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Index:A History Of Mathematical Notations Vol I (1928).djvu

A History Of Mathematical Notations Vol I (1928).djvu

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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