1.P. HISTORY OF 'AATHEMATICAL ound As One 4

2. DOVER BOOKS ON MATHEMATICS A CONCISEHISTORYOF MATHEMATICS,Dirk J. Struik. (60255-9) STATISTICALMETHODFROM THEVIEWPOINTOF QUALITYC O ~ O L ,Walter A. Shewhart. (65232-7)$7.95 VECTORS,TENSORSAND THEBASICEQUATIONSOF FLUIDMECHANICS, Rutherford Aris. (66110-5)$8.95 WTHIRTEENBOOKSOF EUCLID'SE L E M E ~ ,translated with an introduction and commentary by Sir Thomas L. Heath. (60088-2,60089-0,60090-4)$29.85 INTRODUCTIONM PARTIALDIFFERENTIALEQUATIONSw m APPLICATIONS, E.C. Zachmanoglou and Dale W. Thoe. (65251-3)$10.95 NUMERICALMETHODSFOR SCIENTISTSAND ENGINEERS,Richard Hamming. (65241-6)$15.95 ORDINARYDIFFERENTIALEQUATIONS,Morris Tenenbaurn and Hany Pollard. (64940-7)$18.95 TECHMCALCALCULUSw m ANALYTICGEOMETRY,Judith L. Gersting. (67343-X)$13.95 OSCILLATIONSINNONLINEARSYSTEMS,Jack K. Hale. (67362-6)$7.95 GREEKMATHEMATICALTHOUGHTAND THE ORIGINOF AXEBRA,Jacob Klein. (27289-3)$9.95 FINITEDIFFERENCEEQUATIONS,H. Levy & F. Lessman. (67260-3)$7.95 APPLICATIONSOF FINITEGROUPS,J. S. Lomont. 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A HISTORY OF MATHEMATICAL NOTATIONS FLORIAN CAJORI Two klumes Bound As One VolumeI: Notations in Elementary Mathematics VolumeII: Notations Mainly in Higher Mathematics DOVER PUBLICATIONS, INC. Nm Tork 4. Bibliographical Note This Dover edition, first published in 1993, is an unabridged and unaltered republication in one volume of the work first published in two volumes by The Open Court Publishing Com- pany, La Salle, Illinois, in 1928and 1929. Library of Congress Cataloging-in-PublicationData Cajori, Florian, 1859-1930. A history of mathematical notations / by Florian Cajori. p. cm. Originally published: Chicago :Open Court Pub. Co., 1928- 1929. "Two volumes bound as one." Includes indexes. Contents: v. 1.Notations in elementary mathematics - v. 2. Notations mainly in higher mathematics. ISBN 0-486-67766-4(pbk.) 1. Mathematical notation-History. 2. Mathematics-His- tory. 3. Numerals-History. 1. Title. QA41.C32 1993 5101.148-dc20 93-29211 CIP Manufactured in the United Statesof America Dover Publications, Inc., 31East2nd Street,Mineola, N.Y. 11501 5. PREFACE The study of the history of mathematical notations was sug- gested to me by Professor E. H. Moore, of the University of Chicago. To him and to Professor M.W. Haskell, of the University of California, I rm indebted for encouragement in the pursuit of this research. As completed in August, 1925, the present history was intended to be brought out in one volume. To Professor H. E. Slaught, of the Uni- versity 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 synlbols in elementary mathematics, since such a volume would ap- peal 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 desire to record my gratitude to Mrs. Mary IIegeler Carus, president of the Open Court Publishing Company, for undertaking this expen- sive publication from which no financial profits can be expected to accrue. I gratefullyacknowledge the assistance in the reading of the proofs of part of this history rendered by Professor Haskell, of the Uni- versity of California; Professor R. C. Archibald, of Brown University; and Professor L. C. Karpinski, of the University of Michigan. 6. TABLE OF CONTENTS I. INTRODVC~~ON l!AEAOBAPEB . . .11. NUMERALSYMBOLBAND COMBINATIONSOF SYMBOLE 1.- . . . . . . . . . . . . . .Babylonians 1-15 . . . . . . . . . . . . . . .Egyptians 1626 . . . . . . . . . .Phoenicians and Syrians 27-28 . . . . . . . . . . . . . . .Hebrews 29-31 . . . . . . . . . . . . . . . . .Greeks 32-44 Early Arabs . . . . . . . . . . . . . . 45 . . . . . . . . . . . . . . .Romans 4661 . . . .Peruvian and North American Knot Records 62-65 . . . . . . . . . . . . . . . .Aztecs 6667 Maya . . . . . . . . . . . . . . . . 68 . . . . . . . . . . .Chinese and Japanese 69-73 . . . . . . . . . . .Hindu-Arabic Numerals 74-9!) . . . . . . . . . . . . .Introduction 74-77 . . . . . . . . . .Principle of Local Value 78-80 . . . . . . . . . . .Forms of Numerals 81-88 Freak Forms . . . . . . . . . . . . . 89 . . . . . . . . . . .Negative Numerals 90 Grouping of Digits in Numeration . . . . . . . 91 . . . . . . . . . .The Spanish Calderbn 92-93 . . . . . . . . . .The Portuguese Cifrilo 94 Relative Size of Numerals in Tables . . . . . . 95 Fanciful Hypotheses on the Origin of Numeral Forms . 96 . . . . . . . . .A Sporadic Artificial System 97 . . . . . . . . . . . .General Remarks 98 . . . . . . . . . . . .Opinion of Laplace 99 111. SYMBOLS IN ARITHMETICAND ALGEBRA(ELEMENTARYPART) 100 A.Groups of Symbols Used by Individual Writers . 101 Greeks-Diophantus, Third Century A.D. . 101-5 . . . .Hindu-Brahmagupta, Seventh Century 106-8 . . . . .Hindu-The Bakhshiili Manuscript 109 Hindu-Bhaskara, Twelfth Century . 110-14 Arabic-al.Khowkiz~ni. Ninth Century . 115 Arabic-al.Karkhf, Eleventhcentury . 116 Byzantine-Michael Psellus, Eleventh Century . . 117 . . .Arabic-Ibn Albanna, Thirteenth Century 118 Chinese.. Chu ShibChieh, Fourteenth Century . . 119. 120 vii 7. ...vln TABLE OF CONTENTS ByzantineMaximus Planudes. Fourteenth Cent Italian-Leonardo of Pisa. Thirteenth Century French-Nicole Oresme. Fourteenth Century . Arabic-al.Qalasbdi. Fifteenth Century . . German-Regiomontanus. Fifteenth Century . Italian-Earliest Printed Arithmetic. 1478 . . French-Nicolas Chuquet. 1484 . . . . . French-Estienne de la Roche. 1520 . . . Italian-Pietro Borgi. 1484. 1488 . . . . Italian-Luca Pacioli. 1494. 1523 . . . . Italian-F .Ghaligai. 1521. 1548. 1552 . . . Italian-H .Cardan. 1532. 1545. 1570 . . . Italian-Nicolo Tartaglia. 1506-60 . . . . . . . . .Italian-Rafaele Bombelli. 1572 German-Johann Widman. 1489. 1526 . . . . . . .Austrian-Grammateus. 1518. 1535 . . . .German-Christoff Rudolff. 1525 . . .Dutch-Gielis van der Hoecke. 1537 German-Michael Stifel. 1544. 1545. 1553 . . German-Nicolaus Copernicus. 1566 . . . German-Johann Scheubel. 1545. 1551 . . MalteseWil. Klebitius. 1565 . . . . . German-Christophorus Clavius. 1608 . . . Belgium-Simon Stevin. 1585 . . . . . LorraineAlbertGirard. 1629 . . . . . German-Spanish-Marco Aurel. 1552 . . . Portuguese-Spanish-Pedro Nuaez. 1567 . . English-Robert Recorde. 1543(?). 1557 . . . . . . . . .English-John Dee. 1570 English-Leonard and Thomas Digges. 1579 . English-Thomas Masterson. 1592 . . . . . . . . .French-Jacques Peletier. 1554 . . . . . .French-Jean Buteon. 1559 . . . .French-Guillaume Gosselin. 1577 . . . . . .French-Francis Vieta. 1591 . . .Italian-Bonaventura Cavalieri. 1647 English-William Oughtred. 1631. 1632. 1657 . . . . . .English-Thomas Harriot. 1631 . . .French-Pierre HBrigone. 1634. 1644 ScobFrench-James Hume. 1635. 1636 . . French-Renk Descartes . . . . . . . English-Isaac Barrow . . . . . . . English-Richard Rawlinson. 1655-68 . . . Swiss-Johann Heinrich Rahn . . . . . 8. TABLE OF CONTENTS ix PARAQRAPAR . . . .English-John Wallis, 1655, 1657. 1685 195. 196 . . .&tract from Acta eruditorum. Leipzig. 1708 197 Extract from Miscellanea Berolinensial 1710 (Due to G.W.Leibniz) . . . . . . . . . . . 198 Conclusions . . . . . . . . . . . . In9 . . . . .B.Topical Survey of the Use of Notations 200-356 . . . . . .Signs of Addition and Subtraction 200-216 . . . . . . . . . . .Early Symbols 200 . . . . . .Origin and Meaning of the Signs 201-3 . .Spread of the +and Symbols 204 . . . . . . . . .Shapes of the +Sign 205-7 . . . . . . . . .Varieties of Signs 208, 209 . . . . . . .Symbols for "Plus or Minus" 210. 211 Certain Other Specialized Uses of +and . . . 212-14 . . . . . . . . .Four Unusual Signs 215 . . . . . . . . .Composition of Ratios 216 Signs of Multiplication . . . . . . . . 217-34 Early Symbols . . . . . . . . . . 217 Early Uses of the St.Andrew's Cross. but Not as the Symbol of Multiplication of Two Numbers . . 218-30 The Process of Two False Positions . 219 Compound Proportions with Integers . 220 Proportions Involving Fractions . 221 Addition and Subtraction of Fractions . 222 Division of Fractions . . . . . . . 223 Casting Out the 9's. 7's. or 11's . 225 Multiplication of Integers . 226 Reducing Radicals to Radicals of the Same Order 227 Marking the Place for "Thousands" . 228 Place of Multiplication Table above 5X5 . . 229 The St.Andrew's Cross Used as a Symbol of Multi- plication . . . . . . . . . . 231 Unsuccessful Symbols for Multiplication . 232 The Dot for Multiplication . 233 The St.Andrew's Cross in Notation for Transfinite Ordinal Numbers . . . . . . . . . . 234 Signs of Division and Ratio . . 235-47 Early Symbols . . . . . . . . . . . 235. 236 Fhhn's Notation . . . . . . . . . 237 Leibniz's Notations . . . . . . . . . . 238 Relative Position of Divisor and Dividend . 241 Order of Operations in Tenns Containing Both t a n d x . . . . . . . . . . . . . 242 A Critical Estimate of : and + as Symboh . . 243 9. TABLE OF CONTENTS mx.&amms Notations for Geometric Ratio . 244 Division in the Algebra of Complex Numbers . . 247 Signs of Proportion . . . . . . . 248-59 Arithmetical and Geometrical Progression . 248 . . . . . . . .Arithmetic4 Proportion 249 Geometrical Proportion . . . . . . . . 250 Oughtred's Notation . . . . . . . . 251 Struggle in England between Oughtred's and Wing's Notations before 1700 . . . 252 Strugglein England between Oughtred's and Wing's Notationsduring 1700-1750 . 253 . . . . . . . .Sporadic Notations . 254 Oughtred's Notation on the European Continent . 255 Slight Modifications of Oughtred's Notation . . 257 The Notation : :: :in Europe and America . . 258 The Notation of Leibniz . . . . . . . . 259 Signs of Equality . . . . . . . . . . 260-70 Early Symbols . . . . . . . . . . . 260 Recorde's Sign of Eauality . 261 Different Meanings of = . . . . . . . . 262 Competing Symbols . . . . . . . . 263 Descartes' Sign of Equality . 264 Variations in the Form of Descartes' Symbol . . 265 Struggle for Supremacy . . . . . . . . 266 Variation in the Form of Recorde's Symbol . 268 Variation in the Manner of Using It . 269 Nearly Equal . . . . . . . . . . . 270 Signs of Common Fractions . . . 271-75 Early Forms . . . . . . . . . . 271 The Fractional Line . . . . . . . . . . 272 Special Symbolsfor Simple Fractions . 274 The Solidus . . . . . . . . 275 Signs of Decimal Fractions . 276-89 Stevin's Notation . . . . . . . . 276 Other Notations Used before 1617 . 278 Did Pitiscus Use the Decimal Point? . 279 Decimal Comma and Point of Napier . 282 Seventeenth-Century Notations Used after 1617 . 283 Eighteenth-Century Discard of Clumsy Notations . 285 Nineteenth Century : Different Positions for Point and for Comma . . . . . . . . . 286 Signs for Repeating Decimals . . 289 Signs of Powers . . . . . . . . . . .290315 General Remarks . . . . . . . . . . 290 10. TABLE OF CONTENTS PASAGBAPHD . . . . . .Double Significanceof R and 1 291 . . . .Facsimiles of Symbols in Manuscripts 293 . . . .Two General Plans for Marking Powers 294 Early Symbolisms: Abbreviative Plan. Index Plan 295 Notations Applied Only to an Unknown Quantity. the Base Being Omitted . . . . . . . 296 Notations Applied to Any Quantity. the Base Being Designated . . . . . . . . . . . 297 Descartes' Notation of 1637 . 298 Did Stampioen Arrive at Descartes' Notation Inde- pendently? . . . . . . . . . . 299 Notations Used by Descartes before 1637 . 300 Use of H6rigone's Notation after 1637 . 301 Later Use of Hume's Notation of 1636 . 302 Other Exponential Notations Suggested after 1637 . 303 Spread of Descartes' Notation . 307 Negative. Fractional. and Literal Exponents . . 308 Imaginary Exponents . . . . . . . . . 309 Notation for Principal Values . 312 Complicated Exponents . . . . . . . 313 D.F.Gregory's (+Y . . . . . . . . . 314 Conclusions . . . . . . . . . . . . 315 Signs for Roots . . . . . . . . . . 316-38 Early Forms. General Statement . . 316.317 The Sign R. First Appearance . 318 Sixteenth-Century Use of l3 . 319 Seventeenth-Century Useof . 321 The Sign 1 . . . . . . . . . . . 322 Napier's Line Symbolism . 323 The Si@ / . . . . . . . . . . . . 324-38 Origin of / . . . . . . . . . . . 324 Spread of the / . . . . . . . . . 327 RudolfT1sSigns outside of Germany . 328 StevinlsNumeralRoot~Indices. 329 Rudolff and Stifel1sAggregation Signs . 332 Desrstes' Union of Radical Sign and Vinculum . 333 Other Signs of Aggregation of Terms . 334 Redundancy in the Use of Aggregation Signs . 335 Peculiar Dutch Symbolism . 336 . . . . . . . .Principal Root-Values 337 Recommendation of the U.S. National Committee 338 Bigns for Unknown Numben . 339-41 . . . . . . . . . . . .Early Forms 339 11. TABLE OF CONTENTS PIRAQRAPER Crossed Numerals Representing Powers of Un- knowns . . . . . . . . . . . . 340 Descartes' z. y. x . . . . . . . . . . 340 Spread of Descartesl Signs . 341 Signs of Aggregation . . . . . . . . . . 342-56 Introduction . . . . . . . . . . . . 342 Aggregation Expressed by Letters . 343 Aggregation Expressed by Horizontal Bars or Vincu- lums . . . . . . . . . . . . . 344 Aggregation Expressed by Dots . . 345 Aggregation Expressed by Commas . . . 349 Aggregation Expressed by Parentheses . 350 Early Occurrence of Parentheses . 351 Terms in an Aggregate Placed in a I'erticnl Column 353 MarkingBinomialCoefficients . 354 Special Uses of Parentheses . 355 A Star to Mark the Absence of Terms . 356 IV.SYMBOLSIN GEOMETRY(ELEMENTARYPART) . . . . . A. Ordinary Elementary Geometry . . . . . . . Early Use of Pictographs . . . . . . . . . Signs for Angles . . . . . . . . . . . Signs for " Perpendicular" . . . . . . . . . Signs for Triangle. Square. Rectangle. I'arallelograin . The Square as an Operator . . . . . . . . . . . . . . . . . . . .Sign for Circle Signs for Parallel Lines . . . . . . . . . Signs for Equal and Parallel . . . . . . . . Signs for Arcs of Circles . . . . . . . . . Other Pictographs . . . . . . . . . . . Signs for Similarity and Congruence . . . . . . The Sign +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 . . . . . . . . . . 12. ILLUSTRATIONS FIGURE PAIIAORAPHP 1. BABYLONIANTABLETSOF NIPPUR . . . . . . . . . 4 4. MATHEMATICALCUNEIFORMTABLETCBS 8536 IN TIIE MUSEUM OF THE UNIVERSITYOF PENNSYLVANIA. . . . . . . . 26. CHR. RUDOLFF'SNUMERALSAND FRACTIONS. . . . . . . . . . . . . . .27. A CONTRACT.MEXICOCITY.1649 ...Xlll 13. ILLUSTRATIONS PIQUBE PARAGRAPE8 28. REALESTATESALE,MEXICOCITP, 1718 . . . . . . . 94 29. FANCIFULHYPOTHESES . . . . . . . . . . . . 96 . . . . . . . .30. NUMERALSDESCRIBEDBY NOVIOMAGUS 98 31. SANSKRITSYMBOLSFOR THE UNKNOR-N. . . . . . . . 103 32. BAKHSHAL~ARITHMETIC. . . . . . . . . . . 109 33. SR~DHARA~STrisdtikd . . . . . . . . . . . . . 112 34. ORESME'SAlgorismus Proportionurn . . . . . . . . . 123 35. A L - Q A L A S ~ ~ ~ SALGEBRAICSYMBOLS. . . . . . . . . 125 36. COMPUTATIONSOF REGIOMONTANUS. . . . . . . . . 127 37. CALENDAROF REGIOMONTANUS. . . . . . . . . 123 38. FROMEARLIESTPRINTEDARITHMETIC. . . . . . . . 128 39. ~ULTIPLICATIONSINTHE"TREVISO~~ARITHMETIC. 128 40. DE LA ROCHE'SLarismethique. FOLIO60B . 132 41. DE LA ROCHE'SLarismethique. FOLIO66A . . . . . . . 132 . . . . . . .42. PARTOF PAGEIN PACIOLI'SSumma, 1523 138 . . . . . .43. MARGINOF FOLIO123B IN PACIOLI'SSumma 133 44. PARTOF FOLIO72 OF GHALIGAI'SPractica d'arilhmelica, 1552 . 139 45. GHALIGAI'SPraclica d'arithmelica, FOLIO198 . 139 46. CARDAN,Ars magna. ED. 1663. PAGE255 . 141 47. CARDAN.Ars m.agm. ED. 1663. PAGE297 . 141 48. FROMTARTAGLIA'SGen.eru.1Trattato. 1560 . 143 49. FROMTARTAGLIA'SGeneral Trattato, FOLIO4 . . . 144 50. FROMBOMBELLI'SAlgebra. 1572 . . . . . . . . . 144 51. BOMBELLI'SAlgebm (1579 IMPRESSION),PAGE161 . . . . 145 52. FROMTHE MS OF BOMBELLI'SAlgebra IN THE LIBRARYOF BOLOGNA145 53. FROMPAMPHLETNo. 595N IN THE LIBRARYOF TIIE UNIVERSITY OF BOLOGNA. . . . . . . . . . . . . 146 54. WIDMAN'SRechnung. 1526 . . . . . . . . . . . 146 55. FROMTHE ARITHMETICOF GRAMMATEUS. 146 56. FROMTHE ARITHMETICOF GRAMMATEUS,1535 . 147 57. FROMTHE ARITHMETICOF GRAMMATEUS,1518(?) . 147 58. FROMCHR. RUDOLFF'SCOSS,1525 . . . . . . . . . 148 14. RQWE 59. FROMCHR.RUDOLFF'SCOSS.EV. . . . . . . . . 60. FROMVANDER HOECKE'In arithmetica . . . . . . 61. PARTOF PAGEFROM STIFEL'SArithmetica integra. 1544 . . . . .62. FROMSTIFEL'SArithmetica integra. FOLIO31B 63. FROMSTIFEL'BEDITIONOF RUDOLFF'SCOSS.1553 . . . 64. SCHEUBEL.INTRODUCTIONTO EUCLID.PAGE28 . . . . . . . . . . .65. W.KLEBITIUS.BOOKLET.1565 . . . . . . . . ..66 FROMCUVIUS'Algebra. 1608 . . . . . . .67. FROMS.STEVIN'SLe Thiende. 1585 . . . . . . . ..68. FROMS STEVIN'SArithmstiqve . . . . . . . ..69. FROMS STEVIN'SArithmetiqve . . . . . . . . ..70 FROMAUREL'SArithmetica . . . . . ..71. R RECORDE,Whetstone of Witte. 1557 . . . . .75. PROPORTIONIN DEE'SPREFACE . . . . . .76. FROMDIGGES'SStraiioticos . . . . . . . .77. EQUATIONSIN DIGGES . . . . . . . .78. EQUALITYIN DIGGES 79. FROMTHOMASMASTERSON'SArithmeticke. 1592 . . . . . .. .80 J PELETIER'SAlgebra, 1554 81. ALGEBRAICOPERATIONSIN PELETIER'SAlgebra . . . .82. FROMJ.BUTEON.Arithmetica. 1559 . . . . .83. GOSSELIN'SDe arte magna. 1577 84. VIETA.In artem analyticam. 1591 . . . . 85. VIETA.De emendatione aeqvationvm . . . . 86. B.CAVALIERI.Exercitationes. 1647 . . . . 87. FROMTHOMASHARRIOT.1631. PAGE101 . . 88. FROMTHOMASHARRIOT.1631. PAGE65 . . 89. FROMHFRIGONE.Cursus mathematicus. 1644 . 90. ROMANNUMERALSFOR x IN J.HUME.1635 . 15. ILLUSTRATIONS PIOURE PbRAQRAPAB 91. RADICALSIN J.HUME.1635 . . . . . . . . . . . 191 92. R. DESCARTES.Ghomktrie . . . . . . . . . . . 191 93. I. BARROW'SEuclid. LATINEDITION. NOTESBY ISAACXEWTON. 183 94. 1. BARROW'SEuclid. ENGLISHEDITION. 193 95. RICH. RAWLINSON'SSYMBOLS. . . . . . . . . . . 194 96. RAHN'STeutsche Algebra. 1659 . . . . . . . . . 195 97. BRANCKER'STRANSLATIONOF RAHS.~GGS . . . 195 98. J. WALLIS.1657 . . . . . . . . . . . . . . 195 99. FROMTHE HIEROGLYPHICTRANSL.+TIONOF THE AHMESPAPYRUS200 100. MINUSSIGNIN THE GERMANMS C.SO. DRESDENLIBRARY. . 201 101. PLUSAND PIN US SIGNSIN THE LATINMS C. 80. DRESDEN LIBRARY. . . . . . . . . . . . . . . . . 201 102. WIDMANS'R~ARGINALNOTETO MS C.80, DRESDENLIBRARY . 201 103. FROMTIIE ARITHMETICOF BOETHIUS.1458 . . . . . . . 250 104. SIGNSIN GERMANMSS AND EARLYGERMANI~OOKS . . . . 294 105. WRITTENALGEBRAICSYMBOLSFOR POWEIISFROM ['EREZ DE MOYA'SArithmetica . . . . . . . . . . . 294 106. E.WARING'SREPEATEDEXPONENTS . . . . . . . . 313 16. NOTATIONS IN ELEMENTARY MATHEMATICS I INTRODUCTION In this history it has been an aim to give not only the first appear- ance of a symbol and its origin (whenever possible), but also to indi- cate 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 mathe- maticians 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 mathe- matics-if, in other words, this undertaking had no ulterior m o t i v e then indeed the wisdom of preparing and publishing so large a book might be questioned. But the author believes that this history consti- tutes 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. 17. NUMERAL SYhlBOLS AND COMBINATIONS 'OF SYMBOLS BABYLONIANS 1. In the Babylonian notation of numbers a vertical wedge 7 stood for 1, while the characters < and T+ signified 10 and 100, 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 out. Ordinarily, two principles were employed in the Babylonia1 no- tation-the additive and multiplicative. We shall see that limited use was made of a third principle, that of subtraction. 2. Numbers below 200 were expressed ordinarily by symbols whose respective values were to be added. Thus, T+