.
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