel, 1545, 1551
  • Maltese-Wil. Klebitius, 1565
  • German-Christophorus Clavius, 1608
  • Belgium-Simon Stevin, 1585
  • Lorraine-Albert Girard, 1629
  • German-Spanish-Marco Aurel, 1552
  • Portuguese-Spanish-Pedro Nuñez, 1567
  • English-Robert Recorde, 1543(?), 1557
  • English-John Dee, 1570
  • English-Leonard and Thomas Digges, 1579
  • English-Thomas Masterson, 1592
  • French-Jacques Peletier, 1554
  • French-Jean Buteon, 1559
  • French-Guillaume Gosselin, 1577
  • French-Francis Vieta, 1591
  • Italian-Bonaventura Cavalieri, 1647
  • English-William Oughtred, 1631, 1632, 1657
  • English-Thomas Harriot, 1631
  • French-Pierre Hérigone, 1634, 1644
  • Scot-French-James Hume, 1635, 1636
  • French-René Descartes
  • English-Isaac Barrow
  • English-Richard Rawlinson, 1655-68
  • Swiss-Johann Heinrich Rahn
  • English-John Wallis, 1655, 1657, 1685
  • Extract from Acta eruditorum, Leipzig, 1708
  • Extract from Miscellanea Berolinensia, 1710 (Due to G. W. Leibniz)
  • Conclusions
  • B. Topical Survey of the Use of Notations
  • Signs of Addition and Subtraction
  • Early Symbols
  • Origin and Meaning of the Signs
  • Spread of the + and - Symbols
  • Shapes of the + Sign
  • Varieties of - Signs
  • Symbols for "Plus or Minus"
  • Certain Other Specialized Uses of + and -
  • Four Unusual Signs
  • Composition of Ratios
  • Signs of Multiplication
  • Early Symbols
  • Early Uses of the St. Andrew's Cross, but Not as the Symbol of Multiplication of Two Numbers
  • The Process of Two False Positions
  • Compound Proportions with Integers
  • Proportions Involving Fractions
  • Addition and Subtraction of Fractions
  • Division of Fractions
  • Casting Out the 9's, 7's, or 11's
  • Multiplication of Integers
  • Reducing Radicals to Radicals of the Same Order
  • Marking the Place for "Thousands"
  • Place of Multiplication Table above 5×5
  • The St. Andrew's Cross Used as a Symbol of Multiplication
  • Unsuccessful Symbols for Multiplication
  • The Dot for Multiplication
  • The St. Andrew's Cross in Notation for Transfinite Ordinal Numbers
  • Signs of Division and Ratio
  • Early Symbols
  • Rahn's Notation
  • Leibniz's Notations
  • Relative Position of Divisor and Dividend
  • Order of Operations in Terms Containing Both ÷ and ×
  • A Critical Estimate of : and ÷ as Symbols
  • Notations for Geometric Ratio
  • Division in the Algebra of Complex Numbers
  • Signs of Proportion
  • Arithmetical and Geometrical Progression
  • Arithmetical Proportion
  • Geometrical Proportion
  • Oughtred's Notation
  • Struggle in England between Oughtred's and Wing's Notations before 1700
  • Struggle in England between Oughtred's and Wing's Notations during 1700-1750
  • Sporadic Notations
  • Oughtred's Notation on the European Continent
  • Slight Modifications of Oughtred's Notation
  • The Notation : :: : in Europe and America
  • The Notation of Leibniz
  • Signs of Equality
  • Early Symbols
  • Recorde's Sign of Equality
  • Different Meanings of =
  • Competing Symbols
  • Descartes' Sign of Equality
  • Variations in the Form of Descartes' Symbol
  • Struggle for Supremacy
  • Variation in the Form of Recorde's Symbol
  • Variation in the Manner of Using It
  • Nearly Equal
  • Signs of Common Fractions
  • Early Forms
  • The Fractional Line
  • Special Symbols for Simple Fractions
  • The Solidus
  • Signs of Decimal Fractions
  • Stevin's Notation
  • Other Notations Used before 1617
  • Did Pitiscus Use the Decimal Point?
  • Decimal Comma and Point of Napier
  • Seventeenth-Century Notations Used after 1617
  • Eighteenth-Century Discard of Clumsy Notations
  • Nineteenth Century : Different Positions for Point and for Comma
  • Signs for Repeating Decimals
  • Signs of Powers
  • General Remarks
  • Double Significance of R and l
  • Facsimiles of Symbols in Manuscripts
  • Two General Plans for Marking Powers
  • Early Symbolisms: Abbreviative Plan, Index Plan
  • Notations Applied Only to an Unknown Quantity, the Base Being Omitted
  • Notations Applied to Any Quantity, the Base Being Designated
  • Descartes' Notation of 1637
  • Did Stampioen Arrive at Descartes' Notation Independently?
  • Notations Used by Descartes before 1637
  • Use of Hérigone's Notation after 1637
  • Later Use of Hume's Notation of 1636
  • Other Exponential Notations Suggested after 1637
  • Spread of Descartes' Notation
  • Negative, Fractional, and Literal Exponents
  • Imaginary Exponents
  • Notation for Principal Values
  • Complicated Exponents
  • D. F. Gregory's (+)r
  • Conclusions
  • Signs for Roots
  • Early Forms, General Statement
  • The Sign R, First Appearance
  • Sixteenth-Century Use of R
  • Seventeenth-Century Use of R
  • The Sign l
  • Napier's Line Symbolism
  • The Sign V
  • Origin of V
  • Spread of the V
  • Rudolff's Signs outside of Germany
  • Stevin's Numeral Root-Indices
  • Rudolff and Stifel's Aggregation Signs
  • Descartes' Union of Radical Sign and Vinculum
  • Other Signs of Aggregation of Terms
  • Redundancy in the Use of Aggregation Signs
  • Peculiar Dutch Symbolism
  • Principal Root-Values
  • Recommendation of the U.S. National Committee
  • Signs for Unknown Numbers
  • Early Forms
  • Crossed Numerals Representing Powers of Unknowns
  • Descartes' z, y, x
  • Spread of Descartes' Signs
  • Signs of Aggregation
  • Introduction
  • Aggregation Expressed by Letters
  • Aggregation Expressed by Horizontal Bars or Vinculums
  • Aggregation Expressed by Dots
  • Aggregation Expressed by Commas
  • Aggregation Expressed by Parentheses
  • Early Occurrence of Parentheses
  • Terms in an Aggregate Placed in a Vertical Column
  • Marking Binomial Coefficients
  • Special Uses of Parentheses
  • A Star to Mark the Absence of Terms
  • IV. Symbols in Geometry (Elementary Part)
  • A. Ordinary Elementary Geometry
  • Early Use of Pictographs
  • Signs for Angles
  • Signs for "Perpendicular"
  • Signs for Triangle, Square, Rectangle, Parallelogram
  • The Square as an Operator
  • Sign for Circle
  • Signs for Parallel Lines
  • Signs for Equal and Parallel
  • Signs for Arcs of Circles
  • Other Pictographs
  • Signs for Similarity and Congruence
  • The Sign for Equivalence
  • Lettering of Geometric Figures
  • Sign for Spherical Excess
  • Symbols in the Statement of Theorems
  • Signs for Incommensurables
  • Unusual Ideographs in Elementary Geometry
  • Algebraic Symbols in Elementary Geometry
  • B. Past Struggles between Symbolists and Rhetoricians in Elementary Geometry
  • Index
  • Volume II Notations Mainly In Higher Mathematis
  • Preface to the Second Volume
  • Table of Contents
  • Illustrations
  • Introduction to the Second Volume
  • Addenda
  • I. Topical Survey of Symbols in Arithmetic and Algebra (Advanced Part)
  • Letters Representing Magnitudes
  • Greek Period
  • Middle Ages
  • Renaissance
  • Vieta in 1591
  • Descartes in 1637
  • Different Alphabets
  • Astronomical Signs
  • The Letters p and e
  • Early Signs for 3.1415
  • First Occurrence of Sign p
  • Euler's Use of p
  • Spread of Jones's Notation
  • Signs for the Base of Natural Logarithms
  • The Letter e
  • B. Peirce's Signs for 3.141. . and 2.718..
  • The Evolution of the Dollar Mark
  • Different Hypotheses
  • Evidence in Manuscripts and Early Printed Books
  • Modern Dollar Mark in Print
  • Conclusion
  • Signs in the Theory of Numbers
  • Divisors of Numbers, Residues
  • Congruence of Numbers
  • Prime and Relatively Prime Numbers
  • Sums of Numbers
  • Partition of Numbers
  • Figurate Numbers
  • Diophantine Expressions
  • Number Fields
  • Perfect Numbers
  • Mersenne's Numbers
  • Fermat's Numbers
  • Cotes's Numbers
  • Bernoulli's Numbers
  • Euler's Numbers
  • Signs for Infinity and Transfinite Numbers
  • Signs for Continued Fractions and Infinite Series
  • Continued Fractions
  • Tiered Fractions
  • Infinite Series
  • Signs in the Theory of Combinations
  • Binomial Formula
  • Product of Terms of Arithmetical Progression
  • Vandermonde's Symbols
  • Combinatorial School of Hindenburg
  • Kramp on Combinatorial Notations
  • Signs of Argand and Ampère
  • Thomas Jarrett
  • Factorial n
  • Subfactorial N
  • Continued Products
  • Permutations and Combinations
  • Substitutions
  • Groups
  • Invariants and Covariants
  • Dual Arithmetic
  • Chessboard Problem
  • Determinant Notations
  • Seventeenth Century
  • Eighteenth Century
  • Early Nineteenth Century
  • Modern Notations
  • Compressed Notations
  • Jacobian
  • Hessian
  • Cubic Determinants
  • Infinite Determinants
  • Matrix Notations
  • Signs for Logarithms
  • Abbreviation for "Logarithm"
  • Different Meanings of log x, lx, and Lx
  • Power of a Logarithm
  • Iterated Logarithms
  • Marking the Characteristic
  • Marking the Last Digit
  • Sporadic Notations
  • Complex Numbers
  • Exponentiation
  • Dual Logarithms
  • Signs of Theoretical Arithmetic
  • Signs for "Greater" or "Less"
  • Sporadic Symbols for "Greater" or "Less"
  • Improvised Type
  • Modern Modifications
  • Absolute Difference
  • Other Meanings of and ~
  • A Few Other Sporadic Symbols
  • Signs for Absolute Value
  • Zeroes of Different Origin
  • General Combinations between Magnitudes or Numbers
  • Symbolism for Imaginaries and Vector Analysis
  • Symbols for the Square Root of Minus One
  • De Morgan's Comments on -1
  • Notation for a Vector
  • Length of a Vector
  • Equality of Vectors
  • Products of Vectors
  • Certain Operators
  • Rival Vector Systems
  • Attempts at Unification
  • Tensors
  • II. Symbols in Modern Analysis
  • Trigonometric Notations
  • Origin of the Modern Symbols for Degrees, Minutes, and Seconds
  • Signs for Radians
  • Marking Triangles
  • Early Abbreviations of Trigonometric Lines
  • Great Britain during 1602-18
  • European Continent during 1622-32
  • Great Britain during 1624-57
  • Seventeenth-Century English and Continental Practices are Independent
  • England during 1657-1700
  • The Eighteenth Century
  • Trigonometric Symbols of the Eighteenth Century
  • Trigonometric Symbols of the Nineteenth Century
  • Less Common Trigonometric Functions
  • Quaternion Trigonometry
  • Hyperbolic Functions
  • Parabolic Functions
  • Inverse Trigonometric Functions
  • John Herschel's Notation for Inverse Functions
  • Martin Ohm's Notation for Inverse Functions
  • Persistance of Rival Notations for Inverse Functions
  • Inverse Hyperbolic Functions
  • Powers of Trigonometric Functions
  • Survey of Mathematical Symbols Used by Leibniz
  • Introduction
  • Tables of Symbols
  • Remarks on Tables
  • Differential and Integral Calculus
  • 1. Introduction
  • 2. Symbols for Fluxions, Differentials, and Derivatives
  • a) Total Differentiation during the Seventeenth and Eighteenth Centuries. Newton, Leibniz, Landen, Fontaine, Lagrange (1797), Pasquich, Grüson, Arbogast, Kramp
  • b) Criticisms of Eighteenth-Century Notations. Woodhouse, Lacroix, Lagrange
  • c) Total Differentiation during the Nineteenth Century. Barlow, Mitchell, Herschel, Peacock, Babbage, Crelle, Cauchy (1823, 1829), M. Ohm, Cauchy and Moigno (1840), B. Peirce, Carr, Peacock, Fourier
  • d) Partial Differentials and Partial Derivatives. Euler, Karsten, Fontaine, Monge, Condorcet, Legendre, Lagrange (1788), Lacroix, Da Cunha, L'Huilier, Lagrange (1797), Arbogast, Lagrange (1801), Crelle, Barlow, Cauchy, M. Ohm, W. R. Hamilton, W. Bolyai, Cauchy and Moigno, C. G. J. Jacobi, Hesse, B. Peirce, Strauch, Duhamel, Carr, Méray, Muir, Mansion
  • 3. Symbols for Integrals, Leibniz
  • 4. Early Use of Leibnizian Notation in Great Britain
  • 5. Symbols for Fluents: Later Notations in Integral Calculus. Newton, Reyneau, Crelle, Euler, Fourier, Volterra, Peano, E. H. Moore, Cauchy's Residual Calculus
  • 6. Calculus Notations in the United States
  • 7. Symbols for Passing to the Limit. L'Huilier, Weierstrass Oliver, Riemann, Leathem, Dirichlet, Pringsheim, Scheffer, Peano, W. H. Young
  • 9. Concluding Observations
  • Finite Differences
  • Early Notations
  • Later Notations
  • Symbols in Theory of Functions
  • A. Symbols for Functions in General
  • B. Symbols for Some Special Functions
  • Symmetric Functions
  • Gamma and Beta Functions
  • Elliptic Functions
  • Theta Functions
  • Zeta Functions
  • Power Series
  • Laplace, Lamé, and Bessel Functions
  • Logarithm-Integral, Cosine-Integral, etc
  • Symbols in Mathematical Logic
  • Some Early Symbols
  • The Sign for "Therefore"
  • The Sign for "Because"
  • The Program of Leibniz
  • Signs of H. Lambert
  • G. J. von Holland
  • G. F. Castillon
  • J. D. Gergonne
  • Bolyai
  • Bentham
  • A. de Morgan
  • G. Boole
  • W. S. Jevons
  • Macfarlane
  • C. S. Peirce
  • Ladd-Franklin and Mitchell
  • R. G. Grassmann
  • E. Schroeder
  • J. H. MacColl
  • G. Frege
  • G. Peano
  • A. N. Whitehead
  • E. H. Moore
  • Whitehead and Russell
  • P. Poretsky
  • L. Wittgenstein
  • Remarks by Rignano and Jourdain
  • A Question
  • III. Symbols in Geometry (Advanced Part)
  • 1. Recent Geometry of Triangle and Circle, etc.
  • Geometrographie
  • Signs for Polyhedra
  • Geometry of Graphics
  • 2. Projective and Analytical Geometry
  • Signs for Projectivity and Perspectivity
  • Signs for Harmonic and Anharmonic Ratios
  • Descriptive Geometry
  • Analytical Geometry
  • Plücker's Equations
  • The Twenty-seven Lines on a Cubic Surface
  • The Pascal Hexagram
  • IV. The Teachings of History
  • A. The Teachings of History as Interpreted by Various Writers. Individual Judgments
  • Review of D. André
  • Quotations from A. de Morgan
  • J. W. L. Glaisher
  • D. E. Smith
  • A. Savérien
  • C. Maclaurin
  • Ch. Babbage
  • E. Mach
  • B. Branford
  • A. N. Whitehead
  • H. F. Baker
  • H. Burckhardt
  • P. G. Tait
  • O. S. Adams
  • A British Committee
  • B. Empirical Generalizations on the Growth of Mathematical Notations
  • Forms of Symbols
  • Invention of Symbols
  • Nature of Symbols
  • Potency of Symbols
  • Selection and Spread of Symbols
  • State of Flux
  • Defects in Symbolism
  • Individualism a Failure
  • C. Co-operation in Some Other Fields of Scientific Endeavor
  • Electric Units
  • Star Chart and Catalogue
  • D. Group Action Attempted in Mathematics
  • In Vector Analysis
  • In Potential and Elasticity
  • In Actuarial Science
  • E. Agreements To Be Reached by International Committees the Only Hope for Uniformity of Notations
  • Alphabetical Index

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