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A HISTORY OF MATHEMATICAL NOTATIONS

VOLUME I

NOTATIONS IN ELEMENTARY MATHEMATICS

A HISTORY OF

ATHEMATICAL NOTATIONS

BY FLORIAN CAJORJ^H.D.

Professor of the History of Mathematics University of California

VOLUME 1

NOTATIONS IN ELEMENTARY MATHEMATICS

THE OPEN COURT COMPANY.

PUBLISHERS,

86, STRAND, LONDON, W.C.2.

COPYRIGHT 1928 BY

THE OPEN COURT PUBLISHING COMPANY Published September 1928

Composed and Printed By

The University of Chicago Preu

Chicago. Illinois. U.S.A.

PREFACE

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

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

FLORIAN CAJORI UNIVERSITY OF CALIFORNIA

. TABLE OF CONTENTS I. INTRODUCTION

PARAGRAPHS

II. NUMERAL SYMBOLS AND COMBINATIONS OF SYMBOLS . . . 1-99

Babylonians 1-15

Egyptians 16-26

Phoenicians and Syrians 27-28

Hebrews 29-31

Greeks 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 Calderon 92-93

The Portuguese Cifrao 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 Bakhshal! Manuscript 109

Hindu— Bhaskara, Twelfth Century 110-14

Arabic — al-Khow&rizmi, 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 vii

viii TABLE OF CONTENTS

PARAGRAPHS

Byzantine — Maximus Planudes, Fourteenth Century 121

Italian — Leonardo of Pisa, Thirteenth Century . . 122

French — Nicole Oresme, Fourteenth Century . . . 123

Arabic— al-Qalasadi, 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

Italian— 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 — Grarnrnateus, 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 Mastcrson, 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 HSrigone, 1634, 1644 189

Scot-French—James Hume, 1635, 1636 .... 190

French — Rene* Descartes 191

English — Isaac Barrow 192

English— Richard Rawlinson, 1655-68 193

Swiss — Johann Heinrich Rahn 194

TABLE OF CONTENTS ix

PARAGRAPHS

English— John Wallis, 1655, 1657, 1685 .... 195, 196

Extract from Ada eruditorum, Leipzig, 1708 . . . 197 Extract from Miscellanea Berolinensia, 1710 (Duo 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 + 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 ll'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 5X5 . . 229 The St. Andrew's Cross Used as a Symbol of Multi- plication 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 -f-

and X 242

A Critical Estimate of : and •§- as Symbols . . 243

TABLE OF CONTENTS

PABAQRAPH8

Notations for Geometric Ratio 244

Division in the Algebra of Complex Numbers . . 247

Signs of Proportion 248-50

Arithmetical and Geometrical Progression . . . 248

Arithmetical Proportion 249

Geometrical Proportion 250

OughtrecTs 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

TheSolidus 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

TABLE OP CONTENTS »

PARAGRAPHS

Double Significance of R and I 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 Inde- pendently? 299

Notations Used by Descartes before 1637 . . . 300

Use of H6rigone'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 (+)r 314

Conclusions , 315

Signs for Roots 316-38

Early Forms, General Statement 316, 317

The Sign $, First Appearance 318

Sixteenth-Century Use of /J 319

Seventeenth-Century Use of # 321

The Sign I 322

Napier's Line Symbolism 323

The Sign V 324-38

Origin of V 324

Spread of the V 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- Values 337

Recommendation of the U.S. National Committee 338

Signs for Unknown Numbers 339-41

Early Forms 339

xii TABLE OF CONTENTS

PARAGRAPHS

Crossed Numerals Representing Powers of Un- knowns . 340

Descartes' 2, y, x 340

Spread of Descartes' Signs 341

Signs of Aggregation 342-56

Introduction 342

Aggregation Expressed by Letters 343

Aggregation Expressed by Horizontal Bars or Vincu-

lums 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 Verbal 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 f or " Perpendicular" 364

Signs for Triangle, Square, Rectangle, Paiiillclogram . 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 O 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

ILLUSTRATIONS

FIQURB PARAGRAPHS

1. BABYLONIAN TABLETS OF NIPPUR 4

2. PRINCIPLE OF SUBTRACTION IN BABYLONIAN NUMERALS ... 9

3. BABYLONIAN LUNAR TABLES 11

4. MATHEMATICAL CUNEIFORM TABLET CBS 8536 IN THE MUSEUM

OF THE UNIVERSITY OF PENNSYLVANIA 11

5. EGYPTIAN NUMERALS 17

6. EGYPTIAN SYMBOLISM FOR SIMPLE FRACTIONS 18

7. ALGEBRAIC EQUATION IN AHMES 23

8. HIEROGLYPHIC, HIERATIC, AND COPTIC NUMERALS 24

9. PALMYRA (SYRIA) NUMERALS 27

10. SYRIAN NUMERALS 28

11. HEBREW NUMERALS 30

12. COMPUTING TABLE OF SALAMIS 36

13. ACCOUNT OF DISBURSEMENTS OF THE ATHENIAN STATE, 418-

415 B.C. 36

14. ARABIC ALPHABETIC NUMERALS 45

15. DEGENERATE FORMS OF ROMAN NUMERALS 56

16. QUIPU FROM ANCIENT CHANCAY IN PERU 65

17. DIAGRAM OF THE Two RIGHT-HAND GROUPS 65

18. AZTEC NUMERALS 66

19. DRESDEN CODEX OF MAYA 67

20. EARLY CHINESE KNOTS IN STRINGS, REPRESENTING NUMERALS . 70

21. CHINESE AND JAPANESE NUMERALS 74

22. HILL'S TABLE OF BOETHIAN APICES 80

23. TABLE OF IMPORTANT NUMERAL FORMS 80

24. OLD ARABIC AND HINDU-ARABIC NUMERALS 83

25. NUMERALS OF THE MONK NEOPHYTOS 88

26. CHR. RUDOLFF'S NUMERALS AND FRACTIONS 89

27. A CONTRACT, MEXICO CITY, 1649 93

xiv ILLUSTRATIONS

FIGURE PARAGRAPHS

28. REAL ESTATE SALE, MEXICO CITY, 1718 . 94

29. FANCIFUL HYPOTHESES 96

30. NUMERALS DESCRIBED BY NOVIOMAGUS 98

31. SANSKRIT SYMBOLS FOR THE UNKNOWN 108

32. BAKHSHALI ARITHMETIC 109

33. SRIDHARA'S Trisdtika 112

34. ORESME'S Algorismus Proportionum 123

35. AL-QALASADI'S ALGEBRAIC SYMBOLS 125

36. COMPUTATIONS OF REGIOMONTANUS 127

37. CALENDAR OF REGIOMONTANUS 128

38. FROM EARLIEST PRINTED ARITHMETIC 128

39. MULTIPLICATIONS IN THE" TREVISO" ARITHMETIC 128

40. DE LA ROCHE'S Larismethique, FOLIO 605 132

41. DE LA ROCHE'S Larismethique, FOLIO 66A 132

42. PART OF PAGE IN PACIOLI'S Summa, 1523 138

43. MARGIN OF FOLIO 1235 IN PACIOLI'S Summa 139

44. PART OF FOLIO 72 OF GHALIGAI'S Practica d'arithmetica, 1552 . 139

45. GHALIGAI'S Practica d'arithmetica, FOLIO 198 139

46. CARDAN, Ars magna, ED. 1663, PAGE 255 141

47. CARDAN, Ars magna, ED. 1663, PAGE 297 141

48. FROM TARTAGLIA'S General Trattato, 1560 143

49. FROM TARTAGLIA'S General Trattato, FOLIO 4 144

50. FROM BOMBELLI'S Algebra, 1572 144

51. BOMBELLI'S Algebra (1579 IMPRESSION), PAGE 161 .... 145

52. FROM THE MS OF BOMBELLI'S Algebra IN THE LIBRARY OF BOLOGNA 145

53. FROM PAMPHLET No. 595AT IN THE LIBRARY OF THE UNIVERSITY

OF BOLOGNA 146

54. WIDMAN'S Rechnung, 1526 146

55. FROM THE ARITHMETIC OF GRAMMATEUS 146

56. FROM THE ARITHMETIC OF GRAMMATEUS, 1535 147

57. FROM THE ARITHMETIC OF GRAMMATEUS, 1518(?) 147

58. FROM CHR. RUDOLFF'S Coss, 1525 148

ILLUSTRATIONS xv

PARAGRAPHS

59. FROM CHR. RUDOLFF'S Coss, Ev 148

' 60. FROM VAN DER HOECKE' In arithmetica 150

61. PART OF PAGE FROM STIFEL'S Arithmetica intcgra, 1544 . . . 150

62. FROM STIFEL'S Arithmetica Integra, FOLIO 31B 152

63. FROM STIFEL'S EDITION OF RUDOLFF'S Coss, 1553 156

64. SCHEUBEL, INTRODUCTION TO EUCLID, PAGE 28 159

65. W. KLEBITIUS, BOOKLET, 1565 161

66. FROM GLAVIUS' Algebra, 1608 161

67. FROM S. STEVIN'S Le Thiende, 1585 162

68. FROM S. STEVIN'S Arithmetiqve 162

69. FROM S. STEVIN'S Arithmetiqve 164

70. FROM AUREL'S Arithmetica 165

71. R. RECORDS, Whetstone of Witte, 1557 168

72. FRACTIONS IN RECORDE 168

73. RADICALS IN RECORDE 168

74. RADICALS IN DEE'S PREFACE 169

75. PROPORTION IN DEE'S PREFACE 169

76. FROM DIGGES'S Stratioticos 170

77. EQUATIONS IN DIGGES 172

78. EQUALITY IN DIGGES 172

79. FROM THOMAS MASTERSON'S Arithrneticke, 1592 172

80. J. PELETIER'S Algebra, 1554 172

81. ALGEBRAIC OPERATIONS IN PELETIER'S Algebra 172

82. FROM J. BUTEON, Arithmetica, 1559 173

83. GOSSELIN'S De arte magna, 1577 174

84. VIETA, In artem analyticam, 1591 176

85. VIETA, De emendatione aeqvationvm 178

86. B. CAVALIERI, Exercitationes, 1647 179

87. FROM THOMAS HARRIOT, 1631, PAGE 101 189

88. FROM THOMAS HARRIOT, 1631, PAGE 65 189

89. FROM HERIGONE, Cursus mathematicus, 1644 189

90. ROMAN NUMERALS FOR x IN J. HUME, 1635 191

xvi ILLUSTRATIONS

FIGURE PARAGRAPHS

91. RADICALS IN J. HUME, 1635 191

92. R. DESCARTES, Gtomttrie 191

93. I. BARROW'S Euclid, LATIN EDITION. NOTES BY ISAAC NEWTON . 193

94. I. BARROW'S Ewlid, ENGLISH EDITION 193

95. RICH. RAWLINSON'S SYMBOLS 194

96. RAHN'S Teutsche Algebra, 1659 195

97. BRANCKER'S TRANSLATION OF RAHN, 1668 195

98. J. WALLIS, 1657 195

99. FROM THE HIEROGLYPHIC TRANSLATION OF THE AHMES PAPYRUS 200

100. MINUS SIGN IN THE GERMAN MS C. 80, DRESDEN LIBRARY . . 201

101. PLUS AND MINUS SIGNS IN THE LATIN MS C. 80, DRESDEN LIBRARY 201

102. WIDMANS' MARGINAL NOTE TO MS C. 80, DRESDEN LIBRARY . 201

103. FROM THE ARITHMETIC OF BOETHIUS, 1488 250

104. SIGNS IN GERMAN MSS AND EARLY GERMAN BOOKS .... 294

105. WRITTEN ALGEBRAIC SYMBOLS FOR POWERS FROM PEREZ DE MOYA'S Arithmetica 294

106. E. WARING'S REPEATED EXPONENTS 313

INTRODUCTION

In this history it has been an aim to give not only the first appear- ance of a symbol and its origin (whenever possible), but also to indi- cate the competition encountered and the spread of the symbol among writers in different countries. It is the latter part of our program which has given bulk to this history.

The rise of certain symbols, their day of popularity, and their eventual decline constitute in many cases an interesting story. Our endeavor has been to do justice to obsolete and obsolescent notations, as well as to those which have survived and enjoy the favor of mathe- maticians of the present moment.

If the object of this history of notations were simply to present an array of facts, more or less interesting to some students of mathe- matics— if, in other words, this undertaking had no ulterior motive — then indeed the wisdom of preparing and publishing so large a book might be questioned. But the author believes that this history consti- tutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of the notational problems of the present time. |

n

NUMERAL SYMBOLS AND COMBINATIONS OF SYMBOLS

BABYLONIANS

1. In the Babylonian notation of numbers a vertical wedge Y stood for 1, while the characters ^ and Y>- signified 10 and 100, respectively. Grotefend1 believes the character for 10 originally to have been the picture of two hands, as held in prayer, the palms being pressed together, the fingers close to each other, but the thumbs thrust out. Ordinarily, two principles were employed in the Babylonial no- tation— the additive and multiplicative. We shall see that limited use was made of a third principle, that of subtraction.

2. Numbers below 200 were expressed ordinarily by symbols whose respective values were to be added. Thus, Y^XKYYY stands for 123. The principle of multiplication reveals itself in < |>- where the smaller symbol 10, placed before the 100, is to be multiplied by 100, so that this symbolism designates 1,000.

3. These cuneiform symbols were probably invented by the early Sumerians. Their inscriptions disclose the use of a decimal scale of numbers and also of a sexagesimal scale.2

Early Sumerian clay tablets contain also numerals expressed by circles and curved signs, made with the blunt circular end of a stylus, the ordinary wedge-shaped characters being made with the pointed end. A circle • stood for 10, a semicircular or lunar sign stood for 1. Thus, a "round-up" of cattle shows J*DDD> or ^ cows.3

4. The sexagesimal scale was first discovered on a tablet by E. Hincks4 in 1854. It records the magnitude of the illuminated portion

1 His first papers appeared in Gottingische Gelehrte Anzeigen (1802), Stuck 149 und 178; ibid. (1803), Stuck 60 und 117.

2 In the division of the year and of the day, the Babylonians used also the duodecimal plan.

8 G. A. Barton, Haverford Library Collection of Tablets, Part I (Philadelphia, 1905), Plate 3, HCL 17, obverse; see also Plates 20, 26, 34, 35. Allotte de la Fuye, "En-e-tar-zi pate*si de Lagas," H. V. Hilprecht Anniversary Volume (Chi- cago, 1909), p. 128, 133.

4 "On the Assyrian Mythology," Transactions of the Royal Irish Academy. "Polite Literature," Vol. XXII, Part 6 (Dublin, 1855), p. 406, 407.

2

OLD NUMERAL SYMBOLS 3

of the moon's disk for every day from new to full moon, the whole disk being assumed to consist of 240 parts. The illuminated parts during the first five days are the series 5, 10, 20, 40, 1.20, which is a geo- metrical progression, on the assumption that the last number is 80. From here on the series becomes arithmetical, 1.20, 1.36, 1.52, 2.8, 2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4, the common difference being 16. The last number is written in the tablet X^,— and, according to Hincks's interpretation, stood for 4 X 60 = 240.

Obverse. Reverse.

FIG. 1. — Babylonian tablets from Nippur, about 2400 B.C.

5, Hincks's explanation was confirmed by the decipherment of tablets found at Senkereh, near Babylon, in 1854, and called the Tab- lets of Senkereh. One tablet was found to contain a table of square numbers, from I2 to 602, a second one a table of cube numbers from I3 to 323. The tablets were probably written between 2300 and 1600 B.C. Various scholars contributed toward their interpretation. Among them \vere George Smith (1872), J. Oppert, Sir H. Rawlinson, Fr. Lenormant, and finally R. Lepsius.1 The numbers 1, 4, 9, 16, 25, 36,

George Smith, North British Review (July, 1870), p. 332 n.; J. Oppert, Journal asiatique (August-September, 1872; October-November, 1874); J. Oppert, £talon des tnesures assyr. fixe" par les textes cuneiformes (Paris, 1874) ; Sir H. Rawlinson and G. Smith, "The Cuneiform Inscriptions of Western Asia," Vol. IV: A Selection from the Miscellaneous Inscriptions of Assyria (London, 1875), Plate 40; R. Lepsius, "Die Babylonisch-Assyrischen Langenmaasse nach der Tafel von Senkereh," Abhandlungen der Koniglichen Akademie der Wissen- schaften zu Berlin (aus dem Jahre 1877 [Berlin, 1878], Philosophisch-historische Klasse), p. 105-44.

4 A HISTORY OF MATHEMATICAL NOTATIONS

and 49 are given as the squares of the first seven integers, respecti We have next 1.4 = 82, 1.21 = 92, 1.40= 102, etc. This clearly indi the use of the sexagesimal scale which makes 1.4 = 60+4, 1.21 = 21. 1.40 = 60+40, etc. This sexagesimal system marks the ea: appearance of the all-important "principle of position" in wr numbers. In its general and systematic application, this principl quires a symbol for zero. But no such symbol has been found on < Babylonian tablets; records of about 200 B.C. give a symbol for as we shall see later, but it was not used in calculation. The ea: thorough and systematic application of a symbol for zero anc principle of position was made by the Maya of Central America, a the beginning of the Christian Era.

6. An extension of our knowledge of Babylonian mathem was made by H. V. Hilprecht who made excavations at Nuffar ancient Nippur). We reproduce one of his tablets1 in Figure 1.

Hilprecht's transliteration, as given on page 28 of his te as follows:

Line 1. 125 720 Line 9. 2,000

Line 2. IGI-GAL-BI 103,680 Line 10. IGI-GAL-BI (

Line 3. 250 360 Line 11. 4,000

Line 4. IGI-GAL-BI 51,840 Line 12. IGI-GAL-BI

Line 5. 500 180 Line 13. 8,000

Line 6. IGI-GAL-BI 25,920 Line 14. IGI-GAL-BI

Line 7. 1,000 90 Line 15. 16,000

Line 8. IGI-GAL-BI 12,960 Line 16. IGI-GAL-BI

7. In further explanation, observe that in

Line 1. 125 = 2X60+5, 720 = 12X60+0

Line 2. Its denominator, 103,680 = [28 X60+48(?)]X 6

Line 3. 250 = 4X60+10, 360 = 6X60+0

Line 4. Its denominator, 51,840 = [14 X 60+24] X60+(

Line 5. 500 = 8X60+20, 180 = 3X60+0

Line 6. Its denominator, 25,920 = [7 X 60+ 12] X 60+0

Line 7. 1,000=16X60+40, 90=1X60+30

Line 8. Its denominator, 12,960 = [3X60+36]X60+0

1 The Babylonian Expedition of the University of Pennsylvania. Seri "Cuneiform Texts," Vol. XX, Part 1, Mathematical, Metrological and C) logical Tablets from the Temple Library of Nippur (Philadelphia, 1906), Pla No. 25.

OLD NUMERAL SYMBOLS 5

Line 9. 2,000 = 33X60+20, 18=10+8

Line 10. Its denominator, 6,480 = [IX 60+48] X 60+0

Line 11. 4,000 = [1X60+6]X60+40, 9

Line 12. Its denominator, 3,240 = 54 X 60+0

Line 13. 8,000 = [2X60+13]X60+20, 18

Line 14. Its denominator, 1,620 = 27X60+0

Line 15. 16,000 = [4X60+ 26] X 60+40, 9

Line 16. Its denominator, 810=13X60+30

IGI-GAL = Denominator, £/ = Its, i.e., the number 12,960,000 or 604.

We quote from Hilprecht (op. cit., pp. 28-30):

"We observe (a) that the first numbers of all the odd lines (1, 3, 5, 7, 9, 11, 13, 15) form an increasing, and all the numbers of the even lines (preceded by IGI-GAL-BI = (its denominator') a descending geometrical progression; (6) that the first number of every odd line can be expressed by a fraction which has 12,960,000 as its numerator and the closing number of the corresponding even line as its denomi- nator, in other words,

10 12,960,000 . 12,960,000 . Knn_ 12,960,000

1^)"" 103,680 ' 51,840 ' 25,920 '

nn 12,960,000 . 9 mn 12,960,000 . 12,

>m = -l2W- ' 2>°00= 6480 ' 4'000== 3

960,000

6,480 ' ' 3,240 '

12,960,000 . 12,960,000

8,000= lj62Q , 16,000-—^—.

But the closing numbers of all the odd lines (720, 360, 180, 90, 18, 9, 18, 9) are still obscure to me .....

"The question arises, what is the meaning of all this? What in par- ticular is the meaning of the number 12,960,000 ( = 604 or 3,6002) which underlies all the mathematical texts here treated ....?.... This ' geometrical number ' (12,960,000), which he [Plato in his Repub- lic viii. 546#-D] calls 'the lord of better and worse births/ is the arithmetical expression of a great law controlling the Universe. According to Adam this law is 'the Law of Change, that law of in- evitable degeneration to which the Universe and all its parts are sub- ject' — an interpretation from which I arn obliged to differ. On the contrary, it is the Law of Uniformity or Harmony, i.e. that funda- mental law which governs the Universe and all its parts, and which cannot be ignored and violated without causing an anomaly, i.e. with- out resulting in a degeneration of the race." The nature of the "Pla- tonic number" is still a debated question.

6 A HISTORY OF MATHEMATICAL NOTATIONS

8. In the reading of numbers expressed in the Babylonian sexa- gesimal system, uncertainty arises from the fact that the early Baby- lonians had no symbol for zero. In the foregoing tablets, how do we know, for example, that the last number in the first line is 720 and not 12? Nothing in the symbolism indicates that the 12 is in the place where the local value is "sixties" and not "units." Only from the study of the entire tablet has it been inferred that the number in- tended is 12X60 rather than 12 itself. Sometimes a horizontal line was drawn following a number, apparently to indicate the absence of units of lower denomination. But this procedure was not regular, nor carried on in a manner that indicates the number of vacant places.

9. To avoid confusion some Babylonian documents even in early times contained symbols for 1, 60, 3,600, 216,000, also for 10, 600, 36,000.* Thus • was 10, • was 3,600, © was 36,000.

in view of other variants occurring in fchc

mathematical tablets from Nippur, notably the numerous variants of "19,"1 some of which may be merely scribal errors :

They evidently all go back to the form <^}~ ^^^f (20 — 1 = 19).

FIG. 2. — Showing application of the principle of subtraction

10. Besides the principles of addition and multiplication, Baby- lonian tablets reveal also the use of the principle of subtraction, which is familiar to us in the Roman notation XIX (20—1) for the number 19. Hilprecht has collected ideograms from the Babylonian tablets which he has studied, which represent the number 19. We reproduce his symbols in Figure 2. In each of these twelve ideograms (Fig. 2), the' two symbols to the left signify together 20. Of the symbols im- mediately to the right of the 20, one vertical wedge stands for "one" and the remaining symbols, for instance Y^, for LAL or "minus"; the entire ideogram represents in each of the twelve cases the number 20- lor 19.

One finds the principle of subtraction used also with curved signs;2 D • • Y*~~D meant 60+20 — 1, or 79.

1 See Frangois Thureau-Dangin, Recherches sur Vorigine de Vecriture cuntiforme (Paris, 1898), Nos. 485-91, 509-13. See also G. A. Barton, Haverford College Library Collection of Cuneiform Tablets, Part I (Philadelphia, 1905), where the forms are somewhat different; also the Hilprecht Anniversary Volume (Chicago, 1909), p. 128 ff.

2 G. A. Barton, op. cit.t Plate 3, obverse.

OLD NUMERAL SYMBOLS

11. The symbol used about the second century B.C. to designate 5 absence of a number, or a blank space, is shown in Figure 3, con- ning numerical data relating to the moon.1 As previously stated, s symbol, ^ , was not used in computation and therefore performed

FIG. 3. — Babylonian lunar tables, reverse; full moon for one year, about the 1 of the second century B.C.

ly a small part of the functions of our modern zero. The symbol is jn in the tablet in row 10, column 12; also in row 8, column 13. igler's translation of the tablet, given in his book, page 42, is shown low. Of the last column only an indistinct fragment is preserved; 3 rest is broken off.

REVERSE

Niaannu

28°56'30"

19°16' " Librae

3Z 6°45'

4i74ii10ur sik

Airu

28 38 30

175430 Scorpii

321 28

620 30 sik

Simannu

28 20 30

16 15 Arcitenentia

3 31 39

345 30 sik

Dti,zu

28 18 30

14 33 30 Capri

33441

1 10 30 sik

Abu

28 36 30

13 9 Aquarii

32756

1 24 30 bar

Ululu

29 54 30

13 3 30 Piscium

3 1534

1 59 30 num

TiSrltu

29 12 30

11 16 Arietis

258 3

4 34 30 num

Araty-s.

29 30 30

10 46 30 Tauri

24054

6 0 10 num

Kishmu

29 48 30

10 35 Geminorurn

2 29 29

3 25 10 num

Tebitu

29 57 30

10 32 30 Cancri

22430

067 10 num

Sabatu

29 39 30

10 12 Leonis

2 30 53

1 44 50 bar

Addru I

29 21 30

9 33 30 Virginis

2 42 56

2 19 50 sik

Ad6.ru II

29 330

8 36 Librae

3 021

464 50 sik

Nisannu

28 45 30

7 21 30 Scorpii

3 1736

539 50 sik

1 Franz Xaver Kugler, S. J., Die babylonische Mondrechnung (Freiburg im Breis- j, 1900), Plate IV, No. 99 (81-7-6), lower part.

A HISTORY OF MATHEMATICAL NOTATIONS

OBVERSE

W^r-

4

«

^r

EMte-

_^MN

w^

W*-

s^-

31

^¥^-

(THff^-

El^Si^

^^

r-«w-'n..*'

^teh'^tfilii

FIG. 4. —Mathematical cuneiform tablet, CBS 8536, in the Museum of the University of Pennsylvania.

OLD NUMERAL SYMBOLS 9

12. J. Oppert pointed out the Babylonian use of a designation the sixths, viz., |, £, -£, f, $. These are unit fractions or fractions ose numerators are one less than the denominators.1 He also ad- iced evidence pointing to the Babylonian use of sexagesimal frac- is and the use of the sexagesimal system in weights and measures. B occurrence of sexagesimal fractions is shown in tablets recently mined. We reproduce in Figure 4 two out of twelve columns found a tablet described by H. F. Lutz.2 According to Lutz, the tablet innot be placed later than the Cassite period, but it seems more prob- e that it goes back even to the First Dynasty period, ca. 2000 B.C."

13, To mathematicians the tablet is of interest because it reveals orations with sexagesimal fractions resembling modern operations h decimal fractions. For example, 60 is divided by 81 and the >tient expressed sexagesimally. Again, a sexagesimal number with > fractional places, 44 (26) (40), is multiplied by itself, yielding a .duct in four fractional places, namely, [32]55(18)(31)(6)(40). In 3 notation the [32] stands for 32X60 units, and to the (18), (31), , (40) must be assigned, respectively, the denominators 60, 602, , 604.

The tablet contains twelve columns of figures. The first column g. 4) gives the results of dividing 60 in succession by twenty-nine 'erent divisors from 2 to 81. The eleven other columns contain les of multiplication; each of the numbers 50, 48, 45, 44 (26) (40), 36, 30, 25, 24, 22(30), 20 is multiplied by integers up to 20, then by

numbers 30, 40, 50, and finally by itself. Using our modern nu- rals, we interpret on page 10 the first and the fifth columns. They dbit a larger number of fractions than do the other columns. e Babylonians had no mark separating the fractional from the in- ral parts of a number. Hence a number like 44 (26) (40) might be 3rpreted in different ways; among the possible meanings are 44 X +26X60+40, 44X60+26+40X60-1, and 44+26X60~x+40X -2. Which interpretation is the correct one can be judged only by

context, if at all.

The exact meaning of the first two lines in the first column is un- fcain. In this column 60 is divided by each of the integers written the left. The respective quotients are placed on the right.

1 Symbols for such fractions are reproduced also by Thureau-Dangin, op. cit., \. 481-84, 492-508, and by G. A. Barton, Haverford College Library Collection Cuneiform Tablets, Part I (Philadelphia, 1905).

2 "A Mathematical Cuneiform Tablet/' American Journal of Semitic Lan- ges and Literatures, Vol. XXXVI (1920), p. 24^-57.

10

A HISTORY OF MATHEMATICAL NOTATIONS

In the fifth column the multiplicand is 44 (26) (40) or 44 jj. The last two lines seem to mean "602-r-44(26)(40) = 81, 602-r81 = 44(26)(40)."

First Column .... gal (?) -bi 40 -&m Su a- na gal-bi 30 -am

igi 2

30

igi 3

20

igi 4

15

igi 5

12

igi 6

10

igi 8

7(30)

igi 9

6(40)

igi 10

6

igi 12

5

igi 15

4

igi 16

•3(45)

igi 18

3(20)

igi 20

3

igi 24

2(30)

igi 25

2(24)

igi 28*

2(13) (20)

igi 30

2

igi 35*

1(52) (30)

igi 36

1(40)

igi 40

1(30)

igi 45

1(20)

igi 48

1(15)

igi 50

1(12)

igi 54

1(6) (40)

igi 60

1

igi 64

(56) (15)

igi 72

(50)

igi 80

(45)

igi 81

(44) (26) (40)

1 2 3

4

5

6

7

9

10

11

12

13

14

15

16

17

18

19

20

30

40

50

Fifth Column

44(26) (40)

44(26) (40)

[1]28(53)(20)

[2]13(20)

[2]48(56)(40)*

[3]42(13)(20)

[4]26(40)

[6]40 [7]24(26)(40)

[8]8(53)(20)

[8]53(20)

[9]27(46)(40)*

[10]22(13)(20)

[H]6(40)

[12]35(33)(20) [13J20

[14]4(26)(40) [14]48(53)(20) [22]13(20) [29]37(46)(40) [38]2(13)(20)* 44(26) (40)a-na 44(26) (40) [32]55(18)(31)(6)(40) 44 (26) (40) square igi 44(26)(40) 81

igiSl 44 (26) (40)

Numbers that are incorrect are marked by an asterisk (*).

14. The Babylonian use of sexagesimal fractions is shown also in a clay tablet described by A. Ungnad.1 In it the diagonal of a rec- tangle whose sides are 40 and 10 is computed by the approximation

1 Orientalische Literaturzeitung (ed. Peise, 1916), Vol. XIX, p. 363-68. See also Bruno Meissner, Babylonien und Assyrien (Heidelberg, 1925), Vol. II, p. 393.

OLD NUMERAL SYMBOLS 11

40+2X40Xl02-h602, yielding 42(13)(20), and also by the approxi- mation 40+1024- 12X401, yielding 41(15). Translated into the deci- mal scale, the first answer is 42.22+, the second is 41.25, the true value being 41.23+. These computations are difficult to explain, except on the assumption that they involve sexagesimal fractions.

15. From what has been said it appears that the Babylonians had ideograms which, transliterated, are Igi-Gal for "denominator" or "division," and Lai for "minus." They had also ideograms which, transliterated, are Igi-Dua for "division," and A-Du and Ara for "times," as in Ara- 1 18, for "1X18 = 18," Ara- 2 36 for "2 X 18 = 36" ; the Ara was used also in "squaring," as in 3 Ara 3 9 for "3X3 = 9." They had the ideogram Ba-Di-E for "cubing," as in 21 -E 3 Ba-Di-E for "33 = 27"; also Ib-Di for "square," as in 9-# 3 Ib-Di for "32 = 9." The sign A -An rendered numbers "distribu- tive."1

EGYPTIANS

16. The Egyptian number system is based on the scale of 10, al- though traces of other systems, based on the scales of 5, 12, 20, and 60, are believed to have been discovered.2 There are three forms of Egyptian numerals: the hieroglyphic, hieratic, and demotic. Of these the hieroglyphic has been traced back to about 3300 B.C. ;3 it is found mainly on monuments of stone, wood, or metal. Out of the hiero- glyphic sprang a more cursive writing known to us as hieratic. In the beginning the hieratic was simply the hieroglyphic in the rounded forms resulting from the rapid manipulation of a reed-pen as con- trasted with the angular and precise shapes arising from the use of the chisel. About the eighth century B.C. the demotic evolved as a more abbreviated form of cursive writing. It was used since that time down to the beginning of the Christian Era. The important mathematical documents of ancient Egypt were written on papyrus and made use of the hieratic numerals.4

1 Hilprecht. op. til., p. 23; Arno Poebel, Grundzuge der sumerischen Grammatik (Rostock, 1923), p. 115; B. Meissner, op. cit., p. 387-89.

2 Kurt Sethe, Von ZdhLen und Zahlworlen bei den alien Agyptern (Strassburg, 1916), p. 24-29.

3 J. E. Quibell and F. W. Green, Hierakonopolis (London, 1900-1902), Part I, Plate 26B, who describe the victory monument of King Ncr-mr; the number of prisoners taken is given as 120,000, while 400,000 head of cattle and 1,422,000 goats were captured.

4 The evolution of the hieratic writing from the hieroglyphic is explained in G. Moller, Hieratische Palaographie, Vol. I, Nos. 614 ff. The demotic writing

12

A HISTORY OF MATHEMATICAL NOTATIONS

17. The hieroglyphic symbols were I for 1, O for 10, C for 100, I for 1,000, | for 10,000, ^ for 100,000, $ for 1,000,000, Q for 10,000,000. The symbol for 1 represents a vertical staff; that for 1,000 a lotus plant; that for 10,000 a pointing finger; that for 100,000 a burbot; that for 1,000,000 a man in astonishment, or, as more recent

Etner

Zehner

HunJerte

TctusettJe

n

I

I

60

M

nn

A

II

100

01

no n

ODD

nnnn

JiL

Ann

KS

not

8100

AAAA

a

i i

rwvi nnn

II?

CO

FIG. 5. — Egyptian numerals. Hieroglyphic, hieratic, and demotic numeral symbols. (This table was compiled by Kurt Sethc.)

Egyptologists claim, the picture of the cosmic deity Hh.1 The sym- bols for 1 and 10 are sometimes found in a horizontal position.

18. We reproduce in Figures 5 and 6 two tables prepared by Kurt

is explained by F. L. Griffith, Catalogue of the Demotic Papyri in the John Rylands Library (Manchester, 1909), Vol. Ill, p. 415 if., and by H. Brugsch, Grammaire d&motique, §§ 131 ff.

1Sethe, op.cit., p. 11, 12.

OLD NUMERAL SYMBOLS

13

Sethe. They show the most common of the great variety of forms which are found in the expositions given by Moller, Griffith, and Brugsch. Observe that the old hieratic symbol for % was the cross X, sig- nifying perhaps a part obtainable from two sections of a body through the center.

Attaeyyfttiscke BruchiticLcn,

ftrc&ttckt fruchzetJu*

tit

in

M>

IP

X

Mil

mi

III!

**

lift %'/*

tv

%'At

y<

[llllllj

nun

mm

//*

*;»

-90*

FIG. 6. — Egyptian symbolism for simple fractions. (Compiled by Kurt Sethe)

19. In writing numbers, the Egyptians used the principles of addi- tion and multiplication. In applying the additive principle, not more than four symbols of the same kind were placed in any one group. Thus, 4 was written in hieroglyphs 1 1 1 1 ; 5 was not written HIM, but

either 1 1 1 1 1 or , , . There is here recognized the same need which

caused the Romans to write V after IIII, L = 50 after XXXX = 40, D = 500 after CCCC = 400. In case of two unequal groups, the Egyp- tians always wrote the larger group before, or above the smaller group;

thus, seven was written ,, , .

14 A HISTORY OF MATHEMATICAL NOTATIONS

20. In the older hieroglyphs 2,000 or 3,000 was represented by two or three lotus plants grown in one bush. For example, 2,000 was ^ ; correspondingly, 7,000 was designated by 23K £?£ . The later hiero- glyphs simply place two lotus plants together, to represent 2,000, with- out the appearance of springing from one and the same bush.

21. The multiplicative principle is not so old as the additive; it came into use about 1600-2000 B.C. In the oldest example hitherto known,1 the symbols for 120, placed before a lotus plant, signify 120,000. A smaller number written before or below or above a sym- bol representing a larger unit designated multiplication of the larger by the smaller. Mollcr cites a case where 2,800,000 is represented by one burbot, with characters placed beneath it which stand for 28.

22. In hieroglyphic writing, unit fractions were indicated by placing the symbol <o over the number representing the denomina- tor. Exceptions to this arc the modes of writing the fractions | and f ; the old hieroglyph for \ was ^=T, the later was /" ~; of the slightly varying hieroglyphic forms for -|, £ was quite common.2

23. We reproduce an algebraic example in hieratic symbols, as it occurs in the most important mathematical document of antiquity known at the present time — the Rhind papyrus. The scribe, Ahmcs, who copied this papyrus from an older document, used black and red ink, the red in the titles of the individual problems and in writing auxiliary numbers appearing in the computations. The example which, in the Eisenlohr edition of this papyrus, is numbered 34, is hereby shown.3 Hieratic writing was from right to left. To facilitate the study of the problem, we write our translation from right to left and in the same relative positions of its parts as in the papyrus, except that numbers are written in the order familiar to us; i.e., 37 is written in our translation 37, and not 73 as in the papyrus. Ahmes writes unit fractions by placing a dot over the denominator, except in case of

1 Ibid., p. 8.

2 Ibid., p. 92-97, gives detailed information on the forms representing f. The Egyptian procedure for decomposing a quotient into unit fractions is explained by V. V. Bobynin in Abh. Gesch. Math., Vol. IX (1899), p. 3.

8 Ein matkematisches Handbook der alien Agypter (Papyrus Rhind des British Museum) t'ibersetzt und erkldrt (Leipzig, 1877; 2d cd., 1891). The explanation of Problem 34 is given on p. 55, the translation on p. 213, the facsimile reproduction on Plate XIII of the first edition. The second edition was brought out without the plates. A more recent edition of the Ahmes papyrus is due to T. Eric Peet and appears under the title The Rhind Mathematical Papyrus, British Museum, Nos. 10057 and 10058, Introduction, Transcription, and Commentary (London, 1923).

OLD NUMERAL SYMBOLS 15

i> i> £> i> eacn °f which had its own symbol. Some of the numeral symbols in Ahmes deviate somewhat from the forms given in the two preceding tables; other symbols are not given in those tables. For the reading of the example in question we give here the following symbols :

Four — One-fourth X

Five "1 Heap S$t See Fig. 7

Seven Q- The whole |J See Fig. 7

One-half ~7 It gives & See Fig. 7

FIG. 7. — An algebraic equation and its solution in the Ahmes papyrus, 1700 B.C., or, according to recent authorities, 1550 B.C. (Problem 34, Plate XIII in Eisenlohr; p. 70 in Peet; in chancellor Chace's forthcoming edition, p. 76, as R. C. Archibald informs the writer.)

Translation (reading from right to left) :

"10 gives it, whole its, \ its, \ its, Heap No. 34

al4 j I 5 is heap the together 7 4

1 I Proof the of Beginning

-iV 1 1 5

I \- Remainder £ -£ 9 together A izV i i 1 I

14 gives i A -^V •&• TT A I

21 Together .7 gives i 122448"

16

A HISTORY OF MATHEMATICAL NOTATIONS

24. Explanation:

oc or The algebraic equation is 0+4+2= 10

i.e., (l+i+i)*=10

The solution answers the question, By what must (1 ^ |) be multiplied to yield the product 10? The four lines 2-5 contain on the right the following computation :

Twice (1 H) yields 3 fc. Four times (1 -£ \) yields 7. One-seventh of (1 \ -J-) is \.

t° UNITES.

. S1GNES

LETTRES

MIMEIULES

copies.

VALEUR

ik-s

S10NES.

NOMS

OK A01IBBK

<m dialecle tliebaiu.

UlKROCLYriUQUES ,

creux ct plciiis.

UI8AAT1Q1IBS ,

uvcc variantcs.

0 i

') I ? ?

£

t

Vt«l.

00 H

H 'M

&

rt

snau.

DOD HI

^ 04

TT

:t

chonwnt.

DDOD i!

UH -u^ 4

^

A

ftoou.

ODD 00 '"

1 1 1

E

r>

lion.

DOD DOD !!!

t Z

r

(>

soon.

0000000 V"

^t xti ^&i

t.

7

sachf.

flDDOOODO m'i

^=^ =*•

F

8

chinoun.

000 ODD ODD "Si!"

^.^

tf

<)

pxix.

(Continued on facing page]

[i.e., taking (1 \ -J-) once, then four times, together with \ of it, yields only 9; there is lacking 1. The remaining computation is on the four lines 2-5, on the left. Since \ of (1 | {) yields (\ -^ ^8) or -J, lor]

(i A) of (1 i 1), yields |.

And the double of this, namely, (^ -£f) of (1 \ |) yields 1. Adding together 1, 4, | and (fc iV), we obtain Heap = 5^ ^ -^ or 5f , the answer.

OLD NUMERAL SYMBOLS

17

Proof. — 5 ^ \ ilf is multiplied by (1 ^ J) and the partial products are added. In the first line of the proof we have 5 ^ | ^lf, in the second line half of it, in the third line one-fourth of it. Adding at first only the integers of the three partial products and the simpler fractions i> i> !> i> i> the partial sum is 9 -\ \. This is \ I short of 10. In the fourth line of the proof (1. 9) the scribe writes the remaining fractions and, reducing them to the common denominator 56, he writes (in

2° DIZA1NES.

SIGiNES

LETTRES

NUMKRALES

copies.

VALEUR dcs

8IONK8.

NOMS

DE NOMDRB

en dialcctc thebaih.

II1EROGLYPI1IQURS .

crcux el plein.

IIIERATIQUES ,

nvcc varianles.

X X /6

I

10

ment.

ChinVe connmm dos dizaincs :

XX

*

30

sjouoL maab.

mi »« n

^- -*-

H

/JO

hme.

\ *\

1?

5o

taioii.

Jii &.

I

(>o

se.

°x °x

o"

7°

chfe.

>ui4 jiU

TT

80

hmeiie.

^

Ci

9°

ptsldtou.

FIG. 8. — Hieroglyphic, hieratic, and Coptic numerals. (Taken from A. P. Pihan, Expos6 des signes de numeration [Paris, 1860], p. 26, 27.)

red color) in the last line the numerators 8, 4, 4, 2, 2, 1 of the reduced

fractions. Their sum is 21. But ew =

5o

^-=-7 o , which is the exact oo 4 8

amount needed to make the total product 10.

A pair of legs symbolizing addition and subtraction, as found in impaired form in the Ahmes papyrus, are explained in § 200.

25. The Egyptian Coptic numerals are shown in Figure 8. They are of comparatively recent date. The hieroglyphic and hieratic are

18 A HISTORY OF MATHEMATICAL NOTATIONS

the oldest Egyptian writing; the demotic appeared later. The Cop- tic writing is derived from the Greek and demotic writing, and was used by Christians in Egypt after the third century. The Coptic numeral symbols were adopted by the Mohammedans in Egypt after their conquest of that country.

26. At the present time two examples of the old Egyptian solu- tion of problems involving what we now term "quadratic equations"1 are known. For square root the symbol Ir3 has been used in the modern hieroglyphic transcription, as the interpretation of writing in the two papyri; for quotient was used the symbol oo .

PHOENICIANS AND SYRIANS

27. The Phoenicians2 represented the numbers 1-9 by the re- spective number of vertical strokes. Ten was usually designated by a horizontal bar. The numbers 1 1-19 were expressed by the juxtaposi- tion of a horizontal stroke and the required number of vertical ones.

Palmyreaische ZaMzeiebn I X 3; 3, JD'p^ ;,55"7 . '7^3 3 '''CO" Virianten >ei Oruter / -V ; >. 0; , V; >V( >.V ''^0><?'>V'

BtdeuUag 1. 0; 10. 20 100, 110. 1000 JW.

FIG. 9. — Palmyra (Syria) numerals. (From M. Cantor, Kulturleben, etc., Fig. 48)

As Phoenician writing proceeded from right to left, the horizontal stroke signifying 10 was placed farthest to the right. Twenty was represented by two parallel strokes, either horizontal or inclined and sometimes connected by a cross-line as in H, or sometimes by two strokes, thus A- One hundred was written thus |<| or thus | £>| . Phoe- nician inscriptions from which these symbols are taken reach back several centuries before Christ. Symbols found in Palmyra (modern Tadmor in Syria) in the first 250 years of our era resemble somewhat the numerals below 100 just described. New in the Palmyra numer-

1 See H. Schack-Schackcnburg, "Der Berliner Papyrus 6619," Zeitschrift fur dgyptische Sprache und Altertumskunde, Vol. XXXVIII (1900), p. 136, 138, and Vol. XL (1902), p. 6S-66.

2 Our account is taken from Moritz Cantor, Vorlesungen fiber Geschichte der Mathematik, Vol. I (3d ed. ; Leipzig, 1907), p. 123, 124; Mathematische Beitrage zum KuUurleben der Volker (Halle, 1863), p. 255, 256, and Figs. 48 and 49.

OLD NUMERAL SYMBOLS 19

als is 7 for 5. Beginning with 100 the Palmyra numerals contain new forms. Placing a I to the right of the sign for 10 (see Fig. 9) signifies multiplication of 10 by 10, giving 100. Two vertical strokes 1 1 mean 10X20, or 200; three of them, 10X30, or 300.

28. Related to the Phoenician are numerals of Syria, found in manuscripts of the sixth and seventh centuries A.D. Their shapes and their mode of combination are shown in Figure 10. The Syrians em- ployed also the twenty-two letters of their alphabet to represent the numbers 1-9, the tens 10-90, the hundreds 100-400. The following hundreds were indicated by juxtaposition: 500 = 400+100, 600 = 400+200, ____ , 900=400+400+100, or else by writing respectively 50-90 and placing a dot over the letter to express that its value is to be taken tenfold. Thousands were indicated by the letters for 1-9, with a stroke annexed as a subscript. Ten thousands were expressed

I, H - 2, HI - 3, FP- 4,. -*-5. h-* -6

7 HM-8. H^-9 7-io 7-u K7-12 w, HH^ -18, O - 20 70 - M. TI - 100

Syrische Zahlzeiche.n FIG. 10. — Syrian numerals. (From M. Cantor, Kulturleben, etc., Fig. 49)

by drawing a small dash below the letters for one's and ten's. Millions were marked by the letters 1-9 with two strokes annexed as sub- scripts (i.e., 1,000X1,000 = 1,000,000).

HEBREWS

29. The Hebrews used their alphabet of twenty-two letters for the designation of numbers, on the decimal plan, up to 400. Figure 11 shows three forms of characters: the Samaritan, Hebrew, and Rabbinic or cursive. The Rabbinic was used by commentators of the Sacred Writings. In the Hebrew forms, at first, the hundreds from 500 to 800 were represented by juxtaposition of the sign for 400 and a second number sign. Thus, pn stood for 500, ^n for 600, ISO for 700, nn for 800.

30. Later the end forms of five letters of the Hebrew alphabet came to be used to represent the hundreds 500-900. The five letters representing 20, 40, 50, 80, 90, respectively, had two forms; one of

20

A HISTORY OF MATHEMATICAL NOTATIONS

LETTRES

MOMS

NOMS

8AMAIUTAIRES

HEBBAIQUES.

BABBI1UQUBS.

DBS LKTTRB3.

DE NOMBBE.

*

N

(S

aleph , a

t

ekhdd.

a

2

5

bet, b

2

chewing

1

:

J

ghimcl , gh

3

clielochdh.

1

1

1

dalet, d

A

arbd'ah.

*

n

p

y, A

5

khamichdh.

*

i

)

waw, w

6

chichdh.

1»

T

t

zain, z

7

chib'dh.

*

n

p

khel, B

8

chemondh.

*

D

V

t'et', t'

9

tich'dh.

m

'

5

iod, t

10

'asdrdh.

a

3

o

kaph , k

30

'esrim.

4

h

1)

lamed , /

do

chelochim*

»

D

p

mem , m

%0

arbd'im.

A

J

D

noun y n

5o

khamichim.

*

D

D

s'amek i

60

chichim.

v

y

J>

cain , '«

70

chib'im.

3

s

D

ph^, ph

80

chemonim.

Ytt

3J

3

Lsade, to

9<>

tictiim.

1?

p

P

qopli, ^

100

mtdh.

^

*]

•5

rech, r

900

mdtai'm.

JJJL

t^

ft

chin , ch

3oo

cluilvchmttt.

A

n

P

lau, £

W400

arba* mcdt.

FIG. 11. — Hebrew numerals. (Taken from A. P. Pihan, Expos6 des signes de numeration [Paris, 1860], p. 172, 173.)

OLD NUMERAL SYMBOLS 21

the forms occurred when the letter was a terminal letter of a word. These end forms were used as follows:

Y T| T D 1 900 800 700 600 500.

To represent thousands the Hebrews went back to the beginning of their alphabet and placed two dots over each letter. Thereby its value was magnified a thousand fold. Accordingly, £ represented 1,000. Thus any number less than a million could be represented by their system.

31. As indicated above, the Hebrews wrote from right to left. Hence, in writing numbers, the numeral of highest value appeared on the right; )$n meant 5,001, n& meant 1,005. But 1,005 could be written also flK , where the two dots were omitted, for when ^ meant unity, it was always placed to the left of another numeral. Hence when appearing on the right it was interpreted as meaning 1,000. With a similar understanding for other signs, one observes here the beginning of an imperfect application in Hebrew notation of the principle of local value. By about the eighth century A.D., one finds that the signs iTD^n signify 5,845, the number of verses in the laws as given in the Masora. Here the sign on the extreme right means 5,000; the next to the left is an 8 and must stand for a value less than 5,000, yet greater than the third sign representing 40. Hence the sign for 8 is taken here as 800. l

GREEKS

32. On the island of Crete, near Greece, there developed, under Egyptian influence, a remarkable civilization. Hieroglyphic writing on clay, of perhaps about 1500 B.C., discloses number symbols as

follows: ) or I for 1, ))))) or 1 1 1 1 1 or "' for 5, • for 10, \ or / for

100, <> for 1,000, V for i (probably), \\\\: :::))) for 483.2 In thk combination of symbols only the additive principle is employed. Somewhat later,3 10 is represented also by a horizontal dash; the

1 G. H. F. Ncsselmann, Die Algebra der Griechen (Berlin, 1842), p. 72, 494; M. Cantor, Vorlesungen liber Geschichte der Malhematik, Vol. I (3d ed.), P- 126, 127.

2 Arthur J. Evans, Scripla Minoa, Vol. I (1909), p. 258, 256.

8 Arthur J. Evans, The Palace of Minos (London, 1921), Vol. 1, p 646; see also p. 279.

22 A HISTORY OF MATHEMATICAL NOTATIONS

sloping line indicative of 100 and the lozenge-shaped figure used for 1,000 were replaced by the forms O for 100, and <> for 1,000.

OOo o = = = I I I stood for 2>496 •

33. The oldest strictly Greek numeral symbols were the so-called Hcrodianic signs, named after Herodianus, a Byzantine grammarian of about 200 A.D., who describes them. These signs occur frequently in Athenian inscriptions and are, on that account, now generally called Attic. They were the initial letters of numeral adjectives.1 They were used as early as the time of Solon, about GOO B.C., and con- tinued in use for several centuries, traces of them being found as late as the time of Cicero. From about 470 to 350 B.C. this system existed in competition with a newer one to be described presently. The Herodianic signs were

1 Iota for 1 II Eta for 100

II or TI or F Pi for 5 X Chi for 1,000

A Delta for 10 M My for 10,000

34. Combinations of the symbols for 5 with the symbols for 10,100, 1,000 yielded symbols for 50, 500, 5,000. These signs appear on an abacus found in 1847, represented upon a Greek marble monument on the island of Salamis.2 This computing table is represented in Fig- ure 12.

The four right-hand signs I C T X, appearing on the horizontal line below, stand for the fractions -J, ^, -£±, 4*8, respectively. Proceed- ing next from right to left, we have the symbols for 1, 5, 10, 50, 100, 500, 1,000, 5,000, and finally the sign T for 6,000. The group of sym- bols drawn on the left margin, and that drawn above, do not contain the two symbols for 5,000 and 6,000. The pebbles in the columns represent the number 9,823. The four columns represented by the five vertical lines on the right were used for the representation of the fractional values J, -^-5-, ^J4, 4J, respectively.

35. Figure 13 shows the old Herodianic numerals in an Athenian state record of the fifth century B.C. The last two lines are: Ke0<xA(uoj>

1 See, for instance. G. Friedlein, Die Zahlzeichen und das elementarc Rechnen dcr Griechen und Romer (Krlangen, 1869), p. 8; M. Cantor, Vorlesungen uber Geschichte der Mathemaiik, Vol. I (3d ed.), p. 120; II. Ilankel, Zur Geschichte der Mathemalik im Alter thum und Mittelalter (Leipzig, 1874), p. 37.

2 Kubitschek, "Die Salaminische Rechentafel," Numismatische Zeitschrift (Vienna, 1900), Vol. XXXI, p. 393-98; A. Nagl, ibid., Vol. XXXV (1903), p. 131- 43; M. Cantor, Kulturleben der Volker (Halle, 1863), p. 132, 136; M. Cantor, Vor- lesungen uber Geschichte der Mathematik, Vol. I (3d ed.), p. 133.

OLD NUMERAL SYMBOLS

23

ai/a[Xcoarosr] oD eiri r[r?s] apxw HHHPTTT....; i.e., "Total of expenditures during our office three hundred and fifty-three

talents "

36. The exact reason for the displacement of the Herodianic sym- bols by others is not known. It has been suggested that the com- mercial intercourse of Greeks with the Phoenicians, Syrians, and Hebrews brought about the change. The Phoenicians made one im- portant contribution to civilization by their invention of the alpha- bet. The Babylonians and Egyptians had used their symbols to represent whole syllables or words. The Phoenicians borrowed hieratic

X

o

JL <

a.

E. x

TPXPHPAPHCTX

FIG. 12. — The computing table of Salamis

signs from Egypt and assigned them a more primitive function as letters. But the Phoenicians did not use their alphabet for numerical purposes. As previously seen, they represented numbers by vertical and horizontal bars. The earliest use of an entire alphabet for desig- nating numbers has been attributed to the Hebrews. As previously noted, the Syrians had an alphabet representing numbers. The Greeks are supposed by some to have copied the idea from the He- brews. But Moritz Cantor1 argues that the Greek use is the older and that the invention of alphabetic numerals must be ascribed to the Greeks. They used the twenty-four letters of their alphabet, together with three strange and antique letters, ST (old van), 9 (koppa), *) (sampi), and the symbol M. This change was decidedly for the worse, for the old Attic numerals were less burdensome on the memory inas-

1 V&rlesungen uber Geschichte der Mathematik, Vol. I (3d ed., 1907), p. 25.

24

A HISTORY OF MATHEMATICAL NOTATIONS

FIG. 13. — Account of disbursements of the Athenian state, 418-415 B.C., British Museum, Greek Inscription No. 23. (Taken from R. Brown, A History of Accounting and Accountants [Edinburgh, 1905], p. 26.)

OLD NUMERAL SYMBOLS 25

much as they contained fewer symbols. The following are the Greek alphabetic numerals and their respective values:

aftyde^frjBi K X/i v £ o TT 9 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90

P <r r v <p x t « ^ ,a ,/3 ,7, 100 200 300 400 500 600 700 800 900 1,000 2,000 3,000

etc.

P v

M M M, etc.

10,000 20,000 30,000

37. A horizontal line drawn over a number served to distinguish it more readily from words. The coefficient for M was sometimes placed before or behind instead of over the M. Thus 43,678 was written SM^x^- The horizontal line over the Greek numerals can hardly be considered an essential part of the notation ; it does not seem to have been used except in manuscripts of the Byzantine period.1 For 10,000 or myriad one finds frequently the symbol M or Mu, sometimes simply the dot • , as in /3-o5 for 20,074. Often2 the coefficient of the myriad is found written above the symbol /iu.

38. The paradox recurs, Why did the Greeks change from the Herodianic to the alphabet number system? Such a change would not be made if the new did not seem to offer some advantages over the old. And, indeed, in the new system numbers could be written in a more compact form. The Herodianic representation of 1,739 was X HlHIIAAAII MM; the alphabetic was ,a^X0. A scribe might consider the latter a great innovation. The computer derived little aid from either. Some advantage lay, however, on the side of the Herodianic, as Cantor pointed out. Consider HHIIH+HH= SI H,AAAA+AA = S|A; there is an analogy here in the addition of hundred's and of ten's. But no such analogy presents itself in the alphabetic numerals, where the corresponding steps are v+a = x and /Z+K = £; adding the hun- dred's expressed in the newer notation affords no clew as to the sum of the corresponding ten's. But there was another still more impor- tant consideration which placed the Herodianic far above the alpha- betical numerals. The former had only six symbols, yet they afforded an easy representation of numbers below 100,000; the latter demanded twenty-seven symbols for numbers below 1,000! The mental effort

1 Encyc. des stien. math., Tome I, Vol. I (1904), p. 12. 2 Ibid.

26 A HISTORY OF MATHEMATICAL NOTATIONS

of remembering such an array of signs was comparatively great. We are reminded of the centipede having so many legs that it could hardly advance.

39. We have here an instructive illustration of the fact that a mathematical topic may have an amount of symbolism that is a hin- drance rather than a help, that becomes burdensome, that obstructs progress. We have here an early exhibition of the truth that the move- ments of science are not always in a forward direction. Had the Greeks not possessed an abacus and a finger symbolism, by the aid of which computations could be carried out independently of the numeral notation in vogue, their accomplishment in arithmetic and algebra might have been less than it actually was.

40. Notwithstanding the defects of the Greek system of numeral notation, its use is occasionally encountered long after far better systems were generally known. A Calabrian monk by the name of Barlaam,1 of the early part of the fourteenth century, wrote several mathematical books in Greek, including arithmetical proofs of the second book of Euclid's Elements, and six books of Logistic, printed in 1564 at Strassburg arid in several later editions. In the Logistic he de- velops the computation with integers, ordinary fractions, and sexa- gesimal fractions; numbers are expressed by Greek letters. The appearance of an arithmetical book using the Greek numerals at as late a period as the close of the sixteenth century in the cities of Strass- burg and Paris is indeed surprising.

41. Greek writers often express fractional values in words. Thus Archimedes says that the length of a circle amounts to three diameters and a part of one, the size of which lies between one-seventh and ten- seventy-firsts.2 Eratosthenes expresses J J of a unit arc of the earth's meridian by stating that the distance in question "amounts to eleven parts of which the meridian has eighty-three. "3 When expressed in symbols, fractions were often denoted by first writing the numerator marked with an accent, then the denominator marked with two ac- cents and written twice. Thus,4 if KCL" KCL" = |f. Archimedes, Euto- cius, and Diophantus place the denominator in the position of the

1 All our information on Barlaam is drawn from M. Cantor, Vorlesungen liber Geschichte der Matkematik, Vol. I (3d ed.), p. 509, 510; A. G. Kastner, Geschichte der Mathematik (Gottingen, 1796), Vol. I, p. 45; J. C. Hcilbronner, Historia matheseos universae (Lipsiae, 1742), p. 488, 489.

2 Archimedis opera omnia (ed. Heiberg; Leipzig, 1880), Vol. I, p. 262.

3 Ptolemaus, MeyaXij avvrafa (ed. Heiberg), Pars I, Lib. 1, Cap. 12, p. 68.

4 Heron, Stereometrica (ed. Hultsch; Berlin, 1864), Pars I, Par. 8, p. 155.

OLD NUMERAL SYMBOLS 27

modern exponent; thus1 Archimedes and Eutocius use the notation

__ KO! Ka

if or if for ^], and Diophantus (§§ 101-6), in expressing large num- bers, writes (Ariihmetica, Vol. IV, p. 17), — — ^ for -w-L.-,- .

7-/^X/ca 2,704

Here the sign ~ takes the place of the accent. Greek writers, even as late as the Middle Ages, display a preference for unit fractions, which played a dominating role in old Egyptian arithmetic.2 In expressing such fractions, the Greeks omitted the a for the numerator and wrote the denominator only once. Thus ju6//=4V- Unit fractions in juxta- position were added,3 as in f" /cr?" pt/3" o-/c6// = ^+^V+iH+ T*4- ^ne finds also a single accent,4 as in 5' = \. Frequent use of unit fractions is found in Gcminus (first century B.C.), Diophantus (third century A.D.), Eutocius and Proclus (fifth century A.D.). The fraction \ had a mark of its own,5 namely, L or £, but this designation was no more adopted generally among the Greeks than were the other notations of fractions. Ptolemy6 wrote 38°50' (i.e., 380,i |) thus, XT;' £'7'". Hultsch has found in manuscripts other symbols for |, namely, the semicircles £VI, (, and the sign ,S ; the origin of the latter is uncertain. He found also a symbol for §, resembling somewhat the small omega (co).7 Whether these symbols represent late practice, but not early usage, it is difficult to determine with certainty.

42. A table for reducing certain ordinary fractions to the sum of unit fractions is found in a Greek papyrus from Egypt, described by

1 G. II. F. Nessclmarm, Algebra der Gricchcn (Berlin, 1842), p. 114.

2 ,1. Baillct describes a papyrus, "Le papyrus mathematique d'Akhmfm," in Memoires publics par Ics immbrcs de la Mission archeologique fran^aise au Caire (Paris, 1892), Vol. IX, p. 1-89 (8 plates). This papyrus, found at Akhmtrn, in Egypt, is written in Greek, and is supposed to belong to the period between 500 and 800 A.D. It contains a table for the conversion of ordinary fractions into unit frac- tions.

3Fr. Hultsch, Metrologicorum scriplorum reliquiae (1864-66), p. 173-75; M. Cantor, Vorlesungen iiber Geschichte der Mathematik, Vol. I (3d ed.), p. 129.

4 Nesselmann, op. cil., p. 112.

5 Ibid.; James Gow, Short History of Greek Mathematics (Cambridge, 1884), p. 48, 50.

*Geographia (ed. Carolus Mullerus; Paris, 1883), Vol. I, Part I, p. 151.

7 Metrologicorum scriptorum reliquiae (Leipzig, 1804), Vol. I, p. 173, 174. On p. 175 and 176 Hultsch collects the numeral symbols found in three Parisian manu- scripts, written in Greek, which exhibit minute variations in the symbolism. For instance, 700 is found to be ^ ^, \j/f.

28 A HISTORY OF MATHEMATICAL NOTATIONS

L. C. Karpinski,1 and supposed to be intermediate between the Ahmes papyrus and the Akhmim papyrus. Karpinski (p. 22) says: "In the table no distinction is made between integers and the corre- sponding unit fractions; thus 7' may represent either 3 or £, and actually y'y' in the table represents 3^. Commonly the letters used as numerals were distinguished in early Greek manuscripts by a bar placed above the letters but not in this manuscript nor in the Akhmim papyrus.7' In a third document dealing with unit fractions, a Byzan- tine table of fractions, described by Herbert Thompson,2 f is written 1; i, a; J, f (from \ '); |, A/" (from A'); *, e (from e'); i, vf (from H'). As late as the fourteenth century, Nicolas Rhabdas of Smyrna wrote two letters in the Greek language, on arithmetic, containing tables for unit fractions.3 Here letters of the Greek alphabet used as integral numbers have bars placed above them.

43. About the second century before Christ the Babylonian sexa- gesimal numbers were in use in Greek astronomy; the letter omicron, which closely resembles in form our modern zero, was used to desig- nate a vacant space in the writing of numbers. The Byzantines wrote it usually b, the bar indicating a numeral significance as it has when placed over the ordinary Greek letters used as numerals.4

44. The division of the circle into 360 equal parts is found in Hypsicles.5 Hipparchus employed sexagesimal fractions regularly, as did also C. Ptolemy6 who, in his Almagest, took the approximate

8 30 value of TT to be 3+^+™^-^ • ^n ^ne Heiberg edition this value is

written 7 rj X, purely a notation of position. In the tables, as printed by Heiberg, the dash over the letters expressing numbers is omitted. In the edition of N. Halma7 is given the notation 7 ?/ X", which is

1 "The Michigan Mathematical Papyrus No. 621," Isis, Vol. V (1922), p. 20-25.

2 "A Byzantine Table of Fractions," Ancient Egypt, Vol. I (1914), p. 52-54.

3 The letters were edited by Paul Tannery in Notices et extraits des manuscrits de la Bibliotheque Nationale, Vol. XXXII, Part 1 (1886), p. 121-252.

4 C. Ptolemy, Almagest (ed. N. Halma; Paris, 1813), Book I, chap, ix, p. 38 and later; J. L. Heiberg, in his edition of the Almagest (Syntaxis mathematical (Leipzig, 1898; 2d ed., Leipzig, 1903), Book I, does not write the bar over the o but places it over all the significant Greek numerals. This procedure has the ad- vantage of distinguishing between the o which stands for 70 and the o which stands for zero. See Encyc. des scien. math., Tome I, Vol. I (1904), p. 17, n. 89.

5 Ava<£optKos (ed. K. Manitius), p. xxvi.

6 Syntaxis mathematica (ed. Heiberg), Vol. I, Part 1, p. 513.

7 Composition math, de PtoUmee (Paris, 1813), Vol. I, p. 421; see also Encyc. des scien. math., Tome I, Vol. I (1904), p. 53, n. 181.

OLD NUMERAL SYMBOLS

29

probably the older form. Sexagesimal fractions were used during the whole of the Middle Ages in India, and in Arabic and Christian coun- tries. One encounters them again in the sixteenth and seventeenth centuries. Not only sexagesimal fractions, but also the sexagesimal notation of integers, are explained by John Wallis in his Maihesis universalis (Oxford, 1657), page 68, and by V. Wing in his Astronomia Briiannica (London, 1652, 1669), Book I.

EARLY ARABS

45. At the time of Mohammed the Arabs had a script which did not differ materially from that of later centuries. The letters of the early Arabic alphabet came to be used as numerals among the Arabs

1 t

5 6

20 «: so <!

40 r

50 c

60

70

80

90

^

100 d .200 ^

300^ 400 c» 500 & 600 £ 700 o 800 o» 900 Jb

1000 2000 3000 4000 5000 6000 7000 8000 9000

10000 j, 20000 £J 30000 jj 40000 £o 50000 jj 60000 ^ 70000 .«*• 80000 « 90000 «

100000 g 200000 £, 300000 jxi 400000 43 500000 J3 600000 j> 700000 ^ 800000 -£* i 900000 «

FIG. 14. — Arabic alphabetic numerals used before the introduction of the Hindu-Arabic numerals.

as early as the sixth century of our era.1 After the time of Mohammed, the conquering Moslem armies coming in contact with Greek culture acquired the Greek numerals. Administrators and military leaders used them. A tax record of the eighth century contains numbers expressed by Arabic letters and also by Greek letters.2 Figure 14 is a table given by Ruska, exhibiting the Arabic letters and the numerical values which they represent. Taking the symbol for 1,000 twice, on the multiplicative principle, yielded 1,000,000. The Hindu-Arabic

1 Julius Ruska, "Zur altesten arabischen Algebra und Rechenkunst," Sitzungs- berichte d. Heidelberger Akademie der Wissensch. (Philos.-histor. Klasse, 1917; 2. Abhandlung), p. 37.

2 Ibid., p. 40.

30 A HISTORY OF MATHEMATICAL NOTATIONS

numerals, with the zero, began to spread among the Arabs in the nin and tenth centuries, and they slowly displaced the Arabic and Gre< numerals.1

ROMANS

46. We possess little definite information on the origin of tl Roman notation of numbers. The Romans never used the successr letters of their alphabet for numeral purposes in the manner practic< by the Syrians, Hebrews, and Greeks, although (as we shall see) * alphabet system was at one time proposed by a late Roman write Before the ascendancy of Rome the Etruscans, who inhabited ti country nearly corresponding to modern Tuscany and who ruled Rome until about 500 B.C., used numeral signs which resembled lette of their alphabet and also resembled the numeral signs used by tl Romans. Moritz Cantor2 gives the Etrurian and the old Roman sign as follows: For 5, the Etrurian /\ or V, the old Roman V; for 10 tl Etrurian X or +, the old Roman X; for 50 the Etrurian t or I, tl old Roman "f or I or X or 1 or L; for 100 the Etrurian 0, the o Roman © ; for 1,000 the Etrurian #, the old Roman 0. The reser blance of the Etrurian numerals to Etrurian letters of the alphabet seen from the following letters: V, +, I, O, 8. These resemblanc cannot be pronounced accidental. "Accidental, on the other hand says Cantor, "appears the relationship with the later Roman signs, V, X, L, C, M, which from their resemblance to letters transformc themselves by popular etymology into these very letters/7 The origii of the Roman symbols for 100 and 1,000 are uncertain; those for ' and 500 are generally admitted to be the result of a bisection of tl two former. "There was close at hand/' says G. Friedlein,3 "the a breviation of the word centum and mille which at an early age brougl about for 100 the sign C, and for 1,000 the sign M and after Augustu M." A view held by some Latinists6 is that "the signs for 50, 10 1,000 were originally the three Greek aspirate letters which the R« mans did not require, viz., M>, O, 0, i.e., x> 0> *• The *& was writte J_ and abbreviated into L; O from a false notion of its origin made HI

1 Ibid., p. 47.

2 Vorlesungen uber Geschichte der Mathematik, Vol. I (3d ed.), p. 523, and t' table at the end of the volume.

3 Die Zahlzeichen und das elementare Rechnen der Griechen und Homer (E langen, 1869), p. 28.

4 Theodor Mommsen, Die unleritalischen Dialekte (Leipzig, 1840), p. 30. 'Ritschl, Rhein. Mus., Vol. XXIV (1869), p. 12.

OLD NUMERAL SYMBOLS 31

the initial of centum; and 0 assimilated to ordinary letters CIO. The half of 0, viz., D, was taken to be -J- 1,000, i.e., 500; X probably from the ancient form of 0, viz., ®, being adopted for 10, the half of it V was taken for 5."1

47. Our lack of positive information on the origin and early his- tory of the Roman numerals is not due to a failure to advance working hypotheses. In fact, the imagination of historians has been unusually active in this field.2 The dominating feature in the Roman notation is the principle of addition, as seen in II, XII, CC, MDC, etc.

48. Conspicuous also is the frequent use of the principle of sub- traction. If a letter is placed before another of greater value, its value is to be subtracted from that of the greater. One sees this in IV, IX, XL. Occasionally one encounters this principle in the Baby- lonian notations. Remarks on the use of it are made by Adriano Cappelli in the following passage :

"The well-known rule that a smaller number, placed to the left of a larger, shall be subtracted from the latter, as 0|00 = 4,000, etc., was seldom applied by the old Romans and during the entire Middle Ages one finds only a few instances of it. The cases that I have found belong to the middle of the fifteenth century and are all cases of IX, never of IV, and occurring more especially in French and Piedmontese documents. Walther, in his Lexicon diptomaticum, Gottingen, 1745- 47, finds the notation LXL = 90 in use in the eighth century. On the other hand one finds, conversely, the numbers IIIX, VIX with the meaning of 13 and 16, in order to conserve, as Lupi remarks, the Latin terms tertio dedmo and sexto decimo."* L. C. Karpinski points out that the subtractive principle is found on some early tombstones and on a signboard of 130 B.C., where at the crowded end of a line 83 is written XXCIII, instead of LXXXIII.

1 II. J. Roby, A Grammar of the Latin Language from Plaulus to Suetonius (4th ed.; London, 1881), Vol. I, p. 441.

2 Consult, for example, Friedlcin, op. cit., p. 26-31; Ncsselmann, op. tit., p. 86-92; Cantor, Mathematische Beitrdge zum Kulturleben der Volker, p. 155-67; J. C. Heilbronner, Historia Matheseos universae (Lipsiae, 1742), p. 732-35; Grotc- fend, Lateinische Grammatik (3d ed.; Frankfurt, 1820), Vol. II, p. 163, is quoted in the article "Zahlzeichen" in G. S. Kliigel's Malhematisches Worterbuch, continued by C. B. Mollweide and J. A. Grunert (Leipzig, 1831); Mommsen, Hermes, Vol. XXII (1887), p. 596; Vol. XXIII (1888), p. 152. A recent discussion of the history of the Roman numerals is found in an article by Ettore Bortolotti in Bolletino delta Mathesis (Pavia,. 1918), p. 60-66, which is rich in bibliographical references, as is also an article by David Eugene Smith in Scientia (July- August, 1926).

3 Lexicon Abbreviaturarum (Leipzig, 1901), p. xlix.

32 A HISTORY OF MATHEMATICAL NOTATIONS

49. Alexander von Humboldt1 makes the following observations: "Summations by juxtaposition one finds everywhere among the

Etruscans, Romans, Mexicans and Egyptians; subtraction or lessen- ing forms of speech in Sanskrit among the Indians: in 19 or unavinsati; 99 unusata; among the Romans in undeviginti for 19 (unus de viginti), undeoctoginta for 79; duo de quadraginta for 38; among the Greeks tikosi deonta henos 19, and pentekonta duoin deontoin 48, i.e., 2 missing in 50. This lessening form of speech has passed over in the graphics of numbers when the group signs for 5, 10 and even their multiples, for example, 50 or 100, are placed to the left of the characters they modify (IV and IA, XL and XT for 4 and 40) among the Romans and Etrus- cans (Otfried Miiller, Etrusker, II, 317-20), although among the latter, according to Otfried Miiller's new researches, the numerals descended probably entirely from the alphabet. In rare Roman inscriptions which Marini has collected (Iscrizioni della Villa di Albano, p. 193; Hervas, Aritmetica delle nazioni [1786], p. 11, 16), one finds even 4 units placed before 10, for example, IIIIX for 6."

50. There are also sporadic occurrences in the Roman nota- tions of the principle of multiplication, according to which VM does not stand for 1,000 — 5, but for 5,000. Thus, in Pliny's His- toria naturalis (about 77 A.D.), VII, 26; XXXIII, 3; IV praef., one finds2 LXXXIII.M, XCII.M, CX.M for 83,000, 92,000, 110,000, respectively.

51. The thousand-fold value of a number was indicated in some instances by a horizontal line placed above it. Thus, Aelius Lam- pridius (fourth century A.D.) says in one place, "CXX, equitum Persa- rum fudimus: et mox X in bello interemimus," where the numbers designate 120,000 and 10,000. Strokes placed on top and also on the sides indicated hundred thousands; e.g., |X|CLXXXDC stood for 1,180,600. In more recent practice the strokes sometimes occur only on the sides, as in | X | • DC . XC . , the date on the title-page of Sigii- enza's Libra astronomicaj published in the city of Mexico in 1690. In antiquity, to prevent fraudulent alterations, XXXM was written for 30,000, and later still CIO took the place of M.3 According to

1 "liber die bei verschiedenen Volkern ublichen Systeme von Zahlzeichen, etc./' Crclle's Journal fur die reine und angewandte Mathematik (Berlin, 1829), Vol. IV, p. 210, 211.

2 Nesselmann, op. cit., p. 90.

3 Confer, on this point, Theodor Mommsen and J. Marquardt, Manuel des antiquites romaines (trans. G. Humbert), Vol. X by J. Marquardt (trans. A. Vigie"; Paris, 1888), p. 47, 49.

OLD NUMERAL SYMBOLS 33

Cappelli1 "one finds, often in French documents of the Middle Ages, the multiplication of 20 expressed by two small x's which are placed as exponents to the numerals III, VI, VIII, etc., as in IIIIXX = 80, VIXXXI = 131."

52. A Spanish writer2 quotes from a manuscript for the year 1392 the following:

M C "IIII, IIII, LXXIII florins" for 4,473 florins.

M XX "III C IIII III florins" for 3,183 (?) florins.

In a Dutch arithmetic, printed in 1771, one finds8

c c m c

t ffitj for 123, i j£ttj ittj toj for 123,456.

53. For 1,000 the Romans had not only the symbol M, but also I, oo and CIO. According to Priscian, the celebrated Latin grammarian of about 500 A.D., the oo was the ancient Greek sign X for 1,000, but modified by connecting the sides by curved lines so as to distinguish it from the Roman X for 10. As late as 1593 the oo is used by C. Dasypo- dius4 the designer of the famous clock in the cathedral at Strasbourg. The CIO was a I inclosed in parentheses (or apostrophes). When only the right-hand parenthesis is written, 10, the value represented is only half, i.e., 500. According to Priscian,5 "quinque milia per I et duas in dextera parte apostrophes, 100- decem milia per supra dictam formam additis in sinistra parte contrariis duabus notis quam sunt apostrophi, CCIOO." Accordingly, 100 stood for 5,000, CCIOO for 10,000; also 1000 represented 50,000; and CCCIOOO, 100,000; (co), 1,000,000. If we may trust Priscian, the symbols that look like the letters C, or those letters facing in the opposite direction, were not really letters C, but were apostrophes or what we have called

1 Op. cit., p. xlix.

2 Liciniano Saez, Demostracidn Histdrica del verdadero valor de *Todas Las Monedas que corrlan en Castilla durante el reynado del Senor Don Enrique III (Madrid, 1796).

3 De Vernieuwde Cyfferinge van Mf Willem B art j ens. Herstelt, .... door Mr Jan van Dam, .... en van alle voorgaande Fauten gezuyvert door .... Klaas Bosch (Amsterdam, 1771), p. 8.

4 Cunradi Dasypodii Institutionum Mathematicarum voluminis primi Erotemata (1593), p. 23. .

6 "De figuris numerorum," Henrici Keilii Grammaiid Latini (Lipsiae, 1859), Vol. Ill, 2, p. 407.

34 A HISTORY OF MATHEMATICAL NOTATIONS

parentheses. Through Priscian it is established that this notation is at least as old as 500 A.D. ; probably it was much older, but it was not widely used before the Middle Ages.

54. While the Hindu- Arabic numerals became generally known in Europe about 1275, the Roman numerals continued to hold a com- manding place. For example, the fourteenth-century banking-house of Peruzzi in Florence — Compagnia Peruzzi — did not use Arabic numerals in their account-books. Roman numerals were used, but the larger amounts, the thousands of lira, were written out in words; one finds, for instance, "Ib. quindicimilia CXV / V ^ VI in fiorini" for 15,115 lira 5 soldi 6 denari; the specification being made that the lira are lira a fiorino cVoro at 20 soldi and 12 denari. There appears also a symbol much like ? , for thousand.1

Nagl states also: "Specially characteristic is .... during all the Middle Ages, the regular prolongation of the last I in the units, as VI |= VI I, which had no other purpose than to prevent the subsequent addition of a further unit/'

55. In a book by H. Giraua Tarragones2 at Milan the Roman numerals appear in the running text and are usually underlined; in the title-page, the date has the horizontal line above the numerals. The Roman four is 1 1 1 1 . In the tables, columns of degrees and minutes are headed "G.M."; of hour and minutes, "H.M." In the tables, the Hindu-Arabic numerals appear ; the five is printed 3 , without the usual upper stroke. The vitality of the Roman notation is illustrated further by a German writer, Sebastian Frank, of the sixteenth cen- tury, who uses Roman numerals in numbering the folios of his book and in his statistics: "Zimmet kuinpt von Zailon .CC.VN LX. teiitscher meil von Calicut weyter gelegen ..... Die Nagelin kummen von Meluza / fur Calicut hinaussgelegen vij-c. vnd XL. deutscher meyl."3 The two numbers given are 260 and 740 German miles. Pe- culiar is the insertion of vnd ("and")- Observe also the use of the principle of multiplication in vij«c. ( = 700). In Jakob Kobel's Rechenbiechlin (Augsburg, 1514), fractions appear in Roman numerals;

11° thus, c~ Stands f°r *^*

1 Alfred Nagl, Zeitschrift fur Mathematik und Physik, Vol. XXXIV (1889), Historisch-literarische Abthcilung, p. 164.

2 Dos Libros de Cosmographie, complicates nueuamcnte por Hieronymo Giraua Tarragones (Milan, M.D.LVI).

8 Weltbuch I siriegel vnd bildtnis des gantzen Erdtbodens .... von Sebastiano Franco W&rdensi ____ (M.D. XXXIIII), fol. ccxx.

OLD NUMERAL SYMBOLS

35

56. In certain sixteenth-century Portuguese manuscripts on navi- gation one finds the small letter b used for 5, and the capital letter R for 40. Thus, z&iij stands for 18, Rii] for 43.1

to

FIG. 15. — Degenerate forms of Roman numerals in English archives (Common Pleas, Plea Rolls, 637, 701, and 817; also Recovery Roll 1). (Reduced.)

A curious development found in the archives of one or two English courts of the fifteenth and sixteenth centuries2 was a special Roman

1 J. I. de Brito Rcbcllo, Livro de Marinharia (Lisboa, 1903), p. 37, 85-91, 193,

194.

2 Antiquaries Journal (London, 1926), Vol. VI, p. 273, 274.

36

A HISTORY OF MATHEMATICAL NOTATIONS

numeration for the membranes of their Rolls, the numerals assuming a degraded form which in its later stages is practically unreadable. In Figure 15 the first three forms show the number 147 as it was written in the years 1421, 1436, and 1466; the fourth form shows the number 47 as it was written in 1583.

57. At the present time the Roman notation is still widely used in marking the faces of watches and clocks, in marking the dates of books on title-pages, in numbering chapters of books, and on other occasions calling for a double numeration in which confusion might arise from the use of the same set of numerals for both. Often the Roman numerals are employed for aesthetic reasons.

58. A striking feature in Roman arithmetic is the partiality for duodecimal fractions. Why duodecimals and not decimals? We can only guess at the answer. In everyday affairs the division of units into two, three, four, and six equal parts is the commonest, and duodecimal fractions give easier expressions for these parts. Nothing definite is known regarding the time and place or the manner of the origin of these fractions. Unlike the Greeks, the Romans dealt with concrete fractions. The Roman as, originally a copper coin weighing one pound, was divided into 12 unciae. The abstract fraction \-\- was called deuna ( = de unaa, i.e., as [1] less uncia [r2]). Each duodecimal subdivision had its own name and symbol. This is shown in the follow- ing table, taken from Friedlein,1 in which S stands for semis or "half" of an as.

TABLE

as

1

n

f t

I7*

5 T2

i 1 1

SK.

deunx

S r = - or S : : • S = = or S : :

S = - or S r 1 or & : • £- or _ £_ or £: $ — or 6Y< S r — — or — — — or : : • — — or X S or : : — — or — 1 or :• — or z or :

-LL-Ii't

- or • or on bronze abacus ( :>ccur also curved ones /^/.

(de uncia 1— -fy) f(de sextans 1~J) \ (decem unciae) (de quadrans 1— J) (duae assis sc. partes) (scptem unciae)

dextans 1

(decunx)J dodrans

bes

septunx

semis

quincunx

(quinque unciae)

triens . ...

quadrans . .

sextans . . .

sescuncia 1J uncia

^

In place of straight lines - <

1 Op. cit., Plate 2, No. 13; see also p. 35.

OLD NUMERAL SYMBOLS 37

59. Not all of these names and signs were used to the same ex- tent. Since i+i=f, there was used in ordinary life | and £ (semis et triens) in place of $ or \\ (decunx). Nor did the Romans confine them- selves to the duodecimal fractions or their simplified equivalents 1; i> l> 1> etc., but used, for instance, TV in measuring silver, a libella being TV denarius. The uncia was divided in 4 siciliciy and in 24 scripuli etc.1 In the Geometry of Boethius the Roman symbols are omitted and letters of the alphabet are used to represent fractions. Very probably this part of the book is not due to Boethius, but is an inter- polation by a writer of later date.

60. There are indeed indications that the Romans on rare occa- sions used letters for the expression of integral numbers.2 Theodor Mommsen and others discovered in manuscripts found in Bern, Einsiedeln, and Vienna instances of numbers denoted by letters. Tartaglia gives in his General trattato di nvmeri, Part I (1556), folios 4, 5, the following:

A 500 II R 80

B 300 K 51 S 70

C 100 L 50 T 160

D 500 M 1,000 V 5

E 250 N 90 X 10

F 40 0 11 Y 150

G 400 P 400 Z 2,000

H 200 Q 500

61. Gerbert (Pope Sylvestre II) and his pupils explained the Ro- man fractions. As reproduced by Olleris,3 Gerbert's symbol for \ does not resemble the capital letter $, but rather the small letter <J .

1 For additional details and some other symbols used by the Romans, consult Friedlein, p. 33-46 and Plate 3; also H. Hankel, op. tit., p. 57-61, where com- putations with fractions are explained. Consult also Fr. Hultsch, Metrologic. scriplorcs Romani (Leipzig, 1866).

2 Friedlein, op. tit., p. 20, 21, who gives references. In the Standard Dic- tionary of the English Language (New York, 1896), under S, it is stated that 3 stood for 7 or 70.

1 (Euvres de Gerbert (Paris, 1867), p. 343-48, 393-96, 583, 584.

38 A HISTORY OF MATHEMATICAL NOTATIONS

PERUVIAN AND NORTH AMERICAN KNOT RECORDS1 ANCIENT QUIPU

62. 'The use of knots in cords for the purpose of reckoning, and recording numbers" was practiced by the Chinese and some other ancient people; it had a most remarkable development among the Inca of Peru, in South America, who inhabited a territory as large as the United States east of the Rocky Mountains, and were a people of superior mentality. The period of Inca supremacy extended from about the eleventh century A.D. to the time of the Spanish conquest in the sixteenth century. The quipu was a twisted woolen cord, upon which other smaller cords of different colors were tied. The color, length, and number of knots on them and the distance of one from another all had their significance. Specimens of these ancient quipu have been dug from graves.

63. We reproduce from a work by L. Leland Locke a photograph of one of the most highly developed quipu, along with a line diagram of the two right-hand groups of strands. In each group the top strand usually gives the sum of the numbers on the four pendent strands. Thus in the last group, the four hanging strands indicate the numbers 89, 258, 273, 38, respectively. Their sum is 658; it is recorded by the top string. The repetition of units is usually expressed by a long knot formed by tying the overhand knot and passing the cord through the loop of the knot as many times as there are units to be denoted. The numbers were expressed on the decimal plan, but the quipu were not adopted for calculation; pebbles and grains of maize were used in com- puting.

64. Nordenskiold shows that, in Peru, 7 was a magic number; for in some quipu, the sums of numbers on cords of the same color, or the numbers emerging from certain other combinations, are multiples of 7 or yield groups of figures, such as 2777, 777, etc. The quipu dis- close also astronomical knowledge of the Peruvian Indians.2

65. Dr. Leslie Spier, of the University of Washington, sends me the following facts relating to Indians in North America: "The data that I have on the quipu-\ike string records of North-American Indians indicate that there are two types. One is a long cord with knots and

1 The data on Peru knot records given here are drawn from a most interesting work, The Ancient Quipu or Peruvian Knot Record, by L. Leland Locke (American Museum of Natural History, 1923). Our photographs are from the frontispiece and from the diagram facing p. 16. See Figs. 16 and 17.

2Erland Nordenskiold, Comparative Ethnographical Studies, No. 6, Part 1 (1925), p. 36.

OLD NUMERAL SYMBOLS

39

bearing beads, etc., to indicate the days. It is simply a string record. This is known from the Yakima of eastern Washington and some In- terior Salish group of Nicola Valley,1 B.C.

FIG. 16. — A quipu, from ancient Chancay in Peru, now kept in the American Museum of Natural History (Museum No. B8713) in New York City.

1 J. D. Leechman and M. R. Harrington, String Records of the Northwest, Indian Notes and Monographs (1921).

40

A HISTORY OF MATHEMATICAL NOTATIONS

"The other type I have seen in use among the Havasupai and Walapai of Arizona. This is a cord bearing a number of knots to indi- cate the days until a ceremony, etc. This is sent with the messenger who carries the invitation. A knot is cut off or untied for each day that elapses; the last one indicating the night of the dance. This is also

used by the Northern and South- ern Maidu and the Miwok of Cali- fornia.1 There is a mythical ref- erence to these among the Zufii of New Mexico.2 There is a note on its appearance in San Juan Pueblo in the same state in the seventeenth century, which would indicate that its use Was widely known among the Pueblo Indians. 'They directed him (the leader of the Pueblo rebellion of 1680) to make a rope of the palm leaf and tie in it a number of knots to rep- resent the number of days be- fore the rebellion was to take place; that he must send the rope to all the Pueblos in the Kingdom, when each should sig- nify its approval of, and union with, the conspiracy by untying one of the knots/3 The Huichol of Central Mexico also have knot- ted strings to keep count of days, untieing them as the days elapse. They also keep records of their lovers in the same way.4 The Zufii also keep records of days worked in this fashion.6

1 R. B. Dixon, "The Northern Maidu," Bulletin of the American Museum of Natural History, Vol. XVII (1905), p. 228, 271 ; P.-L. Faye, "Notes on the Southern Maidu," University of California Publications of American Archaeology and Ethnology, Vol. XX (1923), p. 44; Stephen Powers, "Tribes of California/' Contri- butions to North American Ethnology, Vol. Ill (1877), p. 352.

2 F. H. Gushing, "Zufti Breadstuff," Indian Notes and Monographs, Vol. VIII (1920), p. 77.

3 Quoted in J. G. Bourke, "Medicine-Men of the Apache," Ninth Annual Report, Bureau of American Ethnology (1892), p. 555.

4 K. Lumholtz, Unknown Mexico, Vol. II, p. 218-30. 6 Leechman and Harrington, op. cit.

<**

FIG. 17. — Diagram of the two right- hand groups of strands in Fig. 16.

OLD NUMERAL SYMBOLS 41

"Bourke1 refers to medicine cords with olivella shells attached among the Tonto and Chiricahua Apache of Arizona and the Zufii. This may be a related form.

"I think that there can be no question the instances of the second type are historically related. Whether the Yakima and Nicola Valley usage is connected with these is not established. "

AZTECS

66. "For figures, one of the numerical signs was the dot (•), which marked the units, and which was repeated either up to 20 or up to the figure 10, represented by a lozenge. The number 2Q was represented by a flag, which, repeated five times, gave the number 100, which was

.:: O P

Xi^Jto 10 &* 6 fVO 100 3t>0

$jjk *§s£ || IP IP u I . "I " 1 1 1 I

FIG. 18. — Aztec numerals

marked by drawing quarter of the barbs of a feather. Half the barbs was equivalent to 200, three-fourths to 300, the entire feather to 400. Four hundred multiplied by the figure 20 gave 8,000, which had a purse for its symbol."2 The symbols were as shown in the first line of Figure 18.

The symbols for 20, 400, and 8,000 disclose the number 20 as the base of Aztec numeration; in the juxtaposition of symbols the additive principle is employed. This is seen in the second line3 of Figure 18, which represents

2X8,000+400+3X20+3X5+3 = 16,478 .

67. The number systems of the Indian tribes of North America, while disclosing no use of a symbol for zero nor of the principle of

1 Op. cit.y p. 550 ff.

2 Lucien Biart, The Aztecs (trans. 3. L. Garner; Chicago, 1905), p. 319.

8 Consult A. F. Pott, Die quindre und vigesimale Zdhlmethode bei Volkern aller Welttheile (Halle, 1847).

42

A HISTORY OF MATHEMATICAL NOTATIONS

»***.*»*» +***

" '

FIG. 19. — From the Dresden Codex, of the Maya, displaying numbers. The second column on the left, from above down, displays the numbers 9, 9, 16, 0, 0, which stand for 9X 144,000+9X7,200+16 X360-fO+0 = 1,366,560. In the third column are the numerals 9, 9, 9, 16, 0, representing 1,364,360. The original appears in black and red colors. (Taken from Morley, An Introduction to the Study of the Maya Hieroglyphs, p. 266.)

OLD NUMERAL SYMBOLS 43

local value, are of interest as exhibiting not only quinary, decimal, and vigesimal systems, but also ternary, quaternary, and octonary sys- tems.1

MAYA

68. The Maya of Central America and Southern Mexico developed hieroglyphic writing, as found on inscriptions and codices, dating apparently from about the beginning of the Christian Era, which dis- closes the use of a remarkable number system and chronology.2 The number system discloses the application of the principle of local value, and the use of a symbol for zero centuries before the Hindus began to use their symbol for zero. The Maya system was vigesimal, except in one step. That is, 20 units (kins, or "days") make 1 unit of the next higher order (uinals, or 20 days), 18 uinals make 1 unit of the third order (tun, or 360 days), 20 tuns make 1 unit of the fourth order (Katun, or 7,200 days), 20 Katuns make 1 unit of the fifth order (cycle, or 144,000 days), and finally 20 cycles make 1 great cycle of 2,880,000 days. In the Maya codices we find symbols for 1-19, expressed by bars and dots. Each bar stands for 5 units, each dot for 1 unit. For instance,

•• — ^- = .

1245 7 11 19

The zero is represented by a symbol that looks roughly like a half- closed eye. In writing 20 the principle of local value enters. It is expressed by a dot placed over the symbol for zero. The numbers are written vertically, the lowest order being assigned the lowest position (see Fig. 19). The largest number found in the codices is 12,489,781.

CHINA AND JAPAN

69. According to tradition, the oldest Chinese representation of number was by the aid of knots in strings, such as are found later among the early inhabitants of Peru. There are extant two Chinese tablets3 exhibiting knots representing numbers, odd numbers being designated by white knots (standing for the complete, as day, warmth,

1 W. C. Eells, "Number-Systems of North-American Indians,7' American Mathematical Monthly, Vol. XX (1913), p. 263-72, 293-99; also Bibliotheca mathe- matica (3d series, 1913), Vol. XIII, p. 218-22.

2 Our information is drawn from S. G. Morley, An Introduction to the Study of the Maya Hieroglyphs (Washington, 1915).

3 Paul Perrty, Grammaire de la langue chinoise orale et ecrite (Paris, 1876), Vol. II, p. 5-7; Cantor, Vorlesungen iiber Geschichte der Mathematik, Vol. I (3d ed.), p. 674.

44

A HISTORY OF MATHEMATICAL NOTATIONS

the sun) while even numbers are designated by black knots (standing for the incomplete, as night, cold, water, earth). The left-hand tablet shown in Figure 20 represents the numbers 1-10. The right-hand tablet pictures the magic square of nine cells in which the sum of each row, column, and diagonal is 15.

70. The Chinese are known to have used three other systems of writing numbers, the Old Chinese numerals, the mercantile numerals, and what have been designated as scientific numerals. The time of the introduction of each of these systems is uncertain.

o 6 o

o

FIG. 20. — Early Chinese knots in strings, representing numerals

71. The Old Chinese numerals were written vertically, from above down. Figure 21 shows the Old Chinese numerals and mercantile numerals, also the Japanese cursive numerals.1

72. The Chinese scientific numerals are made up of vertical and horizontal rods according to the following plan : The numbers 1-9 are represented by the rods |, ||, |||, ||||, |||||, J, JL IJi Iffi; the numbers

10-90 are written thus _ = = = = ._!_=]= = ==• According to the Chinese author Sun-Tsu, units are represented, as just shown, by vertical rods, ten's by horizontal rods, hundred's again by vertical rods, and so on. For example, the number 6,728 was designated by

73. The Japanese make use of the Old Chinese numerals, but have two series of names for the numeral symbols, one indigenous, the other derived from the Chinese language, as seen in Figure 21.

1 See also Ed. Biot, Journal asiatique (December, 1839), p. 497-502; Cantor, Vorlesungen uber Geschichte der Maihematik, Vol. I, p. 673; Biernatzki, Crelle's Journal, Vol. LII (1856), p. 59-94.

HINDU-ARABIC NUMERALS HINDU-ARABIC NUMERALS

45

74. Introduction. — It is impossible to reproduce here all the forms of our numerals which have been collected from sources antedating 1500 or 1510 A.D. G. F. Hill, of the British Museum, has devoted a

CHINOIS

CH1FFKES

VALEUBS.

NOMS DE NOMBttE

JAPOIfAIS

cursifs.

DU COMMERCE.

EN > JAP01UIS PUR.

Elf SINICO-JAPOBTAIS.

—

—

1

1

ftots.

itsi.

^

^

f(

»

foutats.

ni.

H

^5.

W

3

mils.

san.

E3

w^3

^

l\

. yots.

si.

3L

rfS

^r

5

itsouts.

g*

-^

^

_t.

6

mouts.

rok.

-t

-t

j.

7

nanats.

silsi.

A

A

±

8

yatfi.

fats.

A

^l

3

9

kolconots.

kou.

-f*

-f

t

10

towd.

zyou.

lif

"S

^

100

moino.

fakoufyak.

•=f*

^

f

1,000

teidzi.

sen.

M

*

^

10,000

yorodz.

man.

Fie. 21. — Chinese and Japanese numerals. (Taken from A. P. Pihan, Expose des signes de numeration [Paris, 1860], p. 15.)

whole book1 of 125 pages to the early numerals in Europe alone. Yet even Hill feels constrained to remark: "What is now offered, in the shape of just 1,000 classified examples, is nothing more than a vinde-

1 The Development of Arabic Numerals in Europe (exhibited in 64 tables; Oxford, 1915).

46 A HISTORY OF MATHEMATICAL NOTATIONS

miatio prima" Add to the Hill collection the numeral forms, or sup- posedly numeral forms, gathered from other than European sources, and the material would fill a volume very much larger than that of Hill. We are compelled, therefore, to confine ourselves to a few of the more important and interesting forms of our numerals.1

75. One feels the more inclined to insert here only a few tables of numeral forms because the detailed and minute study of these forms has thus far been somewhat barren of positive results. With all the painstaking study which has been given to the history of our numerals we are at the present time obliged to admit that we have not even settled the time and place of their origin. At the beginning of the present century the Hindu origin of our numerals was supposed to have been established beyond reasonable doubt. But at the present time several earnest students of this perplexing question have ex- pressed grave doubts on this point. Three investigators — G. II. Kaye in India, Carra de Vaux in France, and Nicol. Bubnov in Russia — working independently of one another, have denied the Hindu origin.2 However, their arguments are far from conclusive, and the hypothesis of the Hindu origin of our numerals seems to the present writer to explain the known facts more satisfactorily than any of the substitute hypotheses thus far advanced.3

1 The reader who desires fuller information will consult Hill's book which is very rich in bibliographical references, or David Eugene Smith and Louis Charles Karpinski's The Hindu-Arabic Numerals (Boston and London, 1911). See also an article on numerals in English archives by H. Jenkinson in Antiquaries Journal, Vol. VI (1926), p. 263-75. The valuable original researches due to F. Woepcke should be consulted, particularly his great "Mdmoirc sur la propagation des chiffres indiens" published in the Journal asiatique (6th series; Paris, 1863), p. 27- 79, 234-90, 442-529. Reference should be made also to a few other publications of older date, such as G. Friedlein's Zahlzcichen und das elementare Rechnen der Griechen und Homer (Erlangen, 1869), which touches questions relating to our numerals. The reader will consult with profit the well-known histories of mathe- matics by H. Hankel and by Moritz Cantor.

2 G. R. Kaye, "Notes on Indian Mathematics," Journal and Proceedings of the Asiatic Society of Bengal (N.S., 1907), Vol. Ill, p. 475-508; "The Use of the Abacus in Ancient India," ibid., Vol. IV (1908), p. 293-97; "References to Indian Mathe- matics in Certain Mediaeval Works," ibid., Vol. VII (1911), p. 801-13; "A Brief Bibliography of Hindu Mathematics," ibid., p. 679-86; Scientia, Vol. XXIV (1918), p. 54; "Influence grecque dans le de"veloppement des mathc'matiques hindoues," ibid., Vol. XXV (1919), p. 1-14; Carra de Vaux, "Sur 1'origine des chiffres," ibid., Vol. XXI (1917), p. 273-82; Nicol. Bubnov, Arithmetische Selbst- stdndigkeit der europdischen Kultur (Berlin, 1914) (trans, from Russian cd.; Kiev, 1908).

3F. Cajori, "The Controversy on the Origin of Our Numerals," Scientific Monthly, Vol. IX (1919), p. 458-64. See also B. Da'tta in Amer. Math. Monthly, Vol. XXXIII, p. 449; Proceed. Benares Math. Soc.t Vol. VII.

HINDU-ARABIC NUMERALS 47

76. Early Hindu mathematicians, Aryabhata (b. 476 A.D.) and Brahmagupta (b. 598 A.D.), do not give the expected information about the Hindu-Arabic numerals.

Aryabhata's work, called Aryabhatiya, is composed of three parts, in only the first of which use is made of a special notation of numbers. It is an alphabetical system1 in which the twenty-five consonants represent 1-25, respectively; other letters stand for 30, 40, . . . . , 100, etc.2 The other mathematical parts of Aryabhata consists of rules without examples. Another alphabetic system prevailed in Southern India, the numbers 1-19 being designated by consonants, etc.3

In Brahmagupta's Pulverizer, as translated into English by H. T. Colebrooke,4 numbers are written in our notation with a zero and the principle of local value. But the manuscript of Brahmagupta used by Colebrooke belongs to a late century. The earliest commentary on Brahmagupta belongs to the tenth century; Colebrooke's text is later.5 Hence this manuscript cannot be accepted as evidence that Brahmagupta himself used the zero and the principle of local value.

77. Nor do inscriptions, coins, and other manuscripts throw light on the origin of our numerals. Of the old notations the most impor- tant is the Brahmi notation which did not observe place value and in

which 1, 2, and 3 are represented by , , = . The forms of the

Brahmi numbers do not resemble the forms in early place-value nota- tions6 of the Hindu-Arabic numerals.

Still earlier is the Kharoshthi script,7 used about the beginning of the Christian Era in Northwest India and Central Asia. In it the first three numbers are I II III, then X = 4, IX = 5, IIX = 6, XX = 8, 1 = 10, 3 = 20, 33=40, 133 = 50, XI =100. The writing proceeds from right to left.

78. Principle of local value. — Until recently the preponderance of authority favored the hypothesis that our numeral system, with its concept of local value and our symbol for zero, was wholly of Hindu origin. But it is now conclusively established that the principle of

1 M. Cantor, Vorlesungen liber Geschichte der Malhematik, Vol. I (3d ed.), p. 606.

2 G. R. Kaye, Indian Mathematics (Calcutta and Simla, 1915), p. 30, gives full explanation of Aryabhata's notation.

8 M. Cantor, Math. Beitrdge z. Kulturkben der Volkcr (1863), p. 68, 69. 4 Algebra with Arithmetic and Mensuration from the Sanscrit (London, 1817), p. 326 ff.

6 Ibid., p. v, xxxii.

6 See forms given by G. R. Kaye, op. cit., p. 29. 7 Ibid.

48 A HISTORY OF MATHEMATICAL NOTATIONS

local value was used by the Babylonians much earlier than by the Hindus, and that the Maya of Central America used this principle and symbols for zero in a well-developed numeral system of their own and at a period antedating the Hindu use of the zero (§ 68).

79. The earliest-known reference to Hindu numerals outside of India is the one due to Bishop Severus Sebokht of Nisibis, who, living in the convent of Kenneshre on the Euphrates, refers to them in a fragment of a manuscript (MS Syriac [Paris], No. 346) of the year 662 A.D. Whether the numerals referred to are the ancestors of the modern numerals, and whether his Hindu numerals embodied the principle of local value, cannot at present be determined. Apparently hurt by the arrogance of certain Greek scholars who disparaged the Syrians, Sebokht, in the course of his remarks on astronomy and mathematics, refers to the Hindus, " their valuable methods, of cal- culation ; andjjieir computing that surpasses description. Ijwish only to say that this computation is done by means of nine signs."1

80. Some interest attaches to the earliest dates indicating the use of the perfected Hindu numerals. That some kind of numerals with a

earlier than the ninth century is indicated by

Brahmagupta (b. 598 A.D.), who gives rules for computing with a #ero.2 G. Biihler3 believes he has found definite mention of the decimal system and zero m the year 620 A.D. These statements do not neces- sarily imply the use of a decimal" system based on the principle of local value. G. R. Kaye4 points out that the task of the antiquarian is complicated by the existence of forgeries. In the eleventh century in India "there occurred a specially great opportunity to regain con- fiscated endowments and to acquire fresh ones." Of seventeen cita- tions of inscriptions before the tenth century displaying the use of place value in writing numbers, all but two are eliminated as forgeries; these two are for the years 813 and 867 A.D.; Kaye is not sure of the reliability even of these. According to D.JE. Smith _and JLjg.^ Kar- pinski,5 the earliest authentic document unmistakably containing the numerals mttMyh^^r^njMia belongs to the year 876 A.D. The earli-

1 See M. F. Nau, Journal asiatique (10th ser., 1910), Vol. XVI, p. 255; L. C. Karpinski, Science, Vol. XXXV (1912), p. 969-70; J. Ginsburg, Bulletin of the American Mathematical Society, Vol. XXIII (1917), p. 368.

2 Colebrooke, op. cit.r p. 339, 340.

3 "Indische Palaographie," Grundriss d. indogerman. Philologie u. Alieriuma- kunde, Band I, Heft 11 (Strassburg, 1896), p. 78.

4 Journal of the Asiatic Society of Bengal (N.S., 1907), Vol. Ill, p. 482-87.

5 The Hindu-Arabic Numerals (New York, 1911), p. 52.

HINDU-ARABIC NUMERALS

49

est Arabic manuscripts containing the numerals are c>f_874l and 888 A.D. They appear again in a work written at Shiraz in Persia2 in 970 A.D. A church pillar3 not far from the Jeremias Monastery in Egypt has

I

a

3

4

5

6

7

8

9

10

it

la

'3

I

T

I I I 1

i t *

7

I

t

I

1

CD

r

IT

"6 T

T t

S

er

r z & r

M

5

Ih

rt

u

B

fifi

V * Q

b 1?

S

H

O

*r b

b K

b t b L

T V /t

A V

V

V

v V

A X

A

yy,

8 3

a 8

8

sr

8

8 6

8 B &

8 8 8 I x

S

2

/

6

CO

5 S>

976 x

1077

Ixi

XI

XI or XII

| beg.

XII?

XII

XII*

C. 1200 C. 1200

XII

XV

XVI early

FIG. 22. — G. F. Hill's table of early European forms and Boethian apices. (From G. F. Hill, The Development of Arabic Numerals in Europe [Oxford, 1915], p. 28. Mr. Hill gives the MSS from which the various sets of numerals in this table are derived: [1] Codex Vigilanus; [2] St. Gall MS now in Zurich; [3] Vatican MS 3101, etc. The Roman figures in the last column indicate centuries.)

1 Karabacek, Wiener Zeitschriftfur die Kunde des Morgenlandcs, Vol. II (1897), p. 56.

2 L. C. Karpinski, Bibliotheca mathemalica (3d ser., 1910-11), p. 122.

3 Smith and Karpinski, op. cit., p. 138-43.

£ut Ant*. NommliofMui-

50 A HISTORY OF MATHEMATICAL NOTATIONS

the date 349 A.H. ( = 961 A.D.). The oldest definitely dated European manuscript known to contain the Hindu-Arabic numerals is the Codex Vigilanus (see Fig. 22, No. 1), written in the Albelda Cloister in Spain in 976 A.D. The nine characters without the zero are given, as an addition, in a Spanish copy of the Origines by Isidorus of Seville, 992 A.D. A tenth-century manuscript with forms differing materially from those in the Codex Vigilanus was found in the St. Gall manu- script (see Fig. 22, No. 2), now in the University Library at Zurich. [The numerals are contained in a Vatican manuscript of 1077 (see Fig. 22, No. 3), on a Sicilian coin of 1138, in a Regensburg (Bavaria)

•ssi^r £$^uuM>f*n

ApfOM Of Boothia* _. x» ^> r A. x->

*B4 of tho MUdlo | "£* ^k f\ f9 Cl [3 ^/\ g 5 ®

Sr.-r-1 iT7*9-cfrt/'?8cJ>«

Nam.rU, ofthoj p JXfjiortfoOT/Jn V A <\ •

I l>^}afiJVA3°

t^^tf'H^fcr^o

lyonthoJffmwr ^J 9 jf /"• x^A P Q C>

gSKtr- l 2 3 f y * \ * 3 °

JTSJir; i * 3~3*»4<i«5 6 A~7 890

WMHor(T).14«8. ^ v/^l|wv/\» v j ^

From />« ^«rl« -_ .g ^ x *%. > .

s«Pp«tfln<r« byi i. 54-5 6 78 ^10

TODlUU, 1M». ^J< /w/*^

FIG. 23.— Table of important numeral forms. (The first, six lines in this table are copied from a table at the end of Cantor's Vorlesungen liber Geschichte der Mathematik, Vol. 1. The numerals in the Bamberg arithmetic are taken from Friedrich linger, Die Methodik der praktischen Arithmetik in historischer Eni- wickelung [Leipzig, J88S], p. 39.)

chronicle of 1197. The earliest manuscript in French giving the numerals dates about 1275. In the British Museum one English manu- script is of about 1230-50; another is of 1246. The earliest undoubted Hindu-Arabic numerals on a gravestone are at Pforzheim in Baden of 1371 and one at Ulm of 1388. The earliest coins outside of Italy that are dated in the Arabic numerals are as follows: Swiss 1424, Austrian 1484, French 1485, German 1489, Scotch 1539, English 1551. 81. Forms of numerals. — The Sanskrit letters of the second cen- tury A.D. head the list of symbols in the table shown in Figure 23. The implication is that the numerals have evolved from these letters. If such a connection could be really established, the Hindu origin of our numeral forms would be proved. However, a comparison of the forms appearing in that table will convince most observers that an origin

HINDU-ARABIC NUMERALS 61

from Sanskrit letters cannot be successfully demonstrated in that way; the resemblance is no closer than it is to many other alphabets.

The forms of the numerals varied considerably. The 5 was the most freakish. An upright 7 was rare in the earlier centuries. The symbol for zero first used by the Hindus was a dot.1 The symbol for zero (0) of the twelfth anHTEIrteenth centuries i is ^sometiines crossed b^ a horizontal line, or a line slanting upward.2 The Boethian apices, as found in some manuscripts, contain a triangle inscribed in the circular zero. In Athelard of Bath's translation of Al-Madjrltl's re- vision of Al-Khowarizmi's astronomical tables there are in different manuscripts three signs for zero,3 namely ,_JhejQ ( = theta?) referred to above, then T ( = teca),* and 0. In oncToFlKe manuscripts 38 is written severaTtlmes XXXO, and 28 is written XXO, the 0 being intended most likely as the abbreviation for oclo ("eight")-

82. The symbol T for zero is found also in a twelfth-century manuscript5 of N. Ocreatus, addressed to his master Athelard. In that century it appears especially in astronomical tables as an ab- breviation for tcca, which, as already noted, was one of several names for zero;6 it is found in those tables by itself, without connection with other numerals. The symbol occurs in the Alyorixmus vulyaris as- cribed to Sacrobosco.7 C. A. Nallino found o for zero in a manuscript of Escurial, used in the preparation of an edition of Al-Battani. The [symbol © for zero occurs also in printed mathematical books.

The one author who in numerous writings habitually used 6 for zero was the French mathematician Michael Rollc (1652-1719). One finds it in his Traite d'algebre (1690) and in numerous articles in the publications of the French Academy and in the Journal des s^avans.

1 Smith and Karpinski, op. cit., p. 52, 53.

2 Hill, op. cit., p. 30-60.

3 II. Suter, Die astronomischen Tafeln des Muhammed ibn Musd Al-Khwdrizml in der Bearbeitang des Maslama ibn Ahmed Al-Madjrltl und der lateinischen Uebcr- setzung des Alhelhard von Bath (K^benhavn, 1914), p. xxiii.

4 See also M. Curtze, Petri Philomeni de Dacia in Algorismum vulgarem Johannis de Sacrobosco Commentarius (Hauniae, 1897), p. 2, 20.

6 "Prologus N. Ocreati in Helceph ad Adelardum Batcnscm Magistrum suum. Fragment sur la multiplication et la division public* pour la premiere fois par Charles Henry," Abhandlungen zur Geschichle der Mathematik, Vol. Ill (1880), p. 135-38.

6 M. Curtze, Urkunden zur Geschichte der Mathematik im Mittelalter und der Renaissance (Leipzig, 1902), p. 182.

7 M. Curtze, Abhandlungen zur Geschichte der Mathematik, Vol. VIII (Leipzig, 1898), p. 3-27.

52 A HISTORY OF MATHEMATICAL NOTATIONS

Manuscripts of the fifteenth century, on arithmetic, kept in the Ashmolean Museum1 at Oxford, represent the zero by a circle, crossed by a vertical stroke and resembling the Greek letter <£. Such forms for zero arc reproduced by G. F. Hill2 in many of his tables of numer- als.

83. In the fifty-six philosophical treatises of the brothers Ibwan as-safa (about 1000 A.D.) are shown Hindu-Arabic numerals and the corresponding Old Arabic numerals.

The forms of the Hindu-Arabic numerals, as given in Figure 24, have maintained themselves in Syria to the present time. They ap- pear with almost identical form in an Arabic school primer, printed

i«: ?: •* y

1: f r I

f> L: t: *

u 1 A v i 0 f r r t

Fir,. 24. — In the first line are the Old Arabic numerals for 10, 9, 8, 7, 6, 5, 4, 3, 2, 1. In the second line are the Arabic names of the numerals. In the third line are the Hindu-Arabic numerals as given by the brothers Ifrwan as-safa. (Repro- duced from J. Ruska, op. cit., p. 87.)

at Beirut (Syria) in 1920. The only variation is in the 4, which in 1920 assumes more the form of a small Greek epsilon. Observe that 0 is represented by a dot, and 5 by a small circle. The forms used in mod- ern Arabic schoolbooks cannot be recognized by one familiar only with the forms used in Europe.

84. In fifteenth-century Byzantine manuscripts, now kept in the Vienna Library,3 the numerals used are the Greek letters, but the principle of local value is adopted. Zero is 7 or in some places • ; aa means 11, py means 20, ayyy means 1,000. "This symbol 7 for zero means elsewhere 5," says Heiberg, "conversely, o stands for 5 (as now among the Turks) in Byzantine scholia to Euclid ..... In Constanti- nople the new method was for a time practiced with the retention of

1 Robert Stcclc, The Earliest Arithmetics in English (Oxford, 1922), p. 5.

2 Op. cit., Tables III, IV, V, VI, VIII, IX, XI, XV, XVII, XX, XXI, XXII. See also E. Wappler, Zur Geschichte der deutschen Algebra im XV. Jahrhundert (Zwickauer Gymnasialprogramm von 1887), p. 11-30.

3 J. L. Heiberg, "Byzantinische Analekten," Abhandlungen zur Geschichte der Mathematik, Vol. IX (Leipzig, 1899), p. 163, 166, 172. This manuscript in the Vienna Library is marked "Codex Phil. Gr. 65."

HINDU-ARABIC NUMERALS 53

the old letter-numerals, mainly, no doubt, in daily intercourse." At the close of one of the Byzantine manuscripts there is a table of numerals containing an imitation of the Old Attic numerals. The table gives also the Hindu-Arabic numerals, but apparently without recog- nition of the principle of local value; in writing 80, the 0 is placed over the 8. This procedure is probably due to the ignorance of the scribe.

85. A manuscript1 of the twelfth century, in Latin, contains the symbol h for 3 which Curtze and Nagl2 declare to have been found only in the twelfth century. According to Curtze, the foregoing strange symbol for 3 is simply the symbol for tertia used in the nota- tion for sexagesimal fractions which receive much attention in this manuscript.

86. Recently the variations in form of our numerals have been sum- marized as follows: "The form3 of the numerals 1, 6, 8 and 9 has not varied much among the [medieval] Arabs nor among the Christians of the Occident; the numerals of the Arabs of the Occident for 2, 3 and 5 have forms offering some analogy to ours (the 3 and 5 are originally reversed, as well among the Christians as among the Arabs of the Occident); but the form of 4 and that of 7 have greatly modified themselves. The numerals 5, 6, 7, 8 of the Arabs of the Orient differ distinctly from those of the Arabs of the Occident (Gobar numerals). For five one still writes 5 and _J." The use of i for 1 occurs in the first printed arithmetic (Treviso, 1478), presumably because in this early

. stage of printing there was no type for 1. Thus, 9,341 was printed 934;.

87. Many points of historical interest are contained in the fol- lowing quotations from the writings of Alexander von Humboldt. Although over a century old, they still are valuable.

"In the Gobar4 the group signs are dots, that is zeroes, for in India, Tibet and Persia the zeroes and dots are identical. The Gobar symbols, which since the year 1818 have commanded my whole at- tention, were discovered by my friend and teacher, Mr. Silvestre de Sacy, in a manuscript from the Library of the old Abbey St. Germain du Pres. This great orientalist says: 'Le Gobar a un grand rapport

1 Algorithmus-MSS Clm 13021, fols. 27-29, of the Munich Staatsbibliothek. Printed and explained by Maximilian Curtze, Abhandlungen zur Geschichte der Mathematik, Vol. VIII (Leipzig, 1898), p. 3-27.

2 Zeilschrift fur Mathematik und Physik (Hist. Litt. Abth.), Vol. XXXIV (Leipzig, 1889), p. 134.

3 Encyc. des Stien. math., Tome I, Vol. I (1904), p. 20, n. 105, 106.

4 Alexander von Humboldt, Crelle's Journal, Vol. IV (1829), p. 223, 224.

54 A HISTORY OF MATHEMATICAL NOTATIONS

avec le chifYre indien, mais 11 n'a pas de z6ro (S. Gramm. arabe, p. 76, and the note added to PL 8).' I am of the opinion that the zero- symbol is present, but, as in the Scholia of Neophytos on the units, it stands over the units, not by their side. Indeed it is these very zero- symbols or dots, which give these characters the singular name Gobar or dust-writing. At first sight one is uncertain whether one should recognize therein a transition between numerals and letters of the alphabet. One distinguishes with difficulty the Indian 3, 4, 5 and 9. Dal and ha are perhaps ill-formed Indian numerals 6 and 2. The nota- tion by dots is as follows:

3 ' for 30 , 4" for 400, 6 •*• for 6,000 .

These dots remind one of an old-Greek but rare notation (Ducarige, Palacogr., p. xii), which begins with the myriad: a" for 10,000, fl:: for 200 millions. In this system of geometric progressions a single dot, which however is not written down, stands for 100. In Diophantus and Pappus a dot is placed between letter-numerals, instead of the initial Mv (myriad). A dot multiplies what lies to its left by 10,000. .... A real zero symbol, standing for the absence of some unit, is ap- plied by Ptolemy in the descending sexagesimal scale for missing de- grees, minutes or seconds. Delambre claims to have found our sym- bol for zero also in manuscripts of Theon, in the Commentary to the Syntaxis of Ptolemy.1 It is therefore much older in the Occident than the invasion of the Arabs and the work of Planudes on arithmoi indikoi." L. C. Karpinski2 has called attention to a passage in the Arabic biographical work, the Fihrist (987 A.D.), which describes a Hindu notation using dots placed below the numerals; one dot indi- cates tens, two dots hundreds, and three dots thousands.

88. There are indications that the magic power of the principle of local value was not recognized in India from the beginning, and that our perfected Hindu-Arabic notation resulted from gradual evolution. Says Hurnboldt: "In favor of the successive perfecting of the designa- tion of numbers in India testify the Tamul numerals which, by means

1 J. B. J. Delambre, Histoire de Vastron. ancienne, Vol. I, p. 547; Vol. II, p. 10. The alleged passage in the manuscripts of Theon is not found in his printed works. Delambre is inclined to ascribe the Greek sign for zero either as an abbreviation of ouden or as due to the special relation of the numeral omicron to the sexagesimal fractions (op. cit., Vol. II, p. 14, and Journal des sgavans [1817], p. 539).

*Bibliolheca malficmaiica, Vol. XI (1910-11), p. 121-24.

HINDU-ARABIC NUMERALS

55

of the nine signs for the units and by signs of the groups 10, 100, or 1,000, express all values through the aid of multipliers placed on the

r v +3 K^ :i5>ns- xi, -X.jjL.^|.^|

^^^xiiffi

'i

>^-

mi%

•a

o

fl

o

a

H

left. This view is supported also by the singular arithmoi indikoi in the scholium of the monk Neophytosy which is found in the Parisian

A HISTORY OF MATHEMATICAL NOTATIONS

Qit tJifler jiffir werben 0<»ontt<$ mit fren €&«* ractcrn vie btrrwfc rdgulfo gift ribcn/|>abe flUlcfr* volnirvil fonbcrc rcrn>anMun00r0cit ixngcmci* fwnjiffcrn/aufttnomen *>A» f&nfft rn& ffoenfc.

4u$foU&u fbnberlufcnurcftn/ wenn bey cfncr 5i(f«r6rtypon<tOebn/fob«U5a(ftlbi0pag gcrrt fo Ml £?mcr/»n& f tin turt^eil minbcr no$ mcbn

0ie balbcn 2?mcr tr<rt>en allein mit cincr Imi Oder f!ri4>lm 9nrerr<beit>cn.9efi ale off t cin flricfclin btircfr 4<n jiffer gcbi/bcnlmpt re tin balben 2^mer/pnb ba« <jef<fei^t tUtin (>cy 5<n iymcrn pnnb ntcfai t>ein pur/

*1

li<b9<ertcl!mebto^er minber ober bit 0e4 funben £jrmcrbetr/ ba» wirbr bur^ die jtxxy vo!0cnben jty< n/ onb

'!'

fer b~tbtut\u vw .i^tfWeunbbalbfcymer.

• J O-f Jcbcntbalbtr

^ Btbeotbam'cr; -- \ ' let minder. JL* «»«!

Nf*w S

FIG. 26.— From Christoff Ru- dolff's Kunstliche Rechnung mit der Ziffer (Augsburg, 1574[?j).

Library (Cod. Reg., fol. 15), for an account of which I am indebted to Prof. Brandis. The nine digits of Neophytos wholly resemble the Per- sian, except the 4. The digits 1, 2, 3 and 9 are found even in Egyptian number inscriptions (Koscgarten, de Hierogl. AegypL, p. 54). The nine units are enhanced tenfold, 100 fold, 1,000 fold by writing above them one, two or three zeros, as in:

o o o o o°o

2-20, 24 = 24, 5 = 500, 6 = 6,000. If we imagine dots in place of the zero symbols, then we have the arabic Gobar numerals."1 Humboldt copies the scholium of Neophytos. J. L. Heiberg also has called atten- tion to the scholium of Neophytos and to the numbering of scholia to Euclid in a Greek manuscript of the twelfth century (Codex Vindo- bonensis, Gr. 103), in which numer- als resembling the Gobar numerals occur.2 The numerals of the monk Neophytos (Fig. 25), of which Humboldt speaks, have received the special attention of P. Tannery.3

89. Freak forms. — We reproduce herewith from the Augsburg edition of Christoff Rudolff's Kunstliche Rechnung a set of our numerals, and of symbols to represent such fractions

1 Op. tit., p. 227.

2 See J. L. Heiberg's edition of Euclid (Leipzig, 1888), Vol. V; P. Tannery, Revue archeol. (3d scr., 1885), Vol. V, p. 99, also (3d scr., 1886), Vol. VII, p. 355; Encyd des scien. math., Tome I, Vol. I (1904), p. 20, n. 102.

3 Memoir es scientifiques, Vol. IV (Tou- louse and Paris, 1920), p. 22.

HINDU-ARABIC NUMERALS 57

and mixed numbers as were used in Vienna in the measurement of wine. We have not seen the first edition (1526) of Rudolff's book, but Alfred Nagl1 reproduces part of these numerals from the first edition. "In the Viennese wine-cellars," says Hill, "the casks were marked according to their contents with figures of the forms given."2 The symbols for fractions are very curious.

90. Negative numerals. — J. Colson3 in 1726 claimed that, by the use of negative numerals, operations may be performed with "more ease and expedition." If 8605729398715 is to be multiplied by 389175836438, reduce these to small numbers 1414331401315 and 4l 1224244442. Then write the multiplier on a slip of paper and place it in an inverted position, so that its first figure is just over the left-hand figure of the multiplicand. Multiply 4X1=4 and write down 4. Move the multiplier a place to the right and collect the two products, 4X1 + 1X1 = 5; write down 5. Move the multiplier another place to the right, then 4X4+1X1 + 1X1 = 16; write the 1 in the second line. Similarly, the next product is 11, and so on. Similar processes and notations were proposed by A. Cauchy,4 E. Selling,5 and W. B. Ford,6 while J. P. Ballantino7 suggests 1 inverted, thus i, as a sign for negative 1, so that 1X7 = 13 and the logarithm 9 . 69897 - 10 may be written 19 . 69897 or I . 69897. Negative logarithmic charac- teristics are often marked with a negative sign placed over the numeral (Vol. II, §476).

91. Grouping digits in numeration. — In the writing of numbers con- taining many digits it is desirable to have some symbol separating the numbers into groups of, say, three digits. Dots, vertical bars, commas, arcs, and colons occur most frequently as signs of separation.

In a manuscript, Liber algorizmi,8 of about 1200 A.D., there appear

1 Monatsblalt der numismatischen Gesellschaft in Wien, Vol. VII (December, 1906), p. 132.

2 G. F. Hill, op. cU., p. 53.

3 Philosophical Transactions, Vol. XXXIV (1726), p. 161-74; Abridged Trans- actions, Vol. VI (1734), p. 2-4. See also G. Peano, Formulaire mathematique, Vol. IV (1903), p. 49.

4 Comptcs rendus, Vol. XI (1840), p. 796; (Euvres (1st ser.), Vol. V, p. 434-55. 6 Eine mue Rechenrnaschine (Berlin, 1887), p. 16; see also Encyklopddie d.

Math. Wiss., Vol. I, Part 1 (Leipzig, 1898-1904), p. 944.

6 American Mathematical Monthly, Vol. XXXII (1925), p. 302.

7 Op. til., p. 302.

8M. Cantor, Zeitschrift fur Mathematik, Vol. X (1865), p. 3; G. Enestrom, Bibliothcca mathematica (3d ser., 1912-13), Vol. XIII, p. 265.

58 A HISTORY OF MATHEMATICAL NOTATIONS

dots to mark periods of three. Leonardo of Pisa, in his Liber Abbaci (1202), directs that the hundreds, hundred thousands, hundred mil- lions, etc., be marked with an accent above; that the thousands, millions, thousands of millions, etc., be marked with an accent below.

In the 1228 edition,1 Leonardo writes 678 935 784 105 296. Johannes de Sacrobosco (d. 1256), in his Tractatus de arte numerandi, suggests that every third digit be marked with a dot.2 His commentator, Petrus de Dacia, in the first half of the fourteenth century, does the same.3 Directions of the same sort are given by Paolo Dagomari4 of Florence, in his Regoluzze di Maestro Paolo doll Abbaco and Paolo of Pisa,5 both writers of the fourteenth century. Luca Pacioli, in his Summa (1494), folio 196, writes 8 659 421 635 894 676; Georg Peur-

bach (1505),6 "3790528614. Adam Riese7 writes 86789325178. M.

Stifel (1544)8 writes 2329089562800. Gemma Frisius9 in 1540 wrote 24 456 345 678. Adam Riese (1535)10 writes 86 -7 -89 -3 -25 -178. The Dutch writer, Martinus Carolus Creszfeldt,11 in 1557 gives in his Arithmetica the following marking of a number:

"Exempei. || 5 8 7 4 9 3 6 2 5 3 4 || ."

w i w i w i w

1 El liber abbaci di Leonardo Pisano .... da B. Boncompagni (Roma, 1857), p. 4.

2 J. O. Ilalliwcll, Rara malhematica (London, 1839), p. 5; M. Cantor, Vor- lesungen, Vol. II (2d cd., 1913), p. 89.

3 Petri Philomeni de Dacia in Algorismum vulgar em lohannis de Sacrobosco commentarius (ed. M. Curtze; Kopenhagen, 1897), p. 3, 29; J. Tropfke, Geschichte der Nlemcntarmathematik (2d cd., 1921), Vol. I, p. 8.

4 Libri, Histoire des sciences mathematiques en Italic, Vol. Ill, p. 296-301 (Rule 1).

6 Ibid., Vol. II, p. 206, n. 5, and p. 526; Vol. Ill, p. 295; see also Cantor, op. cit., Vol. II (2d ed., 1913), p. 164.

6 Opus alyorithmi (Herbipoli, 1505). See Wildermuth, "Rechnen," Encyklo- paedie des gesammten Erziehungs- und Unterrichtsivesens (Dr. K. A. Schmid, 1885).

7 Hechnung auff der Linien vnnd Federn (1544); Wildermuth, "Reehnen," Encijklopaedie (Schmid, 1885), p. 739.

8 Wildermuth, op. cit., p. 739.

9 Arithmetical practicae methodus facilis (1540) ; F. linger, Die Methodik der praktischen Arithmetik in hislorischer Entwickelung (Leipzig, 1888), p. 25, 71.

10 Rechnung auff d. Linien u. Federn (1535). Taken from H. Hankel, op. cit. (Leipzig, 1874), p. 15.

11 Arithmetica (1557). Taken from Bierens de Haan, Bouwstoffen voor de Ge- schiedenis der Wis-en Natuurkundige Wetenschappent Vol. II (1887), p. 3.

HINDU-ARABIC NUMERALS 59

Thomas Blundeville (1636)1 writes 5|936|649. Tonstall2 writes

. ... 43210

3210987654321. Clavius3 writes 42329089562800. Chr. Rudolff4 writes

23405639567. Johann Caramuel6 separates the digits, as in "34:252,-

Integri. Partes.

341;154,329"; W. Oughtred,6 9!876i543|210l2i345678i9; K. Schott7,

7697432329089562436; N. Barreme,8 254.567.804.652; W. J. G.

Ill II I 0

Karsten,9872 094,826 152,870 364,008; I. A. de Segner,105|329//|870| 325/|743|297°, 174; Thomas Dilworth/1 789 789 789; Nicolas Pike,12

3 2 1

356;809,379;120,406;129,763; Charles Hutton,13 281,427,307; E. Bczout,14 23, 456, 789, 234, 565, 456.

In M. Lcmos' Portuguese encyclopedia15 the population of New

1 Mr. Klundcvil, His Exercises contayning eight Treatises (7th cd., Ro. Hartwell; London, 1636), p. 106.

2 De Artc Svppvtandi, libri qvatvor Cvtheberti Tonstalli (Argentorati), Colophon 1544, p. 5.

3 Christophori Clavii epitome arithmeticae practicac (Romae, 1583), p. 7.

4 Kunstliche Rechnung mil dcr Ziffer (Augsburg, 1574[?J), Aiij B.

6 Joannis Caramvclis mathesis biceps, veins et nova (Cornpaniae [southeast of Naples], 1670), p. 7. The passage is as follows: "Punctum finale ( . ) est, quod poni- tur post unitatem: ut cum scribirnus 23. viginti tria. Comma (,) post millemirium

scribitur . . . . ut cum scribimus, 23,424 Millcriarium & centenario -dis-

tinguere alios populos docent Hispani, qui utuntur hoc charactere \f , . . . . Hypo- colon (;) millioncm a millcnario separat, ut cum scribimus 2;041,311. Duo puncta ponuntur post billioncm, sen millioncm millionum, videlicet 34:252,341;154,329." Caramuel was born in Madrid. For biographical sketch see Rcvista matemdtica Hispano-American, Vol. I (1919), p. 121, 178, 203.

6 Clavis mathematicae (London, 1652), p. 1 (1st ed., 1631).

7 Cursus mathematicus (Herbipoli, 1661), p. 23.

8 Arithm6tique (new ed.; Paris, 1732), p. 6.

9 Mathesis theoretica elementaris atqve svblimior (Rostochii, 1760), p. 195.

10 Elementa arithmelicae gcomelriae et calcvli geometrici (2d ed.; Halle, 1767), p. 13.

11 Schoolmaster's Assistant (22d ed.; London, 1784), p. 3.

12 New and Complete System of Arithmetic (Newburyport, 1788), p. 18.

18 " Numeration," Mathematical and Philosophical Dictionary (London, 1795). 14 Cours de malMmatiques (Paris, 1797), Vol. I, p. 6.

16 ' 'Portugal," Encyclopedia Portugueza ILlustrada . . . . de Maximiano Lemos (Porto).

CO A HISTORY OF MATHEMATICAL NOTATIONS

York City is given as "3 .437:202"; in a recent Spanish encyclopedia,1 the population of America is put down as "150- 979,995."

In the process of extracting square root, two early commentators2 on Bhaskara's Lilavati, namely Rama-Crishna Deva and Gangad'hara (ca. 1420 A.D.), divide numbers into periods of two digits in this man- ner, 8 8 2 0 9. In finding cube roots Rama-Crishna Deva writes

i i — i

1953125.

92. The Spanish "calderon" — In Old Spanish and Portuguese numeral notations there are some strange and curious symbols. In a contract written in Mexico City in 1649 the symbols "7U291e" and "VIIUCCXCIps" each represent 7,291 pesos. The U, which here re- sembles an 0 that is open at the top, stands for "thousands."3 I. B. Richman has seen Spanish manuscripts ranging from 1587 to about 1700, and Mexican manuscripts from 1768 to 1855, all containing symbols for "thousands" resembling U or D, often crossed by one or two horizontal or vertical bars. The writer has observed that after 1600 this U is used freely both with Hindu- Arabic and with Roman numerals; before 1600 the U occurs more commonly with Roman numerals. Karpinski has pointed out that it is used with the Hindu- Arabic numerals as early as 1519, in the accounts of the Magellan voyages. As the Roman notation does not involve the principle of local value, U played in it a somewhat larger role than merely to afford greater facility in the reading of numbers. Thus VIUCXV equals 6X1,000+115. This use is shown in manuscripts from Peru of 1549 and 1543,4 in manuscripts from Spain of 14805 and 1429.6

We have seen the corresponding type symbol for 1,000 in Juan Perez de Moya,7 in accounts of the coming in the Real Casa de Moneda de

1 "America," Encyclopedia illmtrada segui Diccionario universal (Barcelona).

2 Colebrooke, op. cit., p. 9, 12, xxv, xxvii.

3 F. Cajori, "On the Spanish Symbol U for 'thousands/ " Bibliotheca mathe- matica, Vol. XII (1912), p. 133.

4 Carlos de Indias publicalas por primer a vez d Ministerio de Fomento (Madrid, 1877), p. 502, 543, facsimiles X and Y.

5 Jose Gorizalo dc las Casas, Ancdes de la Palcoc/rafia Espanola (Madrid, 1857), Plates 87, 92, 109, 110, 113, 137.

0 Liciniano Saez, Demoslracidn Histdricadel verdadero valor de todas las monedaa que corrian en Caslilla duranle el Reynado del Senor Don Enrique III (Madrid, 1796), p. 447. See also Colomera y Rodriguez, Venancio, Paleoyrajia caslellana (1862).

7 Arilmelica practica (14th ed.; Madrid, 1784), p. 13 (1st ed., 1562).

HINDU-ARABIC NUMERALS 61

Mexico (1787), in eighteenth-century books printed in Madrid,1 in the Gazetas de Mexico of 1784 (p. 1), and in modern reprints of seventeenth-century documents.2 In these publications the printed symbol resembles the Greek sampi 5 for 900, but it has no known connection with it. In books printed in Madrid3 in 1760, 1655, and 1646, the symbol is a closer imitation of the written U, and is curiously made up of the two /small printed letters, I, f, each turned halfway around. The two inverted letters touch each other below, thus \f . Printed symbols representing a distorted U have been found also in some Spanish arithmetics of the sixteenth century, particularly in that of Gaspard de Texeda4 who writes the number 103,075,102,300 in the Castellanean form c.iijU.75qs c.ijU300 and also in the algoristic form 103U075qs 102U300". The Spaniards call this symbol and also the sampi-like symbol a calderon.5 A non-Spanish author who ex- plains the calderdn is Johann Caramuel,6 in 1670.

93. The present writer has been able to follow the trail of this curious symbol U from Spain to Northwestern Italy. In Adriano Cappelli's Lexicon is found the following: "In the liguric documents of the second half of the fifteenth century we found in frequent use, to indicate the multiplication by 1,000, in place of M, an O crossed by a horizontal line."7 This closely resembles some forms of our Spanish symbol U. Cappelli gives two facsimile reproductions8 in

1 Liciniano Saez, op. cit.

2 Manuel Danvila, Boletin de la Real Academia de la Hisloria (Madrid, 1888), Vol. XII, p. 53.

3 Cuentas para lodas, compendia arilhmetico, e Histdrico . . . . su autor D. Manuel Recio, Oficial de la contadurfa general de postos del Rcyno (Madrid, 1760) ; Teatro Eclesidstico de la primitiva Iglesia de las Indias Occidentals .... el M. Gil Gonzalez Davila, su Coronista Mayor de las Indias, y de los Reynos de las dos Castillas (Madrid, 1655), Vol. II; Memorial, y Noticias Sacras, y reales del Imperio de las Indias Occidentales .... Escriuiale por el afio de 1646, Juan Diez de la Calle, Oficial Segundo de la Misma Secretaria.

*Suma de Arithmetica pratica (Valladolid, 1546), fol. iiijr.; taken from D. E. Smith, History of Mathematics, Vol. II (1925), p. 88. The qs means quentos (cuen- tos, "millions")-

5 In Joseph Aladern, Diccionari popular de la Llengua Catalana (Barcelona, 1905), we read under "Caldero": "Among ancient copyists a sign (\/") denoted a thousand."

6 Joannis Caramvelis Mathesis biceps veins et nova (Companiae, 1670), p. 7.

7 Lexicon Abbreviaturarum (Leipzig, 1901), p. 1.

8 Ibid., p. 436, col. 1, Nos. 5 and 6.

62

A HISTORY OP MATHEMATICAL NOTATIONS

which the sign in question is small and is placed in the position of an exponent to the letters XL, to represent the number 40,000. This corresponds to the use of a small c which has been found written to the right of and above the letters XI, to signify 1,100. It follows, there- fore, that the modified U was in use during the fifteenth century in Italy, as well as in Spain, though it is not known which country had the priority.

What is the origin of this calderon? Our studies along this line make it almost certain that it is a modification of one of the Roman

•F '• A

/* * !*'

FIG. 27. — From a contract (Mexi o City, 1649). The right part shows the sum of 7,291 pesos, 4 tomines, 6 granos, ex >ressed in Roman numerals and the calderdn. The left part, from the same contract, shows the same sum in Hindu-Arabic nu- merals and the calderdn.

symbols for 1,000. Besides M, the Romans used for 1,000 the symbols CIO, T, oo, and *f . These symbols are found also in Spanish manu- scripts. It is easy to see how in the hands of successive generations of amanuenses, some of these might assume the forms of the calderdn. If the lower parts of the parentheses in the forms CIO or CIIO are united, we have a close imitation of the U, crossed by one or by two bars.

HINDU-ARABIC NUMERALS

63

94. The Portuguese "cifrao." — Allied to the distorted Spanish U is the Portuguese symbol for 1,000, called the cifrao.1 It looks somewhat like our modern dollar mark, $. But its function in writing numbers was identical with that of the calderon. Moreover, we have seen forms of this Spanish "thousand" which need only to be turned through a right angle to appear like the Portuguese symbol for 1,000. Changes of that sort are not unknown. For instance, the Arabic numeral 5 appears upside down in some Spanish books and manuscripts as late as the eighteenth and nineteenth centuries.

a

FIG. 28. — Real estate sale in Mexico City, 1718. The sum written here is 4,255 pesos.

95. Relative size of numerals in tables. — Andr6 says on this point: "In certain numerical tables, as those of Schron, all numerals are of the same height. In certain other tables, as those of Lalande, of Cal- let, of Houel, of Dupuis, they have unequal heights: the 7 and 9 are prolonged downward; 3, 4, 5, 6 and 8 extend upward; while 1 and 2

do not reach above nor below the central body of the writing

The unequal numerals, by their very inequality, render the long train of numerals easier to read; numerals of uniform height are less legible."2

1 See the word cifrao in Antonio de Moraes Silva, Dice, de Lingua Portuguesa (1877); in Vieira, Grande Dice. Portuguez (1873); in Dice. Comtemp. da Lingua Portuguesa (1881).

2 D. Andre", Des notations math&matiques (Paris, 1909), p. 9.

64 A HISTORY OF MATHEMATICAL NOTATIONS

96. Fanciful hypotheses on the origin of the numeral forms. — A p lem as fascinating as the puzzle of the origin of language relate the evolution of the forms of our numerals. Proceeding on the t assumption that each of our numerals contains within itself, { skeleton so to speak, as many dots, strokes, or angles as it repres units, imaginative writers of different countries and ages have vanced hypotheses as to their origin. Nor did these writers feel i they were indulging simply in pleasing pastime or merely contribu to mathematical recreations. With perhaps only one exception, 1 were as convinced of the correctness of their explanations as are ch squarers of the soundness of their quadratures.

The oldest theory relating to the forms of the numerals is du the Arabic astrologer Aben Ragel1 of the tenth or eleventh cent He held that a circle and two of its diameters contained the rcqu forms as it were in a nutshell. A diameter represents 1; a diam and the two terminal arcs on opposite sides furnished the 2. A glanc Part I of Figure 29 reveals how each of the ten forms may be evol from the fundamental figure.

On the European Continent, a hypothesis of the origin from do the earliest. In the seventeenth century an Italian Jesuit wn Mario Bettini,2 advanced such an explanation which was eag accepted in 1651 by Georg Philipp Harsdorffer3 in Germany, ' said: "Some believe that the numerals arose from points or dots, in Part II. The same idea was advanced much later by Geo Dumesnil4 in the manner shown in the first line of Part III. In cur writing the points supposedly came to be written as dashes, yielc forms resembling those of the second line of Part III. The two hori: tal dashes for 2 became connected by a slanting line yielding the n ern form. In the same way the three horizontal dashes for 3 were joi by two slanting lines. The 4, as first drawn, resembled the 0; but < fusion was avoided by moving the upper horizontal stroke inl

1 J. F. Weidler, De characteribus numerorum vulgaribus dissertatio mathcma critica (Wittembcrgae, 1737), p. 13; quoted from M. Cantor, Kulturleben der V\ (Halle, 1863), p. 60, 373.

2 Apiaria unwersae philosophiae, mathematicae, Vol. II (1642), Apiarium p. 5. See Smith and Karpinski, op. cit., p. 36.

3 Delitae mathematicae et physicae (Niirnberg, 1651). Reference from Sterner, Geschichte der Rechenkunst (Miinchen and Leipzig [1891]), p. 138, 52

4 "Note sur la forme des chifTres usuels," Revue archSologique (3d ser.; P 1890), Vol. XVI, p. 342-48. See also a critical article, "Pretendues notal Pythagoriennes sur Forigine de nos chiffres," by Paul Tannery, in his Mem scientifiques, Vol. V (1922), p. 8.

HINDU-ARABIC NUMERALS

65

vertical position and placing it below on the right. To avoid con- founding the 5 and 6, the lower left-hand stroke of the first 5 was

fc7S?

3 D 5 E B 2 2 J 4 A 7 X

'i •'- H 5 G no % s S a 5 B

I 2 3

o

8 O

S 88'S

o

s 2 O a

FIG. 29. — Fanciful hypotheses

<•

n a

S "5 a o

changed from a vertical to a horizontal position and placed at the top of the numeral. That all these changes were accepted as historical,

66 A HISTORY OF MATHEMATICAL NOTATIONS

without an atom of manuscript evidence to support the different steps in the supposed evolution, is an indication that Baconian inductive methods of research had not gripped the mind of Dumesnil. The origin from dots appealed to him the more strongly because points played a role in Pythagorean philosophy and he assumed that our numeral system originated with the Pythagoreans.

Carlos le-Maur,1 of Madrid, in 1778 suggested that lines joining the centers of circles (or pebbles), placed as shown in the first line of Part IV, constituted the fundamental numeral forms. The explana- tion is especially weak in accounting for the forms of the first three numerals.

A French writer, P. Voizot,2 entertained the theory that originally a numeral contained as many angles as it represents units, as seen in Part V. He did not claim credit for this explanation, but ascribed it to a writer in the Gcnova Catholico Militarite. But Voizot did originate a theory of his own, based on the number of strokes, as shown in Part VI.

Edouard Lucas3 entertains readers with a legend that Solomon's ring contained a square and its diagonals, as shown in Part VII, from which the numeral figures were obtained. Lucas may have taken this explanation from Jacob Leupold4 who in 1727 gave it as widely current in his day.

The historian Moritz Cantor5 tells of an attempt by Anton Miiller6 to explain the shapes of the digits by the number of strokes necessary to construct the forms as seen in Part VIII. An eighteenth-century writer, Georg Wachter,7 placed the strokes differently, somewhat as in Part IX. Cantor tells also of another writer, Piccard,8 who at one time had entertained the idea that the shapes were originally deter-

1 Elcmentos de Matematica pura (Madrid, 1778), Vol. I, chap. i.

J "Lcs chiffres arabes et leur origine," La nature (2d semestre, 1899), Vol. XXVII, p. 222.

3 L' Arithmelique amusanle (Paris, 1895), p. 4. Also M. Cantor, Kulturlcben der Volker (Halle, 1863), p. 60, 374, n. 116; P. Treutlcin, Geschichte unsercr Zahl- zeicJien (Karlsruhe, 1875), p. 16.

4 Theatrvm Arithmetico-Geometricvm (Leipzig, 1727), p. 2 and Table III. 6 Kullurleben der Volker, p. 59, 60.

6 Arilhmetik und Algebra (Heidelberg, 1833). See also a reference to this in P. Treutlein, op. tit. (1875), p. 15.

7 Naturae et Sctipturae Concordia (Lipsiae et Hafniae, 1752), chap. iv.

8 M6moire sur la forme et de la provenance des chiffres, Sociele Vaudoise des sciences nalurelles (stances du 20 Avril et du 4 Mai, 1859), p. 176, 184. M. Cantor reproduces the forms due to Piccard; see Cantor, Kidturleben, etc., Fig. 44.

HINDU-ARABIC NUMERALS 67

mined by the number of strokes, straight or curved, necessary to express the units to be denoted. The detailed execution of this idea, as shown in Part IX, is somewhat different from that of Mliller and some others. But after critical examination of his hypothesis, Pic- card candidly arrives at the conclusion that the resemblances he pointed out are only accidental, especially in the case of 5, 7, and 9, and that his hypothesis is not valid.

This same Piccard offered a special explanation of the forms of the numerals as found in the geometry of Boethius and known as the "Apices of Boethius." He tried to connect these forms with letters in the Phoenician and Greek alphabets (see Part X). Another writer whose explanation is not known to us was J. B. Reveillaud.1

The historian W. W. R. Ball2 in 1888 repeated with apparent ap- proval the suggestion that the nine numerals were originally formed by drawing as many strokes as there are units represented by the respective numerals, with dotted lines added to indicate how the writ- ing became cursive, as in Part XL Later Ball abandoned this ex- planation. A slightly different attempt to build up numerals on the consideration of the number of strokes is cited by W. Lietzmann.3 A still different combination of dashes, as seen in Part XII, was made by the German, David Arnold Crusius, in 1746.4 Finally, C. P. Sherman5 explains the origin by numbers of short straight lines, as shown in Part XI11. "As time went on," he says, "writers tended more and more to substitute the easy curve for the difficult straight line and not to lift the pen from the paper between detached lines, but to join the two — which we will call cursive writing."

These hypotheses of the origin of the forms of our numerals have been barren of results. The value of any scientific hypothesis lies in co-ordinating known facts and in suggesting new inquiries likely to advance our knowledge of the subject under investigation. The hy- potheses here described have done neither. They do not explain the very great variety of forms which our numerals took at different times

1 Essai sur lea chiffrcs arabcs (Paris, 1883). Reference from Smith and Kar- pinski, op. cit., p. 36.

2 A Short Account of the History of Mathematics (London, 1888), p. 147.

3 Lusliges und Merkwurdiges von Zahlen und Formcn (Brcslau, 1922), p. 73, 74. lie found the derivation in Raether, Theorie und Praxis dcs Rcchcnunterrichts (1. Teil, 0. Aufl.; Brcslau, 1920), p. 1, who refers to H. von Jacobs, Das Volk der Sicbener-Zdhler (Berlin, 1896).

4 Anweisung zur Rechen-Kunsl (Halle, 1746), p. 3.

6 Mathematics Teacher, Vol. XVI (1923), p. 398-401.

68 A HISTORY OF MATHEMATICAL NOTATIONS

and in different countries. They simply endeavor to explain the nu- merals as they are printed in our modern European books. Nor have they suggested any fruitful new inquiry. They serve merely as en- tertaining illustrations of the operation of a pseudo-scientific imagina- tion, uncontrolled by all the known facts.

97. A sporadic artificial system. — A most singular system of numeral symbols was described by Agrippa von Nettesheim in his De occulta philosophia (1531) and more fully by Jan Bronkhorst of Nim- wegen in Holland who is named after his birthplace Noviomagus.1 In 1539 he published at Cologne a tract, De numeris, in which he de- scribes numerals composed of straight lines or strokes which, he claims, were used by Chaldaei et Astrologi. Who these Chaldeans are whom he mentions it is difficult to ascertain; Cantor conjectures that they were late Roman or medieval astrologers. The symbols are given again in a document published by M. Host us in 1582 at Antwerp. An examina- tion of the symbols indicates that they enable one to write numbers up into the millions in a very concise form. But this conciseness is at- tained at a great sacrifice of simplicity; the burden on the memory is great. It does not appear as if these numerals grew by successive steps of time; it is more likely that they are the product of some in- ventor who hoped, perhaps, to see his symbols supersede the older (to him) crude and clumsy contrivances.

An examination, in Figure 30, of the symbols for 1, 10, 100, and 1,000 indicates how the numerals are made up of straight lines. The same is seen in 4, 40, 400, and 4,000 or in 5, 50, 500, and 5,000.

98. General remarks. — Evidently one of the earliest ways of re- cording the small numbers, from 1 to 5, was by writing the corre- sponding number of strokes or bars. To shorten the record in express- ing larger numbers new devices were employed, such as placing the bars representing higher values in a different position from the others, or the introduction of an altogether new symbol, to be associated with the primitive strokes on the additive, or multiplicative principle, or in some cases also on the subtractive principle.

After the introduction of alphabets, and the observing of a fixed sequence in listing the letters of the alphabets, the use of these letters

1 See M. Cantor, Vorlesungen uber Geschichte der Mathematik, Vol. II (2d ed.; Leipzig, 1913), p. 410; M. Cantor, Mathemat. Beitrdge zum Kulturleben der Volker (Halle, 1863), p. 166, 167; G. Friedlein, Die Zahlzeichen und das elementare Rechnen der Griechen und Romer (Erlangen, 1869), p. 12; T. H. Martin, Annali di mate- maiica (B. Tortolini; Rome, 1863), Vol. V, p. 298; J. C. Heilbronner, Historia Mathcseos universae (Lipsiae, 1742), p. 735-37; J. Ruska, Archivfiir die Geschichte der Nalurwissenschaflen und Technik, Vol. IX (1922), p. 112-26.

HINDU-ARABIC NUMERALS

69

for the designation of numbers was introduced among the Syrians, Greeks, Hebrews, and the early Arabs. The alphabetic numeral sys- tems called for only very primitive powers of invention; they made

FIG. 30. — The numerals described by Noviomagus in 1539. (Taken from J. C. Heilbronner, Historia malheseos [1742], p. 736.)

70 A HISTORY OF MATHEMATICAL NOTATIONS

unnecessarily heavy demands on the memory and embodied no at- tempt to aid in the processes of computation.

The highest powers of invention were displayed in the systems em- ploying the principle of local value. Instead of introducing new sym- bols for units of higher order, this principle cleverly utilized the posi- tion of one symbol relative to others, as the means of designating different orders. Three important systems utilized this principle: the Babylonian, the Maya, and the Hindu-Arabic systems. These three were based upon different scales, namely, 60, 20 (except in one step), and 10, respectively. The principle of local value applied to a scale with a small base affords magnificent adaptation to processes of computation. Comparing the processes of multiplication and division which we carry out in the Hindu-Arabic scale with_what tEe alpha- beticafsystems or the Roman system afforded places the superiority of the Hindu-Arabic scale in full view. The Greeks resorted to abacal computation, which is simply a primitive way of observing local value in computation. In what way the Maya or the Babylonians used their notations in computation is not evident from records that have come down to us. The scales of 20 or 60 would crJl for large multiplication tables.

The orjgjn_and development of the Hindu-Arabic notation has received Intensive study. Nevertheless, little is known. An" outstand- ing facTis~^Iit~"cIuffng "the past one thousand years no uniformity in the shapes of the numerals has been reached. An American is some- times puzzled by the shape of the number 5 written in France. A European traveler in Turkey would find that what in Europe is a 0 is in Turkey a 5.

99. Opinion of Laplace. — Laplace1 expresses his admiration for the invention of the Hindu-Arabic numerals and notation in this wise: "It is from the Indians that there has come to us the ingenious method of expressing all numbers, in ten characters, by giving them, at the same time, an absolute and a place value; an idea fine and important, which appears indeed so simple, that for this very reason we do not sufficiently recognize its merit. But this very simplicity, and the extreme facility which this method imparts to all calculation, place jour system of arithmetic in the first rank of the useful inventions. How difficult it was to invent such a method one can infer from the fact that it escaped the genius of Archimedes and of Apollonius of Perga, two of the greatest men of antiquity."

1 Exposition du systeme du monde (6th ed.; Paris, 1835), p. 376.

Ill

SYMBOLS IN ARITHMETIC AND ALGEBRA (ELEMENTARY PART)

100. In ancient Babylonian and Egyptian documents occur cer- tain ideograms and symbols which are not attributable to particular individuals and are omitted here for that reason. Among these signs is r~ for square root, occurring in a papyrus found at Kahun and now at University College, London,1 and a pair of walking legs for squaring in the Moscow papyrus.2 These symbols and ideograms will be referred to in our "Topical Survey" of notations.

A. GROUPS OF SYMBOLS USED BY INDIVIDUAL WRITERS GREEK: DIOPHANTUS, THIRD CENTURY A.D.

101. The unknown number in algebra, defined by Diophantus as containing an undefined number of units, is represented by the Greek letter s with an accent, thus s', or in the form s°'. In plural cases the symbol was doubled by the Byzantines and later writers, with the addition of case endings. Paul Tannery holds that the evidence is against supposing that Diophantus himself duplicated the sign.3 G. H. F. Nesselmann4 takes this symbol to be final sigma and remarks that probably its selection was prompted by the fact that it was the only letter in the Greek alphabet which was not used in writing num- bers. Heath favors "the assumption that the sign was a mere tachy- graphic abbreviation and not an algebraical symbol like our x, though discharging much the same function."6 Tannery suggests that the sign is the ancient letter koppa, perhaps slightly modified. Other views on this topic are recorded by Heath.

1 Moritz Cantor, Vorlesungen uber Geschichte der Malhematik, Vol. I, 3d ed., Leipzig, p. 94.

2 B. Touraeff, Ancient Egypt (1917), p. 102.

3 Diophanti Alcxandrini opera omnia cum Graedst commentaries (Lipsiae, 1895), Vol. II, p. xxxiv-xlii; Sir Thomas L. Heath, Diophantus of Alexandria (2d ed.; Cambridge, 1910),. p. 32, 33.

4 Die Algebra der Griechen (Berlin, 1842), p. 290, 291.

5 Op. cit.y p. 34-36.

71

72 A HISTORY OF MATHEMATICAL NOTATIONS

A square, z2, is in Diophantus' Arithmetica AF

A cube, x8, is in Diophantus' Arithmetica KY

A square-square, z4, is in Diophantus' Arithmetica ArA

A square-cube, z5, is in Diophantus' Arithmetica AKr

A cube-cube, x6, is in Diophantus' Arithmetica KrK

In place of the capital letters kappa and delta, small letters are some- times used.1 Heath2 comments on these symbols as follows: "There is no obvious connection between the symbol Ay and the symbol s of which it is the square, as there is between x2 and x, and in this lies the great inconvenience of the notation. But upon this notation no advance was made even by late editors, such as Xylander, or by Bachet and Fermat. They wrote N (which was short for Numerus) for the s of Diophantus, Q (Quadratus) for A F, C (Cubus) for K y , so that we find, for example, 1Q+5JV = 24, corresponding to z2+5z = 24.3 Other symbols were however used even before the publication of Xylander's Diophantus, e.g., in Bombelli's Algebra"

102. Diophantus has no symbol for multiplication; he writes down the numerical results of multiplication without any preliminary step which would necessitate the use of a symbol. Addition is expressed

1 From Format's edition of Bachct;s Diophantus (Toulouse, 1670), p. 2, Definition II, we quote: "Appellatvr igitur Quadratus, Dynamis, & est illius nota 5' superscriptum habens u sic S«>. Qui autem sit ex quadrato in suum latus cubus est, cuius nota est \, superscriptum habens v hoc pacto «w. Qui autem sit ex quad- rato in seipsum multiplicato, quadrato-quadratus est, cuius nota est geminum 5' habens superscriptum i>, hac ratione 55". Qui sit quadrato in cubum qui ab eodem latere profectus est, ducto, quadrato-cubus nominatur, nota eius 5/c superscriptum habens u sic 8i<y. Qui ex cubo in se ducto nascitur, cubocubus vocatur, & est eius nota geminum K superscriptum habens v, hoc pacto KK". Cui vero nulla harum proprietatum obtigit, sed constat multitudine vnitatem rationis experte, nurnerus vocatur, nota eius V Est et aliud signum immutabile definitorum, vnitas, cuius nota Jj. superscriptum habens 6 sic /z°." The passage in Bachet's edition of 1621 is the same as this.

2 Op. tit., p. 38.

8 In Fermat's edition of Bachet's Diophantus (Toulouse, 1670), p. 3, Definition II, we read: "Haec ad verbum exprimenda esse arbitratus sum potius quam cum Xilandro nescio quid aliud comminisci. Quamuis enim in reliqua versione nostra notis ab eodem Xilandro excogitatis libenter vsus sim, quas tradam infra. Hie tamen ab ipso Diophanto longius recedere nolui, quod hac definitione notas ex- plicet quibus passim libris istis vtitur ad species omnes compcndio designandas, & qui has ignoret ne quidem Graeca Diophanti legere possit. Porr6 quadrat urn Dy- namin vocat, quae vox potestatem sonat, quia videlicet quadratus est veluti potestas cuius libet lineae, & passim ab Euclide, per id quod potest linea, quadratus illius designatur. Itali, Hispanique eadem ferd de causa Censum vocant, quasi

INDIVIDUAL WRITERS 73

y mere juxtaposition. Thus the polynomial X3+13z2+5:r+2 would

o _ o

e in Diophantine symbols K FdA YLyseMp, where M is used to repre- jnt units and shows that fi or 2 is the absolute term and not a part f the coefficient of s or x. It is to be noted that in Diophantus' square-cube" symbol for a;5, and "cube-cube" symbol for x6, the Iditive principle for exponents is employed, rather than the multipli- itive principle (found later widely prevalent among the Arabs and ^alians), according to which the "square-cube" power would mean x* id the "cube-cube" would mean #9.

103. Diophantus' symbol for subtraction is "an inverted ^ with le top shortened, A." Heath pertinently remarks: "As Diophantus scd no distinct sign for +, it is clearly necessary, in order to avoid mfusion, that all the negative terms in an expression, should be laced together after all the positive terms. And so in fact he does lace them."1 As regards the origin of this sign /jv, Heath believes lat the explanation which is quoted above from the Diophantine ixt as we have it is not due to Diophantus himself, but is "an explana- on made by a scribe of a symbol which he did not understand." eath2 advances the hypothesis that the symbol originated by placing I within the uncial form A> thus yielding A . Paul Tannery,3 on the ;her hand, in 1895 thought that the sign in question was adapted om the old letter sampi !), but in 1904 he4 concluded that it was ,ther a conventional abbreviation associated with the root of a cer- in Greek verb. His considerations involve questions of Greek gram- ar and were prompted by the appearance of the Diophantine sign

ms rcdditum, prouentumque, qudd a latere seu radice, tanquam a feraci solo ladratus oriatur. Inde factum vt Gallorum nonnulli & Cermanorum corrupto cabulo zerizum appellarint. Numerum autem indeterminatum & ignotum, qui aliarum omnium potestatum latus esse intelligitur, Numerum sirnpliciter Dio- antus appellat. Alij passim Radicem, vel latus, vel rein dixerunt, Itali patrio cabulo Cosam. Caeterum nos in versione nostra his notis N. Q. C. QQ. QC. CC. signabimus Numerum, Quadratum, Cubum, Quadratoquadratum, Quadrato- bum, Cubocubum. Nam quod ad vnitates certas & determinatas spectat, eis tarn aliquam adscribere superuacaneum duxi, qu6d hae seipsis absque vlla ibiguitate sese satis indicent. Ecquis enim cum audit numerum 6. non statim ^itat sex vnitates? Quid ergo necesse est sex vnitates dicere, cum sufficiat dicere, c? . . . . " This passage is the same as in Bachet's edition of 1621.

1 Heath, op. cit., p. 42.

2 Ibid., p. 42, 43.

3 Tannery, op. cit., Vol. II, p. xli.

4 Bibliolheca mathematica (3d ser.), Vol. V, p. 5-8.

74 A HISTORY OF MATHEMATICAL NOTATIONS

of subtraction in the critical notes to Schone's edition1 of the Metrica of Heron.

For equality the sign in the archetypal manuscripts seems to have been i°\ "but copyists introduced a sign which was sometimes con- fused with the sign l|" (Heath).

104. The notation for division comes under the same head as the notation for fractions (see § 41). In the case of unit fractions, a double accent is used with the denominator: thus y" = %. Sometimes a simple accent is used; sometimes it appears in a somewhat modified

form as ^, or (as Tannery interprets it) as X •' thus y^~ J . For \- appear the symbols Z' and ^, the latter sometimes without the dot. Of fractions that are not unit fractions, f has a peculiar sign U7 of its own, as was the case in Egyptian notations. "Curiously enough," says Heath, "it occurs only four times in Diophantus." In some old manuscripts the denominator is written above the numerator, in some rare cases. Once we find ie8 = *45 , the denominator taking the position where we place exponents. Another alternative is to write the numerator first and the denominator after it in the same line, marking the denominator with a submultiple sign in some form : thus, =f y = | .2 The following are examples of fractions from Diophantus :

From v. 10: l^ = ~ From v. 8, Lemma: 0ZV = 2 -J- J

l £* \-£

8 V 250

From iv. 3: sX*/ = ~ From iv. 15: r^ —

Fromvi. 12: ^M

= (60z2+2,520)/(z4+900-60z2) .

105. The fact that Diophantus had only one symbol for unknown quantity affected considerably his mode of exposition. Says Heath: "This limitation has made his procedure often very different from our modern work." As we have seen, Diophantus used but few symbols. Sometimes he ignored even these by describing an operation in words, when the symbol would have answered as well or better. Considering the amount of symbolism used, Diophantus' algebra may be desig- nated as "syncopated."

1 Heronis Alexandrini opera, Vol. Ill (Leipzig, 1903), p. 156, 1. 8, 10. The manuscript reading is novkbuv oSriS', the meaning of which is 74 — jV

2 Heath, op. oil., p. 45, 47.

INDIVIDUAL WRITERS 75

HINDU: BRAHMAGUPTA, SEVENTH CENTURY A.D.

106. We begin with a quotation from H. T. Colebrooke on Hindu algebraic notation:1 "The Hindu algebraists use abbreviations and initials for symbols: they distinguish negative quantities by a dot, but have not any mark, besides the absence of the negative sign, to discriminate a positive quantity. No marks or symbols (other than abbreviations of words) indicating operations of addition or multipli- cation, etc., are employed by them: nor any announcing equality2

or relative magnitude (greater or less) A fraction is indicated

by placing the divisor under the dividend, but without a line of sepa- ration. The two sides of an equation are ordered in the same manner,

one under the other The symbols of unknown quantity are not

confined to a single one: but extend to ever so great a variety of denominations: and the characters used are the initial syllables of the names of colours, excepting the first, which is the initial of ydvat- tdvat, as much as."

107. In Brahmagupta,3 and later Hindu writers, abbreviations occur which, when transliterated into our alphabet, are as follows:

ru for rupa, the absolute number ya for ydvat-tdvat, the (first) unknown ca for calaca (black), a second unknown ni for nilaca (blue), a third unknown pi for pitaca (yellow), a fourth unknown pa for pandu (white), a fifth unknown lo for lohita (red), a sixth unknown c for caranij surd, or square root ya v for x2, the v being the contraction for varga, square number

108. In Brahmagupta,4 the division of ru 3 c 450 c 75 c 54 by c 18 c 3 (i.e., 3+ V/450+l/ 75+1/54 by 1/18+1/3) is carried out as follows: "Put c 18 c 3. The dividend and divisor, multiplied by this, make ru 75 c 625. The dividend being then divided by the single surd

ru 15 constituting the divisor, the quotient is ru 5 c 3."

1 H. T. Colebrooke, Algebra, with Arithmetic and Mensuration from the Sanscrit of Bramegupta and Bhdscara (London, 1817), p. x, xi.

2 The Bakhshali MS (§ 109) was found after the time of Colebrooke and has an equality sign.

*Ibid., p. 339 ff.

4 Brahme-sphuta-sidd'hdnta, chap. xii. Translated by H. T. Colebrooke in op. cit. (1817), p. 277-378; we quote from p. 342.

A HISTORY OF MATHEMATICAL NOTATIONS

| In modern symbols, the statement is, substantially: Multipl Mend and divisor by 1/18— 1/3; the products are 75+1/675 an 15; divide the former by the latter, 5+1/3.

"Question 16.1 When does the residue of revolutions of the sur less one, fall, on a Wednesday, equal to the square root of two leg than the residue of revolutions, less one, multiplied by ten and aug mented by two?

"The value of residue of revolutions is to be here put square c ydvat-tdvat with two added : ya v 1 ru 2 is the residue of revolutions

Sanskrit character

or letters, by which the Hindus denote the unknown quan- tities in their notation, are the following: TJJ,. offf,

FIG. 31. — Sanskrit symbols for unknowns. (From Charles Hutton, Mai In matical Tracts, II, 167.) The first symbol, pa, is the contraction for "white"; th second, ca, the initial for "black"; the third, ni, the initial for "blue"; the fourti pi, the initial for "yellow"; the fifth, lo, for "red."

This less two isyavl; the square root of which isyal. Less one, it i ya 1 ru 1; which multiplied by ten is ya 10 ru 10; and augmented fy two, ya 10 ru 8. It is equal to the residue of revolutions yavl ru 2 less

1 a*. *. x f i xi • i ya v 0 ya 10 ru 8 ,,

one; viz. yav I ru 1. Statement of both sides ' ~ ., . li/qua

J ya v 1 ya 0 ru 1

subtraction being made conformably to rule 1 there arises ya v I

ya 10

Now, from the absolute number (9), multiplied by four times the [co efficient of the] square (36), and added to (100) the square of the [coefficient of the] middle term (making consequently 64), the square root being extracted (8), and lessened by the [coefficient of the] middle term (10), the remainder is 18 divided by twice the [coefficient of the square (2), yields the value of the middle term 9. Substituting with this in the expression put for the residue of revolutions, the answei comes out, residue of revolutions of the sun 83. Elapsed period ol days deduced from this, 393, must have the denominator in leasl terms added so often until it fall on Wednesday."

1 Colebrooke, op. ciL, p. 346. The abbreviations ru, c, ya, ya v, ca, ni, etc., an

f rar»o1if r»ro f innu r\f fV«r* /irki'»«r>c'r\/~kr»/4ir»rr IriffoTO \r\ fl»o Sianalrrif alnVial-kftt

INDIVIDUAL WRITERS 77

Notice that ya V ® ya J? m J signifies Oz2+10z-8 = z2+0;r+l. 7/a v 1 ya 0 ru 1

Brahmagupta gives1 the following equation in three unknown quantities and the expression of one unknown in terms of the other two:

"ya 197 ca 1644 nil ru 0 ya 0 ca 0 ni 0 ru 6302.

Equal subtraction being made, the value of ydvat-tdvat is ca 1644 ni 1 ru 6302 ."

M 197 In modern notation:

whence,

_1644?/+;g+6302 X~~ ~ 197

HINDU: THE BAKHSHALI MS

109. The so-called Bakhshali MS, found in 1881 buried in the earth near the village of Bakhshali in the northwestern frontier of India, is an arithmetic written on leaves of birch-bark, but has come down in mutilated condition. It is an incomplete copy of an older manuscript, the copy having been prepared, probably about the eighth, ninth, or tenth century. "The system of notation/' says A. F. Rudolph Hoernle,2 "is much the same as that employed in the arith- metical works of Brahmagupta and Bhaskara. There is, however, a very important exception. The sign for the negative quantity is a cross (+). It looks exactly like our modern sign for the positive quantity, but it is placed after the number which it qualifies. Thus

12 7 I

means 12—7 (i.e. 5). This is a sign which I have not met with

in any other Indian arithmetic ..... The sign now used is a dot placed over the number to which it refers. Here, therefore, there appears to be a mark of great antiquity. As to its origin I am not able to suggest any satisfactory explanation ..... A whole number, when it occurs in an arithmetical operation, as may be seen from the above given ex- ample, is indicated by placing the number 1 under it. This, however, is

1 Colebrooke, op. cit., p. 352.

2 "The Bakhshali Manuscript," Indian Antiquary, Vol. XVII (Bombay, 1888), p. 33-48, 275-79; see p. 34.

78

A HISTORY OF MATHEMATICAL NOTATIONS

a practice which is still occasionally observed in India The

following statement from the first example of the twenty-fifth siitra affords a good example of the system of notation employed in the Bakhshall arithmetic:

1

1 1 1 bhd 32 1 1 1

3+ 3+ 3+

phalarh 108

Here the initial dot is used much in the same way as we use the letter x to denote the unknown quantity, the value of which is sought. The number 1 under the dot is the sign of the whole (in this case, unknown) number. A fraction is denoted by placing one number under the other

without any line of separation; thus Q is ,,, i.e. one-third. A mixed

o o

number is shown by placing the three numbers under one another;

1

,1

1

thus 1 is 1+,, or 10, i.e. one and one-third. Hence 1

3+

means 1 — ,3

o

i.e. - ). Multiplication is usually indicated by placing the numbers side by side; thus

& 39.

phalam 20

5 32

8 1

1

o means 0X32 = 20. Similarly 1

222

means .-;X.;X,-, or 333

8

3+ 3+ 3+

i.e. ~. Bhd is an abbreviation of bhdga, 'part/ and means that the number preceding it is to be treated as a denominator. Hence

111 8 27

111 bhd means 1 4- ~= or -^-. The whole statement, therefore,

3+ 3+ 3+ Z7 *

1 1 1

1

1

1

1 bhd 32

3+ 3+ 3+

phalam 108 ,

27,

means Q X 32 = 108, and may be thus explained, — 'a certain number is

o

g

found by dividing with ^ and multiplying with 32; that number is 108.' The dot is also used for another purpose, namely as one of the

INDIVIDUAL WRITERS 79

ten fundamental figures of the decimal system of notation, or the zero (0123456789). It is still so used in India for both purposes, to

indicate the unknown quantity as well as the naught The

Indian dot, unlike our modern zero, is not properly a numerical figure at all. It is simply a sign to indicate an empty place or a hiatus. This is clearly shown by its name sdnya, 'empty/ .... Thus the two fig- ures 3 and 7, placed in juxtaposition (37), mean 'thirty-seven/ but with an 'empty space' interposed between them (3 7), they mean 'three hundred and seven/ To prevent misunderstanding the presence of the 'empty space' was indicated by a dot (3.7); or by what is now the zero (307). On the other hand, oc-

v ; . '

currmg in the statement of a problem, the 'empty place7

could be filled up, and here the -n .J0 ^ „ , l l .,, .,, , . ^' FIG. 32. — From Bakhshah arithmetic

dot which marked its presence (G. R. Kay6j Indian Mathematics [1915], signified a 'something' which p. 26; R. Hoernle, op. tit., p. 277). was to be discovered and to

be put in the empty place In its double signification, which

still survives in India, we can still discern an indication of that

country as its birthplace The operation of multiplication

alone is not indicated by any special sign. Addition is indicated by yu (for yuta), subtraction by + (ka for kanitaf) and division by bhd (for bhdga). The whole operation is commonly enclosed be- tween lines (or sometimes double lines), and the result is set down outside, introduced by pha (for phald)." Thus, pha served as a sign of equality.

The problem solved in Figure 32 appears from the extant parts to have been : Of a certain quantity of goods, a merchant has to pay, as duty, £, \, and J on three successive occasions. The total duty is 24. What was the original quantity of his goods? The solution ap- pears in the manuscript as follows: "Having subtracted the series from one," we get f , f , $ ; these multiplied together give | ; that again, subtracted from 1 gives f; with this, after having divided (i.e., in- verted, f), the total duty (24) is multiplied, giving 40; that is the original amount. Proof: £ multiplied by 40 gives 16 as the remainder. Hence the original amount is 40. Another proof: 40 multiplied by 1 — £ and 1 — } and 1 — J gives the result 16; the deduction is 24; hence the total is 40.

80 A HISTORY OF MATHEMATICAL NOTATIONS

HINDU: BHASKARA, TWELFTH CENTURY A.D.

110. Bhaskara speaks in his Lilavati1 of squares and cubes of numbers and makes an allusion to the raising of numbers to higher powers than the cube. Ganesa, a sixteenth-century Indian commen- tator of Bhaskara, specifics some of them. Taking the words varga for square of a number, and g'hana for cube of a number (found in Bhas- kara and earlier writers), Ganesa explains2 that the product of four like numbers is the square of a square, varga-varga; the product of six like numbers is the cube of a square, or square of a cube, varga-g'hana or g'hana-varga; the product of eight numbers gives varga-varga-varga; of nine, gives the cube of a cube, g'hana-g'hana. The fifth power was called varga-g'hana-ghdta; the seventh, varga-varga-g'hana-ghdta.

111. It is of importance to note that the higher powers of the unknown number are built up on the principle of involution, except the powers whose index is a prime number. According to this prin- ciple, indices are multiplied. Thus g'hana-varga does not mean n3-n2 = n5, but (tt3)2 = n6. Similarly, g'hana-g'hana does not mean n3«n3=n6, but (n3)3 = n9. In the case of indices that are prime, as in the fifth and seventh powers, the multiplicative principle became inoperative and the additive principle was resorted to. This is indicated by the word ghdta ("product")- Thus, varga-g'hana-ghdta means n2'n3 = n5.

In the application, whenever possible, of the multiplicative prin- ciple in building up a symbolism for the higher powers of a number, we see a departure from Diophantus. With Diophantus the symbol for x2, followed by the symbol for x3, meant x5; with the Hindus it meant x6. We shall see that among the Arabs and the Europeans of the thirteenth to the seventeenth centuries, the practice was divided, some following the Hindu plan, others the plan of Diophantus.

112. In Bhaskara, when unlike colors (dissimilar unknown quanti- ties, like x and y) are multiplied together, the result is called bhavita ("product"), and is abbreviated bha. Says Colebrooke: "The prod- uct of two unknown quantities is denoted by three letters or syllables, as ya.ca bha, ca.ni bha, etc. Or, if one of the quantities be a higher power, more syllables or letters are requisite; for the square, cube, etc., are likewise denoted by the initial syllables, va, gha, va-va, va-gha, gha-gha,* etc. Thus ya va • ca gha bha will signify the square of the

1 Colebrooke, op. cit., p. 9, 10. *Ibid., p. 10, n.3;p. 11.

8 Gha-gha for the sixth, instead of the ninth, power, indicates the use here of the additive principle.

INDIVIDUAL WRITERS

81

first unknown quantity multiplied by the cube of the second. A dot is, in some copies of the text and its commentaries, interposed between the factors, without any special direction, however, for this notation."1 Instead of ya va one finds in Brahmagupta and BhiLskara also the severer contraction ya v; similarly, one finds cav for the square of the second unknown.2

It should be noted also that "equations are not ordered so as to put all the quantities positive; nor to give precedence to a positive term in a compound quantity: for the negative terms are retained, and even preferably put in the first place."3

According to N. Raman ujacharia and G. R. Kaye,4 the content of the part of the manuscript shown in Figure 33 is as follows: The

*rtrv

,-*

jSg^g^aS

^^»^**^ S&iEHE^')^aft

[b^Uf\lifr&%Jd&uJU^I«fc

©i^tih^^^g^t 8feUtfi%it / ' ._ ^*^kk^^^?i^^''^/6^>g

FIG. 33. — Sridhara's Trisdtika. Sridhara was born 991 A.D. Ho is cited by Bhaskara; he explains the "Hindu method of completing the square" in solving quadratic equations.

circumference of a circle is equal to the square root of ten times the square of its diameter. The area is the square root of the product of ten with the square of half the diameter. Multiply the quantity whose square root cannot be found by any large number, take the square root of the product, leaving out of account the remainder. Divide it by the square root of the factor. To find the segment of a circle, take the sum of the chord and arrow, multiply it by the arrow, and square the product. Again multiply it by ten-ninths and extract its square root. Plane figures other than these areas should be calculated by considering them to be composed of quadrilaterals, segments of circles, etc.

1Op. cit., p. 140, n. 2; p. 141. In this quotation we omitted, for simplicity, some of the accents found in Colebrooke's transliteration from the Sanskrit.

2 Ibid., p. 63, 140, 346.

3 Ibid., p. xii.

4 Bibliotheca mathematica (3d ser.), Vol. XIIT (1912-13), p. 206, 213, 214.

82 A HISTORY OF MATHEMATICAL NOTATIONS

113. Bhaskara Achdbrya, "Lilavati,"1 11 50 A D— "Example: Tell me the fractions reduced to a common denominator which answer to three and a fifth, and one-third, proposed for addition; and those which correspond to a sixty-third and a fourteenth offered for sub- traction. Statement:

3 1 1 1 5 3

Answer: Reduced to a common denominator

45 3 5 G 53 15 15 15' bumi5-

Statement of the second example:

1 1

63 14 '

Answer: The denominator being abridged, or reduced to least terms, by the common measure seven, the fractions become

1 1 9 2*

Numerator and denominator, multiplied by the abridged denomina- p

2 9

tors, give respectively . ^p anc^ i or • Subtraction being made, the

difference is .™ .

114. Bhaskara Achdbrya, "Vija-Ganita."2— "Example: Tell quickly the result of the numbers three and four, negative or affirma- tive, taken together: .... The characters, denoting the quantities known and unknown, should be first written to indicate them gener- ally; and those, which become negative, should be then marked with a dot over them. Statement:3 3*4. Adding them, the sum is found 7. Statement: 3»4. Adding them, the sum is 7. Statement: 3*4. Tak- ing the difference, the result of addition comes out 1.

" 'So much as' and the colours 'black, blue, yellow and red/4 and others besides these, have been selected by venerable teaShers for names of values of unknown quantities, for the purpose of reckoning therewith.

1 Colebrooke, op. cit., p. 13, 14. 2 Ibid., p. 131.

3 In modern notation, 3+4 = 7, (-3) + (-4) = -7, 3 + (-4) = -1.

4 Colebrooke, op. cit., p. 139.

INDIVIDUAL WRITERS 83

"Example:1 Say quickly, friend, what will affirmative one un- known with one absolute, and affirmative pair unknown less eight absolute, make, if addition of the two sets take place? .... State- ment :2

ya 1 ru 1

ya 2 ru 8

Answer: The sum is ya 3 ru 7.

"When absolute number and colour (or letter) are multiplied one by the other, the product will be colour (or letter). When two, three or more homogeneous quantities are multiplied together, the product will be the square, cube or other [power] of the quantity. But, if unlike quantities be multiplied, the result is their (bhdvita) 'to be' product or factum.

"23. Example:3 Tell directly, learned sir, $he product of the multiplication of the unknown (ydvat-tdvat) five, less the absolute num- ber one, by the unknown (ydvat-tdvat) thrfce joined with the absolute two: .... Statement:4

ya 5 ru 1 ^ , . . _ _ •

0 0 Product: ya v 15 ya 1 ru 2 . ya 3 ru 2 J u

"Example:5 'So much as' three, 'black' five, 'blue' seven, all affirmative: how many do they make with negative two, three, and one of the same respectively, added to or subtracted from them? Statement:6

ya 3 ca 5 ni 7 Answer: Sum ya I ca 2 ni 6 . ya 2 ca 3 ni 1 Difference ya 5 ca 8 ni 8 .

"Example:7 Say, friend, [find] the sum and difference of two ir- rational numbers eight and two: .... after full consideration, if thou be acquainted with the sixfold rule of surds. Statement :8 c 2 c 8.

1 Ibid. 2 In modern notation, x-\- 1 and 2x— 8 have the sum 3x— 7.

3 Colebrookc, op. cit., p. 141, 142.

4 In modern notation (5x - 1) (3x +2) = 15x2 +7x -2.

5 Colcbrooke, op. cit., p. 144.

6 In modern symbols, 3x -\-5y-\-7z and — 2x— 3y—z have the sum z-f 2y+Gz, and the difference 5z-f 8?/-f 8z.

7 Colebrooke, op. cit., p. 146.

8 In modern symbols, the example is l/8+/2 = vT8, T/8-1/^/2. The same example is given earlier by Brahmagupta in his Brahme-sputOrsidd'hdnta, chap, xviii, in Colebrooke, op. cit., p. 341.

84 A HISTORY OF MATHEMATICAL NOTATIONS

Answer: Addition being made, the sum is c 18. Subtraction taking place, the difference is c 2."

ARABIC: aL-KHOWARizMi, NINTH CENTURY A.D.

115. In 772 Indian astronomy became known to Arabic scholars. As regards algebra, the early Arabs failed to adopt either the Dio- phantine or the Hindu notations. The famous Algebra of al-Khow£r- izmi of Bagdad was published in the original Arabic, together with an English translation, by Frederic Rosen,1 in 1831. He used a manu- script preserved in the Bodleian Collection at Oxford. An examination of this text shows that the exposition was altogether rhetorical, i.e., devoid of all symbolism. "Numerals arc in the text of the work al- ways expressed by words: [Hindu-Arabic] figures are only used in some of the diagrams, arid in a few marginal notes."2 As a specimen of al-Khowarizmi's exposition we quote the following from his Algebra, as translated by Rosen:

"What must be the amount of a square, which, when twenty-one dirhems are added to it, becomes equal to the equivalent of ten roots of that square? Solution: Halve the number of the roots; the moiety is five. Multiply this by itself; the product is twenty-five. Subtract from this the twenty-one which are connected with the square; the remainder is four. Extract its root; it is two. Subtract this from the moiety of the roots, which is five; the remainder is three. This is the root of the square which you required, and the square is nine. Or you may add the root to the moiety of the roots; the sum is seven; this is the root of the square which you sought for, and the square itself is forty-nine."3

By way of explanation, Rosen indicates the steps in this solution, expressed in modern symbols, as follows: Example:

ARABIC: aL-KARKHi, EARLY ELEVENTH CENTURY A.D. 116. It is worthy of note that while Arabic algebraists usually build up the higher powers of the unknown quantit}' on the multiplica- tive principle of the Hindus, there is at least one Arabic writer, al- Karkhi of Bagdad, who followed the Diophantine additive principle.4

1 The Algebra of Mohammed Ben Musa (cd. and trans. Frederic Rosen; London, 1831). See also L. C. Karpinski, Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi (1915).

2 Rosen, op. dt.., p. xv. 3 Ibid., p. 11.

4 See Cantor, op. cit., Vol. I (3d ed.), p. 767, 768; Heath, op. dt.t p. 41.

INDIVIDUAL WRITERS 85

In al-Kharki's work, the Fakhri, the word mal means x2, kacb means a3; the higher powers are mal mal for x4, mdl kacb for x5 (not for x6), kaLb kacb for x6 (not for x9), wdZ moZ kacb for x7 (not for x12), and so on. Cantor1 points out that there are cases among Arabic writers where mdl is made to stand for x, instead of x2, and that this ambiguity is reflected in the early Latin translations from the Arabic, where the word census sometimes means x, and not x2.2

BYZANTINE: MICHAEL PSELLUS, ELEVENTH CENTURY A.D.

117. Michael Psellus, a Byzantine writer of the eleventh century who among his contemporaries enjoyed the reputation of being the first of philosophers, wrote a letter3 about Diophantus, in which he gives the names of the successive powers of the unknown, used in Egypt, which are of historical interest in connection with the names used some centuries later by Nicolas Chuquet and Luca Pacioli. In Psellus the successive powers are designated as the first number, the second n umber (square), etc. This nomenclature appears to have been borrowed, through the medium of the commentary by Hypatia, from Anatolius, a contemporary of Diophantus.4 The association of the successive powers of the unknown with the series of natural numbers is perhaps a partial recognition of exponential values, for which there existed then, and for several centuries that followed Psellus, no ade- quate notation. The next power after the fourth, namely, x5, the Egyptians called "the first undescribed," because it is neither a square nor a cube; the sixth power they called the "cube-cube"; but the seventh was "the second undescribed," as being the product of the square and the "first undescribed." These expressions for x6 and x7 are closely related to Luca Pacioli's primo relato and secondo relato, found in his Summa of 1494.5 Was Pacioli directly or indirectly in- fluenced by Michael Psellus?

ARABIC: IBN ALBANNA, THIRTEENTH CENTURY A.D.

118. While the early Arabic algebras of the Orient are character- ized by almost complete absence of signs, certain later Arabic works on

1 Op. tit., p. 768. See also Karpinski, op. tit., p. 107, n. 1.

2 Such translations are printed by G. Libri, in his Histoire des stiences matht- matiques, Vol. I (Paris, 1838), p. 276, 277, 305.

3 Reproduced by Paul Tannery, op. tit., Vol. II (1895), p. 37-42.

4 See Heath, op. tit., p. 2, 18.

5 See ibid., p. 41; Cantor, op. tit., Vol. II (2d ed.), p. 317.

86 A HISTORY OF MATHEMATICAL NOTATIONS

algebra, produced in the Occident, particularly that of al-Qalasadi of Granacla, exhibit considerable symbolism. In fact, as early as the thirteenth century symbolism began to appear; for example, a nota- tion for continued fractions in al-Ha$sar (§391). Ibn Khaldun1 states that Ibn Albanna at the close of the thirteenth century wrote a book when under the influence of the works of two predecessors, Ibn Almuncim and Alahdab. "He [Ibn Albanna] gave a summary of the demonstrations of these two works and of other things as well, con- cerning the technical employment of symbols2 in the proofs, which serve at the same time in the abstract reasoning and the representa- tion to the eye, wherein lies the secret and essence of the explication of theorems of calculation with the aid of signs." This statement of Ibn Khaldun, from which it would seem that symbols were used by Arabic mathematicians before the thirteenth century, finds apparent confirmation in the translation of an Arabic text into Latin, effected by Gerard of Cremona (1114-87). This translation contains symbols for x and x2 which we shall notice more fully later. It is, of course, quite possible that these notations were introduced into the text by the translator and did not occur in the original Arabic. As regards Ibn Albanna, many of his writings have been lost and none of his extant works contain algebraic symbolism.

CHINESE: cnu SHIH-CHIEH (1303 A.D.)

119. Chu Shih-Chieh bears the distinction of having been "in- strumental in the advancement of the Chinese abacus algebra to the highest mark it has ever attained."3 The Chinese notation is interest- ing as being decidedly unique. Chu Shih-Chieh published in 1303 a treatise, entitled Szu-yuen Yii-chien, or "The Precious Mirror of the Four Elements," from which our examples are taken. An expression like a+6+c+d, and its square, a2+62+c2+d2+2a6+2ac+2ad+

1 Consult F. Woepcke, "Rechcrches sur Fhistoire dcs sciences mathematiques chez les orientaux," Journal asiatique (5th ser.), Vol. IV (Paris, 1854), p. 369-72; Woepckc quotes the original Arabic and gives a translation in French. See also Cantor, op, cit., Vol. I (3d ed,), p. 805.

2 Or, perhaps, letters of the alphabet.

3 Yoshio Mikami, The Development of Mathematics in China and Japan (Leip- zig, 1912), p. 89. All our information relating to Chinese algebra is drawn from this book, p. 89-98.

INDIVIDUAL WRITERS

87

2bc+2bd+2cd, were represented as shown in the following two illus- trations:

1

1 202

2 1*1 1 0 -X- 0 1

2 1 202

1

Where we have used the asterisk in the middle, the original has the character t'ai ("great extreme"). We may interpret this symbolism by considering a located one space to the right of the asterisk (•#),& above, c to the left, and d below. In the symbolism for the square of a+b+c+d, the O's indicate that the terms a, 6, c, d do not occur in the expression. The squares of these letters are designated by the 1's two spaces from -K-. The four 2's farthest from -)£ stand for 2ab, 2ac, 2bc, 2bd, respectively, while the two 2's nearest to -)f stand for 2ac and 2bd. One is impressed both by the beautiful symmetry and by the extreme limitations of this notation.

120. Previous to Chu Shih-Chieh's time algebraic equations of only one unknown number were considered ; Chu extended the process to as many as four unknowns. These unknowns or elements were called the "elements of heaven, earth, man, and thing." Mikami states that, of these, the heaven clement was arranged below the known quantity (which was called "the great extreme"), the earth clement to the left, the man element to the right, and the thing ele- ment above. Letting -)f stand for the great extreme, and x, y, z, u, for heaven, earth, man, thing, respectively, the idea is made plain by the following representations :

Mikami gives additional illustrations:

0

0 1

1

0-2

#

0

1

0

1

+2yz

xz+z2

88

A HISTORY OF MATHEMATICAL NOTATIONS

Using the Hindu-Arabic numerals in place of the Chinese calculating pieces or rods, Mikami represents three equations, used by Chu, in the following manner:

In our notation, the four equations are, respectively,

a) 2x —

6) z2+ i/2- 22 = 0,

c) 2x +2y - u = 0.

No sign of equality is used here. All terms appear on one side of the equation. Notwithstanding the two-dimensional character of the notation, which permits symbols to be placed above and below the starting-point, as well as to left and right, it made insufficient pro- vision for the representation of complicated expressions and for easy methods of computation. The scheme does not lend itself easily to varying algebraic forms. It is difficult to see how, in such a system, the science of algebra could experience a rapid and extended growth. The fact that Chinese algebra reached a standstill after the thirteenth century may be largely due to its inelastic and faulty notation.

BYZANTINE: MAXIMUS PLANUDES, FOURTEENTH CENTURY A.D.

121. Maximus Planudes, a monk of the first half of the fourteenth century residing in Constantinople, brought out among his various compilations in Greek an arithmetic,1 and also scholia to the first two books of Diophantus' Arithmetical These scholia are of interest to us, for, while Diophantus evidently wrote his equations in the running text and did not assign each equation a separate line, we find in Planudes the algebraic work broken up so that each step or each equation is assigned a separate line, in a manner closely resembling modern practice. To illustrate this, take the problem in Diophantus (i. 29),

1 Das Recheribuch des Maximus Planudes (Halle: herausgegeben von C. I. Gcrhardt, 1865).

2 First printed in Xylander's Latin translation of Diophantus' Arilhmetica (Basel, 1575). These scholia in Diophantus are again reprinted in P. Tannery, Diophanti Alexandrini opera omnia (Lipsiae, 1895), Vol. II, p. 123-255; the ex- ample which we quote is from p. 201.

INDIVIDUAL WRITERS 89

"to find two numbers such that their sum and the difference of their squares are given numbers." We give the exposition of Planudes and its translation.

Planudes Translation

K TT [Given the numbers], 20, 80

eK& • sd/x°Z M0*Asa Putting for the numbers, x+10,

10-z rerp • AFdss/c^°p A Yd{jL°p A ss/c . . .Squaring, £2+20#+100,

z2+100-20z

virepox* ssju lff" JU°TT Taking the difference, 40# = 80

/xep • sd I9* jjpp Dividing, x = 2

for • M°^ AM?' Result, 12, 8

ITALIAN: LEONARDO OF PISA (1202 A.D.)

122. Leonardo of Pisa's mathematical writings are almost wholly rhetorical in mode of exposition. In his Liber abbaci (1202) he used the Hindu-Arabic numerals. To a modern reader it looks odd to see expressions like -£$ "A t 42, the fractions written before the integer in the case of a mixed number. Yet that mode of writing is his invariable practice. Similarly, the coefficient of x is written after the name for x, as, for example,1 — "radices ^12" for I2%x. A computation is indi- cated, or partly carried out, on the margin of the page, and is inclosed in a rectangle, or some irregular polygon whose angles are right angles. The reason for the inverted order of writing coefficients or of mixed numbers is due, doubtless, to the habit formed from the study of Arabic works; the Arabic script proceeds from right to left. Influ- enced again by Arabic authors, Leonardo <