Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.

Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.

Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.

Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.

Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.

Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.

Written as a teaching aid for graduate and undergraduate math students, Florian Cajori's comprehensive 1896 survey of mathematics from Babylonian to modern times makes for a fascinating read. (Did you know that the decimal system is based on our having ten fingers and toes?) Beginning with the number systems of antiquity, continuing through the Hindu and Arabic influence on medieval thought, and concluding with an overview of trends in modern mathematical teaching, this is an invaluable work not only for students and educators but for readers of the history of human thought as well. Swiss-American author, educator, and mathematician FLORIAN CAJORI (1859-1930) was one of the world's most distinguished mathematical historians. Appointed to a specially created chair in the history of mathematics at the University of California, Berkeley, he also wrote An Introduction to the Theory of Equations, A History of Mathematical Notations, and The Chequered Career of Ferdinand Rudolph Hassler.

Originally published in 1893. PREFACE: AN increased interest in the history of the exact sciences manifested in recent years by teachers everywhere, and the attention given to historical inquiry in the mathematical class-rooms and seminaries of our leading universities, cause me to believe that a brief general History of Mathematics will be found acceptable to teachers and students. The pages treating necessarily in a very condensed form of the progress made during the present century, are put forth with great diffidence, although I have spent much time in the effort to render them accurate and reasonably complete. Many valuable suggestions and criti cisms on the chapter on quot B ecent Times quot have been made by, I r. E. W. Davis, of the University of Nebraska. ...FLORIAN CAJOBL COLORADO COLLEGE, December, 1893. Contents include: PAGE INTRODUCTION 1, ANTIQUITY 5 THE BABYLONIANS 5 THE EGYPTIANS 9 THE GREEKS 16 Greek Geometry 16 The Ionic School 17 The School of Pythagoras 19 The Sophist School 23 The Platonic School 29 The First Alexandrian School 34 The Second Alexandrian School 54 Greek Arithmetic 63 TUB ROMANS 77 MIDDLE AGES 84 THE HINDOOS 84 THE ARABS 100 EtJBOPE DURING THE MIDDLE AOES 117 Introduction of Roman Mathematics 117 Translation of Arabic Manuscripts 124 The First Awakening and its Sequel 128 MODERN EUROPE 138 THE RENAISSANCE . . . . 189 VIETA TO DJCSOARTES DBSGARTES TO NEWTON 183 NEWTON TO EULER 199 EULER, LAGRANGE, AND LAPLACE 246 The Origin of Modern Geometry 285 RECENT TIMES 291 SYNTHETIC GEOMETRY 293 ANALYTIC GEOMETRY 307 ALGEBRA 315 ANALYSIS 331 THEORY OP FUNCTIONS 347 THEORY OF NUMBERS 362 APPLIED MATHEMATICS 373 INDEX 405 BOOKS OF REFEKENCE. The following books, pamphlets, and articles have been used in the preparation of this history. Reference to any of them is made in the text by giving the respective number. Histories marked with a star are the only ones of which extensive use has been made. 1. GUNTHER, S. Ziele tmd Hesultate der neueren Mathematisch-his torischen JForschung. Erlangen, 1876. 2. CAJTOEI, F. The Teaching and History of Mathematics in the U. S. Washington, 1890. 3. CANToit, MORITZ. Vorlesungen uber Gfeschichte der MathematiJc. Leipzig. Bel I., 1880 Bd. II., 1892. 4. EPPING, J. Astronomisches aus Babylon. Unter Mitwirlcung von P. J. K. STUASSMAIER. Freiburg, 1889. 5. BituTHOHNKiDfflR, C. A. Die Qeometrie und die G-eometer vor Eukli des. Leipzig, 1870. 6. Gow, JAMES. A Short History of Greek Mathematics. Cambridge, 1884. 7. HANKBL, HERMANN. Zur Gfeschichte der MathematiJc im Alterthum und Mittelalter. Leipzig, 1874. 8. ALLMAN, G. J. G-reek G-eometr y from Thales to JEuclid. Dublin, 1889. 9. DB MORGAN, A. quot Euclides quot in Smith s Dictionary of Greek and Itoman Biography and Mythology. 10.

Described even today as "unsurpassed," this history of mathematical notation stretching back to the Babylonians and Egyptians is one of the most comprehensive written. In two impressive volumes-first published in 1928-9-distinguished mathematician Florian Cajori shows the origin, evolution, and dissemination of each symbol and the competition it faced in its rise to popularity or fall into obscurity. Illustrated with more than a hundred diagrams and figures, this "mirror of past and present conditions in mathematics" will give students and historians a whole new appreciation for "1 + 1 = 2." Swiss-American author, educator, and mathematician FLORIAN CAJORI (1859-1930) was one of the world's most distinguished mathematical historians. Appointed to a specially created chair in the history of mathematics at the University of California, Berkeley, he also wrote An Introduction to the Theory of Equations, A History of Elementary Mathematics, and The Chequered Career of Ferdinand Rudolph Hassler.

A HISTORY OF MATHEMATICAL NOTATIONS BY FLORIAN CAJORJ H. D. Professor of the History of Mathematics University of California VOLUME 1 NOTATIONS IN ELEMENTARY MATHEMATICS 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 am 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 symbols 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 Hegeler 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 gratefully acknowledge 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. FLORIAN CAJORI UNIVERSITY OF CALIFORNIA . TABLE OF CONTENTS I. INTRODUCTION PARAGRAPHS II. NUMERAL SYMBOLS ANDCOMBINATIONS OF SYMBOLS . . . 1-99 Babylonians 1-15 Egyptians 16-26 Phoenicians and Syrians 27-28 Hebrews 29-31 Greeks 32-44 Early Arabs 45 Romans 46-61 Peruvian and North American Knot Records .... 62-65 Aztecs 66-67 Maya 68 Chinese and Japanese 69-73 Hindu-Arabic Numerals 74-99 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 Calderon 92-93 The Portuguese Cifrao 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 III. SYMBOLS IN ARITHMETIC AND ALGEBRA ELEMENTARY PART 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 Bakhshal Manuscript 109 Hindu Bhaskara, Twelfth Century 110-14 Arabic al-Khow rizmi, Ninth Century .... 115 Arabic al-Karkhf, Eleventh Century 116 Byzantine Michael Psellus, Eleventh Century . . 117 Arabic Ibn Albanna, Thirteenth Century ... 118 Chinese Chu Shih-Chieh, Fourteenth Century . .119, 120 vii viii TABLE OF CONTENTS PARAGRAPHS Byzantine Maximus Planudes, Fourteenth Century 121 Italian Leonardo of Pisa, Thirteenth Century . . 122 French Nicole Oresme, Fourteenth Century . . . 123 Arabic al-Qalasadi, Fifteenth Century .... 124 German Regiomontanus, Fifteenth Century . . . 125-27 ItalianEarliest Printed Arithmetic, 1478 . . . . 128 French Nicolas Chuquet, 1484 129-31 French Estienne de la Roche, 1520 132 Italian Pietro Borgi, 1484, 1488 133 Italian Luca Pacioli, 1494, 1523 134-38 Italian F. Ghaligai, 1521, 1548, 1552 139 Italian H.Cardan, 1532, 1545, 1570 140, 141 Italian Nicolo Tartaglia, 1506-60 142, 143 Italian Rafaele Bombelli, 1572 144, 145 German Johann Widman, 1489, 1526 146 Austrian Grarnrnateus, 1518, 1535 147 German Christoff Rudolff, 1525 148, 149 Dutch Gielis van der Hoecke, 1537 150 German Michael Stifel, 1544, 1545, 1553 ......