P. HISTORY OF ' AATHEMATICAL - School of Mathematics

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64 A HISTORY OF MATHEMATICAL NOTATIONS 96. Fann'ful hypotheses on the origin of the numeral forms.-A problem as fascinating as the puzzle of the origin of language relates to the evolution of the forms of our numerals. Proceeding on the tacit assumption that each of our numerals contains within itself, as a skeleton so to speak, as many dots, strokes, or angles as it represents units, imaginative writers of different countries and ages have advanced hypotheses as to their origin. Nor did these writers feel that they were indulging simply in pleasing pastime or merely contributing to mathematical recreations. With perhaps only one exception, they were as convinced of the correctness of their explanations as are circlesquarers of the soundness of their quadratures. The oldest theory relating to the forms of the numerals is due to the Arabic astrologer Aben Ragel1 of the tenth or eleventh century. He held that a circle and two of its diameters contained the required forms as it were in a nutshell. A diameter represents 1; a diameter and the two terminal arcs on opposite sides furnished the 2. A glance at Part I of Figure 29 reveals how each of the ten forms may be evolved from the fundamental figure. On the European Continent, a hypothesis of the origin from dots is the earliest. In the seventeenth century an Italian Jesuit writer, Mario Bettini,2 advanced such an explanation which was eagerly accepted in 1651 by Georg Philipp Harsdijrffer3 in Germany, who said: "Some believe that the numerals arose from points or dots," as in Part II. The same idea was advanced much later by Georges Dumesni14 in the manner shown in the first line of Part 111. In cursive writing the points supposedly came to be written as dashes, yielding forms resembling those of the second line of Part 111. The two horizontal dashes for 2 became connected by a slanting line yielding the modern form. In the same way the three horizontal dashes for 3 were joined by two slanting lines. The 4, as first drawn, resembled the 0; but confusion was avoided by moving the upper horizontal stroke into a J. F. Weidler, De characleribus numerorum vulgaribus disserlatio malhematicacrilica (Wittembergae, 1737), p. 13; quoted from M. Cantor, Kullurleben der Volker (Halle, 1863), p. 60, 373. Apiaria unwersae philosophiae, malhematicae, Vol. I1 (1642), Apiarium XI, p. 5. See Smith and Karpinski, op. cil., p. 36. Delitae malhemalicae el physicae (Niirnberg, 1651). Reference from M. Sterner, Geschichle der Rechenkunst (Miinchen and Leipzig [1891]), p. 138, 524. "Note sur la forme des chiffres usuels," Revue archbologique (3d ser.; Paris, 1890), Vol. XVI, p. 34248. See also a critical article, "PrBtendues notations Pythagoriennes sur l'origine de nos chiffres," by Paul Tannery, in his Mhoires seientijiques, Vol. V (1922), p. 8.
HINDU-ARABIC NUMERALS 65 vertical position and placing it below on the right. To avoid confounding the 5 and 6, the lower left-hand stroke of the first 5 was b e ~ n : l 23 Y 5 6 7 8 9 o IFbm8:1?,34 b $ 4 0 II: i 2345G"I E& 000 @ @ @ O @ @ O a o {- ; 3 o,~bs6 Q 0 4 FIG. 29.-Fanciful hypotheses changed from a vertical to a horizontal position and placed at the top of the numeral. That all these changes were accepted as historical,
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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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10 A HISTORY 6F' MATHEMATICAL NOTAT
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176 A HISTORY OF MATEFEMATICAL NOTA
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182 A HISTORY OF MATREMATICAL NOTAT
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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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200 A HISTORY OF MATHEMATICAL 'NOTA
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A HISTORY OF MATHEMATICAL NOTATIONS
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208 A HISTORY OF MATHJ3MATICAL NOTA
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216 ,A HISTORY OF MATHEMATICAL NOTA
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268 A HISTORY OF MATEIEMATICAL 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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