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