A History Of Mathematical Notations/Volume 1



A HISTORY
OF
MATHEMATICAL
NOTATIONS


FLORIAN CAJORI


A HISTORY OF
MATHEMATICAL NOTATIONS

Volume I

NOTATIONS IN ELEMENTARY
MATHEMATICS

A HISTORY OF
MATHEMATICAL
NOTATIONS


By
FLORIAN CAJORI, Ph.D.

Professor of the History of Mathematics
University of California


Volume I

NOTATIONS IN ELEMENTARY MATHEMATICS


The Open Court Publishing Company
LA SALLE · ILLINOIS

Copyright 1928 By
The Open Court Publishing Company
Published September 1928


Printed in the United States of America
By Paquin Printers, Chicago

PREFACE

The study of the history of mathematical notations was suggested to me by Professor E. H. Moore, of the University of Chicago. To him and to Professor M. W. Haskell, of the University of California, I am indebted for encouragement in the pursuit of this research. As completed in August, 1925, the present history was intended to be brought out in one volume. To Professor H. E. Slaught, of the University of Chicago, I owe the suggestion that the work be divided into two volumes, of which the first should limit itself to the history of symbols in elementary mathematics, since such a volume would appeal to a wider constituency of readers than would be the case with the part on symbols in higher mathematics. To Professor Slaught I also owe generous and vital assistance in many other ways. He examined the entire manuscript of this work in detail, and brought it to the sympathetic attention of the Open Court Publishing Company. I desire to record my gratitude to Mrs. Mary Hegeler Carus, president of the Open Court Publishing Company, for undertaking this expensive publication from which no financial profits can be expected to accrue.

I gratefully acknowledge the assistance in the reading of the proofs of part of this history rendered by Professor Haskell, of the University of California; Professor R. C. Archibald, of Brown University; and Professor L. C. Karpinski, of the University of Michigan.

Florian Cajori

University of California

TABLE OF CONTENTS

  1. I.Introduction
  2. PARAGRAPHS
  3. II.
    Numeral Symbols and Combination of Symbols
    ................................................................................................................................................................................................................................................................................................................................................................................................
    1–99
  4. ................................................................................................................................................................................................................................................................................................................................................................................................
    1–15
  5. ................................................................................................................................................................................................................................................................................................................................................................................................
    16–26
  6. ................................................................................................................................................................................................................................................................................................................................................................................................
    27–28
  7. ................................................................................................................................................................................................................................................................................................................................................................................................
    29–31
  8. ................................................................................................................................................................................................................................................................................................................................................................................................
    32–44
  9. Early Arabs
    ................................................................................................................................................................................................................................................................................................................................................................................................
    45
  10. Romans
    ................................................................................................................................................................................................................................................................................................................................................................................................
    46–61
  11. Peruvian and North American Knot Records
    ................................................................................................................................................................................................................................................................................................................................................................................................
    62–65
  12. Aztecs
    ................................................................................................................................................................................................................................................................................................................................................................................................
    66–67
  13. Maya
    ................................................................................................................................................................................................................................................................................................................................................................................................
    68
  14. Chinese and Japanese
    ................................................................................................................................................................................................................................................................................................................................................................................................
    69–73
  15. Hindu-Arabic Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    74–99
  16. Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    74–77
  17. Principle of Local Value
    ................................................................................................................................................................................................................................................................................................................................................................................................
    78–80
  18. Forms of Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    81–88
  19. Freak Forms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    89
  20. Negative Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    90
  21. Grouping of Digits in Numeration
    ................................................................................................................................................................................................................................................................................................................................................................................................
    91
  22. The Spanish Calderón
    ................................................................................................................................................................................................................................................................................................................................................................................................
    92–93
  23. The Portuguese Cifrão
    ................................................................................................................................................................................................................................................................................................................................................................................................
    94
  24. Relative Size of Numerals in Tables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    95
  25. Fanciful Hypotheses on the Origin of Numeral Forms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    96
  26. A Sporadic Artificial System
    ................................................................................................................................................................................................................................................................................................................................................................................................
    97
  27. General Remarks
    ................................................................................................................................................................................................................................................................................................................................................................................................
    98
  28. Opinion of Laplace
    ................................................................................................................................................................................................................................................................................................................................................................................................
    99
  29. III.
    Symbols in Arithmetic and Algebra (Elementary Part)
    ................................................................................................................................................................................................................................................................................................................................................................................................
    100
  30. A.Groups of Symbols Used by Individual Writers101
  31. Greeks—Diophantus, Third Century A.D.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    101–5
  32. Hindu—Brahmagupta, Seventh Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    106–8
  33. Hindu—The Bakhshālī Manuscript
    ................................................................................................................................................................................................................................................................................................................................................................................................
    109
  34. Hindu—Bhāskara, Twelfth Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    110–14
  35. Arabic—al-Khowârizmî, Ninth Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    115
  36. Arabic—al-Karkhf, Eleventh Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    116
  37. Byzantine—Michael Psellus, Eleventh Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    117
  38. Arabic—Ibn Albanna, Thirteenth Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    118
  39. Chinese—Chu Shih-Chieh, Fourteenth Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    119, 120
  40. Byzantine—Maximus Planudes, Fourteenth Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    121
  41. Italian—Leonardo of Pisa, Thirteenth Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    122
  42. French—Nicole Oresme, Fourteenth Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    123
  43. Arabic—al-Qalasâdî, Fifteenth Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    124
  44. German—Regiomontanus, Fifteenth Century
    ................................................................................................................................................................................................................................................................................................................................................................................................
    125–27
  45. Italian—Earliest Printed Arithmetic, 1478
    ................................................................................................................................................................................................................................................................................................................................................................................................
    128
  46. French—Nicolas Chuquet, 1484
    ................................................................................................................................................................................................................................................................................................................................................................................................
    129-31
  47. French—Estienne de la Roche, 1520
    ................................................................................................................................................................................................................................................................................................................................................................................................
    132
  48. Italiaim-Pietro Borgi, 1484, 1488
    ................................................................................................................................................................................................................................................................................................................................................................................................
    133
  49. Italian—Luca Pacioli, 1494, 1523
    ................................................................................................................................................................................................................................................................................................................................................................................................
    134–38
  50. Italian—F. Ghaligai, 1521, 1548, 1552
    ................................................................................................................................................................................................................................................................................................................................................................................................
    139
  51. Italian—H. Cardan, 1532, 1545, 1570
    ................................................................................................................................................................................................................................................................................................................................................................................................
    140, 141
  52. Italian—Nicolo Tartaglia, 1506–60
    ................................................................................................................................................................................................................................................................................................................................................................................................
    142, 143
  53. Italian—Rafaele Bombelli, 1572
    ................................................................................................................................................................................................................................................................................................................................................................................................
    144, 145
  54. German—Johann Widman, 1489, 1526
    ................................................................................................................................................................................................................................................................................................................................................................................................
    146
  55. Austrian—Grammateus, 1518, 1535
    ................................................................................................................................................................................................................................................................................................................................................................................................
    147
  56. German—Christoff Rudolff, 1525
    ................................................................................................................................................................................................................................................................................................................................................................................................
    148, 149
  57. Dutch—Gielis van der Hoecke, 1537
    ................................................................................................................................................................................................................................................................................................................................................................................................
    150
  58. German—Michael Stifel, 1544, 1545, 1553
    ................................................................................................................................................................................................................................................................................................................................................................................................
    151–56
  59. German—Nicolaus Copernicus, 1566
    ................................................................................................................................................................................................................................................................................................................................................................................................
    157
  60. German—Johann Scheubel, 1545, 1551
    ................................................................................................................................................................................................................................................................................................................................................................................................
    158,159
  61. Maltese—Wil. Klebitius, 1565
    ................................................................................................................................................................................................................................................................................................................................................................................................
    160
  62. German—Christophorus Clavius, 1608
    ................................................................................................................................................................................................................................................................................................................................................................................................
    161
  63. Belgium—Simon Stevin, 1585
    ................................................................................................................................................................................................................................................................................................................................................................................................
    162,163
  64. Lorraine—Albert Girard, 1629
    ................................................................................................................................................................................................................................................................................................................................................................................................
    164
  65. German-Spanish—Marco Aurel, 1552
    ................................................................................................................................................................................................................................................................................................................................................................................................
    165
  66. Portuguese-Spanish—Pedro Nunez, 1567
    ................................................................................................................................................................................................................................................................................................................................................................................................
    166
  67. English—Robert Recorde, 1543(?), 1557
    ................................................................................................................................................................................................................................................................................................................................................................................................
    167–68
  68. English—John Dee, 1570
    ................................................................................................................................................................................................................................................................................................................................................................................................
    169
  69. English—Leonard and Thomas Digges, 1579
    ................................................................................................................................................................................................................................................................................................................................................................................................
    170
  70. English—Thomas Masterson, 1592
    ................................................................................................................................................................................................................................................................................................................................................................................................
    171
  71. French—Jacques Peletier, 1554
    ................................................................................................................................................................................................................................................................................................................................................................................................
    172
  72. French—Jean Buteon, 1559
    ................................................................................................................................................................................................................................................................................................................................................................................................
    173
  73. French—Guillaume Gosselin, 1577
    ................................................................................................................................................................................................................................................................................................................................................................................................
    174
  74. French—Francis Vieta, 1591
    ................................................................................................................................................................................................................................................................................................................................................................................................
    176-78
  75. Italian—Bonaventura Cavalieri, 1647
    ................................................................................................................................................................................................................................................................................................................................................................................................
    179
  76. English—William Oughtred, 1631, 1632, 1657
    ................................................................................................................................................................................................................................................................................................................................................................................................
    180–87
  77. English—Thomas Harriot, 1631
    ................................................................................................................................................................................................................................................................................................................................................................................................
    188
  78. French—Pierre Herigone, 1634, 1644
    ................................................................................................................................................................................................................................................................................................................................................................................................
    189
  79. Scot-French—James Hume, 1635, 1636
    ................................................................................................................................................................................................................................................................................................................................................................................................
    190
  80. French—René Descartes
    ................................................................................................................................................................................................................................................................................................................................................................................................
    191
  81. English—Isaac Barrow
    ................................................................................................................................................................................................................................................................................................................................................................................................
    192
  82. English—Richard Rawlinson, 1655-68
    ................................................................................................................................................................................................................................................................................................................................................................................................
    193
  83. Swiss—Johann Heinrich Rahn
    ................................................................................................................................................................................................................................................................................................................................................................................................
    194
  84. English—John Wallis, 1655, 1657, 1685
    ................................................................................................................................................................................................................................................................................................................................................................................................
    195, 196
  85. Extract from Acta eruditorum, Leipzig, 1708
    ................................................................................................................................................................................................................................................................................................................................................................................................
    197
  86. Extract from Miscellanea Berolinensia, 1710 (Due to G. W. Leibniz)
    ................................................................................................................................................................................................................................................................................................................................................................................................
    198
  87. Conclusions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    199
  88. B.
    Topical Survey of the Use of Notations.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    200–356
  89. Signs of Addition and Subtraction.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    200–216
  90. Early Symbols
    ................................................................................................................................................................................................................................................................................................................................................................................................
    200
  91. Origin and Meaning of the Signs.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    201–3
  92. Spread of the + and — Symbols.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    204
  93. Shapes of the -p Sign
    ................................................................................................................................................................................................................................................................................................................................................................................................
    205–7
  94. Varieties of — Signs
    ................................................................................................................................................................................................................................................................................................................................................................................................
    208, 209
  95. Symbols for "Plus or Minus"
    ................................................................................................................................................................................................................................................................................................................................................................................................
    210, 211
  96. Certain Other Specialized Uses of + and —
    ................................................................................................................................................................................................................................................................................................................................................................................................
    212–14
  97. Four Unusual Signs
    ................................................................................................................................................................................................................................................................................................................................................................................................
    215
  98. Composition of Ratios
    ................................................................................................................................................................................................................................................................................................................................................................................................
    216
  99. Signs of Multiplication
    ................................................................................................................................................................................................................................................................................................................................................................................................
    217–34
  100. Early Symbols
    ................................................................................................................................................................................................................................................................................................................................................................................................
    217
  101. Early Uses of the St. Andrew's Cross, but Not as the Symbol of Multiplication of Two Numbers
    ................................................................................................................................................................................................................................................................................................................................................................................................
    218–30
  102. The Process of Two False Positions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    219
  103. Compound Proportions with Integers
    ................................................................................................................................................................................................................................................................................................................................................................................................
    220
  104. Proportions Involving Fractions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    221
  105. Addition and Subtraction of Fractions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    222
  106. Division of Fractions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    223
  107. Casting Out the 9's, 7's, or 11's
    ................................................................................................................................................................................................................................................................................................................................................................................................
    225
  108. Multiplication of Integers
    ................................................................................................................................................................................................................................................................................................................................................................................................
    226
  109. Reducing Radicals to Radicals of the Same Order
    ................................................................................................................................................................................................................................................................................................................................................................................................
    227
  110. Marking the Place for "Thousands"
    ................................................................................................................................................................................................................................................................................................................................................................................................
    228
  111. Place of Multiplication Table above 5×5
    ................................................................................................................................................................................................................................................................................................................................................................................................
    229
  112. The St. Andrew’s Cross Used as a Symbol of Multiplication
    ................................................................................................................................................................................................................................................................................................................................................................................................
    231
  113. Unsuccessful Symbols for Multiplication
    ................................................................................................................................................................................................................................................................................................................................................................................................
    232
  114. The Dot for Multiplication
    ................................................................................................................................................................................................................................................................................................................................................................................................
    233
  115. The St. Andrew's Cross in Notation for Transfinite Ordinal Numbers
    ................................................................................................................................................................................................................................................................................................................................................................................................
    234
  116. Signs of Division and Ratio
    ................................................................................................................................................................................................................................................................................................................................................................................................
    235–47
  117. Early Symbols
    ................................................................................................................................................................................................................................................................................................................................................................................................
    235, 236
  118. Rahn's Notation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    237
  119. Leibniz’s Notations
    ................................................................................................................................................................................................................................................................................................................................................................................................
    238
  120. Relative Position of Divisor and Dividend
    ................................................................................................................................................................................................................................................................................................................................................................................................
    241
  121. Order of Operations in Terms Containing Both ÷ and ×
    ................................................................................................................................................................................................................................................................................................................................................................................................
    242
  122. A Critical Estimate of : and ÷ as Symbols
    ................................................................................................................................................................................................................................................................................................................................................................................................
    243
  123. Notations for Geometric Ratio
    ................................................................................................................................................................................................................................................................................................................................................................................................
    244
  124. Division in the Algebra of Complex Numbers
    ................................................................................................................................................................................................................................................................................................................................................................................................
    247
  125. Signs of Proportion
    ................................................................................................................................................................................................................................................................................................................................................................................................
    248–59
  126. Arithmetical and Geometrical Progression
    ................................................................................................................................................................................................................................................................................................................................................................................................
    248
  127. Arithmetical Proportion
    ................................................................................................................................................................................................................................................................................................................................................................................................
    249
  128. Geometrical Proportion
    ................................................................................................................................................................................................................................................................................................................................................................................................
    250
  129. Oughtred's Notation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    251
  130. Struggle in England between Oughtred's and Wing's Notations before 1700
    ................................................................................................................................................................................................................................................................................................................................................................................................
    252
  131. Struggle in England between Oughtred's and Wing's Notations during 1700–1750
    ................................................................................................................................................................................................................................................................................................................................................................................................
    253
  132. Sporadic Notations
    ................................................................................................................................................................................................................................................................................................................................................................................................
    254
  133. Oughtred's Notation on the European Continent
    ................................................................................................................................................................................................................................................................................................................................................................................................
    255
  134. Slight Modifications of Oughtred's Notation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    257
  135. The Notation:::: in Europe and America
    ................................................................................................................................................................................................................................................................................................................................................................................................
    258
  136. The Notation of Leibniz
    ................................................................................................................................................................................................................................................................................................................................................................................................
    259
  137. Signs of Equality
    ................................................................................................................................................................................................................................................................................................................................................................................................
    260–70
  138. Early Symbols
    ................................................................................................................................................................................................................................................................................................................................................................................................
    260
  139. Recorde's Sign of Equality
    ................................................................................................................................................................................................................................................................................................................................................................................................
    261
  140. Different Meanings of =
    ................................................................................................................................................................................................................................................................................................................................................................................................
    262
  141. Competing Symbols
    ................................................................................................................................................................................................................................................................................................................................................................................................
    263
  142. Descartes' Sign of Equality
    ................................................................................................................................................................................................................................................................................................................................................................................................
    264
  143. Variations in the Form of Descartes' Symbol
    ................................................................................................................................................................................................................................................................................................................................................................................................
    265
  144. Struggle for Supremacy
    ................................................................................................................................................................................................................................................................................................................................................................................................
    266
  145. Variation in the Form of Recorde's Symbol
    ................................................................................................................................................................................................................................................................................................................................................................................................
    268
  146. Variation in the Manner of Using It
    ................................................................................................................................................................................................................................................................................................................................................................................................
    269
  147. Nearly Equal
    ................................................................................................................................................................................................................................................................................................................................................................................................
    270
  148. Signs of Common Fractions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    271–75
  149. Early Forms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    271
  150. The Fractional Line
    ................................................................................................................................................................................................................................................................................................................................................................................................
    272
  151. Special Symbols for Simple Fractions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    274
  152. The Solidus
    ................................................................................................................................................................................................................................................................................................................................................................................................
    275
  153. Signs of Decimal Fractions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    276–89
  154. Stevin's Notation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    276
  155. Other Notations Used before 1617
    ................................................................................................................................................................................................................................................................................................................................................................................................
    278
  156. Did Pitiscus Use the Decimal Point?
    ................................................................................................................................................................................................................................................................................................................................................................................................
    279
  157. Decimal Comma and Point of Napier
    ................................................................................................................................................................................................................................................................................................................................................................................................
    282
  158. Seventeenth-Century Notations Used after 1617
    ................................................................................................................................................................................................................................................................................................................................................................................................
    283
  159. Eighteenth-Century Discard of Clumsy Notations
    ................................................................................................................................................................................................................................................................................................................................................................................................
    285
  160. Nineteenth Century: Different Positions for Point and for Comma
    ................................................................................................................................................................................................................................................................................................................................................................................................
    286
  161. Signs for Repeating Decimals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    289
  162. Signs of Powers
    ................................................................................................................................................................................................................................................................................................................................................................................................
    290–315
  163. General Remarks
    ................................................................................................................................................................................................................................................................................................................................................................................................
    290
  164. Double Significance of R and l
    ................................................................................................................................................................................................................................................................................................................................................................................................
    291
  165. Facsimiles of Symbols in Manuscripts
    ................................................................................................................................................................................................................................................................................................................................................................................................
    293
  166. Two General Plans for Marking Powers
    ................................................................................................................................................................................................................................................................................................................................................................................................
    294
  167. Early Symbolisms: Abbreviative Plan, Index Plan
    ................................................................................................................................................................................................................................................................................................................................................................................................
    295
  168. Notations Applied Only to an Unknown Quantity, the Base Being Omitted
    ................................................................................................................................................................................................................................................................................................................................................................................................
    296
  169. Notations Applied to Any Quantity, the Base Being Designated
    ................................................................................................................................................................................................................................................................................................................................................................................................
    297
  170. Descartes’ Notation of 1637
    ................................................................................................................................................................................................................................................................................................................................................................................................
    298
  171. Did Stampioen Arrive at Descartes’ Notation Independently?
    ................................................................................................................................................................................................................................................................................................................................................................................................
    299
  172. Notations Used by Descartes before 1637
    ................................................................................................................................................................................................................................................................................................................................................................................................
    300
  173. Use of Hérigone's Notation after 1637
    ................................................................................................................................................................................................................................................................................................................................................................................................
    301
  174. Later Use of Hume’s Notation of 1636
    ................................................................................................................................................................................................................................................................................................................................................................................................
    302
  175. Other Exponential Notations Suggested after 1637
    ................................................................................................................................................................................................................................................................................................................................................................................................
    303
  176. Spread of Descartes' Notation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    307
  177. Negative, Fractional, and Literal Exponents
    ................................................................................................................................................................................................................................................................................................................................................................................................
    308
  178. Imaginary Exponents
    ................................................................................................................................................................................................................................................................................................................................................................................................
    309
  179. Notation for Principal Values
    ................................................................................................................................................................................................................................................................................................................................................................................................
    312
  180. Complicated Exponents
    ................................................................................................................................................................................................................................................................................................................................................................................................
    313
  181. D. F. Gregory’s
    ................................................................................................................................................................................................................................................................................................................................................................................................
    314
  182. Conclusions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    315
  183. Signs for Roots
    ................................................................................................................................................................................................................................................................................................................................................................................................
    316–38
  184. Early Forms, General Statement
    ................................................................................................................................................................................................................................................................................................................................................................................................
    316, 317
  185. The Sign , First Appearance
    ................................................................................................................................................................................................................................................................................................................................................................................................
    318
  186. Sixteenth-Century Use of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    319
  187. Seventeenth-Century Use of
    ................................................................................................................................................................................................................................................................................................................................................................................................
    321
  188. The Sign l
    ................................................................................................................................................................................................................................................................................................................................................................................................
    322
  189. Napier’s Line Symbolism
    ................................................................................................................................................................................................................................................................................................................................................................................................
    323
  190. The Sign √
    ................................................................................................................................................................................................................................................................................................................................................................................................
    324–38
  191. Origin of √
    ................................................................................................................................................................................................................................................................................................................................................................................................
    324
  192. Spread of the √
    ................................................................................................................................................................................................................................................................................................................................................................................................
    327
  193. Rudolff’s Signs outside of Germany
    ................................................................................................................................................................................................................................................................................................................................................................................................
    328
  194. Stevin’s Numeral Root-Indices
    ................................................................................................................................................................................................................................................................................................................................................................................................
    329
  195. Rudolff and Stifel's Aggregation Signs
    ................................................................................................................................................................................................................................................................................................................................................................................................
    332
  196. Descartes' Union of Radical Sign and Vinculum
    ................................................................................................................................................................................................................................................................................................................................................................................................
    333
  197. Other Signs of Aggregation of Terms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    334
  198. Redundancy in the Use of Aggregation Signs
    ................................................................................................................................................................................................................................................................................................................................................................................................
    335
  199. Peculiar Dutch Symbolism
    ................................................................................................................................................................................................................................................................................................................................................................................................
    336
  200. Principal Root-Value
    ................................................................................................................................................................................................................................................................................................................................................................................................
    337
  201. Recommendation of the U.S. National Committee
    ................................................................................................................................................................................................................................................................................................................................................................................................
    338
  202. Signs for Unknown Numbers.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    339–41
  203. Early Forms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    339
  204. Crossed Numerals Representing Powers of Unknowns
    ................................................................................................................................................................................................................................................................................................................................................................................................
    340
  205. Descartes' z, y, x
    ................................................................................................................................................................................................................................................................................................................................................................................................
    340
  206. Spread of Descartes' Signs.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    341
  207. Signs of Aggregation
    ................................................................................................................................................................................................................................................................................................................................................................................................
    342–50
  208. Introduction
    ................................................................................................................................................................................................................................................................................................................................................................................................
    342
  209. Aggregation Expressed by Letters
    ................................................................................................................................................................................................................................................................................................................................................................................................
    343
  210. Aggregation Expressed by Horizontal Bars or Vinculums
    ................................................................................................................................................................................................................................................................................................................................................................................................
    344
  211. Aggregation Expressed by Dots
    ................................................................................................................................................................................................................................................................................................................................................................................................
    348
  212. Aggregation Expressed by Commas
    ................................................................................................................................................................................................................................................................................................................................................................................................
    349
  213. Aggregation Expressed by Parentheses
    ................................................................................................................................................................................................................................................................................................................................................................................................
    350
  214. Early Occurrence of Parentheses
    ................................................................................................................................................................................................................................................................................................................................................................................................
    351
  215. Terms in an Aggregate Placed in a Vertical Column
    ................................................................................................................................................................................................................................................................................................................................................................................................
    353
  216. Marking Binomial Coefficients
    ................................................................................................................................................................................................................................................................................................................................................................................................
    354
  217. Special Uses of Parentheses
    ................................................................................................................................................................................................................................................................................................................................................................................................
    355
  218. A Star to Mark the Absence of Terms
    ................................................................................................................................................................................................................................................................................................................................................................................................
    356
  219. IV.
    Symbols in Geometry (Elementary Part)
    ................................................................................................................................................................................................................................................................................................................................................................................................
    357–85
  220. A.
    Ordinary Elementary Geometry
    ................................................................................................................................................................................................................................................................................................................................................................................................
    357
  221. Early Use of Pictographs
    ................................................................................................................................................................................................................................................................................................................................................................................................
    357
  222. Signs for Angles
    ................................................................................................................................................................................................................................................................................................................................................................................................
    360
  223. Signs for "Perpendicular"
    ................................................................................................................................................................................................................................................................................................................................................................................................
    364
  224. Signs for Triangle, Square, Rectangle, Parallelogram
    ................................................................................................................................................................................................................................................................................................................................................................................................
    365
  225. The Square as an Operator
    ................................................................................................................................................................................................................................................................................................................................................................................................
    366
  226. Sign for Circle
    ................................................................................................................................................................................................................................................................................................................................................................................................
    367
  227. Signs for Parallel Lines
    ................................................................................................................................................................................................................................................................................................................................................................................................
    368
  228. Signs for Equal and Parallel
    ................................................................................................................................................................................................................................................................................................................................................................................................
    369
  229. Signs for Arcs of Circles
    ................................................................................................................................................................................................................................................................................................................................................................................................
    370
  230. Other Pictographs
    ................................................................................................................................................................................................................................................................................................................................................................................................
    371
  231. Signs for Similarity and Congruence
    ................................................................................................................................................................................................................................................................................................................................................................................................
    372
  232. The Sign ≎ for Equivalence
    ................................................................................................................................................................................................................................................................................................................................................................................................
    375
  233. Lettering of Geometric Figures
    ................................................................................................................................................................................................................................................................................................................................................................................................
    376
  234. Sign for Spherical Excess
    ................................................................................................................................................................................................................................................................................................................................................................................................
    380
  235. Symbols in the Statement of Theorems
    ................................................................................................................................................................................................................................................................................................................................................................................................
    381
  236. Signs for Incommensurables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    382
  237. Unusual Ideographs in Elementary Geometry
    ................................................................................................................................................................................................................................................................................................................................................................................................
    383
  238. Algebraic Symbols in Elementary Geometry
    ................................................................................................................................................................................................................................................................................................................................................................................................
    384
  239. B.
    Past Struggles between Symbolists and Rhetoricians in Elementary Geometry
    ................................................................................................................................................................................................................................................................................................................................................................................................
    385
  240. Index

ILLUSTRATIONS

  1. FIGUREPARAGRPAHS
  2. 1.
    Babylonian Tablets of Nippur.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    4
  3. 2.
    Principle of Subtraction in Babylonian Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    9
  4. 3.
    Babylonian Lunar Tables
    ................................................................................................................................................................................................................................................................................................................................................................................................
    11
  5. 4.
    Mathematical Cuneiform Tablet CBS 8536 in the Museum of the University of Pennsylvania
    ................................................................................................................................................................................................................................................................................................................................................................................................
    11
  6. 5.
    Egyptian Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    17
  7. 6.
    Egyptian Symbolism for Simple Fractions
    ................................................................................................................................................................................................................................................................................................................................................................................................
    18
  8. 7.
    Algebraic Equation in Ahmes
    ................................................................................................................................................................................................................................................................................................................................................................................................
    23
  9. 8.
    Hieroglyphic, Hieratic, and Coptic Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    24
  10. 9.
    Palmyra (Syria) Numeral
    ................................................................................................................................................................................................................................................................................................................................................................................................
    27
  11. 10.
    Syrian Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    28
  12. 11.
    Hebrew Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    30
  13. 12.
    Computing Table of Salamis
    ................................................................................................................................................................................................................................................................................................................................................................................................
    36
  14. 13.
    Account of Disbursements of the Athenian State, 418-415 B.C.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    36
  15. 14.
    Arabic Alphabetic Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    45
  16. 15.
    Degenerate Forms of Roman Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    56
  17. 16.
    Quipu from Ancient Chancay in Peru
    ................................................................................................................................................................................................................................................................................................................................................................................................
    65
  18. 17.
    Diagram of the Two Right-Hand Groups
    ................................................................................................................................................................................................................................................................................................................................................................................................
    65
  19. 18.
    Aztec Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    66
  20. 19.
    Dresden Codex of Maya
    ................................................................................................................................................................................................................................................................................................................................................................................................
    67
  21. 20.
    Early Chinese Knots in Strings, Representing Numerals
    ................................................................................................................................................................................................................................................................................................................................................................................................
    70
  22. 21.
    Chinese and .Japanese Numerals.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    74
  23. 22.
    Hill’s Table of Boethian Apices.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    80
  24. 23.
    Table of Important Numeral Forms.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    80
  25. 24.
    Old Arabic and Hindu-Arabic Numerals.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    83
  26. 25.
    Numerals of the Monk Neophytos.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    88
  27. 26.
    Chr. Rudolff’s Numerals and Fractions.
    ................................................................................................................................................................................................................................................................................................................................................................................................
    89
  28. 27.
    A Contract, Mexico City, 1649
    ................................................................................................................................................................................................................................................................................................................................................................................................
    93
    1. FIGUREPARAGRPAHS
    2. 28.
      Real Estate Sale, Mexico City, 1718
      ................................................................................................................................................................................................................................................................................................................................................................................................
      94
    3. 29.
      Fanciful Hypotheses
      ................................................................................................................................................................................................................................................................................................................................................................................................
      96
    4. 30.
      Numerals Described by Noviomagus
      ................................................................................................................................................................................................................................................................................................................................................................................................
      98
    5. 31.
      Sanskrit Symbols for the Unknown
      ................................................................................................................................................................................................................................................................................................................................................................................................
      108
    6. 32.
      Bakhshālī Arithmetic.
      ................................................................................................................................................................................................................................................................................................................................................................................................
      109
    7. 33.
      Śrīdhara's Trisátikā
      ................................................................................................................................................................................................................................................................................................................................................................................................
      112
    8. 34.
      Oresme's Algorismus Proportionum
      ................................................................................................................................................................................................................................................................................................................................................................................................
      123
    9. 35.
      al-Qalâsadî's Algebraic Symbols.
      ................................................................................................................................................................................................................................................................................................................................................................................................
      125
    10. 36.
      Computations of Regiomontanus
      ................................................................................................................................................................................................................................................................................................................................................................................................
      127
    11. 37.
      Calendar of Regiomontanus.
      ................................................................................................................................................................................................................................................................................................................................................................................................
      128
    12. 38.
      From Earliest Printed Arithmetic.
      ................................................................................................................................................................................................................................................................................................................................................................................................
      128
    13. 39.
      Multiplications in the "Treviso" Arithmetic.
      ................................................................................................................................................................................................................................................................................................................................................................................................
      128
    14. 40.
      De la Roche's Larismethique, Folio 60B
      ................................................................................................................................................................................................................................................................................................................................................................................................
      132
    15. 41.
      De la Roche's Larismethique, Folio 66A
      ................................................................................................................................................................................................................................................................................................................................................................................................
      132
    16. 42.
      Part of Page in Pacioli's Summa, 1523
      ................................................................................................................................................................................................................................................................................................................................................................................................
      138
    17. 43.
      Margin of Folio 123B in Pacioli's Summa
      ................................................................................................................................................................................................................................................................................................................................................................................................
      139
    18. 44.
      Part of Folio 72 of Ghaligai's Practica d'arithmetica, 1552
      ................................................................................................................................................................................................................................................................................................................................................................................................
      139
    19. 45.
      Ghaligai's Practica d' arithmetica, Folio 198
      ................................................................................................................................................................................................................................................................................................................................................................................................
      139
    20. 46.
      Cardan, Ars magna, Ed. 1663, Page 255
      ................................................................................................................................................................................................................................................................................................................................................................................................
      141
    21. 47.
      Cardan, Ars magna, Ed. 1663, Page 297
      ................................................................................................................................................................................................................................................................................................................................................................................................
      141
    22. 48.
      From Tartaglia's General Trattato, 1560
      ................................................................................................................................................................................................................................................................................................................................................................................................
      143
    23. 49.
      From Tartaglia's General Trattato, Folio 4
      ................................................................................................................................................................................................................................................................................................................................................................................................
      144
    24. 50.
      From Bombelli's Algebra, 1572
      ................................................................................................................................................................................................................................................................................................................................................................................................
      144
    25. 51.
      Bombelli's Algebra (1579 Impression), Page 161
      ................................................................................................................................................................................................................................................................................................................................................................................................
      145
    26. 52.
      From the MS of Bombelli's Algebra in the Library of Bologna
      ................................................................................................................................................................................................................................................................................................................................................................................................
      145
    27. 53.
      From Pamphlet No. 595 A in the Library of the University of Bologna.
      ................................................................................................................................................................................................................................................................................................................................................................................................
      146
    28. 54.
      Widman's Rechnung, 1526
      ................................................................................................................................................................................................................................................................................................................................................................................................
      146
    29. 55.
      From the Arithmetic of Grammateus.
      ................................................................................................................................................................................................................................................................................................................................................................................................
      146
    30. 56.
      From the Arithmetic of Grammateus, 1535
      ................................................................................................................................................................................................................................................................................................................................................................................................
      147
    31. 57.
      From the Arithmetic of Grammateus, 1518(?)
      ................................................................................................................................................................................................................................................................................................................................................................................................
      147
    32. 58.
      From Chr. Rudolff's Goss, 1525
      ................................................................................................................................................................................................................................................................................................................................................................................................
      148
    33. 59.
      From Chr. Rudolff's Coss, Ev.
      ................................................................................................................................................................................................................................................................................................................................................................................................
      148
    34. 60.
      From Van der Hoecke' In arithmetica
      ................................................................................................................................................................................................................................................................................................................................................................................................
      150
    35. 61.
      Part of Page from Stifel's Arithmetica integra, 1544
      ................................................................................................................................................................................................................................................................................................................................................................................................
      150
    36. 62.
      From Stifel's Arithmetica integra, Folio 31B
      ................................................................................................................................................................................................................................................................................................................................................................................................
      152
    37. 63.
      From Stifel's Edition of Rudolff's Coss, 1553
      ................................................................................................................................................................................................................................................................................................................................................................................................
      156
    38. 64.
      Scheubel, Introduction to Euclid, Page 28
      ................................................................................................................................................................................................................................................................................................................................................................................................
      159
    39. 65.
      W. Klebitius, Booklet, 1565
      ................................................................................................................................................................................................................................................................................................................................................................................................
      161
    40. 66.
      From Clavius' Algebra, 1608
      ................................................................................................................................................................................................................................................................................................................................................................................................
      161
    41. 67.
      From S. Stevin's Le Thiende, 1585
      ................................................................................................................................................................................................................................................................................................................................................................................................
      162
    42. 68.
      From S. Stevin's Arithmetiqve
      ................................................................................................................................................................................................................................................................................................................................................................................................
      162
    43. 69.
      From S. Stevin's Arithmetiqve
      ................................................................................................................................................................................................................................................................................................................................................................................................
      164
    44. 70.
      From Aurel's Arithmetica
      ................................................................................................................................................................................................................................................................................................................................................................................................
      165
    45. 71.
      R. Recorde,Whetstone of Witte, 1557
      ................................................................................................................................................................................................................................................................................................................................................................................................
      168
    46. 72.
      Fractions in Recorde
      ................................................................................................................................................................................................................................................................................................................................................................................................
      168
    47. 73.
      Radicals in Recorde
      ................................................................................................................................................................................................................................................................................................................................................................................................
      168
    48. 74.
      Radicals in Dee's Preface
      ................................................................................................................................................................................................................................................................................................................................................................................................
      169
    49. 75.
      Proportion in Dee's Preface
      ................................................................................................................................................................................................................................................................................................................................................................................................
      169
    50. 76.
      From Digges's Stratioticos
      ................................................................................................................................................................................................................................................................................................................................................................................................
      170
    51. 77.
      Equations in Digges
      ................................................................................................................................................................................................................................................................................................................................................................................................
      172
    52. 78.
      Equality in Digges
      ................................................................................................................................................................................................................................................................................................................................................................................................
      172
    53. 79.
      From Thomas Masterson's Arithmeticke, 1592
      ................................................................................................................................................................................................................................................................................................................................................................................................
      172
    54. 80.
      J. Peletier's Algebra, 1554
      ................................................................................................................................................................................................................................................................................................................................................................................................
      172
    55. 81.
      Algebraic Operations in Peletier's Algebra
      ................................................................................................................................................................................................................................................................................................................................................................................................
      172
    56. 82.
      From J. Buteon, Arithmetica, 1559
      ................................................................................................................................................................................................................................................................................................................................................................................................
      173
    57. 83.
      Gosselin's De arte magna, 1577
      ................................................................................................................................................................................................................................................................................................................................................................................................
      174
    58. 84.
      Vieta, In artem analyticam, 1591
      ................................................................................................................................................................................................................................................................................................................................................................................................
      176
    59. 85.
      Vieta, De emendatione aeqvationvm
      ................................................................................................................................................................................................................................................................................................................................................................................................
      178
    60. 86.
      B. Cavalieri, Exercitationes, 1647
      ................................................................................................................................................................................................................................................................................................................................................................................................
      179
    61. 87.
      From Thomas Harriot, 1631, Page 101
      ................................................................................................................................................................................................................................................................................................................................................................................................
      183
    62. 88.
      From Thomas Harriot, 1631, Page 65
      ................................................................................................................................................................................................................................................................................................................................................................................................
      180
    63. 89.
      From Hérigone, Cursus mathematicus, 1644
      ................................................................................................................................................................................................................................................................................................................................................................................................
      189
    64. 90.
      Roman Numerals for x in J. Hume, 1635
      ................................................................................................................................................................................................................................................................................................................................................................................................
      191
    65. FIGUREPARAGRPAHS
    66. 91.
      Radicals in J. Hume, 1635
      ................................................................................................................................................................................................................................................................................................................................................................................................
      191
    67. 92.
      R. Descartes, Geometrie
      ................................................................................................................................................................................................................................................................................................................................................................................................
      191
    68. 93.
      I. Barrow's Euclid, Latin Edition. Notes by Isaac Newton
      ................................................................................................................................................................................................................................................................................................................................................................................................
      193
    69. 94.
      I. Barrow's Euclid, English Edition
      ................................................................................................................................................................................................................................................................................................................................................................................................
      193
    70. 95.
      Rich. Rawlinson’s Symbols
      ................................................................................................................................................................................................................................................................................................................................................................................................
      194
    71. 96.
      Rahn's Teutsche Algebra, 1659
      ................................................................................................................................................................................................................................................................................................................................................................................................
      195
    72. 97.
      Brancker’s Translation of Rahn, 1668
      ................................................................................................................................................................................................................................................................................................................................................................................................
      195
    73. 98.
      J. Wallis, 1657
      ................................................................................................................................................................................................................................................................................................................................................................................................
      195
    74. 99.
      From the Hieroglyphic Translation of the Ahmes Papyrus
      ................................................................................................................................................................................................................................................................................................................................................................................................
      200
    75. 100.
      Minus Sign in the German MS C. 80, Dresden Library
      ................................................................................................................................................................................................................................................................................................................................................................................................
      201
    76. 101.
      Plus and Minus Signs in the Latin MS C. 80, Dresden Library
      ................................................................................................................................................................................................................................................................................................................................................................................................
      201
    77. 102.
      Widmans' Marginal Note to MS C. 80, Dresden Library
      ................................................................................................................................................................................................................................................................................................................................................................................................
      201
    78. 103.
      From the Arithmetic of Boethius, 1488
      ................................................................................................................................................................................................................................................................................................................................................................................................
      250
    79. 104.
      Signs in German MSS and Early German Books
      ................................................................................................................................................................................................................................................................................................................................................................................................
      291
    80. 105.
      Written Algebraic Symbols for Powers from Perez de Moya's Arithmetica
      ................................................................................................................................................................................................................................................................................................................................................................................................
      294
    81. 106.
      E. Waring's Repeated Exponents.313
      ................................................................................................................................................................................................................................................................................................................................................................................................