P. HISTORY OF ' AATHEMATICAL - School of Mathematics

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Bibliographical Note This Dover edition, first published in 1993, is an unabridged and unaltered republication in one volume of the work first published in two volumes by The Open Court Publishing Company, La Salle, Illinois, in 1928 and 1929. Library of Congress Cataloging-in-Publication Data Cajori, Florian, 1859-1930. A history of mathematical notations / by Florian Cajori. p. cm. Originally published: Chicago : Open Court Pub. Co., 1928- 1929. "Two volumes bound as one." Includes indexes. Contents: v. 1. Notations in elementary mathematics - v. 2. Notations mainly in higher mathematics. ISBN 0-486-67766-4 (pbk.) 1. Mathematical notation-History. 2. Mathematics-History. 3. Numerals-History. 1. Title. QA41.C32 1993 5101.148-dc20 93-29211 CIP Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
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 rm 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 synlbols 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 IIegeler 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.
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......................... 192 A HIS
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@ De 290 A HISTORY OF MATHEMATICAL
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@ B. 292 A HISTORY OF MATHEMATICAL
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@ G. 404 A HISTORY OF MATmMATICAL N
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Abacus, 39, 75, 119 Abu Kamil, 273;
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ALPHABETICAL INDEX Birks, John, omi
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ALPHABETICAL INDEX 437 Curtze, M.,
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ALPWETICAL INDEX Fol!inus, H., 208,
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ALPHABETICAL INDEX Hindenburg, C. F
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ALPHABETICAL INDEX 443 Local value
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ALPWETICAL INDEX Oliver, Wait, and
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ALPHABETICAL INDFZ Rawlinson, H., 5
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Smith Eugene R.: area, 370; congrue
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ALPHABETICAL INDEX 451 Wachter, G.,
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h o ~ u c m TO o THE ~ SECOND VOL-
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TABLE OF CONTENTS PAEAQBbPHB Expone
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TABLE OF CONTENT'% Finite Differenc
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TABLE OF CONTENTS PIIIAQBAPBB C. Co
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INTRODUCTION TO THE SECOND VOLUME I
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TOPICAL SURVEY OF SYMBOLS IN ARITHM
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LE I'TERS REPRESENTING MAGNITUDES .
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LETTERS REPRESENTING MAGNITUDES 5 d
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LETTERS REPRESENTING MAGNITUDES Y'
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LETTERS n AND e 9 William Oughtred1
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LETTERS r AND e 11 he began to use
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LETTERS r AND e 13 mate ratio of th
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DOLLAR MARK 15 The use of the lette
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DOLLAR MARK 17 "dollar" and to have
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DOLLAR MARK 19 lines being marks of
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DOLLAR MARK 21 Ivan Vasquez de Sern
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DOLLAR MARK 18, 19 are from the Dra
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DOLLAR MARK 25 tion of the conventi
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DOLLAR MARK 27 114 and 115. It will
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THEORY OF NUMBERS 29 405. Conclusio
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THEORY OF NUMBERS complex integers
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THEORY OF NUMBERS 33 +'(n) is half
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THEORY OF NUMBERS 35 deux. Ce signe
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INFINITY AND TRANSFINITE NUMBERS 45
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INFINITY AND TRANSFINITE NUMBERS 47
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CONTINUED FRACTIONS, INFINITE SERIE
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THEORY OF COMBINATIONS 61 438. The
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THEORY OF COMBINATIONS 63 expansion
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THEORY OF COMBINATIONS 65 in the po
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THEORY OF COMBINATIONS 67 The polyn
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THEORY OF COMBINATIONS 69 0, 1, ...
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THEORY OF COMBINATIONS 2;""3 C, •
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THEORY OF COMBINATIONS 73 A new des
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THEORY OF COMBINATIONS 75 universal
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THEORY OF COMBINATIONS 77 Of course
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THEORY OF COMBINATIONS 79 ubi k nat
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THEORY OF COMBINATIONS 81 the eleme
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THEORY OF COMBINATIONS 83 applying
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THEORY OF COMBINATIONS brevity, ~(n
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DETERMINANTS 87 DETERMINANT NOTATIO
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DETERMINANTS 89 rum bb-ac, a cuius
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DETERMINANTS 91 Jacobi! chose the r
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DETERMINANTS and failure to represe
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DETERMINANTS 95 side the array. The
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DETERMINANTS 97 and in general the
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DETERMINANTS 99 may be written! (~,
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DETERMINANTS 101 Sheppard' follows
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DETERMINANTS 103 The notation for f
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LOGARITHMS 105 Most general single-
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LOGARITHMS 107 The capital letter "
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LOGARITHMS 109 in Napier's Descript
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LOGARITHMS 111 Morgan states that "
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LOGARITHMS 113 en general, aNsera l
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THEORETICAL ARITHMETIC 115 stand fo
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THEORETICAL ARITHMETIC 117 argues t
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THEORETICAL ARITHMETIC 119 485. Imp
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THEORETICAL ARITHMETIC 121 er." Pro
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THEORETICAL ARITHMETIC 123 Littlewo
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THEORETICAL ARITHMETIC 125 494. Gen
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IMAGINARIES AND VECTOR ANALYSIS 127
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and laws of combination, without gi
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TRIGONOMETRY 143 512. A different d
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TRIGONOMETRY 145 Gioseppe' uses Gra
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TRIGONOMETRY 147 A novel mark is ad
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TRIGONOMETRY 149 tion (p. 47) a str
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TRIGONOMETRY 151 (P. 360) Rad. sin.
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TRIGONOMETRY 153 As far as the form
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TRIGONOMETRY 155 lettering is emplo
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TRIGONOMETRY 157 edition of the boo
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TRIGONOMETRY 159 also his Trigonome
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TRIGONOMETRY 161 During the latter
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TRIGONOMETRY 163 his lack of assert
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TRIGONOMETRY 165 We close with a re
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••••••••••• .
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21. J. A. Segner, Cur.... matMmatic
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TRIGONOMETRY 171 9. G. U. A. Vieth,
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TRIGONOMETRY 173 J. Houel! in 1864
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TRIGONOMETRY 175 tan [(ct>, x)] =ta
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TRIGONOMETRY 177 for the calculus,
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TRIGONOMETRY 179 formula "4 sin.f1=
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SYMBOLS USED BY LEIBNIZ 181 sponded
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SYMBOLS USED BY LEIBNIZ 183 by two
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SYMBOLS USED BY LEIBNIZ 185 desuetu
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SYMBOLS USED BY LEIBNIZ 187 544. Si
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SYMBOLS USED BY LEIBNIZ 189 Sign Me
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SYMBOLS USED BY LEIllNIZ 191 552. P
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SYMBOLS USED BY LEIBNIZ 193 558. Fu
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SYMBOLS USED BY LEIBNIZ 195 BC, or
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DIFFERENTIAL CALCULUS 197 2. SYMBOL
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DIFFERENTIAL CALCULUS 199 few other
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DIFFERENTIAL CALCULUS 201 duced Flu
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DIFFERENTIAL CALCULUS 203 admirable
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DIFFERENTIAL CALCULUS 205 did he su
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DIFFERENTIAL CALCULUS 207 Here the
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DIFFERENTIAL CALCULUS 209 1797 edit
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DIFFERENTIAL CALCULUS 211 Kramp say
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DIFFERENTIAL CALCULUS 213 580. S. F
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DIFFERENTIAL CALCULUS 215 Nor had L
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DIFFERENTIAL CALCULUS 217 The varia
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DIFFERENTIAL CALCULUS 219 the book
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DIFFERENTIAL CALCULUS 221 l'Hospita
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DIFFERENTIAL CALCULUS 223 different
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DIFFERENTIAL CALCULUS 225 x and y,
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DIFFERENTIAL CALCULUS . 'l2.7 shoul
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DIFFERENTIAL CALCULUS 229 represent
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DIFFERENTIAL CALCULUS 231 cannot be
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DIFFERENTIAL CALCULUS 233 607. M. O
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DIFFERENTIAL CALCULUS 235 610. Cauc
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DIFFERENTIAL CALCULUS 237 values ac
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DIFFERENTIAL CALCULUS 239 "The stud
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DIFFERENTIAL CALCULUS 241 618. T. M
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INTEGRAL CALCULUS 243 In Leibnizens
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INTEGRAL CALCULUS 245 nizian notati
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INTEGRAL CALCULUS 247 In Newton's P
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INTEGRAL CALCULUS 249 624. A. L. Cr
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INTEGRAL CALCULUS 251 claiming for
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INTEGRAL CALCULUS 253 6. CALCULUS N
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INTEGRAL CALCULUS 255 and "Lim. ~~.
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INTEGRAL CALCULUS 257 limit, when .
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INTEGRAL CALCULUS 259 introduced th
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INTEGRAL CALCULUS 261 Diderot's Enc
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FINITE DIFFERENCES 263 FINITE DIFFE
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FINITE DIFFERENCES 265 Wide accepta
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THEORY OF FUNCTIONS 267 introductio
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THEORY OF FUNCTIONS 269 1747 D'Alem
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THEORY OF FUNCTIONS 271 by Cauchy'
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THEORY OF FUNCTIONS 273 Hence this
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THEORY OF FUNCTIONS 275 where the a
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THEORY OF FUNCTIONS 277 he employed
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THEORY OF FUNCTIONS 279 in a series
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