# A History of Mathematical Notations/Volume 1

A HISTORY OF MATHEMATICAL NOTATIONS
FLORIAN CAJORI
A HISTORY OF MATHEMATICAL NOTATIONS
Volume I
NOTATIONS IN ELEMENTARY MATHEMATICS
A HISTORY OF MATHEMATICAL NOTATIONS
By FLORIAN CAJORI, Ph.D.
Professor of the History of Mathematics University of California
Volume I
NOTATIONS IN ELEMENTARY MATHEMATICS
The Open Court Publishing Company LA SALLE · ILLINOIS
Copyright 1928 By The Open Court Publishing Company Published September 1928
Printed in the United States of America By Paquin Printers, Chicago
PREFACE
The study of the history of mathematical notations was suggested to me by Professor E. H. Moore, of the University of Chicago. To him and to Professor M. W. Haskell, of the University of California, I am indebted for encouragement in the pursuit of this research. As completed in August, 1925, the present history was intended to be brought out in one volume. To Professor H. E. Slaught, of the University of Chicago, I owe the suggestion that the work be divided into two volumes, of which the first should limit itself to the history of symbols in elementary mathematics, since such a volume would appeal to a wider constituency of readers than would be the case with the part on symbols in higher mathematics. To Professor Slaught I also owe generous and vital assistance in many other ways. He examined the entire manuscript of this work in detail, and brought it to the sympathetic attention of the Open Court Publishing Company. I desire to record my gratitude to Mrs. Mary Hegeler Carus, president of the Open Court Publishing Company, for undertaking this expensive publication from which no financial profits can be expected to accrue.
I gratefully acknowledge the assistance in the reading of the proofs of part of this history rendered by Professor Haskell, of the University of California; Professor R. C. Archibald, of Brown University; and Professor L. C. Karpinski, of the University of Michigan.
Florian Cajori
TABLE OF CONTENTS
* I.Introduction
* PARAGRAPHS
* II.Numeral Symbols and Combination 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 Calderón................................................................................................................................................................................................................................................................................................................................................................................................92–93
* The Portuguese Cifrão................................................................................................................................................................................................................................................................................................................................................................................................94
* Relative Size of Numerals in Tables................................................................................................................................................................................................................................................................................................................................................................................................95
* Fanciful Hypotheses on the Origin of Numeral Forms................................................................................................................................................................................................................................................................................................................................................................................................96
* A Sporadic Artificial System................................................................................................................................................................................................................................................................................................................................................................................................97
* General Remarks................................................................................................................................................................................................................................................................................................................................................................................................98
* Opinion of Laplace................................................................................................................................................................................................................................................................................................................................................................................................99
* III.Symbols in Arithmetic and Algebra (Elementary Part)................................................................................................................................................................................................................................................................................................................................................................................................100
* A.Groups of Symbols Used by Individual Writers101
* Greeks—Diophantus, Third Century A.D.................................................................................................................................................................................................................................................................................................................................................................................................101–5
* Hindu—Brahmagupta, Seventh Century................................................................................................................................................................................................................................................................................................................................................................................................106–8
* Hindu—The Bakhshālī Manuscript................................................................................................................................................................................................................................................................................................................................................................................................109
* Hindu—Bhāskara, Twelfth Century................................................................................................................................................................................................................................................................................................................................................................................................110–14
* Arabic—al-Khowârizmî, Ninth Century................................................................................................................................................................................................................................................................................................................................................................................................115
* Arabic—al-Karkhf, Eleventh Century................................................................................................................................................................................................................................................................................................................................................................................................116
* Byzantine—Michael Psellus, Eleventh Century................................................................................................................................................................................................................................................................................................................................................................................................117
* Arabic—Ibn Albanna, Thirteenth Century................................................................................................................................................................................................................................................................................................................................................................................................118
* Chinese—Chu Shih-Chieh, Fourteenth Century................................................................................................................................................................................................................................................................................................................................................................................................119, 120
* Byzantine—Maximus Planudes, Fourteenth Century................................................................................................................................................................................................................................................................................................................................................................................................121
* Italian—Leonardo of Pisa, Thirteenth Century................................................................................................................................................................................................................................................................................................................................................................................................122
* French—Nicole Oresme, Fourteenth Century................................................................................................................................................................................................................................................................................................................................................................................................123
* Arabic—al-Qalasâdî, Fifteenth Century................................................................................................................................................................................................................................................................................................................................................................................................124
* German—Regiomontanus, Fifteenth Century................................................................................................................................................................................................................................................................................................................................................................................................125–27
* Italian—Earliest Printed Arithmetic, 1478................................................................................................................................................................................................................................................................................................................................................................................................128
* French—Nicolas Chuquet, 1484................................................................................................................................................................................................................................................................................................................................................................................................129-31
* French—Estienne de la Roche, 1520................................................................................................................................................................................................................................................................................................................................................................................................132
* Italiaim-Pietro Borgi, 1484, 1488................................................................................................................................................................................................................................................................................................................................................................................................133
* Italian—Luca Pacioli, 1494, 1523................................................................................................................................................................................................................................................................................................................................................................................................134–38
* Italian—F. Ghaligai, 1521, 1548, 1552................................................................................................................................................................................................................................................................................................................................................................................................139
* Italian—H. Cardan, 1532, 1545, 1570................................................................................................................................................................................................................................................................................................................................................................................................140, 141
* Italian—Nicolo Tartaglia, 1506–60................................................................................................................................................................................................................................................................................................................................................................................................142, 143
* Italian—Rafaele Bombelli, 1572................................................................................................................................................................................................................................................................................................................................................................................................144, 145
* German—Johann Widman, 1489, 1526................................................................................................................................................................................................................................................................................................................................................................................................146
* Austrian—Grammateus, 1518, 1535................................................................................................................................................................................................................................................................................................................................................................................................147
* German—Christoff Rudolff, 1525................................................................................................................................................................................................................................................................................................................................................................................................148, 149
* Dutch—Gielis van der Hoecke, 1537................................................................................................................................................................................................................................................................................................................................................................................................150
* German—Michael Stifel, 1544, 1545, 1553................................................................................................................................................................................................................................................................................................................................................................................................151–56
* German—Nicolaus Copernicus, 1566................................................................................................................................................................................................................................................................................................................................................................................................157
* German—Johann Scheubel, 1545, 1551................................................................................................................................................................................................................................................................................................................................................................................................158,159
* Maltese—Wil. Klebitius, 1565................................................................................................................................................................................................................................................................................................................................................................................................160
* German—Christophorus Clavius, 1608................................................................................................................................................................................................................................................................................................................................................................................................161
* Belgium—Simon Stevin, 1585................................................................................................................................................................................................................................................................................................................................................................................................162,163
* Lorraine—Albert Girard, 1629................................................................................................................................................................................................................................................................................................................................................................................................164
* German-Spanish—Marco Aurel, 1552................................................................................................................................................................................................................................................................................................................................................................................................165
* Portuguese-Spanish—Pedro Nunez, 1567................................................................................................................................................................................................................................................................................................................................................................................................166
* English—Robert Recorde, 1543(?), 1557................................................................................................................................................................................................................................................................................................................................................................................................167–68
* English—John Dee, 1570................................................................................................................................................................................................................................................................................................................................................................................................169
* English—Leonard and Thomas Digges, 1579................................................................................................................................................................................................................................................................................................................................................................................................170
* English—Thomas Masterson, 1592................................................................................................................................................................................................................................................................................................................................................................................................171
* French—Jacques Peletier, 1554................................................................................................................................................................................................................................................................................................................................................................................................172
* French—Jean Buteon, 1559................................................................................................................................................................................................................................................................................................................................................................................................173
* French—Guillaume Gosselin, 1577................................................................................................................................................................................................................................................................................................................................................................................................174
* French—Francis Vieta, 1591................................................................................................................................................................................................................................................................................................................................................................................................176-78
* Italian—Bonaventura Cavalieri, 1647................................................................................................................................................................................................................................................................................................................................................................................................179
* English—William Oughtred, 1631, 1632, 1657................................................................................................................................................................................................................................................................................................................................................................................................180–87
* English—Thomas Harriot, 1631................................................................................................................................................................................................................................................................................................................................................................................................188
* French—Pierre Herigone, 1634, 1644................................................................................................................................................................................................................................................................................................................................................................................................189
* Scot-French—James Hume, 1635, 1636................................................................................................................................................................................................................................................................................................................................................................................................190
* French—René Descartes................................................................................................................................................................................................................................................................................................................................................................................................191
* English—Isaac Barrow................................................................................................................................................................................................................................................................................................................................................................................................192
* English—Richard Rawlinson, 1655-68................................................................................................................................................................................................................................................................................................................................................................................................193
* Swiss—Johann Heinrich Rahn................................................................................................................................................................................................................................................................................................................................................................................................194
* English—John Wallis, 1655, 1657, 1685................................................................................................................................................................................................................................................................................................................................................................................................195, 196
* Extract from Acta eruditorum, Leipzig, 1708................................................................................................................................................................................................................................................................................................................................................................................................197
* Extract from Miscellanea Berolinensia, 1710 (Due to G. W. Leibniz)................................................................................................................................................................................................................................................................................................................................................................................................198
* Conclusions................................................................................................................................................................................................................................................................................................................................................................................................199
* B.Topical Survey of the Use of Notations.................................................................................................................................................................................................................................................................................................................................................................................................200–356
* Signs of Addition and Subtraction.................................................................................................................................................................................................................................................................................................................................................................................................200–216
* Early Symbols................................................................................................................................................................................................................................................................................................................................................................................................200
* Origin and Meaning of the Signs.................................................................................................................................................................................................................................................................................................................................................................................................201–3
* Spread of the + and — Symbols.................................................................................................................................................................................................................................................................................................................................................................................................204
* Shapes of the -p Sign................................................................................................................................................................................................................................................................................................................................................................................................205–7
* Varieties of — Signs................................................................................................................................................................................................................................................................................................................................................................................................208, 209
* Symbols for "Plus or Minus"................................................................................................................................................................................................................................................................................................................................................................................................210, 211
* Certain Other Specialized Uses of + and —................................................................................................................................................................................................................................................................................................................................................................................................212–14
* Four Unusual Signs................................................................................................................................................................................................................................................................................................................................................................................................215
* Composition of Ratios................................................................................................................................................................................................................................................................................................................................................................................................216
* Signs of Multiplication................................................................................................................................................................................................................................................................................................................................................................................................217–34
* Early Symbols................................................................................................................................................................................................................................................................................................................................................................................................217
* Early Uses of the St. Andrew's Cross, but Not as the Symbol of Multiplication of Two Numbers................................................................................................................................................................................................................................................................................................................................................................................................218–30
* The Process of Two False Positions................................................................................................................................................................................................................................................................................................................................................................................................219
* Compound Proportions with Integers................................................................................................................................................................................................................................................................................................................................................................................................220
* Proportions Involving Fractions................................................................................................................................................................................................................................................................................................................................................................................................221
* Addition and Subtraction of Fractions................................................................................................................................................................................................................................................................................................................................................................................................222
* Division of Fractions................................................................................................................................................................................................................................................................................................................................................................................................223
* Casting Out the 9's, 7's, or 11's................................................................................................................................................................................................................................................................................................................................................................................................225
* Multiplication of Integers................................................................................................................................................................................................................................................................................................................................................................................................226
* Reducing Radicals to Radicals of the Same Order................................................................................................................................................................................................................................................................................................................................................................................................227
* Marking the Place for "Thousands"................................................................................................................................................................................................................................................................................................................................................................................................228
* Place of Multiplication Table above 5×5................................................................................................................................................................................................................................................................................................................................................................................................229
* The St. Andrew’s Cross Used as a Symbol of Multiplication................................................................................................................................................................................................................................................................................................................................................................................................231
* Unsuccessful Symbols for Multiplication................................................................................................................................................................................................................................................................................................................................................................................................232
* The Dot for Multiplication................................................................................................................................................................................................................................................................................................................................................................................................233
* The St. Andrew's Cross in Notation for Transfinite Ordinal Numbers................................................................................................................................................................................................................................................................................................................................................................................................234
* Signs of Division and Ratio................................................................................................................................................................................................................................................................................................................................................................................................235–47
* Early Symbols................................................................................................................................................................................................................................................................................................................................................................................................235, 236
* Rahn's Notation................................................................................................................................................................................................................................................................................................................................................................................................237
* Leibniz’s Notations................................................................................................................................................................................................................................................................................................................................................................................................238
* Relative Position of Divisor and Dividend................................................................................................................................................................................................................................................................................................................................................................................................241
* Order of Operations in Terms Containing Both ÷ and ×................................................................................................................................................................................................................................................................................................................................................................................................242
* A Critical Estimate of : and ÷ as Symbols................................................................................................................................................................................................................................................................................................................................................................................................243
* Notations for Geometric Ratio................................................................................................................................................................................................................................................................................................................................................................................................244
* Division in the Algebra of Complex Numbers................................................................................................................................................................................................................................................................................................................................................................................................247
* Signs of Proportion................................................................................................................................................................................................................................................................................................................................................................................................248–59
* Arithmetical and Geometrical Progression................................................................................................................................................................................................................................................................................................................................................................................................248
* Arithmetical Proportion................................................................................................................................................................................................................................................................................................................................................................................................249
* Geometrical Proportion................................................................................................................................................................................................................................................................................................................................................................................................250
* Oughtred's Notation................................................................................................................................................................................................................................................................................................................................................................................................251
* Struggle in England between Oughtred's and Wing's Notations before 1700................................................................................................................................................................................................................................................................................................................................................................................................252
* Struggle in England between Oughtred's and Wing's Notations during 1700–1750................................................................................................................................................................................................................................................................................................................................................................................................253
* Sporadic Notations................................................................................................................................................................................................................................................................................................................................................................................................254
* Oughtred's Notation on the European Continent................................................................................................................................................................................................................................................................................................................................................................................................255
* Slight Modifications of Oughtred's Notation................................................................................................................................................................................................................................................................................................................................................................................................257
* The Notation:::: in Europe and America................................................................................................................................................................................................................................................................................................................................................................................................258
* The Notation of Leibniz................................................................................................................................................................................................................................................................................................................................................................................................259
* Signs of Equality................................................................................................................................................................................................................................................................................................................................................................................................260–70
* Early Symbols................................................................................................................................................................................................................................................................................................................................................................................................260
* Recorde's Sign of Equality................................................................................................................................................................................................................................................................................................................................................................................................261
* Different Meanings of =................................................................................................................................................................................................................................................................................................................................................................................................262
* Competing Symbols................................................................................................................................................................................................................................................................................................................................................................................................263
* Descartes' Sign of Equality................................................................................................................................................................................................................................................................................................................................................................................................264
* Variations in the Form of Descartes' Symbol................................................................................................................................................................................................................................................................................................................................................................................................265
* Struggle for Supremacy................................................................................................................................................................................................................................................................................................................................................................................................266
* Variation in the Form of Recorde's Symbol................................................................................................................................................................................................................................................................................................................................................................................................268
* Variation in the Manner of Using It................................................................................................................................................................................................................................................................................................................................................................................................269
* Nearly Equal................................................................................................................................................................................................................................................................................................................................................................................................270
* Signs of Common Fractions................................................................................................................................................................................................................................................................................................................................................................................................271–75
* Early Forms................................................................................................................................................................................................................................................................................................................................................................................................271
* The Fractional Line................................................................................................................................................................................................................................................................................................................................................................................................272
* Special Symbols for Simple Fractions................................................................................................................................................................................................................................................................................................................................................................................................274
* The Solidus................................................................................................................................................................................................................................................................................................................................................................................................275
* Signs of Decimal Fractions................................................................................................................................................................................................................................................................................................................................................................................................276–89
* Stevin's Notation................................................................................................................................................................................................................................................................................................................................................................................................276
* Other Notations Used before 1617................................................................................................................................................................................................................................................................................................................................................................................................278
* Did Pitiscus Use the Decimal Point?................................................................................................................................................................................................................................................................................................................................................................................................279
* Decimal Comma and Point of Napier................................................................................................................................................................................................................................................................................................................................................................................................282
* Seventeenth-Century Notations Used after 1617................................................................................................................................................................................................................................................................................................................................................................................................283
* Eighteenth-Century Discard of Clumsy Notations................................................................................................................................................................................................................................................................................................................................................................................................285
* Nineteenth Century: Different Positions for Point and for Comma................................................................................................................................................................................................................................................................................................................................................................................................286
* Signs for Repeating Decimals................................................................................................................................................................................................................................................................................................................................................................................................289
* Signs of Powers................................................................................................................................................................................................................................................................................................................................................................................................290–315
* General Remarks................................................................................................................................................................................................................................................................................................................................................................................................290
* Double Significance of R and l................................................................................................................................................................................................................................................................................................................................................................................................291
* Facsimiles of Symbols in Manuscripts................................................................................................................................................................................................................................................................................................................................................................................................293
* Two General Plans for Marking Powers................................................................................................................................................................................................................................................................................................................................................................................................294
* Early Symbolisms: Abbreviative Plan, Index Plan................................................................................................................................................................................................................................................................................................................................................................................................295
* Notations Applied Only to an Unknown Quantity, the Base Being Omitted................................................................................................................................................................................................................................................................................................................................................................................................296
* Notations Applied to Any Quantity, the Base Being Designated................................................................................................................................................................................................................................................................................................................................................................................................297
* Descartes’ Notation of 1637................................................................................................................................................................................................................................................................................................................................................................................................298
* Did Stampioen Arrive at Descartes’ Notation Independently?................................................................................................................................................................................................................................................................................................................................................................................................299
* Notations Used by Descartes before 1637................................................................................................................................................................................................................................................................................................................................................................................................300
* Use of Hérigone's Notation after 1637................................................................................................................................................................................................................................................................................................................................................................................................301
* Later Use of Hume’s Notation of 1636................................................................................................................................................................................................................................................................................................................................................................................................302
* Other Exponential Notations Suggested after 1637................................................................................................................................................................................................................................................................................................................................................................................................303
* Spread of Descartes' Notation................................................................................................................................................................................................................................................................................................................................................................................................307
* Negative, Fractional, and Literal Exponents................................................................................................................................................................................................................................................................................................................................................................................................308
* Imaginary Exponents................................................................................................................................................................................................................................................................................................................................................................................................309
* Notation for Principal Values................................................................................................................................................................................................................................................................................................................................................................................................312
* Complicated Exponents................................................................................................................................................................................................................................................................................................................................................................................................313
* D. F. Gregory’s $(+)^{r}$................................................................................................................................................................................................................................................................................................................................................................................................314
* Conclusions................................................................................................................................................................................................................................................................................................................................................................................................315
* Signs for Roots................................................................................................................................................................................................................................................................................................................................................................................................316–38
* Early Forms, General Statement................................................................................................................................................................................................................................................................................................................................................................................................316, 317
* The Sign ℞, First Appearance................................................................................................................................................................................................................................................................................................................................................................................................318
* Sixteenth-Century Use of ℞................................................................................................................................................................................................................................................................................................................................................................................................319
* Seventeenth-Century Use of ℞................................................................................................................................................................................................................................................................................................................................................................................................321
* The Sign l................................................................................................................................................................................................................................................................................................................................................................................................322
* Napier’s Line Symbolism................................................................................................................................................................................................................................................................................................................................................................................................323
* The Sign √................................................................................................................................................................................................................................................................................................................................................................................................324–38
* Origin of √................................................................................................................................................................................................................................................................................................................................................................................................324
* Spread of the √................................................................................................................................................................................................................................................................................................................................................................................................327
* Rudolff’s Signs outside of Germany................................................................................................................................................................................................................................................................................................................................................................................................328
* Stevin’s Numeral Root-Indices................................................................................................................................................................................................................................................................................................................................................................................................329
* Rudolff and Stifel's Aggregation Signs................................................................................................................................................................................................................................................................................................................................................................................................332
* Descartes' Union of Radical Sign and Vinculum................................................................................................................................................................................................................................................................................................................................................................................................333
* Other Signs of Aggregation of Terms................................................................................................................................................................................................................................................................................................................................................................................................334
* Redundancy in the Use of Aggregation Signs................................................................................................................................................................................................................................................................................................................................................................................................335
* Peculiar Dutch Symbolism................................................................................................................................................................................................................................................................................................................................................................................................336
* Principal Root-Value................................................................................................................................................................................................................................................................................................................................................................................................337
* Recommendation of the U.S. National Committee................................................................................................................................................................................................................................................................................................................................................................................................338
* Signs for Unknown Numbers.................................................................................................................................................................................................................................................................................................................................................................................................339–41
* Early Forms................................................................................................................................................................................................................................................................................................................................................................................................339
* Crossed Numerals Representing Powers of Unknowns................................................................................................................................................................................................................................................................................................................................................................................................340
* Descartes' z, y, x................................................................................................................................................................................................................................................................................................................................................................................................340
* Spread of Descartes' Signs.................................................................................................................................................................................................................................................................................................................................................................................................341
* Signs of Aggregation................................................................................................................................................................................................................................................................................................................................................................................................342–50
* Introduction................................................................................................................................................................................................................................................................................................................................................................................................342
* Aggregation Expressed by Letters................................................................................................................................................................................................................................................................................................................................................................................................343
* Aggregation Expressed by Horizontal Bars or Vinculums................................................................................................................................................................................................................................................................................................................................................................................................344
* Aggregation Expressed by Dots................................................................................................................................................................................................................................................................................................................................................................................................348
* Aggregation Expressed by Commas................................................................................................................................................................................................................................................................................................................................................................................................349
* Aggregation Expressed by Parentheses................................................................................................................................................................................................................................................................................................................................................................................................350
* Early Occurrence of Parentheses................................................................................................................................................................................................................................................................................................................................................................................................351
* Terms in an Aggregate Placed in a Vertical Column................................................................................................................................................................................................................................................................................................................................................................................................353
* Marking Binomial Coefficients................................................................................................................................................................................................................................................................................................................................................................................................354
* Special Uses of Parentheses................................................................................................................................................................................................................................................................................................................................................................................................355
* A Star to Mark the Absence of Terms................................................................................................................................................................................................................................................................................................................................................................................................356
* IV.Symbols in Geometry (Elementary Part)................................................................................................................................................................................................................................................................................................................................................................................................357–85
* A.Ordinary Elementary Geometry................................................................................................................................................................................................................................................................................................................................................................................................357
* Early Use of Pictographs................................................................................................................................................................................................................................................................................................................................................................................................357
* Signs for Angles................................................................................................................................................................................................................................................................................................................................................................................................360
* Signs for "Perpendicular"................................................................................................................................................................................................................................................................................................................................................................................................364
* Signs for Triangle, Square, Rectangle, Parallelogram................................................................................................................................................................................................................................................................................................................................................................................................365
* The Square as an Operator................................................................................................................................................................................................................................................................................................................................................................................................366
* Sign for Circle................................................................................................................................................................................................................................................................................................................................................................................................367
* Signs for Parallel Lines................................................................................................................................................................................................................................................................................................................................................................................................368
* Signs for Equal and Parallel................................................................................................................................................................................................................................................................................................................................................................................................369
* Signs for Arcs of Circles................................................................................................................................................................................................................................................................................................................................................................................................370
* Other Pictographs................................................................................................................................................................................................................................................................................................................................................................................................371
* Signs for Similarity and Congruence................................................................................................................................................................................................................................................................................................................................................................................................372
* The Sign ≎ for Equivalence................................................................................................................................................................................................................................................................................................................................................................................................375
* Lettering of Geometric Figures................................................................................................................................................................................................................................................................................................................................................................................................376
* Sign for Spherical Excess................................................................................................................................................................................................................................................................................................................................................................................................380
* Symbols in the Statement of Theorems................................................................................................................................................................................................................................................................................................................................................................................................381
* Signs for Incommensurables................................................................................................................................................................................................................................................................................................................................................................................................382
* Unusual Ideographs in Elementary Geometry................................................................................................................................................................................................................................................................................................................................................................................................383
* Algebraic Symbols in Elementary Geometry................................................................................................................................................................................................................................................................................................................................................................................................384
* B.Past Struggles between Symbolists and Rhetoricians in Elementary Geometry................................................................................................................................................................................................................................................................................................................................................................................................385
* Index
ILLUSTRATIONS
* FIGUREPARAGRPAHS
* 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) Numeral................................................................................................................................................................................................................................................................................................................................................................................................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
* FIGUREPARAGRPAHS
* 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.Bakhshālī Arithmetic.................................................................................................................................................................................................................................................................................................................................................................................................109
* 33.Śrīdhara's Trisátikā................................................................................................................................................................................................................................................................................................................................................................................................112
* 34.Oresme's Algorismus Proportionum................................................................................................................................................................................................................................................................................................................................................................................................123
* 35.al-Qalâsadî'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 60B................................................................................................................................................................................................................................................................................................................................................................................................132
* 41.De la Roche's Larismethique, Folio 66A................................................................................................................................................................................................................................................................................................................................................................................................132
* 42.Part of Page in Pacioli's Summa, 1523................................................................................................................................................................................................................................................................................................................................................................................................138
* 43.Margin of Folio 123B 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. 595 A 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 Goss, 1525................................................................................................................................................................................................................................................................................................................................................................................................148
* 59.From Chr. Rudolff's Coss, Ev.................................................................................................................................................................................................................................................................................................................................................................................................148
* 60.From Van der Hoecke' In arithmetica................................................................................................................................................................................................................................................................................................................................................................................................150
* 61.Part of Page from Stifel's Arithmetica integra, 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 Clavius' 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. Recorde,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 Arithmeticke, 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................................................................................................................................................................................................................................................................................................................................................................................................183
* 88.From Thomas Harriot, 1631, Page 65................................................................................................................................................................................................................................................................................................................................................................................................180
* 89.From Hérigone, Cursus mathematicus, 1644................................................................................................................................................................................................................................................................................................................................................................................................189
* 90.Roman Numerals for x in J. Hume, 1635................................................................................................................................................................................................................................................................................................................................................................................................191
* FIGUREPARAGRPAHS
* 91.Radicals in J. Hume, 1635................................................................................................................................................................................................................................................................................................................................................................................................191
* 92. R. Descartes, Geometrie................................................................................................................................................................................................................................................................................................................................................................................................191
* 93.I. Barrow's Euclid, Latin Edition. Notes by Isaac Newton................................................................................................................................................................................................................................................................................................................................................................................................193
* 94.I. Barrow's Euclid, 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................................................................................................................................................................................................................................................................................................................................................................................................291
* 105.Written Algebraic Symbols for Powers from Perez de Moya's Arithmetica................................................................................................................................................................................................................................................................................................................................................................................................294
* 106.E. Waring's Repeated Exponents................................................................................................................................................................................................................................................................................................................................................................................................313