P. HISTORY OF ' AATHEMATICAL - School of Mathematics

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136 A HISTORY OF MATHEMATICAL NOTATIONS the second one by the small letter q, an abbreviation for quantita, which Pacioli had used for the second unknown.' Interesting at this early period is the following use of the letters a, c, and d to represent ordinary numbers (folio Giija) : "Nim 3 solchs collects 1 setz es auff ein ort ( dz werd von lere wegen c genennt. Dar- nach subtrahier das c vom a / das iibrig werd gesprochen d. Nun sag ich dz &+l/d ist quadrata radix des ersten binomij." ("Take 3 this sum, assume for it a position, which, being empty, is called c. Then subtract c from a, what remains call d. Now I say that I/;+& is the square root of the first bin~mial.")~ 149. Rudolff was convinced that development of a science is dependent upon its symbols. In the Preface to the second part of Rudolff's Coss he states: "Das bezeugen alte biicher nit vor wenig jaren von der coss geschriben, in welchen die quantitetn, als dragma, res, substantia etc. nit durch character, sunder durch gantz geschribne wort dargegeben sein, vnd sunderlich in practicirung eines' yeden exempels die frag gesetzt, ein ding, mit solchen worten, ponatur vna res." In translation: "This is evident from old books on algebra, written many years ago, in which quantities are represented, not by characters, but by words written out in full, 'drachm,' 'thing,' 'substance,' etc., and in the solution of each special example the statement was put, 'one thing,' in such words as ponatur, una res, etc."a In another place Rudolff says: "Lernt die zalen der coss aussprechen vnnd durch ire charakter erkennen vnd schreiben."' ("Learn to pronounce the numbers of algebra and to recognize and write them by their characters.") DUTCH: QIELIS VAN DER HOECKE (1537) 150. An early Dutch algebra was published by Gielis van der Hoecke which appeared under the title, In arithmetica een sonderlinge ezcelEt boeck (Antwerp [1537]).6 We see in this book the early appear- 1 Chr. Rudolff, Behend vnnd Hubsch Rechnung (Stramburg, 1525), fol. Ria. Quoted by Enestrom, Bibliolheca mathemalica, Vol. XI (1910-ll), p. 357. 2 Quoted from Rudolff by Enestrom, ibid., Vol. X (1909-lo), p. 61. a Quoted by Gerhardt, op. cit. (1870), p. 153. This quotation is taken from the second part of Gerhardt's article; the firat part appeared in 'the same publication, for the year 1867, p. 38-54. 4 Op. cil., Buch I, Kap. 5, B1. Dijro; quoted by Tropfke, op. cil. (2. ed.), Vol. 11, p. 7. On the date of publication, see Enestrom, op. cit., Vol. VII (1906-7), p. 211; Vol. X (190(r10), p. 87.
INDIVIDUAL WRITERS 137 ance of the plus and minus signs in Holland. As the symbols for powers one finds here the notation of Grammateus, N., pri., se., 3a, 4", 5", etc., though occasionally, to fill out a space on a line, one en- FIG. 60.-From Gielis van der Hoecke's In arithmetica (1537). Multiplication of fractions by regule cos. counters numerw, num., or nu. in place of N.; also secu. in place of se. For pri. he uses a few times p. The translation of matter shown in Figure 60 is as follows: "[In order to multiply fractions simply multiply numerators by numera-
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P. HISTORY OF ' AATHEMATICAL ound A
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A HISTORY OF MATHEMATICAL NOTATIONS
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PREFACE The study of the history of
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... vln TABLE OF CONTENTS Byzantine
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TABLE OF CONTENTS mx.&amms Notation
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TABLE OF CONTENTS PIRAQRAPER Crosse
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ILLUSTRATIONS PIQUBE 28 . REAL ESTA
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PIOURE 91 . RADICALS ILLUSTRATIONS
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NUMERAL SYhlBOLS AND COMBINATIONS '
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4 A HISTORY OF MATHEMATICAL NOTATIO
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18 A HISTORY OF MATHEMATICAL KOTATI
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' 1652 190 A HISTORY OF MATHEMATICA
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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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298 A HISTORY OF MATEIEMATICAL NOTA
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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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