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

Title	A History Of Mathematical Notations, 1
Author	Florian Cajori
Year	1928
Publisher	The Open Court Company, Publishers
Source	djvu
Progress	To be proofread
Transclusion	Index not transcluded or unreviewed
Volumes	Volume 1 - Volume 2

Pages (key to Page Status)

- - - - i ii iii iv v - vii viii ix x xi xii xiii xiv xv xvi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 - 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 - - - - -

TABLE OF CONTENTS

1. I.

   Introduction

2. PARAGRAPHS

3. II.

   Numeral Symbols and Combination of Symbols ................................................................................................................................................................................................................................................................................................................................................................................................

   1–99

4. Babylonians ................................................................................................................................................................................................................................................................................................................................................................................................

   1–15

5. Egyptians ................................................................................................................................................................................................................................................................................................................................................................................................

   16–26

6. Phoenicians and Syrians ................................................................................................................................................................................................................................................................................................................................................................................................

   27–28

7. Hebrews ................................................................................................................................................................................................................................................................................................................................................................................................

   29–31

8. Greeks ................................................................................................................................................................................................................................................................................................................................................................................................

   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 Writers

    101

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 $(+)^{r}$ ................................................................................................................................................................................................................................................................................................................................................................................................

     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