P. HISTORY OF ' AATHEMATICAL - School of Mathematics

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118 A HISTORY OF MATHEMATICAL NOTATIONS and m: . For example, in 1545 one finds (5+ 1/=) (5 - )/--i5) = 25- (-15) =40 printed in this form: " 5p: R m: 15 5m: R m: 15 25m:m: 15 ijd est 40 ," while in 1663 the same passage appears in the form: " 5. p. R. %. 15. 5. %. 8. m. 15. 25. m. m. 15. quad. est 40. "I 141. Cardan uses R to mark square root. He employs2 Pacioli's radix vniversalis to binomials and polynomials, thus "R.V.7. p R. 4. vel sic (R) 13.p R. 9." for or d13+d6; "R.V.lO.p.R.16.p.3.p R.64." for d 10 + d16+3+d64. Cardan proceeds to new notations. He introduces the radix ligata to express the roots of each of the terms of a binomial; he writes: "LR. 7. pR. for d5+ dG. This L would seem superfluous, but was introduced to distinguish between the foregoing form and the radix distincta, as in "R.D. 9 p. R. 4.," which signified 3 and 2 taken separately. Accordingly, "R.D. 4. p. @. 9.," multiplied into itself, gives 4+9 or 13, while the "R.L. 4. p. R. 9.," multiplied into itself, gives 13+d==25. In later passages Cardan seldom uses the radix ligata and radix distinct~. In squaring binomials involving radicalslSlike "R.V.L. R. 5. P. 8. 1. % R. V.L. 8.5. % R. I.," he sometimes writes the binomial a second time, beneath the first, with the capital letter X between the two binomials, to indicate cross-multiplication.' Of interest is the following passage in the Regula aliza which Cardan brought out in 1570: "Rp: est p: R m: quadrata nulla est iuxta usum communem" ("The square root of a positive number is positive; the square root of a negative number is not proper, according to the common acceptation").s 1 See Tropfke, op. cit., Vol. I11 (1922), p. 134, 135. Cardan, op. eit., p. 14, 16, of the Practiea arithmeticae of 1539. a Zbid., p. 16. Zbid., p. 194. 'Op. cit. (Basel, 1570), p. 15. Reference taken from Enestrom, Bibliotheca mathematics, Vol. XI11 (1912-13), p. 163.
INDIVIDUAL WRITERS 119 However, in the Ars magna1 Cardan solves the problem, to divide 10 into two parts, whose product is 40, and writes (as shown above) : " I 5. @ l$. fi. 15. 5. 7% R. m.. 15. 1 25 m.m.. 15. quad. est 40 . 7, Fxa. 46.-Part of a page (255) from the Ars magnu as reprinted in H. Cardan'a Operum tomvs quurtvs (Lvgdvni, 1663). The Ars magna was first published in 1545. Opmm tomvs quartvs, p. 287.
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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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34 A HISTORY OF MATmMATICAL NOTATIO
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' 1652 190 A HISTORY OF MATHEMATICA
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......................... 192 A HIS
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196 A HISTORY OF ,MATHEMATICAL NOTA
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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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@ 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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266 . A HISTORY OF MATHEMATICAL NOT
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