Synopses & Reviews
'The historian of mathematics will find much to interest him here . . . while the casual reader is likely to be intrigued by the author's superior narrative ability." — Library Journal
This book is not only a fascinating introduction to the concept of number and to numbers themselves, hut a multifaceted linguistic and historical analysis of how numbers have developed and evolved in many different cultures. Drawing on evidence from history, literature, philosophy and ethnology, noted German scholar Karl Menninger. recounts the development of numbers both as they are spoken (and written as words) and as symbolic abstract numerals that can he readily manipulated and combined.
Despite the immense erudition the author brings to the topic, he maintains a light tone throughout, presenting much of the information in anecdotal form. Moreover, almost 300 illustrations (photographs and drawings) and many comparative language tables serve to enhance the text. The author begins with a lucid treatment of number sequence and number language, including the formation of number words in both Indo-European and non-IndoEuropean languages, hidden number words and the evolution of the number sequence. He then turns to written numerals and computations: finger counting, folk symbols for numbers, alphabetical numerals, the "German" Roman numerals, the abacus and more. The final section concerns the development of our modem decimal system, with its place notation and zero, based on the Indian number system, and its introduction to the West through the work of the Italian mathematician Fibonacci. The author concludes with a review of spoken numbers and number symbols in China and Japan.
"The book is especially good on early counting and calculating devices: primitive tally sticks, the knotted cords of ancient Peru, the elaborate finger symbols once used for numbers, counting boards with movable counters, and of course the abacus." — Martin Gardner, Book World
Synopsis
Classic study discusses number sequence and language and explores written numerals and computations in many cultures. "The historian of mathematics will find much to interest him here both in the contents and viewpoint, while the casual reader is likely to be intrigued by the author's superior narrative ability." — Library Journal. 282 illustrations.
Synopsis
Classic study discusses number sequence and number language, then explores written numerals and computations in a wide range of cultures. 282 illustrations. "Superior narrative ability." — Library Journal.
Synopsis
This book is not only a fascinating introduction to the number concept and to numbers themselves but also a multifaceted linguistic and historic analysis of how numbers have evolved in many cultures. A noted scholar draws upon evidence from history, literature, philosophy, and ethnology to recount the oral, written, and symbolic development of numbers. "The historian of mathematics will find much to interest him." — Library Journal.
Table of Contents
Preface
List of Phonetic Symbols
NUMBER SEQUENCE AND NUMBER LANGUAGE
Introduction
The Number Sequence
The Abstract Number Sequence
The Number Sequence Use Concretely
NUMBERS WITHOUT WORDS
NUMBERS AS ATTRIBUTES
NUMBERS AS ADJECTIVES
Expansion of the Number Sequence by Means of Supplementary Quantities
NUMBERING BY FINGERS AND TOES
Principles of the Number Sequence
SUCCESSION AND GROUP
GROUPINGS OF HIGHER RANK
GRADATION BY STEPS
GROUPINGS IN THE GRADUATED NUMBER SEQUENCE
"20-GROUP "MAN"
NUMBER LANGUAGE AND NUMBER SYMBOLS
Historical Gradations
REPRESENTING THE GRADATION
NUMBER SYSTEMS BASED ON GRADATIONS OF 20
VIGESIMAL GRADATIONS IN EUROPEAN NUMBER SEQUENCES
THE AINU NUMBER SEQUENCE
Laws Governing the Formation of Number Words
SUCCESSION OF MAGNITUDES
BACK-COUNTING
OVERCOUNTING
ADDITIONAL WAYS OF FORMING NUMBER WORDS
THE FIRST STEPS BEYOND TEN
Our Number Words
Indo-European Family Languages
SURVEY OF LANGUAGES AND NUMBER WORDS
THE ORIGINAL INDO-EUROPEAN LANGUAGE
THE INDIVIDUAL INDO-EUROPEAN LANGUAGES
The Non-Indo-European Languages
The Meaning of Our Number Words
IMAGES BEHIND THE CONCEPTS OF SIZE AND NUMBER
SEMANTIC FADING
OUR INDO-EUROPEAN NUMBER WORDS
Was There a Babylonian Influence on Our Number Sequence?
THE BREAK AFTER SIXTY
THE ROLE PLAYED BY THE NUMBER SIXTY IN CULTURAL HISTORY
GREAT HUNDRED
THE NUMBER TWELVE AS THE BASIC UNIT OF THE GREAT HUNDRED
THE ROMAN DUODECIMAL FRACTIONS
THE BABYLONIAN SEXAGESIMAL SYSTEM
THE BREAK AFTER SIXTY IN THE GERMANIC NUMBER SEQUENCE
Hidden Number Words
ONE
TWO
THREE
FOUR
FIVE
SIX
SEVEN
NINE
TEN
TWENTY
FORTY
FIFTY
SEVENTY
HUNDRED
THOUSAND
NUMBER
The Evolution of the Number Sequence
FOREIGN LOAN WORDS
WRITTEN NUMERALS AND COMPUTATIONS
Preface
Introduction
Finger Counting
Finger Counting
THE VENERABLE BEDE AND HIS FINGER COUNTING
FINGER COUNTING IN ANTIQUITY
FINGER COUNTING IN ARABIC AND EAST AFRICAN COMMERCE TRADE
ROMAN FINGER COUNTING IN THE WEST
COMPUTATIONS ON THE FINGERS
SOME FORMS OF FINGER COUNTING USE BY PEOPLES OF OTHER CULTURES
Folk Symbols for Numbers
Tally Sticks
EARLY WRITING AND READING
TALLY STICKS WERE UNIVERSAL
VARIETIES OF TALLY STICKS
THE BRITISH EXCHEQUER TALLIES
THE NUMBERS ON THE TALLY STICKS
ROMAN NUMERALS
CHINESE HAN STICKS
Peasant Numerals
Knots Used as Numerals
Alphabetical Numerals
The Gothic Numerals
Letters and Numbers
HISTORY OF THE ALPHABET
The Two Greek Sets of Numerals
Some Other Connections Between Letters and Numbers
"The "German" Roman Numerals"
Roman Numerals in Cursive Form
MEDIEVAL NUMBERS
THE GRADUAL PENETRATION OF THE NEW INDIAN NUMERALS
THE ACCOUNT BOOKS OF THE IMPERIAL FREE CITY OF AUGSBURG
WRITTEN NUMERALS AND COMPUTATIONS
The Abacus
The Nature of the Counting Board
The Counting Board in Ancient Civilizations
THE SALAMIS TABLET
DARIUS VASE
THE ETRUSCAN CAMEO
THE ROMAN HAND ABACUS
THE HAND ABACUS IN ASIA
THE ROMAN COUNTING BOARD WITH LOOSE COUNTERS
The Counting Board in the Early Middle Ages
THE WEST
THE MONASTIC ABACUS
The Counting Board in the Later Middle Ages
THE EVIDENCE FOR ITS USE
THE NEW RECKONING BOARD
THE NAMES OF THE COUNTING BOARDS
"RECKONING "ON THE LINES"
THE COUNTING BOARD IN EVERYDAY LIFE
THE COUNTING BOARD AND WRITTEN NUMERALS
The Counters
Our Own Numerals
Place-Value Notation
The Antecedents of Our Numerals
THE KHAROSTHI NUMERALS
THE BRAHMI NUMERALS
PLACE-VALUE NOTATION
The Westward Migration of the Indian Numerals
THE ZERO
ALEXANDRIA
THE INDIAN NUMERALS IN ARAB HANDS
THE FAMILY TREE OF INDIAN NUMERALS
The Indian Numerals in Western Europe
ITALY
LEONARDO OF PISA
THE GERMAN ARITHMETICIANS
"THE "NEW" NUMERALS"
RETROSPECT
Spoken Numbers and Number Symbols in China and Japan
Far Eastern Number Systems
Spoken Numbers
THE CHINESE NUMBER WORDS
JAPANESE NUMBER WORDS
KOREAN NUMBER WORDS
SUMMARY
WRITTEN NUMERALS
RETROSPECT: SPOKEN LANGUAGE?WRITING?WRITTEN NUMERALS
Index