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K 64083
A HISTORY OF MATHEMATICAL NOTATIONS VOLUME
I
NOTATIONS IN ELEMENTARY
MATHEMATICS
A HISTORY OF
ATHEMATICAL NOTATIONS BY
FLORIAN CAJORJ^H.D. Professor of the History of Mathematics
University of California
VOLUME
1
NOTATIONS IN ELEMENTARY
MATHEMATICS
THE OPEN COURT COMPANY. PUBLISHERS, 86,
STRAND, LONDON,
W.C.2.
COPYRIGHT 1928 BY THE OPEN COURT PUBLISHING COMPANY Published September 1928
Composed and Printed By The University of Chicago Preu Chicago.
Illinois.
U.S.A.
PREFACE The study gested to
me by
of the history of mathematical notations was sugProfessor E. H. Moore, of the University of Chicago.
to Professor M.W. Haskell, of the University of California, indebted for encouragement in the pursuit of this research. As completed in August, 1925, the present history was intended to be
To him and I
am
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 exam-
ined the entire manuscript of this work in detail, and brought it to the sympathetic attention of the Open Court Publishing Company. I
my gratitude to
desire to record of the
Mrs.
Mary
Open Court Publishing Company,
sive publication
from which no financial
Hegeler Carus, president
for undertaking this expenprofits
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
.
I.
TABLE OF CONTENTS
INTRODUCTION PARAGRAPHS
II.
NUMERAL SYMBOLS AND COMBINATIONS OF SYMBOLS
.
.
.
Egyptians Phoenicians and Syrians
Hebrews Greeks Early Arabs
Romans Peruvian and North American Knot Records Aztecs
....
Maya Chinese and Japanese Hindu- Arabic Numerals Introduction Principle of Local Value of Numerals
Forms
Freak Forms Negative Numerals
Grouping of Digits in Numeration The Spanish Calderon
The Portuguese
Cifrao
Fanciful Hypotheses on the Origin of Artificial
Sporadic General Remarks
Numeral Forms
.
System
Opinion of Laplace III.
...
Hindu Hindu Hindu Arabic
Arabic
Diophantus, Third Century A.D Brahmagupta, Seventh Century The Bakhshal! Manuscript Bhaskara, Twelfth Century al-Khow&rizmi, Ninth Century al-Karkhf, Eleventh Century
....
....
Byzantine Michael Psellus, Eleventh Century Arabic Ibn Albanna, Thirteenth Century
.
.
...
Chinese
Chu
92-93 94 95 96 97 98 99
SYMBOLS IN ARITHMETIC AND ALGEBRA (ELEMENTARY PART) A. Groups of Symbols Used by Individual Writers Greeks
27-28 29-31 32-44 45 46-61 62-65 66-67 68 69-73 74-99 74-77 78-80 81-88 89 90 91
Relative Size of Numerals in Tables
A
1-99
1-15 16-26
Babylonians
Shih-Chieh, Fourteenth Century vii
.
100 101
101-5 106-8 109 110-14 115 116
117 118
.119, 120
TABLE OF CONTENTS
viii
PARAGRAPHS
Byzantine Maximus Planudes, Fourteenth Century Italian Leonardo of Pisa, Thirteenth Century French Nicole Oresme, Fourteenth Century
.
122
.
.
123
.
.
.
125-27
.
.
.
.
Arabic
121
.
....
Fifteenth Century Regiomontanus, Fifteenth Century ItalianEarliest Printed Arithmetic, 1478 French Nicolas Chuquet, 1484 French Estienne de la Roche, 1520 Pietro Borgi, 1484, 1488 Italian Luca Pacioli, 1494, 1523 Italian al-Qalasadi,
German
.
124 128
129-31 132 133
134-38
Italian
F. Ghaligai, 1521, 1548, 1552
Italian
140, 141
Italian
H. Cardan, 1532, 1545, 1570 Nicolo Tartaglia, 1506-60
Italian
Rafaele Bombelli, 1572
144, 145
German
Dutch
142, 143
Johann Widman, 1489, 1526
Austrian
German
139
146
Grarnrnateus, 1518, 1535
Christoff Rudolff, 1525 Gielis van der Hoecke, 1537
German German German
Michael
Maltese
Wil. Klebitius, 1565
German Belgium
Christophorus Clavius, 1608 Simon Stevin, 1585
Lorraine
Albert Girard, 1629
Stifel, 1544, 1545, 1553 Nicolaus Copernicus, 1566 Johann Scheubel, 1545, 1551
German-SpanishMarco
Aurel, 1552
147 148, 149
.... ....
161 162, 163
164
.... ....
English
French French
Jacques Peletier, 1554 Jean Buteon, 1559
French French
Guillaume Gosselin, 1577 Francis Vieta, 1591
Italian
Bonaventura Cavalieri, 1647
English English English
.
.
.
166
167-68 169 170 171
173
174
176-78 179 .
.
.
....
Rene* Descartes
165
172
English William Oughtred, 1631, 1632, 1657 English Thomas Harriot, 1631 French Pierre HSrigone, 1634, 1644 Scot-FrenchJames Hume, 1635, 1636
French
157 158, 159
160
Pedro Nunez, 1567 Robert Recorde, 1543(?), 1557 John Dee, 1570 Leonard and Thomas Digges, 1579 Thomas Mastcrson, 1592
Portuguese-Spanish
150
151-56
180-87 188 189
190 191
Barrow
192
English Richard Rawlinson, 1655-68 Swiss Johann Heinrich Rahn
194
English
Isaac
193
TABLE OF CONTENTS
ix
PARAGRAPHS
....
195, 196 English John Wallis, 1655, 1657, 1685 197 Extract from Ada eruditorum, Leipzig, 1708 Extract from Miscellanea Berolinensia, 1710 (Duo to 198 G. W. Leibniz) 199 Conclusions .
.
.
B. Topical Survey of the Use of Notations Signs of Addition and Subtraction
200-356 200-216 200 201-3 204 205-7 208, 209
Early Symbols Origin and Meaning of the Signs and Symbols Spread of the
Shapes of the Varieties of
-
+ +
Sign
Signs " Symbols for Plus or Minus" Certain Other Specialized Uses of
+
and
.
.
Four Unusual Signs Composition of Ratios Signs of Multiplication
Early Symbols Early Uses of the
St. Andrew's Cross, but Not Symbol of Multiplication of Two Numbers The Process of Two False Positions
Proportions with Integers Proportions Involving Fractions Addition and Subtraction of Fractions Division of Fractions Casting Out the 9's, 7's, or ll's
Compound
Multiplication of Integers Reducing Radicals to Radicals of the " Marking the Place for Thousands"
as the .
.
.
.
.... .... .
Same Order
....
Place of Multiplication Table above 5X5 The St. Andrew's Cross Used as a Symbol of Multi.
.
plication
Unsuccessful Symbols for Multiplication The Dot for Multiplication The St. Andrew's Cross in Notation for Transfinite Ordinal Numbers Signs of Division and Ratio .
Early Symbols Rahn's Notation Leibniz's Notations
.
A
.
...
Relative Position of Divisor and Dividend Order of Operations in Terms Containing Both
and
Estimate of
:
and
-
as Symbols
.
218-30 219 220 221 222 223
225 226 227 228 229 231 232 233
234 235-47
235,236 237 238 241
-f-
X
Critical
210,211 212-14 215 216 217-34 217
.
242 243
TABLE OF CONTENTS PABAQRAPH8
Notations for Geometric Ratio
244
Numbers
Division in the Algebra of Complex
Signs of Proportion Arithmetical and Geometrical Progression Arithmetical Proportion
.
.
.
247
.
.
248-50 248 249
Geometrical Proportion OughtrecTs Notation
251
Struggle in England between Oughtred's and Wing's Notations before 1700
252
250
Struggle in England between Oughtred's and Wing's
Notations during 1700-1750 Sporadic Notations Oughtred's Notation on the European Continent Slight Modifications of Oughtred's Notation The Notation : : : : in Europe and America
The Notation
253 254 .
.
.
.
.
of Leibniz
259
260-70 260
Signs of Equality
Early Symbols Recorde's Sign of Equality Different
Meanings of Competing Symbols
261
=
Descartes' Sign of Equality Variations in the Form of Descartes' Symbol
Struggle for Supremacy Variation in the Form of Recorde's
Variation in the
Manner
Symbol
.
.
.
.
.
of Using It
Nearly Equal Signs of
Common
Fractions
Early Forms
The
255 257 258
262 263 264 265 266 268 269 270 271-75 271 272 274 275
Fractional Line
Special Symbols for Simple Fractions TheSolidus Signs of Decimal Fractions
276-89
Stevin's Notation
Other Notations Used before 1617 Did Pitiscus Use the Decimal Point? Decimal Comma and Point of Napier Seventeenth-Century Notations Used after 1617 Eighteenth-Century Discard of Clumsy Notations Nineteenth Century Different Positions for Point
.... .... .
.
276 278 279 282 283 285
:
and
for
Comma
Signs for Repeating Decimals Signs of Powers
General Remarks
286 289
290-315 290
TABLE OP CONTENTS PARAGRAPHS
Double Significance of R and I Facsimiles of Symbols in Manuscripts Two General Plans for Marking Powers
.... ....
Early Symbolisms: Abbreviative Plan, Index Plan Notations Applied Only to an Unknown Quantity, the Base Being Omitted Notations Applied to Any Quantity, the Base Being
291 293 294 295
296 297 298
Designated Descartes' Notation of 1637
Did Stampioen Arrive
at Descartes' Notation Independently? Notations Used by Descartes before 1637 Use of H6rigone's Notation after 1637 Later Use of Hume's Notation of 1636 Other Exponential Notations Suggested after 1637 Spread of Descartes' Notation .
.
.
.... ....
.
Negative, Fractional, and Literal Exponents
.
.
299 300 301 302 303 307 308
309 312 313 314 Conclusions 315 316-38 Signs for Roots Early Forms, General Statement 316, 317 The Sign $, First Appearance 318 319 Sixteenth-Century Use of /J Seventeenth-Century Use of # 321 The Sign I 322 323 Napier's Line Symbolism The Sign V 324-38 324 Origin of V 327 Spread of the V 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 334 335 Redundancy in the Use of Aggregation Signs Peculiar Dutch Symbolism 336 337 Principal Root- Values Recommendation of the U.S. National Committee 338 for Unknown Numbers 339-41 Signs 339 Early Forms
Imaginary Exponents Notation for Principal Values Complicated Exponents D. F. Gregory's (+) r
,
.... ...... .
.
.
.
.
.
..
.
.
TABLE OF CONTENTS
xii
PARAGRAPHS
Numerals Representing knowns
Crossed
Powers of Un340 340
.
Descartes'
2, y,
x
341
Spread of Descartes' Signs Signs of Aggregation Introduction
Aggregation Expressed by Letters Aggregation Expressed by Horizontal Bars or Vincu-
lums Aggregation Expressed by Dots Aggregation Expressed by Commas Aggregation Expressed by Parentheses Early Occurrence of Parentheses Terms in an Aggregate Placed in a Verbal Column
....
Marking Binomial Coefficients Special Uses of Parentheses A Star to Mark the Absence of Terms IV.
....
A, Ordinary Elementary Geometry
Early Use of Pictographs Signs for Angles " Signs f or Perpendicular" Signs for Triangle, Square, Rectangle, Paiiillclogram
.
The Square
as an Operator for Circle Sign Signs for Parallel Lines
Signs for Equal and Parallel Signs for Arcs of Circles Other Pictographs
The Sign
and Congruence
O for Equivalence
Lettering of Geometric Figures Sign for Spherical Excess Symbols in the Statement of Theorems
Signs for Incommensurables
Unusual Ideographs in Elementary Geometry Algebraic Symbols in Elementary Geometry B. Past Struggles between Symbolists and Rhetoricians in Elementary Geometry
INDEX
344 348 349 350 351
353 354 355 356
357-85
SYMBOLS IN GEOMETRY (ELEMENTARY PART)
Signs for Similarity
342-56 342 343
.
.
.
.
.
.
357 357 360 364 365 366 367 368 369 370 371 372 375 376 380 381 382 383 384
.385
ILLUSTRATIONS FIQURB 1.
PARAGRAPHS
BABYLONIAN TABLETS OF NIPPUR
4
2.
PRINCIPLE OF SUBTRACTION IN BABYLONIAN NUMERALS
3.
BABYLONIAN LUNAR TABLES
4.
MATHEMATICAL CUNEIFORM TABLET CBS 8536
...
9 11
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)
NUMERALS
27
10.
SYRIAN NUMERALS
28
11.
HEBREW NUMERALS
30
12.
COMPUTING TABLE OF SALAMIS
36
13.
ACCOUNT OF DISBURSEMENTS OF THE ATHENIAN STATE, 418415 B.C.
36
14.
ARABIC ALPHABETIC NUMERALS
45
15.
DEGENERATE FORMS OF ROMAN NUMERALS
56
16. 17.
QUIPU FROM ANCIENT CHANCAY IN PERU 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 CHINESE AND JAPANESE NUMERALS
21.
22. HILL'S
TABLE OF BOETHIAN APICES
65
.
70 74
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
93
CONTRACT, MEXICO
CITY, 1649
ILLUSTRATIONS
xiv FIGURE
PARAGRAPHS
28.
REAL ESTATE SALE, MEXICO
29.
FANCIFUL HYPOTHESES
30.
NUMERALS DESCRIBED BY NOVIOMAGUS
98
31.
SANSKRIT SYMBOLS FOR THE
UNKNOWN
108
32.
BAKHSHALI ARITHMETIC
109
33.
SRIDHARA'S Trisdtika
112
34.
ORESME'S Algorismus Proportionum
123
35.
AL-QALASADI'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 605
132
41.
DE
LA ROCHE'S Larismethique, FOLIO 66A
42.
PART OF PAGE
43.
MARGIN OF FOLIO 1235
44.
PART OF FOLIO 72 OF GHALIGAI'S
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
51.
BOMBELLI'S Algebra (1579 IMPRESSION), PAGE 161
....
145
52.
FROM THE MS OF BOMBELLI'S Algebra IN THE LIBRARY OF BOLOGNA
145
FROM PAMPHLET No. OF BOLOGNA
146
53.
CITY, 1718
.
94 96
IN PACIOLI'S
Summa,
IN PACIOLI'S
132
1523
138
Summa
139
Practica d'arithmetica, 1552
.
Algebra, 1572
139
144
595AT IN THE LIBRARY OF THE UNIVERSITY
54.
WIDMAN'S Rechnung, 1526
55.
FROM THE ARITHMETIC OF GRAMMATEUS
56. 57.
FROM THE ARITHMETIC OF GRAMMATEUS, FROM THE ARITHMETIC OF GRAMMATEUS,
58.
FROM CHR. RUDOLFF'S
Coss, 1525
146 146
1535
147
1518(?)
147
148
ILLUSTRATIONS
xv PARAGRAPHS
59.
FROM CHR. RUDOLFF'S
60.
FROM VAN DER HOECKE' In
61.
PART OF PAGE FROM STIFEL'S
62.
FROM
STIFEL'S Arithmetica Integra, FOLIO
63.
FROM
STIFEL'S EDITION OF RUDOLFF'S Coss, 1553
64.
SCHEUBEL, INTRODUCTION TO EUCLID, PAGE 28
159
65.
W. KLEBITIUS, BOOKLET,
161
Ev
Coss,
148
'
arithmetica
150
Arithmetica intcgra, 1544
31B
1565
.
.
.
150 152
156
66.
FROM GLAVIUS'
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
71.
R. RECORDS, Whetstone of Witte, 1557
168
RECORDE
168
72. FRACTIONS IN
Algebra, 1608
161
Arithmetica
165
73.
RADICALS IN RECORDE
168
74.
RADICALS IN DEE'S PREFACE
169
75.
PROPORTION IN DEE'S PREFACE
169
76.
FROM
77.
EQUATIONS IN DIGGES
172
78.
EQUALITY IN DIGGES
172
79.
FROM THOMAS MASTERSON'S
80. J.
DIGGES'S
Stratioticos
170
Arithrneticke,
1592
PELETIER'S Algebra, 1554
172
172
81.
ALGEBRAIC OPERATIONS IN PELETIER'S Algebra
172
82.
FROM
173
83.
GOSSELIN'S De
84.
VIETA, In artem analyticam, 1591
85.
VIETA, De emendatione aeqvationvm
178
CAVALIERI, Exercitationes, 1647
179
86. B.
J.
BUTEON,
Arithmetica, 1559
arte
magna, 1577
87.
FROM THOMAS HARRIOT,
88.
FROM THOMAS
89.
FROM HERIGONE,
90.
ROMAN NUMERALS FOR
1631,
PAGE
174 176
101
HARRIOT, 1631, PAGE 65 Cursus mathematicus, 1644
x IN
J.
HUME, 1635
189
189 189 191
ILLUSTRATIONS
xvi
PARAGRAPHS
FIGURE
191
1635
91.
RADICALS IN
92.
R. DESCARTES, Gtomttrie
93.
I.
BARROW'S
94.
I.
BARROW'S Ewlid, ENGLISH EDITION
95. RICH.
J.
HUME,
Euclid,
191
LATIN EDITION. NOTES BY ISAAC
194
RAWLINSON'S SYMBOLS
195
RAHN'S Teutsche
97.
BRANCKER'S TRANSLATION OF RAHN, 1668
99.
Algebra, 1659
195
200
FROM THE HIEROGLYPHIC TRANSLATION OF THE AHMES PAPYRUS THE GERMAN
MS
DRESDEN LIBRARY
100.
MINUS SIGN
101.
PLUS AND MINUS SIGNS IN THE LATIN LIBRARY
C. 80,
MS
102.
WIDMANS' MARGINAL NOTE TO
103.
FROM THE ARITHMETIC OF BOETHIUS, 1488
C. 80,
MS
C. 80,
.
.
201
DRESDEN 201
DRESDEN LIBRARY
.
201
250
....
294
WRITTEN ALGEBRAIC SYMBOLS FOR POWERS FROM PEREZ DE MOYA'S Arithmetica
294
104. SIGNS IN 105.
195
WALLIS, 1657
IN
193 193
96.
98. J.
NEWTON
.
106. E.
GERMAN MSS AND EARLY GERMAN BOOKS
WARING'S REPEATED EXPONENTS
313
INTRODUCTION In this history it has been an aim to give not only the first appearance of a symbol and its origin (whenever possible), but also to indicate the competition encountered and the spread of the symbol among writers in different countries. It
is
the latter part of our program
which has given bulk to this history.
The rise of certain symbols, their day of popularity, and their eventual decline constitute in many cases an interesting story. Our endeavor has been to do justice to obsolete and obsolescent notations, as well as to those which have survived and enjoy the favor of mathematicians of the present moment. If the object of this history of notations were simply to present an array of facts, more or less interesting to some students of mathematics if, in other words, this undertaking had no ulterior motive
then indeed the wisdom of preparing and publishing so large a book might be questioned. But the author believes that this history constitutes a mirror of past and present conditions in mathematics which
can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of the notational problems of the present time.
|
n NUMERAL SYMBOLS AND COMBINATIONS OF SYMBOLS BABYLONIANS 1.
In the Babylonian notation of numbers a vertical wedge Y 1, while the characters ^ and Y>- signified 10 and 100,
stood for
1 respectively. Grotefend believes the character for 10 originally to have been the picture of two hands, as held in prayer, the palms being pressed together, the fingers close to each other, but the thumbs thrust
two principles were employed in the Babylonial noand multiplicative. We shall see that limited use the additive tation was made of a third principle, that of subtraction. out. Ordinarily,
2. Numbers below 200 were expressed ordinarily by symbols whose respective values were to be added. Thus, Y^XKYYY stands for 123. The principle of multiplication reveals itself in < |>- where
the smaller symbol 10, placed before the 100, 100, so that this symbolism designates 1,000.
is
to be multiplied
by
3. These cuneiform symbols were probably invented by the early Sumerians. Their inscriptions disclose the use of a decimal scale of
numbers and also of a sexagesimal scale. 2 Early Sumerian clay tablets contain also numerals expressed by circles and curved signs, made with the blunt circular end of a stylus, the ordinary wedge-shaped characters being made with the pointed A circle stood for 10, a semicircular or lunar sign stood for 1.
end.
Thus, a "round-up" of cattle shows 4.
The sexagesimal
Hincks 4 1
His
und 178; 2
scale
was
in 1854. It records the first
J*DDD>
first
or
^
cows. 3
discovered on a tablet
magnitude
by E.
of the illuminated portion
papers appeared in Gottingische Gelehrte Anzeigen (1802), Stuck 149 Stuck 60 und 117.
ibid. (1803),
In the division of the year and of the day, the Babylonians used also the
duodecimal plan. G. A. Barton, Haverford Library Collection of Tablets, Part I (Philadelphia, Allotte de la 17, obverse; see also Plates 20, 26, 34, 35. 1905), Plate 3, Fuye, "En-e-tar-zi pate*si de Lagas," H. V. Hilprecht Anniversary Volume (Chicago, 1909), p. 128, 133. 8
HCL
4
"On
the Assyrian Mythology," Transactions of the Royal Irish Academy. XXII, Part 6 (Dublin, 1855), p. 406, 407.
"Polite Literature," Vol.
2
OLD NUMERAL SYMBOLS
3
moon's disk for every day from new to full moon, the whole disk assumed to consist of 240 parts. The illuminated parts during being the first five days are the series 5, 10, 20, 40, 1.20, which is a geometrical progression, on the assumption that the last number is 80. From here on the series becomes arithmetical, 1.20, 1.36, 1.52, 2.8, of the
2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4, the
The
last
number
is
common
written in the tablet
difference being 16.
X^,
and, according to
Hincks's interpretation, stood for 4 X 60 = 240. Obverse.
FIG.
1.
Reverse.
Babylonian tablets from Nippur, about 2400
B.C.
Hincks's explanation was confirmed by the decipherment of
5,
tablets found at Senkereh, near Babylon, in 1854, and called the Tablets of Senkereh. One tablet was found to contain a table of square 2 3 2 numbers, from I to 60 a second one a table of cube numbers from I to 32 3 The tablets were probably written between 2300 and 1600 B.C. Various scholars contributed toward their interpretation. Among them \vere George Smith (1872), J. Oppert, Sir H. Rawlinson, Fr. 1 Lenormant, and finally R. Lepsius. The numbers 1, 4, 9, 16, 25, 36, ,
.
George Journal Oppert,
Smith, North British Review (July, 1870), p. 332 n.;
J.
Oppert,
asiatique (August-September, 1872; October-November, 1874); J. talon des tnesures assyr. fixe" par les textes cuneiformes (Paris, 1874) ; Sir
H. Rawlinson and G. Smith, "The Cuneiform Inscriptions of Western Asia," Vol. IV: A Selection from the Miscellaneous Inscriptions of Assyria (London, 1875), Plate 40; R. Lepsius, "Die Babylonisch-Assyrischen Langenmaasse nach der Tafel von Senkereh," Abhandlungen der Koniglichen Akademie der Wissenschaften zu Berlin (aus
Klasse), p. 105-44.
dem
Jahre 1877 [Berlin, 1878], Philosophisch-historische
A HISTORY OF MATHEMATICAL NOTATIONS
4
and 49 are given as the squares of the first seven integers, respecti We have next 1.4 = 8 2 1.21 = 9 2 1.40= 102 etc. This clearly indi the use of the sexagesimal scale which makes 1.4 = 60+4, 1.21 = 21. 1.40 = 60+40, etc. This sexagesimal system marks the ea: appearance of the all-important "principle of position" in wr numbers. In its general and systematic application, this principl quires a symbol for zero. But no such symbol has been found on Babylonian tablets; records of about 200 B.C. give a symbol for as we shall see later, but it was not used in calculation. The ea: thorough and systematic application of a symbol for zero anc principle of position was made by the Maya of Central America, a ,
,
,
12,960,000
12,960,000
8,000=
Knn _ 12,960,000 25,920
.
'
51,840 .
'
.
,
lj62Q
12,960,000
.
'
9 2
>
mn 00= 12,960,000 6480 6,480
.
'
4 000== '
'
12, 960,000
3 3,240
'
12,960,000
16,000-^.
closing numbers of all the odd lines (720, 360, 180, 90, 18, 9, still obscure to me ..... are 18, 9) "The question arises, what is the meaning of all this? What in par-
But the
ticular is the meaning of the number 12,960,000 ( = 60 or 3,600 2 ) which underlies all the mathematical texts here treated ....?.... This geometrical number (12,960,000), which he [Plato in his Republic viii. 546#-D] calls 'the lord of better and worse births/ is the arithmetical expression of a great law controlling the Universe. According to Adam this law is 'the Law of Change, that law of inevitable degeneration to which the Universe and all its parts are suban interpretation from which I arn obliged to differ. On the ject' 4
'
'
it is the Law of Uniformity or Harmony, i.e. that fundamental law which governs the Universe and all its parts, and which cannot be ignored and violated without causing an anomaly, i.e. without resulting in a degeneration of the race." The nature of the "Platonic number" is still a debated question.
contrary,
A HISTORY OF MATHEMATICAL NOTATIONS
6
8. In the reading of numbers expressed in the Babylonian sexagesimal system, uncertainty arises from the fact that the early Babylonians had no symbol for zero. In the foregoing tablets, how do we know, for example, that the last number in the first line is 720 and
not 12? Nothing in the symbolism indicates that the 12 is in the place where the local value is "sixties" and not "units." Only from the study of the entire tablet has it been inferred that the number intended is 12X60 rather than 12 itself. Sometimes a horizontal line was drawn following a number, apparently to indicate the absence
But this procedure was not regular, nor carried on in a manner that indicates the number of vacant places. 9. To avoid confusion some Babylonian documents even in early times contained symbols for 1, 60, 3,600, 216,000, also for 10, 600, of units of lower denomination.
36,000.*
Thus
was
10,
was
3,600,
was 36,000.
in view of other variants occurring in fchc mathematical tablets from Nippur, notably the numerous variants of "19," some of which may be merely scribal errors : 1
They
evidently all go back
FIG. 2.
to the form
Showing application
i> !> i> i> the partial sum is 9 -\ \. This is \ I short of 10. In the it,
fourth line of the proof
(1.
it.
9) the scribe writes the
and, reducing them to the
remaining fractions 56, he writes (in
common denominator 2
DIZA1NES.
FIG. 8. Hieroglyphic, hieratic, and Coptic numerals. (Taken from A. P. Pihan, Expos6 des signes de numeration [Paris, 1860], p. 26, 27.)
red color) in the last line the numerators fractions.
Their
amount needed
sum to
is 21.
make
But e w = 5o
8, 4, 4, 2, 2, 1 of
^-=-74 8 oo o
,
which
the reduced is
the exact
the total product 10.
A
pair of legs symbolizing addition and subtraction, as found in impaired form in the Ahmes papyrus, are explained in 200. 25.
The Egyptian Coptic numerals The
are of comparatively recent date.
shown
in Figure 8. They hieroglyphic and hieratic are
are
A HISTORY OF MATHEMATICAL NOTATIONS
18
the oldest Egyptian writing; the demotic appeared later.
The Cop-
derived from the Greek and demotic writing, and was writing used by Christians in Egypt after the third century. The Coptic tic
is
Mohammedans
numeral symbols were adopted by the
in
Egypt
after
their conquest of that country. 26.
At the present time two examples of the old Egyptian soluwhat we now term "quadratic equations" 1
tion of problems involving
3
are known. For square root the symbol
Ir
has been used in the modern
hieroglyphic transcription, as the interpretation of writing in the two papyri; for quotient
was used the symbol
oo
.
PHOENICIANS AND SYRIANS The Phoenicians 2 represented the numbers 1-9 by the respective number of vertical strokes. Ten was usually designated by a horizontal bar. The numbers 1 1-19 were expressed by the juxtaposition of a horizontal stroke and the required number of vertical ones. 27.
Palmyreaische ZaMzeiebn
Virianten
>ei
I
Oruter
/
BtdeuUag FIG.
9.
1.
Palmyra
3
X -V 0;
(Syria) numerals.
;
>. 10.
,
;
20
V >V ;
100,
110.
(
.
>.V
''^0>
