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A HISTORY OF MATHEMATICAL NOTATIONS BY FLORIAN CAJORJ H. D. Professor of the History of Mathematics University of California VOLUME 1 NOTATIONS IN ELEMENTARY MATHEMATICSPREFACE 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 AND COMBINATIONS 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 ...... ... Read more

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This is a reproduction of a book published before 1923.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process.We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide.We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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This volume is produced from digital images created through the University of Michigan University Library's large-scale digitization efforts. The Library seeks to preserve the intellectual content of items in a manner that facilitates and promotes a variety of uses. The digital reformatting process results in an electronic version of the original text that can be both accessed online and used to create new print copies. The Library also understands and values the usefulness of print and makes reprints available to the public whenever possible. This book and hundreds of thousands of others can be found in the HathiTrust, an archive of the digitized collections of many great research libraries. For access to the University of Michigan Library's digital collections, please see http://www.lib.umich.edu and for information about the HathiTrust, please visit http://www.hathitrust.org ... Read more

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long out of print
Dover has done historians and students of mathematics a favour by publishing this long out of print text, from 1917. It was a seminal study of how mathematics developed in the ancient world. Cajori focused on the classical realm of the Mediterranean world. But then also in more recent times, when Europe was in its Dark Ages. He explains how the Arabs and Indians gave their contributions. Especially the role of the Arabs in, albeit inadvertantly, nurturing the maths knowledge painfully accrued in past millenia.

Unlike recent texts, there are no snazzy diagrams or whatnot to enliven the narrative. Figures in books were very expensive to arrange in the 1910s. But the writing itself remains lucid. ... Read more

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This scarce antiquarian book is a selection from Kessinger Publishing's Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the world's literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more