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