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Title: A Short Account of the History of Mathematics
Author: W. W. Rouse Ball
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Language: English
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*** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICS ***
A SHORT ACCOUNT
OF THE
HISTORY OF MATHEMATICS
BY
W. W. ROUSE BALL
FELLOW OF TRINITY COLLEGE, CAMBRIDGE
DOVER PUBLICATIONS, INC.
NEW YORK
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This new Dover edition, first published in 1960, is an unabridged and
unaltered republication of the author’s last revision—the fourth
edition which appeared in 1908.
International Standard Book Number: 0-486-20630-0
Library of Congress Catalog Card Number: 60-3187
Manufactured in the United States of America
Dover Publications, Inc.
180 Varick Street
New York, N. Y. 10014
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Produced by Greg Lindahl, Viv, Juliet Sutherland, Nigel Blower and the
Online Distributed Proofreading Team at
Transcriber’s Notes
A small number of minor typographical errors and inconsistencies have been
corrected. References to figures such as “on the next page” have been replaced with text such as “below” which is more suited to an eBook.
Such changes are documented in the LATEX source: %[**TN: text of note]
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PREFACE.
The subject-matter of this book is a historical summary of the
development of mathematics, illustrated by the lives and discoveries of
those to whom the progress of the science is mainly due. It may serve as
an introduction to more elaborate works on the subject, but primarily
it is intended to give a short and popular account of those leading facts
in the history of mathematics which many who are unwilling, or have
not the time, to study it systematically may yet desire to know.
The first edition was substantially a transcript of some lectures
which I delivered in the year 1888 with the object of giving a sketch of
the history, previous to the nineteenth century, that should be intelligible to any one acquainted with the elements of mathematics. In the
second edition, issued in 1893, I rearranged parts of it, and introduced
a good deal of additional matter.
The scheme of arrangement will be gathered from the table of contents at the end of this preface. Shortly it is as follows. The first chapter
contains a brief statement of what is known concerning the mathematics of the Egyptians and Phoenicians; this is introductory to the history
of mathematics under Greek influence. The subsequent history is divided into three periods: first, that under Greek influence, chapters ii
to vii; second, that of the middle ages and renaissance, chapters viii
to xiii; and lastly that of modern times, chapters xiv to xix.
In discussing the mathematics of these periods I have confined myself to giving the leading events in the history, and frequently have
passed in silence over men or works whose influence was comparatively
unimportant. Doubtless an exaggerated view of the discoveries of those
mathematicians who are mentioned may be caused by the non-allusion
to minor writers who preceded and prepared the way for them, but in
all historical sketches this is to some extent inevitable, and I have done
my best to guard against it by interpolating remarks on the progress
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v
PREFACE
of the science at different times. Perhaps also I should here state that
generally I have not referred to the results obtained by practical astronomers and physicists unless there was some mathematical interest
in them. In quoting results I have commonly made use of modern notation; the reader must therefore recollect that, while the matter is
the same as that of any writer to whom allusion is made, his proof is
sometimes translated into a more convenient and familiar language.
The greater part of my account is a compilation from existing histories or memoirs, as indeed must be necessarily the case where the works
discussed are so numerous and cover so much ground. When authorities disagree I have generally stated only that view which seems to me
to be the most probable; but if the question be one of importance, I
believe that I have always indicated that there is a difference of opinion
about it.
I think that it is undesirable to overload a popular account with
a mass of detailed references or the authority for every particular fact
mentioned. For the history previous to 1758, I need only refer, once for
all, to the closely printed pages of M. Cantor’s monumental Vorlesungen
u
ăber die Geschichte der Mathematik (hereafter alluded to as Cantor),
which may be regarded as the standard treatise on the subject, but
usually I have given references to the other leading authorities on which
I have relied or with which I am acquainted. My account for the period
subsequent to 1758 is generally based on the memoirs or monographs
referred to in the footnotes, but the main facts to 1799 have been also
enumerated in a supplementary volume issued by Prof. Cantor last year.
I hope that my footnotes will supply the means of studying in detail
the history of mathematics at any specified period should the reader
desire to do so.
My thanks are due to various friends and correspondents who have
called my attention to points in the previous editions. I shall be grateful
for notices of additions or corrections which may occur to any of my
readers.
W. W. ROUSE BALL.
TRINITY COLLEGE, CAMBRIDGE.
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NOTE.
The fourth edition was stereotyped in 1908, but no material changes
have been made since the issue of the second edition in 1893, other
duties having, for a few years, rendered it impossible for me to find
time for any extensive revision. Such revision and incorporation of
recent researches on the subject have now to be postponed till the cost
of printing has fallen, though advantage has been taken of reprints to
make trivial corrections and additions.
W. W. R. B.
TRINITY COLLEGE, CAMBRIDGE.
vi
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vii
TABLE OF CONTENTS.
Preface .
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Table of Contents
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vii
Chapter I. Egyptian and Phoenician Mathematics.
The history of mathematics begins with that of the Ionian Greeks
Greek indebtedness to Egyptians and Phoenicians
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Knowledge of the science of numbers possessed by the Phoenicians
Knowledge of the science of numbers possessed by the Egyptians
Knowledge of the science of geometry possessed by the Egyptians
Note on ignorance of mathematics shewn by the Chinese .
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2
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4
7
First Period. Mathematics under Greek Influence.
This period begins with the teaching of Thales, circ. 600 b.c., and ends with the
capture of Alexandria by the Mohammedans in or about 641 a.d. The
characteristic feature of this period is the development of geometry.
Chapter II. The Ionian and Pythagorean Schools.
Circ. 600 b.c.–400 b.c.
Authorities
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The Ionian School
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Thales, 640–550 b.c.
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His geometrical discoveries
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His astronomical teaching .
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Anaximander. Anaximenes. Mamercus. Mandryatus
The Pythagorean School
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Pythagoras, 569–500 b.c.
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The Pythagorean teaching .
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The Pythagorean geometry
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viii
TABLE OF CONTENTS
The Pythagorean theory of numbers .
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Epicharmus. Hippasus. Philolaus. Archippus. Lysis .
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Archytas, circ. 400 b.c. .
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His solution of the duplication of a cube
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Theodorus. Timaeus. Bryso
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Other Greek Mathematical Schools in the Fifth Century b.c.
Oenopides of Chios
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Zeno of Elea. Democritus of Abdera .
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23
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Chapter III. The Schools of Athens and Cyzicus.
Circ. 420–300 b.c.
Authorities
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Mathematical teachers at Athens prior to 420 b.c.
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Anaxagoras. The Sophists. Hippias (The quadratrix).
Antipho
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Three problems in which these schools were specially interested
Hippocrates of Chios, circ. 420 b.c.
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Letters used to describe geometrical diagrams
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Introduction in geometry of the method of reduction
The quadrature of certain lunes .
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The problem of the duplication of the cube
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Plato, 429–348 b.c.
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Introduction in geometry of the method of analysis
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Theorem on the duplication of the cube
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Eudoxus, 408–355 b.c.
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Theorems on the golden section .
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Introduction of the method of exhaustions
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Pupils of Plato and Eudoxus
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Menaechmus, circ. 340 b.c.
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Discussion of the conic sections .
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His two solutions of the duplication of the cube
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Aristaeus. Theaetetus
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Aristotle, 384–322 b.c.
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Questions on mechanics. Letters used to indicate magnitudes
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40
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44
47
Chapter IV. The First Alexandrian School.
Circ. 300–30 b.c.
Authorities
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Foundation of Alexandria .
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The Third Century before Christ
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Euclid, circ. 330–275 b.c.
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Euclid’s Elements
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The Elements as a text-book of geometry .
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The Elements as a text-book of the theory of numbers
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ix
TABLE OF CONTENTS
Euclid’s other works .
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Aristarchus, circ. 310–250 b.c. .
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Method of determining the distance of the sun .
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Conon. Dositheus. Zeuxippus. Nicoteles
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Archimedes, 287–212 b.c.
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His works on plane geometry
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His works on geometry of three dimensions
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His two papers on arithmetic, and the “cattle problem”
His works on the statics of solids and fluids
.
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His astronomy
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The principles of geometry assumed by Archimedes
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Apollonius, circ. 260–200 b.c.
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His conic sections
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His other works .
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His solution of the duplication of the cube
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Contrast between his geometry and that of Archimedes
Eratosthenes, 275–194 b.c.
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The Sieve of Eratosthenes .
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The Second Century before Christ
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Hypsicles (Euclid, book xiv). Nicomedes. Diocles
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Perseus. Zenodorus
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Hipparchus, circ. 130 b.c.
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Foundation of scientific astronomy
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Foundation of trigonometry
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Hero of Alexandria, circ. 125 b.c.
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Foundation of scientific engineering and of land-surveying
Area of a triangle determined in terms of its sides
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Features of Hero’s works
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The First Century before Christ
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Theodosius
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Dionysodorus
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End of the First Alexandrian School .
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Egypt constituted a Roman province
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49
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79
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80
Chapter V. The Second Alexandrian School.
30 b.c.–641 a.d.
Authorities
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The First Century after Christ .
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Serenus. Menelaus
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Nicomachus
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Introduction of the arithmetic current in medieval Europe
The Second Century after Christ
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Theon of Smyrna. Thymaridas .
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Ptolemy, died in 168
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The Almagest
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Ptolemy’s astronomy .
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TABLE OF CONTENTS
Ptolemy’s geometry
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The Third Century after Christ
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Pappus, circ. 280
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The Συναγωγή, a synopsis of Greek mathematics
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The Fourth Century after Christ
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Metrodorus. Elementary problems in arithmetic and algebra
Three stages in the development of algebra
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Diophantus, circ. 320 (?)
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Introduction of syncopated algebra in his Arithmetic
The notation, methods, and subject-matter of the work
His Porisms
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Subsequent neglect of his discoveries .
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Iamblichus
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Theon of Alexandria. Hypatia .
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Hostility of the Eastern Church to Greek science
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The Athenian School (in the Fifth Century)
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Proclus, 412–485. Damascius. Eutocius
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Roman Mathematics .
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Nature and extent of the mathematics read at Rome
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Contrast between the conditions at Rome and at Alexandria
End of the Second Alexandrian School
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The capture of Alexandria, and end of the Alexandrian Schools
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97
97
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100
Chapter VI. The Byzantine School. 641–1453.
Preservation of works of the great Greek Mathematicians
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Hero of Constantinople. Psellus. Planudes. Barlaam. Argyrus .
Nicholas Rhabdas, Pachymeres. Moschopulus (Magic Squares)
Capture of Constantinople, and dispersal of Greek Mathematicians
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Chapter VII. Systems of Numeration and Primitive
Arithmetic.
Authorities
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Methods of counting and indicating numbers among primitive races
Use of the abacus or swan-pan for practical calculation
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Methods of representing numbers in writing
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The Roman and Attic symbols for numbers
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The Alexandrian (or later Greek) symbols for numbers
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Greek arithmetic
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Adoption of the Arabic system of notation among civilized races
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xi
TABLE OF CONTENTS
Second Period. Mathematics of the Middle Ages
and of the Renaissance.
This period begins about the sixth century, and may be said to end with the
invention of analytical geometry and of the infinitesimal calculus. The
characteristic feature of this period is the creation or development of modern
arithmetic, algebra, and trigonometry.
Chapter VIII. The Rise Of Learning In Western Europe.
Circ. 600–1200.
Authorities
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Education in the Sixth, Seventh, and Eighth Centuries
The Monastic Schools
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Boethius, circ. 475–526
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Medieval text-books in geometry and arithmetic
Cassiodorus, 490–566. Isidorus of Seville, 570–636
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The Cathedral and Conventual Schools
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The Schools of Charles the Great
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Alcuin, 735–804
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Education in the Ninth and Tenth Centuries
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Gerbert (Sylvester II.), died in 1003 .
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Bernelinus
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The Early Medieval Universities
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Rise during the twelfth century of the earliest universities
Development of the medieval universities .
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Outline of the course of studies in a medieval university
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Chapter IX. The Mathematics Of The Arabs.
Authorities
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Extent of Mathematics obtained from Greek Sources .
The College of Scribes
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Extent of Mathematics obtained from the (Aryan) Hindoos
Arya-Bhata, circ. 530
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His algebra and trigonometry (in his Aryabhathiya)
Brahmagupta, circ. 640 .
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His algebra and geometry (in his Siddhanta)
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Bhaskara, circ. 1140
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The Lilavati or arithmetic; decimal numeration used
The Bija Ganita or algebra
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Development of Mathematics in Arabia
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¯ rizm¯i, circ. 830 .
Alkarismi or Al-Khwa
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His Al-gebr we’ l mukabala
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His solution of a quadratic equation .
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TABLE OF CONTENTS
Introduction of Arabic or Indian system of numeration
Tabit ibn Korra, 836–901; solution of a cubic equation
Alkayami. Alkarki. Development of algebra
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Albategni. Albuzjani. Development of trigonometry .
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Alhazen. Abd-al-gehl. Development of geometry
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Characteristics of the Arabian School
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134
134
Chapter X. Introduction of Arabian Works into Europe.
Circ. 1150–1450.
The Eleventh Century
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Moorish Teachers. Geber ibn Aphla. Arzachel .
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The Twelfth Century
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Adelhard of Bath
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Ben-Ezra. Gerard. John Hispalensis .
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The Thirteenth Century
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Leonardo of Pisa, circ. 1175–1230
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The Liber Abaci, 1202
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The introduction of the Arabic numerals into commerce
The introduction of the Arabic numerals into science
The mathematical tournament
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Frederick II., 1194–1250
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Jordanus, circ. 1220
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His De Numeris Datis; syncopated algebra
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Holywood .
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Roger Bacon, 1214–1294
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Campanus
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The Fourteenth Century
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Bradwardine
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Oresmus
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The reform of the university curriculum
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The Fifteenth Century
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Beldomandi
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136
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147
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148
149
149
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151
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Chapter XI. The Development Of Arithmetic.
Circ. 1300–1637.
Authorities
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The Boethian arithmetic
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Algorism or modern arithmetic .
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The Arabic (or Indian) symbols: history of
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Introduction into Europe by science, commerce, and calendars
Improvements introduced in algoristic arithmetic
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(i) Simplification of the fundamental processes .
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(ii) Introduction of signs for addition and subtraction
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TABLE OF CONTENTS
(iii) Invention of logarithms, 1614
(iv) Use of decimals, 1619 .
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162
163
Chapter XII. The Mathematics of the Renaissance.
Circ. 1450–1637.
Authorities
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Effect of invention of printing. The renaissance
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Development of Syncopated Algebra and Trigonometry
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Regiomontanus, 1436–1476
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His De Triangulis (printed in 1496)
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Purbach, 1423–1461. Cusa, 1401–1464. Chuquet, circ. 1484
Introduction and origin of symbols + and −
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Pacioli or Lucas di Burgo, circ. 1500
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His arithmetic and geometry, 1494
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Leonardo da Vinci, 14521519 .
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Dă
urer, 14711528. Copernicus, 14731543
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Record, 15101558; introduction of symbol for equality
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Rudolff, circ. 1525. Riese, 1489–1559
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Stifel, 1486–1567
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His Arithmetica Integra, 1544
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Tartaglia, 1500–1557
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His solution of a cubic equation, 1535
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His arithmetic, 1556–1560 .
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Cardan, 1501–1576 .
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His Ars Magna, 1545; the third work printed on algebra.
His solution of a cubic equation .
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Ferrari, 1522–1565; solution of a biquadratic equation
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Rheticus, 1514–1576. Maurolycus. Borrel. Xylander
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Commandino. Peletier. Romanus. Pitiscus. Ramus. 1515–1572
Bombelli, circ. 1570 .
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Development of Symbolic Algebra
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Vieta, 1540–1603
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The In Artem; introduction of symbolic algebra, 1591
Vieta’s other works
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Girard, 1595–1632; development of trigonometry and algebra
Napier, 1550–1617; introduction of logarithms, 1614
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Briggs, 1561–1631; calculations of tables of logarithms
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Harriot, 1560–1621; development of analysis in algebra .
Oughtred, 1574–1660
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The Origin of the more Common Symbols in Algebra
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165
165
166
166
167
170
171
173
173
176
176
177
178
178
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180
181
182
183
184
186
186
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187
188
189
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195
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xiv
TABLE OF CONTENTS
Chapter XIII. The Close of the Renaissance.
Circ. 1586–1637.
Authorities
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Development of Mechanics and Experimental Methods
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Stevinus, 1548–1620
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Commencement of the modern treatment of statics, 1586
Galileo, 1564–1642 .
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Commencement of the science of dynamics
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Galileo’s astronomy
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Francis Bacon, 1561–1626. Guldinus, 1577–1643
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Wright, 1560–1615; construction of maps .
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Snell, 1591–1626
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Revival of Interest in Pure Geometry
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Kepler, 1571–1630 .
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His Paralipomena, 1604; principle of continuity .
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His Stereometria, 1615; use of infinitesimals
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Kepler’s laws of planetary motion, 1609 and 1619
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Desargues, 1593–1662
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His Brouillon project; use of projective geometry
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Mathematical Knowledge at the Close of the Renaissance .
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202
202
202
203
205
205
206
208
209
210
210
210
211
212
212
213
213
214
Third period. Modern Mathematics.
This period begins with the invention of analytical geometry and the infinitesimal
calculus. The mathematics is far more complex than that produced in either of the
preceding periods: but it may be generally described as characterized by the
development of analysis, and its application to the phenomena of nature.
Chapter XIV. The History of Modern Mathematics.
Treatment of the subject .
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Invention of analytical geometry and the method of indivisibles
Invention of the calculus
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Development of mechanics
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Application of mathematics to physics
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Recent development of pure mathematics .
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217
218
218
219
219
220
Chapter XV. History of Mathematics from Descartes to
Huygens. Circ. 1635–1675.
Authorities
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Descartes, 1596–1650
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His views on philosophy
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221
221
224
xv
TABLE OF CONTENTS
His invention of analytical geometry, 1637 .
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His algebra, optics, and theory of vortices .
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Cavalieri, 1598–1647
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The method of indivisibles .
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Pascal, 1623–1662
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His geometrical conics
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The arithmetical triangle
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Foundation of the theory of probabilities, 1654 .
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His discussion of the cycloid
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Wallis, 1616–1703
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The Arithmetica Infinitorum, 1656
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Law of indices in algebra
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Use of series in quadratures
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Earliest rectification of curves, 1657
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Wallis’s algebra .
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Fermat, 1601–1665 .
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His investigations on the theory of numbers
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His use in geometry of analysis and of infinitesimals .
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Foundation of the theory of probabilities, 1654 .
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Huygens, 1629–1695
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The Horologium Oscillatorium, 1673 .
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The undulatory theory of light .
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Other Mathematicians of this Time .
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Bachet
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Mersenne; theorem on primes and perfect numbers
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Roberval. Van Schooten. Saint-Vincent
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Torricelli. Hudde. Fr´enicle
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De Laloub`ere. Mercator. Barrow; the differential triangle
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Brouncker; continued fractions .
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James Gregory; distinction between convergent and divergent series
Sir Christopher Wren
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Hooke. Collins .
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Pell. Sluze. Viviani
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Tschirnhausen. De la Hire. Roemer. Rolle
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224
227
229
230
232
234
234
235
236
237
238
238
239
240
241
241
242
246
247
248
249
250
251
252
252
253
254
254
257
258
259
259
260
261
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263
263
264
265
266
267
267
268
269
272
Chapter XVI. The Life and Works of Newton.
Authorities
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Newton’s school and undergraduate life
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Investigations in 1665–1666 on fluxions, optics,
His views on gravitation, 1666
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Researches in 1667–1669
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Elected Lucasian professor, 1669
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Optical lectures and discoveries, 1669–1671
Emission theory of light, 1675
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The Leibnitz Letters, 1676
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Discoveries on gravitation, 1679
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and gravitation
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xvi
TABLE OF CONTENTS
Discoveries and lectures on algebra, 1673–1683 .
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Discoveries and lectures on gravitation, 1684
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The Principia, 1685–1686 .
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The subject-matter of the Principia .
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Publication of the Principia
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Investigations and work from 1686 to 1696
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Appointment at the Mint, and removal to London, 1696
Publication of the Optics, 1704 .
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Appendix on classification of cubic curves .
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Appendix on quadrature by means of infinite series
Appendix on method of fluxions
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The invention of fluxions and the infinitesimal calculus
Newton’s death, 1727
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List of his works
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Newton’s character
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Newton’s discoveries .
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272
274
275
276
278
278
279
279
279
281
282
286
286
286
287
289
Chapter XVII. Leibnitz and the Mathematicians of the
First Half of the Eighteenth Century.
Authorities
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Leibnitz and the Bernoullis
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Leibnitz, 1646–1716
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His system of philosophy, and services to literature
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The controversy as to the origin of the calculus .
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His memoirs on the infinitesimal calculus
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His papers on various mechanical problems
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Characteristics of his work .
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James Bernoulli, 1654–1705 .
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John Bernoulli, 1667–1748
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The younger Bernouillis
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Development of Analysis on the Continent
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L’Hospital, 1661–1704
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Varignon, 1654–1722. De Montmort. Nicole
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Parent. Saurin. De Gua. Cramer, 1704–1752
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Riccati, 1676–1754. Fagnano, 1682–1766
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Clairaut, 1713–1765
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D’Alembert, 1717–1783 .
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Solution of a partial differential equation of the second order
Daniel Bernoulli, 1700–1782
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English Mathematicians of the Eighteenth Century
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David Gregory, 1661–1708. Halley, 1656–1742 .
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Ditton, 1675–1715
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Brook Taylor, 1685–1731
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Taylor’s theorem
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Taylor’s physical researches
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Cotes, 1682–1716
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291
291
291
293
293
298
299
301
301
302
303
304
304
305
305
306
307
308
309
311
312
312
313
313
314
314
315
xvii
TABLE OF CONTENTS
Demoivre, 1667–1754; development of trigonometry
Maclaurin, 1698–1746
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His geometrical discoveries
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The Treatise of Fluxions
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His propositions on attractions .
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Stewart, 1717–1785. Thomas Simpson, 1710–1761
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315
316
317
318
318
319
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322
323
323
324
326
326
326
327
329
330
330
331
334
337
338
338
339
339
340
340
341
341
342
343
344
344
345
346
346
347
348
348
349
349
350
350
351
Chapter XVIII. Lagrange, Laplace, and their
Contemporaries. Circ. 1740–1830.
Characteristics of the mathematics of the period
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Development of Analysis and Mechanics
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Euler, 1707–1783
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The Introductio in Analysin Infinitorum, 1748
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The Institutiones Calculi Differentialis, 1755
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The Institutiones Calculi Integralis, 1768–1770 .
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The Anleitung zur Algebra, 1770
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Euler’s works on mechanics and astronomy
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Lambert, 1728–1777 .
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B´ezout, 1730–1783. Trembley, 1749–1811. Arbogast, 1759–1803
Lagrange, 1736–1813
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Memoirs on various subjects
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The M´ecanique analytique, 1788
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The Th´eorie and Calcul des fonctions, 1797, 1804
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The R´esolution des ´equations num´eriques, 1798.
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Characteristics of Lagrange’s work
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Laplace, 1749–1827
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Memoirs on astronomy and attractions, 1773–1784
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Use of spherical harmonics and the potential
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Memoirs on problems in astronomy, 1784–1786 .
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The M´ecanique c´eleste and Exposition du syst`eme du monde
The Nebular Hypothesis
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The Meteoric Hypothesis
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The Th´eorie analytique des probabilit´es, 1812
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The Method of Least Squares
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Other researches in pure mathematics and in physics
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Characteristics of Laplace’s work
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Character of Laplace .
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Legendre, 1752–1833
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His memoirs on attractions
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The Th´eorie des nombres, 1798 .
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Law of quadratic reciprocity
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The Calcul int´egral and the Fonctions elliptiques
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Pfaff, 1765–1825
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Creation of Modern Geometry
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Monge, 1746–1818
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Lazare Carnot, 1753–1823. Poncelet, 1788–1867
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xviii
TABLE OF CONTENTS
Development of Mathematical Physics
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Cavendish, 1731–1810
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Rumford, 1753–1815. Young, 1773–1829
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Dalton, 1766–1844
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Fourier, 1768–1830 .
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Sadi Carnot; foundation of thermodynamics
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Poisson, 1781–1840 .
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Amp`ere, 1775–1836. Fresnel, 1788–1827. Biot, 1774–1862
Arago, 1786–1853
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Introduction of Analysis into England
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Ivory, 1765–1842
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The Cambridge Analytical School
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Woodhouse, 1773–1827
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Peacock, 1791–1858. Babbage, 1792–1871. John Herschel,
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1792–1871
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353
353
353
354
355
356
356
358
359
360
360
361
361
362
Chapter XIX. Mathematics of the Nineteenth Century.
Creation of new branches of mathematics .
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Difficulty in discussing the mathematics of this century
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Account of contemporary work not intended to be exhaustive .
Authorities
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Gauss, 1777–1855
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Investigations in astronomy
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Investigations in electricity
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The Disquisitiones Arithmeticae, 1801
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His other discoveries .
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Comparison of Lagrange, Laplace, and Gauss
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Dirichlet, 1805–1859 .
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Development of the Theory of Numbers
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Eisenstein, 1823–1852
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Henry Smith, 1826–1883
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Kummer, 1810–1893 .
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Notes on other writers on the Theory of Numbers
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Development of the Theory of Functions of Multiple Periodicity
Abel, 1802–1829. Abel’s Theorem
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Jacobi, 1804–1851
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Riemann, 1826–1866
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Notes on other writers on Elliptic and Abelian Functions .
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Weierstrass, 1815–1897
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Notes on recent writers on Elliptic and Abelian Functions
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The Theory of Functions
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Development of Higher Algebra
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Cauchy, 1789–1857 .
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Argand, 1768–1822; geometrical interpretation of complex numbers
Sir William Hamilton, 1805–1865; introduction of quaternions
Grassmann, 1809–1877; his non-commutative algebra, 1844
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Boole, 1815–1864. De Morgan, 1806–1871
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365
365
365
366
367
368
369
371
372
373
373
374
374
374
377
377
378
379
380
381
382
382
383
384
385
385
387
387
389
389
xix
TABLE OF CONTENTS
Galois, 1811–1832; theory of discontinuous substitution groups
Cayley, 1821–1895 .
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Sylvester, 1814–1897
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Lie, 1842–1889; theory of continuous substitution groups .
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Hermite, 1822–1901
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Notes on other writers on Higher Algebra .
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Development of Analytical Geometry
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Notes on some recent writers on Analytical Geometry
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Line Geometry .
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Analysis. Names of some recent writers on Analysis .
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Development of Synthetic Geometry .
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Steiner, 1796–1863
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Von Staudt, 1798–1867
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Other writers on modern Synthetic Geometry .
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Development of Non-Euclidean Geometry .
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Euclid’s Postulate on Parallel Lines
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Hyperbolic Geometry. Elliptic Geometry
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Congruent Figures
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Foundations of Mathematics. Assumptions made in the subject
Kinematics
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Development of the Theory of Mechanics, treated Graphically
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Development of Theoretical Mechanics, treated Analytically
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Notes on recent writers on Mechanics
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Development of Theoretical Astronomy
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Bessel, 1784–1846
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Leverrier, 1811–1877. Adams, 1819–1892 .
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Notes on other writers on Theoretical Astronomy
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Recent Developments
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Development of Mathematical Physics
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Index
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390
390
391
392
392
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409
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410
1
CHAPTER I.
egyptian and phoenician mathematics.
The history of mathematics cannot with certainty be traced back to
any school or period before that of the Ionian Greeks. The subsequent
history may be divided into three periods, the distinctions between
which are tolerably well marked. The first period is that of the history
of mathematics under Greek influence, this is discussed in chapters ii
to vii; the second is that of the mathematics of the middle ages and
the renaissance, this is discussed in chapters viii to xiii; the third is
that of modern mathematics, and this is discussed in chapters xiv to
xix.
Although the history of mathematics commences with that of the
Ionian schools, there is no doubt that those Greeks who first paid attention to the subject were largely indebted to the previous investigations
of the Egyptians and Phoenicians. Our knowledge of the mathematical attainments of those races is imperfect and partly conjectural, but,
such as it is, it is here briefly summarised. The definite history begins
with the next chapter.
On the subject of prehistoric mathematics, we may observe in the
first place that, though all early races which have left records behind
them knew something of numeration and mechanics, and though the
majority were also acquainted with the elements of land-surveying, yet
the rules which they possessed were in general founded only on the
results of observation and experiment, and were neither deduced from
nor did they form part of any science. The fact then that various
nations in the vicinity of Greece had reached a high state of civilisation
does not justify us in assuming that they had studied mathematics.
The only races with whom the Greeks of Asia Minor (amongst whom
our history begins) were likely to have come into frequent contact were
those inhabiting the eastern littoral of the Mediterranean; and Greek
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CH. I]
EGYPTIAN AND PHOENICIAN MATHEMATICS
2
tradition uniformly assigned the special development of geometry to the
Egyptians, and that of the science of numbers either to the Egyptians
or to the Phoenicians. I discuss these subjects separately.
First, as to the science of numbers. So far as the acquirements of
the Phoenicians on this subject are concerned it is impossible to speak
with certainty. The magnitude of the commercial transactions of Tyre
and Sidon necessitated a considerable development of arithmetic, to
which it is probable the name of science might be properly applied. A
Babylonian table of the numerical value of the squares of a series of
consecutive integers has been found, and this would seem to indicate
that properties of numbers were studied. According to Strabo the Tyrians paid particular attention to the sciences of numbers, navigation,
and astronomy; they had, we know, considerable commerce with their
neighbours and kinsmen the Chaldaeans; and Băockh says that they
regularly supplied the weights and measures used in Babylon. Now the
Chaldaeans had certainly paid some attention to arithmetic and geometry, as is shown by their astronomical calculations; and, whatever was
the extent of their attainments in arithmetic, it is almost certain that
the Phoenicians were equally proficient, while it is likely that the knowledge of the latter, such as it was, was communicated to the Greeks. On
the whole it seems probable that the early Greeks were largely indebted
to the Phoenicians for their knowledge of practical arithmetic or the art
of calculation, and perhaps also learnt from them a few properties of
numbers. It may be worthy of note that Pythagoras was a Phoenician;
and according to Herodotus, but this is more doubtful, Thales was also
of that race.
I may mention that the almost universal use of the abacus or swanpan rendered it easy for the ancients to add and subtract without any
knowledge of theoretical arithmetic. These instruments will be described later in chapter vii; it will be sufficient here to say that they
afford a concrete way of representing a number in the decimal scale,
and enable the results of addition and subtraction to be obtained by a
merely mechanical process. This, coupled with a means of representing
the result in writing, was all that was required for practical purposes.
We are able to speak with more certainty on the arithmetic of the
Egyptians. About forty years ago a hieratic papyrus,1 forming part
1
See Ein mathematisches Handbuch der alten Aegypter, by A. Eisenlohr, second
edition, Leipzig, 1891; see also Cantor, chap. i; and A Short History of Greek Mathematics, by J. Gow, Cambridge, 1884, arts. 12–14. Besides these authorities the
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CH. I]
EGYPTIAN AND PHOENICIAN MATHEMATICS
3
of the Rhind collection in the British Museum, was deciphered, which
has thrown considerable light on their mathematical attainments. The
manuscript was written by a scribe named Ahmes at a date, according to Egyptologists, considerably more than a thousand years before
Christ, and it is believed to be itself a copy, with emendations, of a treatise more than a thousand years older. The work is called “directions
for knowing all dark things,” and consists of a collection of problems
in arithmetic and geometry; the answers are given, but in general not
the processes by which they are obtained. It appears to be a summary
of rules and questions familiar to the priests.
The first part deals with the reduction of fractions of the form
2/(2n + 1) to a sum of fractions each of whose numerators is unity:
1
1
1
1
2
is the sum of 24
, 58
, 174
, and 232
;
for example, Ahmes states that 29
1
1
1
2
and 97 is the sum of 56 , 679 , and 776 . In all the examples n is less than
50. Probably he had no rule for forming the component fractions, and
the answers given represent the accumulated experiences of previous
writers: in one solitary case, however, he has indicated his method,
for, after having asserted that 23 is the sum of 12 and 16 , he adds that
therefore two-thirds of one-fifth is equal to the sum of a half of a fifth
1
1
and a sixth of a fifth, that is, to 10
+ 30
.
That so much attention was paid to fractions is explained by the
fact that in early times their treatment was found difficult. The Egyptians and Greeks simplified the problem by reducing a fraction to the
sum of several fractions, in each of which the numerator was unity,
the sole exception to this rule being the fraction 23 . This remained the
Greek practice until the sixth century of our era. The Romans, on
the other hand, generally kept the denominator constant and equal to
twelve, expressing the fraction (approximately) as so many twelfths.
The Babylonians did the same in astronomy, except that they used
sixty as the constant denominator; and from them through the Greeks
the modern division of a degree into sixty equal parts is derived. Thus
in one way or the other the difficulty of having to consider changes in
both numerator and denominator was evaded. To-day when using decimals we often keep a fixed denominator, thus reverting to the Roman
practice.
After considering fractions Ahmes proceeds to some examples of the
fundamental processes of arithmetic. In multiplication he seems to have
papyrus has been discussed in memoirs by L. Rodet, A. Favaro, V. Bobynin, and
E. Weyr.
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CH. I]
EGYPTIAN AND PHOENICIAN MATHEMATICS
4
relied on repeated additions. Thus in one numerical example, where he
requires to multiply a certain number, say a, by 13, he first multiplies
by 2 and gets 2a, then he doubles the results and gets 4a, then he again
doubles the result and gets 8a, and lastly he adds together a, 4a, and
8a. Probably division was also performed by repeated subtractions,
but, as he rarely explains the process by which he arrived at a result,
this is not certain. After these examples Ahmes goes on to the solution
of some simple numerical equations. For example, he says “heap, its
seventh, its whole, it makes nineteen,” by which he means that the
object is to find a number such that the sum of it and one-seventh of
it shall be together equal to 19; and he gives as the answer 16 + 12 + 18 ,
which is correct.
The arithmetical part of the papyrus indicates that he had some
idea of algebraic symbols. The unknown quantity is always represented
by the symbol which means a heap; addition is sometimes represented
by a pair of legs walking forwards, subtraction by a pair of legs walking
−.
backwards or by a flight of arrows; and equality by the sign <
The latter part of the book contains various geometrical problems
to which I allude later. He concludes the work with some arithmeticoalgebraical questions, two of which deal with arithmetical progressions
and seem to indicate that he knew how to sum such series.
Second, as to the science of geometry. Geometry is supposed to have
had its origin in land-surveying; but while it is difficult to say when the
study of numbers and calculation—some knowledge of which is essential in any civilised state—became a science, it is comparatively easy to
distinguish between the abstract reasonings of geometry and the practical rules of the land-surveyor. Some methods of land-surveying must
have been practised from very early times, but the universal tradition
of antiquity asserted that the origin of geometry was to be sought in
Egypt. That it was not indigenous to Greece, and that it arose from
the necessity of surveying, is rendered the more probable by the derivaη, the earth, and μετρέω, I measure. Now
tion of the word from γ˜
the Greek geometricians, as far as we can judge by their extant works,
always dealt with the science as an abstract one: they sought for theorems which should be absolutely true, and, at any rate in historical
times, would have argued that to measure quantities in terms of a unit
which might have been incommensurable with some of the magnitudes
considered would have made their results mere approximations to the
truth. The name does not therefore refer to their practice. It is not,
however, unlikely that it indicates the use which was made of geomewww.pdfgrip.com
CH. I]
EGYPTIAN AND PHOENICIAN MATHEMATICS
5
try among the Egyptians from whom the Greeks learned it. This also
agrees with the Greek traditions, which in themselves appear probable;
for Herodotus states that the periodical inundations of the Nile (which
swept away the landmarks in the valley of the river, and by altering its
course increased or decreased the taxable value of the adjoining lands)
rendered a tolerably accurate system of surveying indispensable, and
thus led to a systematic study of the subject by the priests.
We have no reason to think that any special attention was paid to
geometry by the Phoenicians, or other neighbours of the Egyptians. A
small piece of evidence which tends to show that the Jews had not paid
much attention to it is to be found in the mistake made in their sacred
books,1 where it is stated that the circumference of a circle is three
times its diameter: the Babylonians2 also reckoned that π was equal to
3.
Assuming, then, that a knowledge of geometry was first derived by
the Greeks from Egypt, we must next discuss the range and nature
of Egyptian geometry.3 That some geometrical results were known
at a date anterior to Ahmes’s work seems clear if we admit, as we
have reason to do, that, centuries before it was written, the following
method of obtaining a right angle was used in laying out the groundplan of certain buildings. The Egyptians were very particular about
the exact orientation of their temples; and they had therefore to obtain
with accuracy a north and south line, as also an east and west line. By
observing the points on the horizon where a star rose and set, and taking
a plane midway between them, they could obtain a north and south line.
To get an east and west line, which had to be drawn at right angles to
this, certain professional “rope-fasteners” were employed. These men
used a rope ABCD divided by knots or marks at B and C, so that the
lengths AB, BC, CD were in the ratio 3 : 4 : 5. The length BC was
placed along the north and south line, and pegs P and Q inserted at the
knots B and C. The piece BA (keeping it stretched all the time) was
then rotated round the peg P , and similarly the piece CD was rotated
round the peg Q, until the ends A and D coincided; the point thus
indicated was marked by a peg R. The result was to form a triangle
P QR whose sides RP , P Q, QR were in the ratio 3 : 4 : 5. The angle of
1
I. Kings, chap. vii, verse 23, and II. Chronicles, chap. iv, verse 2.
See J. Oppert, Journal Asiatique, August 1872, and October 1874.
3
See Eisenlohr; Cantor, chap. ii; Gow, arts. 75, 76; and Die Geometrie der alten
Aegypter, by E. Weyr, Vienna, 1884.
2
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