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HISTORY &P MAM EM AT I CS

VOLUME I

GENERAL SURVEY OF THE HISTORY OF
ELEMENTARY MATHEMATICS

BY
DAVID EUGENE SMITH

DOVER PUBLICATIONS, INC.
NEW YORK NEW YORK

This new Dover edition, first published in
1958, is an unaltered and unabridged re-
publication of the last edition. It is published
through special arrangement with GINN
AND COMPANY.

PRINTED IN THE UNITED STATES OF AMERICA
357.3

PREFACE

iis work has been written for the purpose of supplying teachers
ana students with a usable textbook on the history of elementary
mathematics, that is, of mathematics through the first steps in the
calculus. The subject has come to be recognized as an important
one in the preparation of teachers of mathematics and in the liberal
education of students in colleges and high schools. Although several
^p||$ upon the history of mathematics are already available, the
ainiior feel? that a book written from somewhat different stand-
points will be found helpful to those who are beginning the study of
^he subject in our universities, colleges, and normal schools, and
because of this belief the present work has been prepared.

A history of mathematics may be constructed on several general

plans, each of which may be justified by the purpose in mind. For

example, it may be arranged solely with a view to the chronological

sequel of events, or as a series of biographies, or according to the

branches of mathematics, or as a source book of material

ucK, or with respect to national or racial achievements, or in

us jther ways, each of which may have certain advantages.

The general plan adopted in the preparation of this work is that

itf presenting the subject from two distinct standpoints, the first, as

f |n Volume I, leading to a survey of the growth of mathematics by

Chronological periods, with due consideration to racial achievements ;

jbid the second, as in Volume II, leading to a discussion of the evolu-

ion of certain important topics. To attempt to fuse these two fea-

refc ind thus to carry them along together has often been attempted.

Sjjjj^iracterizes, for example, the monumental treatise of Montucla

and, to a large extent, that of Cantor. For the teacher, however, this

plan is not satisfactory, and the excellent work of Tropfke is an

example of the tendency to break away from the mere chronological

recital of facts. Long experience in teaching the subject in colleges

has convinced the author that a general historical presentation is

iii

iv PREFACE

desirable for the purpose of relating the development of mathematics
to the development of the race, of revealing the science as a great
stream rather than a static mass, and of emphasizing the human
element, but that this ought to lead to a topical presentation by
which the student may understand something of the life history of
the special subject which he may be studying, whether it be the
elementary theory of numbers, the methods of calculation, the
solution of equations, the functions of trigonometry, the common
symbolism in use, the various types of elementary geometry, the
early steps in the calculus, or one of the various other important
topics of elementary mathematics. The general plan can best be
understood by a glance at the table of contents in each volume.

Perhaps the chief objection to the general arrangement set forth
in Volume I is that the reader may occasionally feel that a mere
statement of the subjects in which some particular mathematician
was interested is not very illuminating, and that a more extended
statement of his achievements would have greater significance. In
most cases, however, a further elaboration of the record would destroy
the possibility of successfully carrying out the plan of showing the
growth of the several leading branches of elementary mathematics
by themselves, as in Volume II, at least without a large amount of
wearisome repetition. Of the two evils the lesser has been chosen.

In Volume I, which forms the general survey by periods, attention
has been given to geographical and racial considerations as well as
to chronological sequence. While it is evident that no race or
country has any monopoly of genius, and while the limits of suc-
cessive centuries are only artificial boundaries with no significance
in the creation of the masterpieces of any science, nevertheless lin-
guistic and racial influences tend to develop tastes in mathematics as
they do in art and in letters, and certain centuries stand out with
interesting prominence.

The student will therefore find it to his advantage to give some
attention to the geographical distribution of scholars as well as to
the general periods in which they lived. While it is impossible to
grade countries according to any definite scale of excellence, and
while the world has always seen more or less of the migration of
scholars from one country to another, it is possible in a general way

PREFACE v

to give prominent positions to those national groups which have
contributed most to the advancement of the science in each period
under discussion.

In this treatment of the subject an attempt has been made to seek
out the causes of the advance or the retardation of mathematics in
different centuries and with different races, but always with the con-
sciousness that the world has no certain prescription for the creation
of the genius and that the causes of any series of historical events are
usually very intangible. The effort has also been made to introduce
enough of the anecdote to relieve the monotony of mere historical
statement and to reveal the mathematician as a human being like
others of his race.

While the footnote is often condemned as merely an apology for
obscurity or as an exhibition of pedantry, it would be difficult, in a
work of this kind, to dispense with its aid. There are two principal
justifications for such a device : first, it enables an author to place
the responsibility for a statement that may be open to question;
and second, it encourages many students to undertake further study,
either from secondary sources or, what is more important, from the
original writings of the men who rank among the creators of mathe-
matics. With these two points in mind, footnotes have been intro-
duced in such a way as to be used by readers who wish for further
aid, and to be neglected by those who wish merely a summary of
historical facts. For the student who seeks an opportunity to study
original sources, a sl'ght introduction has been made to this field.
The text of the book contains almost no quotations in foreign
languages, the result being that the reader will not meet with
linguistic difficulties in the general narrative. In the notes, however,
it is frequently desirable to quote the precise words of an author,
and this has been done with reference to such European languages
as are more or less familiar. It is not necessary to translate literally
all these extracts, since the text itself sets forth the general meaning.
Students who have some general knowledge of Latin, French, or
German will have little difficulty, and in many cases will have much
interest, in seeing various statements in their original form. For
special reasons a few notes have been given in Greek, but in every
case the meaning is evident from the text.

vi PREFACE

The footnotes have also permitted of the insertion of various bio-
graphical items which would merely burden the text, but which
have considerable value to the student. In a general way it may be
said that it is a matter of no moment where a man was born or on
what day ; but a work of this nature must be more than a book to be
read, it must be a work for future reference, and for this reason
there may properly be made available, to be used if thought neces-
sary, certain material which will aid the student in his later research.
It would be possible to place all such supplementary material at the
end of the book, but this would be merely an invitation to ignore
it entirely.

No selection of names is ever satisfactory, even to the writer who
makes it. In this work there are often included in one period names
which would not be considered a century later; while others are
omitted, particularly in the last three centuries, that would have
been given prominence had their possessors lived at an earlier date.
The criterion of selection has been the contribution of the individual
to the development of elementary mathematics, his reputation as a
scholar, and in particular his work in the creating of tendencies to
further the study of particular branches of the subject. For the
latter reason certain names have been included which would not
otherwise have been considered. In Volume II a few minor names of
arithmeticians have been mentioned in connection with the peculiar
use of certain terms and the like, with no biographical notes, the
latter being of little or no consequence.

In connection with dates before the Christian era the letters B. c.
are used; in connection with dates after the beginning of this era
no distinguishing letters are added except in a few cases near the
beginning of the period, in which the conventional letters A. D. have
occasionally been inserted to avoid ambiguity. With some hesita-
tion, but for a purpose which seems valid, dates are frequently given
in parentheses after proper names. It is well recognized that a
precise date, like 1202 after the name Fibonacci, is of no particular
value in itself. It makes no difference, in ordinary cases, whether
Fibonacci wrote his Liber Abaci in 1202, in 1180, or in 1220, or
whether abacus is spelled abbacus, as in some manuscripts, or in
the more correct Latin form. On the other hand, two things are

PREFACE vii

accomplished by a free use of such dates. In the first place, a reader
is furnished with a convenient measuring instrument; he does not have
to look in the index or a chronological table in order to see approxi-
mately where the particular writer belongs in the world's progress.
The casual reader may well be pardoned if he does not recall where
Bede, Alcuin, Gerbert, Jordanus, Fibonacci, and Roger Bacon stood
chronologically with respect to one another, and in reading a tech-
nical history of this kind there is no reason why he should not be
relieved of the trouble of consulting an index when he meets with
one of these names in the text. In the second place, it needs no
psychologist to confirm the familiar principle that the mind comes,
without conscious effort, to associate in memory those things which
the eye has frequently associated in reading. At the risk, therefore,
of disturbing the minds of those who are chiefly interested in the
literary aspect of a general statement of the progress of mathe-
matics, many important dates have been repeated, especially where
they have not appeared in the pages immediately preceding.

In quoting from other writers the rule has been followed of making
the quotation exact in spelling, punctuation, and phraseology. In
carrying out this rule it is inevitable that errors should occasionally
enter into the transcription, particularly in the case of old dialects ;
but the effort has been made to give the language precisely as it
appears in the original. This accounts for the fact that certain French
words in a quotation will sometimes appear without the modern
accent, and that a word like Lilavati may appear with any one of
three spellings, depending upon the translator to whose work refer-
ence is made, or upon the author using the word.

Use has been made of such international symbols as s.a. (sine anno,
without date of publication), s.l. (sine loco, without place of publi-
cation), s.l. a. (without place or date), c. (circa, about), and seq.
(sequens y following), and of the abbreviations ed. (edition, edited by,
or edition of), vol. (volume), and p. (page).

At the close of each chapter there has been given a page of topics
for discussion, so arranged as to command more attention than
they would have received had they been given in scattered form.
These topics are not limited to questions to be answered from the
text, but have purposely been made general, suggesting somewhat

viii PREFACE

more extended fields for study. The student will find it to his
advantage if he is thus led to consult encyclopedias, general histories,
and such works as are suggested in the bibliographical notes and as
are available in libraries to which he may have access. It is by no
means expected that an elementary work like this should contain the
material for an extended study of any of these topics.

In the selection of illustrations the general plan has been to include
only such as will be helpful to the reader or likely to stimulate his
interest. It would be undesirable to attempt to give, even if this
were possible, illustrations from all the important sources, for this
would tend to weary the reader. On the other hand, where the stu-
dent has no access to a classic that is being described or even to a
work which is mentioned as having contributed to the world 's prog-
ress in some humbler manner, a page in facsimile is often of value.

In general the illustrations have been made from the original
books or manuscripts in the well-known and extensive library of
George A. Plimpton, Esq., who has generously allowed this material
to be used for the purpose ; from the author's collection of books,
manuscripts, mathematical portraits and medals, and early math-
ematical instruments ; from manuscripts in various other libraries ;
and from such works as those by Professor Breasted.

Long experience in the use of books of reference has led the author
to believe that a single index is more convenient than a series of
indexes by names, subjects, and titles. Furthermore, readers who
iiave used works like those of Cantor and Tropfke, for example,
know the annoyance of a long list of page references after a given
name, many of them of no particular significance. In this work,
therefore, only a single index is given in each volume, and in each
entry the page references are only such as the reader will find of
particular value. In each case the first reference after a proper
name relates to the biography of the individual, if one is given ; the
others relate to his leading contributions and are arranged approxi-
mately in order of importance.

DAVID EUGENE SMITH

CONTENTS

( MATTER PAftK

BIBLIOGRAPHY xiii

PRONUNCIATION, TRANSLITERATION, AND SPELLING OF

PROPER NAMES xvif

I. PREHISTORIC MATHEMATICS i

1. IN THE BEGINNING i

2. PRIMITIXE COUNTING 6

3. GEOMETRIC ORNAMENT 15

4. MYSTICISM 16

TOPICS FOR DISCUSSION 19

II. THE HISTORIC PERIOD DOWN TO 1000 B.C 20

1. GENERAL VIEW 20

2. CHINA 22

3- INDIA 33

4. BABYLON 35

5. EGYPT 41

TOPICS FOR DISCUSSION 53

III. THE PERIOD FROM 1000 B.C. TO 300 B.C 54

1. THE OCCIDENT IN GENERAL 54

2. THE GREEKS 55

3. ORIGINS OF GREEK MATHEMATICS 63

4. FROM PYTHAGORAS TO PLATO 69

5. INFLUENCE OF PLATO AND ARISTOTLE 87

6. THE ORIENT 95

TOPICS FOR DISCUSSION 101

x CONTENTS

CHAPTER PAGE

IV. THE PERIOD FROM 300 B.C. TO 500 A.D 102

1. THE SCHOOL OF ALEXANDRIA 102

2. EUCLID 9 103

3. ERATOSTHENES AND ARCHIMEDES 108

4. APOLLONIUS AND HIS SUCCESSORS 116

5. PERIOD OF MENELAUS 125

6. PTOLEMY AND HIS SUCCESSORS 130

7. DlOPHANTUS AND HIS SUCCESSORS 133

8. THE ORIENT 138

TOPICS FOR DISCUSSION 147

V. THE PERIOD FROM 500 TO 1000 148

1. CHINA 148

2. JAPAN 151

3. INDIA 152

4. PERSIA AND ARABIA 164

5. THE CHRISTIAN WEST 177

6. THE CHRISTIAN EAST IQO

7. SPAIN 192

TOPICS FOR DISCUSSION 193

VI. THE OCCIDENT FROM 1000 TO 1500 194

1. CHRISTIAN EUROPE FROM 1000 TO 1200 194

2. ORIENTAL CIVILIZATION IN THE WEST 205

3. CHRISTIAN EUROPE FROM 1200 TO 1300 211

4. CHRISTIAN EUROPE FROM 1300 TO 1400 230

5. CHRISTIAN EUROPE FROM 1400 TO 1500 242

TOPICS FOR DISCUSSION 265

CONTENTS xi

CHAPTER 1'AGB

VII. THE ORIENT FROM 1000 TO 1500 266

1. CHINA 266

2. JAPAN 273

3. INDIA 274

4. PERSIA AND ARABIA . . . . 283

TOPICS FOR DISCUSSION 291

VIII. THE SIXTEENTH CENTURY 292

1. GENERAL CONDITIONS 292

2. ITALY 294

3. FRANCE 306

4. ENGLAND 314

5. GERMANY 324

6. THE NETHERLANDS 338

7- SPAIN 343

8. OTHER ETROPEAN COUNTRIES 346

q. THE ORIENT 350

10. THE NEW WORLD 353

TOPICS FOR DISCUSSION 357

IX. THE SEVENTEENTH CENTURY 358

1. GENERAL CONDITIONS 358

2. ITALY 361

3. FRANCE 370

4. GREAT BRITAIN 387

5. GERMANY 416

6. THE NETHERLANDS 422

7. OTHER EUROPEAN COUNTRIES 426

8. THE ORIENT 435

TOPICS FOR DISCUSSION 443

xii CONTENTS

_HAJ'TKR PAGE

X. THE EIGHTEENTH CENTURY AND AFTER 444

1. GENERAL CONDITIONS 444

2. GREAT BRITAIN 4^6

3. FRANCE 470

4. GERMANY 501

5. ITALY 511

6. SWITZERLAND 519

7. OTHER EUROPEAN COUNTRIES . 526

8. UNITED STATES 531

g. THE ORIENT 533

10. THE HISTORIANS OF MATHEMATICS 539

TOPICS FOR DISCUSSION 547

CHRONOLOGICAL TABLE 549

INDEX 571

BIBLIOGRAPHY

The extent of a bibliography in a work of this kind is a matter of
judgment. It can easily run to great length if the writer is a
bibliophile, or it may have but little attention. The purpose of giv-
ing lists of books for further study is that the student may have
access to information which the author has himself used and which
he believes will be of service to the reader. For this reason the
secondary sources mentioned in this work are such as may be avail-
able, and in many cases are sure to be so, in the libraries connected
with our universities, while the original sources are those which are of
importance in the development of elementary mathematics or which
may be of assistance in showing certain tendencies.

The first time a book is mentioned the title, date, and place of publi-
cation are given, together, whenever it seems necessary, with the
abbreviated title which will thereafter be used. To find the com-
plete title at any time, the reader has only to turn to the index, where
he will find given the first reference to the book. The abbreviation
loc. cit. (for loco citato, in the place cited) is used only where the
work has been cited a little distance back, since any more general
use of the term would be confusing. The symbolism "I, 7" has been
used for "Vol. I, p. 7" in order to conserve space, although excep-
tions have been made in certain ambiguous cases, as in the refer-
ences to Heath's Euclid, references to Euclid being commonly by
book and proposition, as in the case of Euclid, I, 47.

Although the number of works and articles on the history of
mathematics is very great, the student will be able, in the initial stages
of his investigation, to consult relatively few. For his convenience
the books that he may most frequently use are here listed, special
reference being made to those in English, French, and German
which are likely to be found in college, university, and city libraries.
The student will also find it advantageous to consult the leading
encyclopedias.

xiv BIBLIOGRAPHY

Allman, G. J., Greek Geometry from Tliales to Euclid, Dublin, 1889. Re-
ferred to as Allman, Greek Geom.

Ball, W. W. Rouse, A Short Account of the History of Mathematics, 6th ed.,
London, 1915. A readable survey of the general field. Referred to as Bali
History.

Bretschneider, C. A., Die Geometrie und die Geometer i>or Euklcides, Leip
zig, 1870, Referred to as Bretschneider, Die Geometrie.

Cajori, F., A History of Elementary Mathematics, rev. ed., New York, 1917
Referred to as Cajori, Elem. Math.

A History of Mathematics, 2d ed., New York, 1919. Referred to as

Cajori, History.

Cantor, M ., Mathematische Beitrdge sum Kulturleben derVolker, Halle, 1 863.
Referred to as Cantor, Beitrdge.

Vorlesungen iiber Geschichte derMathematik, 4 vols., Leipzig, 1 880- 1 908,

with various revisions. The standard general history of mathematics. Re-
ferred to as Cantor, Geschichte.

Encyklopddie der Mathematischen }Vissenschaf1ei>, Leipzig, 1898-, with a
French translation. Referred to as Encyklopddie.

Gow, James, A Short History of Greek Mathematics, Cambridge, 1884.
Referred to as Gow, Greek Math.

G thither, S., and Wieleitner, H., Geschichte der Mathematik, 2 vols., Leipzig,
1908-1921. Referred to as Gunthcr- Wieleitner, Geschichte. The second
volume is the work of Dr. Wieleitner.

Hankel, H., Zur Geschichle der Mathemalik in Alterthum und Mitt el-
alter, Leipzig, 1874. Referred to as Hankel, Geschichte.

Heath, Sir Thomas Little, A History of Greek Mathematics, 2 vols., Cam-
bridge, 1921. Referred to as Heath, History. Although the following
special works by the same author are referred to in the footnotes, they are so
important that it seems advisable to include them in this general bibliography.

Apollonius of Perga, Cambridge, 1896. Referred to as Heath, Apol-

lonius.

Archimedes, Cambridge, 1897. Referred to as Heath, Archimedes.

Aristarchus ofSamos, Oxford, 1913. Referred to as Heath, Aristarchus.

Aristarchus of Samos. The Copernicus of Antiquity, London, 1920.

Referred to as Heath, A ristarchits (abridged).

Diophantits of Alexandria, ad ed., Cambridge, 1910. Referred to as

Heath, Diophantits.

Euclid in Greek, Book /, Cambridge, 1920. Referred to as Heath,

Euclid in Greek.

The Thirteen Books of Euclid's Elements, 3 vols., Cambridge, 1908.

Referred to as Heath, Euclid.

Greek Mathematics and Science, pamphlet, Cambridge, 1921. Referred
to as Heath, Address.

BIBLIOGRAPHY xv

Hilprecht, H. V., Mathematical, Metrological, and Chronological Tablets
from the Temple Library of Nippur. Philadelphia, 1906. Referred to as
Hilprecht, Tablets.

Libri, G., Histoire ties Sciences Mathematiques en Italie, 4 vols., Paris,
1838-1841. Valuable on account of its style and its extensive notes. Re-
ferred to as Libri, Histoire.

Loria, G., Guida allo Studio del la Storia delle Matematiche, Milan, 1916.
Very valuable for its bibliography of the history of mathematics.

Marie, M., Histoire des Sciences Mathematiques et Physiques, 12 vols., Paris,
1883-1888. Biographical, convenient for reference, but inaccurate. Re-
ferred to as Marie, Histoire.

Mikami, Y., The Development of Mathematics in China and Japan, Leipzig,

1913. Referred to as Mikami, China. See also Smith-Mikami.

Miller, G. A., Historical Introduction to Mathematical Literature, New
York, 1 9 r 6. Serves a purpose in English similar to that of Loria's Guida
in Italian. Referred to as Miller, Introduction.

Montucla, J. E., Histoire des Mathematiques, 2d ed., 4 vols., Paris, 1 799-1802.
Although written in the 1 8th century, it is a classic that is well worth con-
sulting, particularly for its style. Referred to as Montucla, Histoire.

Pauly (A.)-Wissowa (G.), Real-Encyclopddie der Classischen Altertumsivis-
senschaft, Stuttgart, 1894-. Best reference work for classical biography
and antiquities. Referred to as Pauly-Wissowa.

Poggendorff, J. C, Handworterbuch sur Geschichte der exact en IVissen-
schaften, 4 vols., Leipzig, 1863-1904. Referred to as Poggendorff.

Smith, David Eugene, Our Debt to Greece and Rome. Mathematics, Boston,
1922. Referred to as Smith, Greece and Rome.

Rara Arithmetica, Boston, 1908. being a bibliography of early arithmetics.

Referred to as Rara Arithmetica.

Smith, D. E., and Karpinski, L. C., The Hindu-Arabic Numerals, Boston,
1911. Referred to as Smith- Karpinski.

Smith, D. E., and Mikami, Y., History of Japanese Mathematics, Chicago,

1914. Referred to as Smith-Mikami.

Tannery, P., La Gcome'trie Grecque, Paris, 1887. Referred to as Tannery,

Gt f om. Grecque.
Afemoires Scientifiques, edited by J. L. Heiberg and H. G. Zeuthen,

2 vols., Paris, 1912.
Pwr r Histoire de la Science Hellbie de Thalh a Emptdocle, Paris,

1587. Referred to as Tannery, Histoire.
Tropfke, J., Geschichte der Elementar-Mathematik in systematischer Darstel

lung, 2 vols., Leipzig, 1902, 1903; 2d ed., 1921-. The best history o)

elementary mathematics. Referred to as Tropfke, Geschichte.
Zeuthen, H. G., Histoire des Mathematiques dans r Antiquite 1 et le Moyen Agt

translated by J. Mascart. Paris, 1902. Referred to as Zeuthen, Histoi'-'f

xvi BIBLIOGRAPHY

In matters of biography the student will find in W. Smith, Dic-
tionary of Greek and Roman Biography, London, 3 vols., 1862-1864
(referred to as Smith's Diet, of Greek and Roman Biog.), a work of
exceptional value, particularly with reference to the Greek mathe-
maticians ; and in the Dictionary of National Biography he will find
the British mathematicians treated in a scholarly manner. For
French biographies, Michaud's Biographie Universelle (1854-1865),
Hoefer's Nouvelle Biographie Generate (1857-1866), La Grande
Encyclopedie, and Larousse's Grand Dictionnaire Utiiversel du
XIX siecle jranqais are very satisfactory. For German biographies
the Allgemeine Deutsche Biographie, the Brockhaus Conversations-
Lexikon, and the Meyer Grosses Konversations-Lcxikon are helpful.
Various earlier and less frequently used works are referred to in
the footnotes.

Such special works as those of Matthiessen, Braunmuhl, and Dick-
son are mentioned in the notes from time to time, as well as various
other sources of information that may be found in the larger libraries.

Of the journals devoted to the history of mathematics those to
which the student will most frequently refer are Boncompagni's
Bullettino di Bibliografia e Storia delle Scienze Matcmatiche e
Fisiche, Rome, 1868-1887 (referred to as Boncompagni's Bullettino),
and Enestrom's Bibliotheca Mathematica ? Leipzig, three series,
1885-1915 (referred to as Bibl. Math.).

PRONUNCIATION, TRANSLITERATION, AND
SPELLING OF PROPER NAMES

General Question. The question of the spelling and translit-
eration of proper names is always an annoying one for a writer of
history. There is no precise rule that can be followed to the satis-
faction of all readers. In general it may be said that in this work
a man's name has been given as he ordinarily spelled it, if this spell-
ing can be ascertained. To this rule there is the exception that
where a name has been definitely anglicized, the English form has
been adopted. For example, it would be mere pedantry to use, in
a work in English, such forms as Platon and Strabon, although it is
proper to speak of Antiphon and Bryson instead of Antipho and
Bryso. When in doubt, as in the case of Heron, the preference has
been given to the transliteration which most clearly represents the
spelling that the man himself used.

In many cases this rule becomes a matter of compromise, and then
the custom of a writer's modern compatriots is followed. An example
is seen in the case of Leibniz. This spelling seems to be gaining
ground in our language, and it has therefore been adopted instead
of Leibnitz, even though the latter shows the English pronunciation
better than the former. Leibniz himself wrote in Latin, and the
family spelled the name variously in the vernacular. There seems,
therefore, to be no better plan than to conform to the spelling of
those recent German writers who appear to be setting the standard
that is likely to be followed.

There is also the difficulty of finding a satisfactory solution in the
case of men who were themselves polyglots, who lived in polyglot
towns, or who made their homes in more than a single country. This
is seen, for example, in the case of the Bernoullis. Jacques Bernoulli
lived in Basel, a Swiss city where German was chiefly spoken and
where the common spelling of the name of the place is the one

1 xvii

xviii PROPER NAMES

here given. He was of Belgian descent, but he usually wrote either
in Latin, in which his first name was spelled Jacobus, or in French,
in which he would naturally use the name of Jacques. To call him
James, as various English writers have done, would merely confuse an
American reader, while to adopt the German Jakob would be to use a
form which Bernoulli himself did not adopt in writing. The fact that
he preferred to use French as his means of correspondence, when not
writing in Latin, makes it desirable to speak of him as Jacques and
to follow a similar usage with respect to his brother, Jean Bernoulli.

Another difficulty arises when we consider the Graeco-Latin forms
of names in the Renaissance period. In general, if a man commonly
used such a form, as was the case with Grammateus, Regiomontanus,
and Dasypodius, this form has been used in the text, with the family
name given in a footnote. In a case on the border line, like that
of Schoner, however, the vernacular form, spelled as the man seems
himself to have preferred, has been adopted. It must also be under-
stood that early writers were often not uniform in spelling their
own names. Thus, we have Recorde and Record, Widman and Wid-
mann, and Scheubel and Scheybel, and in these cases all that the
historian can do is to endeavor to choose that spelling which the
writer seems himself to have most commonly used.

A further difficulty is encountered with certain names in regard
to which the possessor was himself undecided as to his preference.
A typical case is that of Leonardo Pisano Fibonacci, Leonardo the
Pisan, son of Fibonacci, or of Bonacci, or of Bonacius. It would be
proper to write his name "1. pisano," since this form appears in one
of the early manuscripts ; or Bigollo, since he used this nickname ;
or Leonardo of Pisa, although this combines Italian and English ;
but the form Fibonacci has been chosen for general use, chiefly
because Fibonacci's Series is so frequently mentioned in mathe-
matics. It would not be difficult to show a lack of consistency in
many cases, as when the common form of Gemma Frisius (Gemma
the Frieslander) is preferred to the family name of Renier, with
various spellings. In the case of a name like that of Pacioli, where
different forms are used in the various works of the individual, the
one seemingly preferred by the majority of historians has been
chosen. In the case of a name like Joannes or Johannes the effort

PROPER NAMES xix

has been made to use the form which the possessor used, or at least
the one which was the more commonly employed by his contem-
poraries when referring to him.

The greatest difficulty in transliteration arises with respect to
oriental names. In the first place, we have no international system
of transliteration that is generally accepted ; and in the second place,
it is difficult to know the name which the writer himself preferred.
An Arab scholar may have as many as a dozen parts to his name ;
a Japanese or Hindu writer may have an intimate name and also an
official name ; a Chinese mathematician may be known only by an
ideogram, the pronunciation of the name being lost or varying in
different parts of his own country ; and certain of these names may
have found their way into medieval Latin and have been distorted
almost beyond recognition.

If a name is fairly familiar in English, like Omar Khayyam, it
has been retained, even if the form is open to criticism. If it ha?
taken an English form but is not so familiar, as in the case of
Savasorda, the attempt has been made to use the distorted name
and also to adopt the best modern transliteration of the real name
from which this is derived. In the cases of such less familiar names
as seem to deserve mention, these will neither be read aloud nor be
kept in mind by most readers, and hence an abridged form has been
given in the text, in as good transliteration as seems possible, the
full form being placed in a footnote. The Arabic al- has been used
instead of el- or /-, simply because it is the most common form in
English. As a matter of fact, the Arabic pronunciation, like that of
the Chinese, is by no means standardized.

The pronunciation of proper names has been given in cases where
it is likely to be helpful to the student, and in many cases the accent
has been indicated when the name first appears in the text. In such
cases the English pronunciation has been taken whenever the name
has become thoroughly anglicized, but otherwise the pronunciation
has been given as nearly as possible as it stands in the vernacular.
In the case of Greek names the original form has usually been
given in the notes, partly because of the differences in accent and
partly because the Greek alphabet is well enough known to allow
the original and frequently interesting form to be understood.

xx PROPER NAMES

Arabic Names. The standard authority on the transliteration
and pronunciation of Arabic names is Suter, a Swiss writer, whose
"Die Mathematiker und Astronomen der Araber und ihre Werke"
appeared in Volume X of theAbhandlungen zur Geschichte der Mat he-
matlschen Wisscnschajten, and in "Das Mathematiker- Verzeichniss
im Fihrist," in Volume VI of the same work. The rules given by this
writer have, in general, been followed, except that j has been used
f r g> y f r Jj v f r w > kh for ch, and al- for el-, to conform to Eng-
lish pronunciations and custom. Although the reader will seldom
need to pronounce the names, it will be helpful to be able to do so
if necessary. The following is a summary of the scheme of trans-
literation and pronunciation employed :

b, d, f, g, h, j, 1, m, n, p, s, sh, t, th, w, x, z as in English.

a as in ask; & as mfaf/ier, the form d being used instead of a in Arabic words,
partly to conform to the Suter list.

e as in bed.

i as in pin ; i as in pique.

o as in obey.

u as in //// ; fi as in rule.

d, s, t, z as in English but made with the tongue spread so that the sounds
are produced largely against the side teeth.

n is generally pronounced by Europeans as simple ;/.

d like /// in that ; t like th in thin.

g is a voiced consonant formed below the vocal chords ; it is sometimes com-
pared to a guttural g and sometimes to a guttural r.

h retains its consonant sound at the end of a word.

h may be compared to the German hard ch, as in nach.

k as in English ; kh is the hard German ch, as in nach.

q like c or k in cook.

r stronger than in English.

V like the English w\ y as in you.

' represents the spititus lenis and may be taken simply as separating dis-
tinctly two vowels, like the break between the *'s in reentrant.

A final vowel is shortened before al (which then becomes V) or ibn (whose / is
then silent).

In al the final / often takes the sound of a following consonant, as in
al-Rashid (ar-Rashid).

The accent is on the last syllable containing a long vowel or a vowel followed
by two consonants, except that a final long vowel is not usually accented.
Otherwise the accent falls on the first syllable.

PROPER NAMES xxi

Hindu Names. The transliteration of Hindu names has changed
greatly within a century, and even yet is not internationally stand-
ardized. In general, in quoting from earlier English writers, the
forms which they used have been followed. Thus, there will be
found in the notes various references to Taylor's Lilawati, this being
the name of the book as the translator used it ; or, when the actual
title is mentioned, to Colebrooke's translation of the Lildvati, this
being the form which this author used; but the modern form
Lilavati appears in the text. The effort h^.s been made to follow
the best current practice of English orientalists, and in determining
the form and pronunciation of Sanskrit words the following equiva-
lents have been used :

b, d, f, g, h, j, 1, m, n, p, v, w, x, z as in English.

a like u in but; thus, pandit, pronounced pundit \ a as in father, the form it

being used instead of d in Hindu words.
e as in they.

i as in pin ; i as in pique.
as in so.

u as in put ; u as in rule.
c like ch in church (Italian c in cento}.
$, n, s, t like d, n, sh, t made with the tip of the tongue turned up and back

into the dome of the palate.
h preceded by b, c, /, / does not form a single sound with these letters but is a

more or less distinct sound following them, somewhat as in abhor \ h is

final consonant h.
k as in kick.

m, h like the French final /// or ;/, nasalizing the preceding vowel,
s", Knglish sh.
y as in you.
' in some transliterations is used to indicate the spiritus lenis, a break between

two letters.
The accent is as in Latin : if the penult is long, it is accented ; if it is short,

the antepenult is accented.

Japanese Names. Modern Japanese scholars have carefully trans-
literated into the Roman alphabet the names of all their leading
mathematicians. The letters are pronounced as in English except
that i is pronounced like e in feel ; e as in grey ; ai as in aisle ; and
ei like long a ; but i and e also take a short sound as in English.
Japanese names have only a slight accent.

xxii PROPER NAMES

Chinese Names, There is no uniform system of transliterating
and pronouncing Chinese names and terms. The author's colleague,
Professor Hirth, in his Ancient History of China, followed in general
the plan adopted by the Royal Geographical Society of London and
the United States Board on Geographic Names, and the present
text follows in the main the rules which he has laid down. Briefly
stated, the scheme of pronunciation is as follows :

a as in father.

e, 6 as in men. The accent simply shows that it does not form part of a

diphthong.

i as in pique. When followed by n or by a vowel it is short as in pin.

i, used when / is intonated with the adjoining consonant, as in /*, or is but
faintly heard, when it follows , as in lei.

o as in mote.

o like the French eu mjeu or like the German o.

u like oo in boot. When preceding ;/, a, or o it is short.

ii like the French //. When preceding //, , or / it is short.

ai like i in ice.

au like ow in how.

ei somewhat like ey in they.

6u a diphthong with the two vowels distinctly intonated.

ui like ooi contracted into a diphthong.

The initials k, p, t, ch, ts, and tz are not so hard as in English. When pro-
nounced as hard as possible they are followed by (') as in Pan.

ch like ch in church. When followed by i\ the vowel blends with it.

f , h, 1, m, n, sh, w, ng as in English.
j like/ in French.

33 like ss in mess. When followed by /', the vowel disappears,
y like y in you.

Names from Other Languages. In the case of Russian names
there has been chosen the transliteration which represents most
effectively the English equivalent sounds. For example, the spelling
Lobachevsky has been preferred to the German form, Lobatschew-
ski, or to other forms which are not appropriate to our language.
The same may be said with respect to other foreign names where
the Roman alphabet is not in use or is supplemented by other letters,

EARLY ART

3. GEOMETRIC ORNAMENT

Early Art A further prehistoric stage of mathematical
development is seen in the use of such simple geometric forms
as were suggested by the plaiting of rushes, the first step in
the textile art. From this there developed those forms used
in clothing, tent cloths, rugs, and drapery which are usually
found among primitive peoples.
Since the earliest trace of hu-
man art that we have thus far
found is seen in representations
of animals, these being drawn
on bone in the Early Stone Age,
one might expect to find such
figures in early mural decora-
tions, and this is not only the
case but is one means of dating
the latter with some degree of
approximation. The geometric
ornament, however, became in
due time a favorite one among
nearly all early peoples. This
may have been because the
plaiting of rushes furnishes an
easy medium for the representa-
tion of geometric forms, but at any rate such forms as the
swastika and the Greek key developed at an early period.
Such decorations are not confined to the textiles of the people ;
they are equally prominent in architecture in all parts of the
world. They are found on the early monuments of Mexico,
on the architectural remains of Peru; on the huts of the savage,
and on the early buildings of the historic period in various
parts of the Old World, especially on those devoted to the
commemoration of the dead or to the worship of the gods.

The same instinct that leads to geometric decoration of reli-
gious structures shows itself in the decoration of personal
ornaments and of articles intended for domestic use. This
is seen in the handicraft of the Stone Age, it is found in the

EGYPTIAN POTTERY OF THE
PREDYNASTIC PERIOD

It shows the earliest stage of geo-
metric ornament on pottery. The
Predynastic Period extended from
c. 4000 to c. 3400 B.C. From the
Metropolitan Museum, New York

MYSTICISM

rich gold work of early Egypt, and it is equally in evidence in
most of the jewelry of modern times. It is not merely the
instinct of symmetry that we find in these petrified thoughts of
the race ; it is quite as much a desire to fathom the mystery
and grasp the meaning of the beauty of geometric form.

Early Pottery. The early pot-
tery of Egypt and Cyprus shows
very clearly the progressive stages
of geometric ornament, from rude
figures involving parallels to more
carefully drawn figures in which
geometric design plays a more im-
portant part and in which such
mystic symbols as the swastika
are found. Art was preparing the
way for geometry.

4. MYSTICISM

Religious Mysticism. The be-
ginning of an appreciation of
the wonders of mathematics is
closely connected with the be-
ginning of religious mysticism.
Man wondered at the heavens
above him; he wondered at life
and he wondered even more at
death; all was a mystery. He
likewise wondered at the pecu-
liarities of geometric forms and at the strange properties
of such numbers as three and seven, the two primes within his
limited number realm that were not connected with his common
scales of counting. The mystery of form and the mystery of
number he connected with the mystery of the universe about
him, the universe in which he felt himself a mere mote in the
sunbeam. His sense of wonder at the potency of the sun led
him to the orientation of his religious structures ; his recogni-

CYPRUS JUG OF THE PERIOD
3000-2000 B.C.

Pottery of the Early Bronze Age,
showing the second stage in geo-
metric ornament. From the Metro-
politan Museum, New York

RELIGIOUS MYSTICISM

tion of a pole star led him to consider a fourfold division of
his horizon, and to speak of the four corners of the earth;
and it is not impossible that the swastika and the various other
cruciform figures of the ancient civilization are a recognition of
this tendency. The number four was looked upon as peculiarly
significant by certain American
aborigines as well as by the early
peoples of Asia, Australia, and
Africa, and we may have a relic
of this attitude of mind when
we speak of "a square man"
or one who acts "squarely."

Architecture. Just as we find
an instinctive appreciation of
the beauties of geometric forms
as applied to personal orna-
ments, so we find it as applied
to architecture, not merely with
respect to decoration as already
mentioned, but in the general
structure of temples, of altars,
and of tombs. In early India,
for example, there seems to have
been no study of geometry
as such except in connection
with forms used in the temple,
and this was probably the case
in other parts of the earth. A
desire to adapt symmetry to
architecture is seen in the ter-
raced pyramids of Mexico as well as in those of Egypt; and
while these buildings are not prehistoric, they doubtless are
the outgrowth of prehistoric forms.

Observations of the Stars. As already mentioned, the prim-
itive man seems to have felt that the secret of the stars was
closely bound up with the secret of his destiny. It was

PAINTED JUG FROM CYPRUS
1000-750 B.C.

Pottery of the Early Iron Age in
Cyprus, showing a third stage in the
use of geometric ornament, with
the swastika. The Geometric Period
of decoration closed, for the Medi-
terranean countries, just before the
time of Thales. From the Metro-
politan Museum, New York

1 8 MYSTICISM

this that led the Babylonian shepherd and the desert nomad to
observe the stars, to speculate upon their meaning, and to take
the first steps in what developed into a priest lore in the
temples along the Nile and in the land of Mesopotamia. It
was this, too, that led the early philosophers and poets to
consider the stars as lighted lamps suspended in a vast material
vault, or as golden nails fixed in a crystal sphere, ideas
perfectly suited to the childhood of the race. When it was
that these observations of the heavens led to angle measure,
to the recording of such celestial phenomena as eclipses, and
to a naming of the signs of the zodiac and the constellations,
we cannot say. One writer of prominence 1 places a recognition
of the common constellations as early as 17000 B.C., and while
this date seems to be very improbable, even though supported
by certain historico-astronomical considerations, it is doubt-
less true that the period of this recognition and of the observ-
ance of certain celestial phenomena is very remote. While
there is good reason for thinking that these early steps in
astronomy were taken in Mesopotamia, the proof is not suf-
ficiently strong to enable us to say that this was unquestionably
the case, nor are we able to fix upon the period within any
particular century or even within any particular millennium.
Similarly, we are unable to state the time or place in which the
early peoples began to recognize the constellations or to give
them fanciful names. After attaining a certain degree of suc-
cess in our research, we are lost in the prehistoric clouds.

Lengthen the story all that we can, it is not possible to extend
it back more than an imperceptible distance on the great clock
face. For if we represent the period of all life on our planet
by one revolution of the minute hand, the period of human
life will be covered by only half a minute, and recorded history
will be represented by less than two seconds. What we defi-
nitely know of the history of mathematics covers a period in
world development so short as to seem almost infinitesimal.

a G. Schlegei, Uranograpkie Chinoise, 2 vols., II, 796 (Leyden, 1875). For a
recent discussion of the whole question, see Leopold de Saussure, " Les engines
de 1'astroiiomie chinoise," Toung Pao, Vols. X seq. (Leyden).

DISCUSSION 19

TOPICS FOR DISCUSSION

1. Geometric forms that were in existence before the advent of
life on the planet.

2. Laws of motion that entered into the formation and perpetua-
tion of our solar system.

3. Geometric forms that appear prominently in the vegetable
world and in the bodily structure of certain animals.

4. Geometric forms that appear prominently in the products
of the labor of the lower animals, with the question of maximum
efficiency in any of these cases.

5. The question of animal counting or pseudo-counting as
discussed by psychologists,

6. Evidence of primitive counting without any scale.

7. The world's use of scales below five as shown by a study of
our language and of savage tribes.

8. Reasons why the scales of five, ten, and twenty were the
leading favorites.

9. Reasons why the scale of twelve would have been a par-
ticularly good one.

10. Reasons why three and seven have been particularly notable as
mystic numbers, with several illustrations.

11. Circumstances which developed a high degree of skill in
counting among certain peoples.

12. Reasons which led primitive peoples to the use of geometric
forms in ornament.

13. The effect of religious mysticism upon primitive mathematics.

14. Various stages of geometric ornament in Cyprus, Crete, and
the mainland of Greece.

15. Possible influence of geometric decoration upon the study of
geometry as a science.

1 6. Geometric decoration that has persisted in all ages, with a
study of the probable causes for this persistence.

17. Causes leading to an interest in astronomy among primitive
peoples. Features of the ancient astronomy that are still found
either in our present study of the science or in folklore.

1 8. Evidence of the antiquity of astronomical ideas, particularly
in Mesopotamia, Egypt, and China, with probable evidence in the
case of India and other parts of the East.

CHAPTER II

THE HISTORIC PERIOD DOWN TO 1000 B.C.
i. GENERAL VIEW

Sources of our Knowledge. The period down to the arbitrarily
selected date 1000 B.C. overlaps the prehistoric period men-
tioned in Chapter I, the prehistoric gradually merging into the
historic in certain parts of the world but not reaching this stage
in certain other parts. Various facts which might properly
have found place in Chapter I will therefore be related in this
chapter, but in a general way we shall now pass to that period
in the evolution of the race in which less use need be made of
conjecture in the recital of the story of mathematics, although
it cannot be wholly eliminated.

The sources of our knowledge are no longer mere tradition,
nor does inference from the study of savage tribes constitute
so important a basis for our statements. The sources are
now, in general, the relics of human activity, largely in the
form of inscriptions or manuscripts which actually date from
remote centuries, or of copies of such evidences.

Countries Considered. There are four countries which have
left to posterity such an abundance of historical material prior
to the beginning of the first millennium of the pre-Christian era
as to warrant our special consideration. These countries, con-
sidered geographically instead of politically, are Egypt, Meso-
potamia, China, and India. Each claims for itself a high
degree of antiquity, each claims to have been a pioneer in mathe-
matical development, each is ethnographically somewhat of a
unit ; and in certain respects the claims of each have reasonable
foundations. Each had, at least in considerable areas, a
salubrious climate in the warm intervals between the several

SOURCES OF KNOWLEDGE 21

descents of the ice from the north, descents which character-
ize what is commonly called the Glacial Epoch, and hence
each was able to develop an early civilization. Each flourished
along one or more important rivers, which not only furnished
water for navigation and for domestic purposes but also
afforded opportunities for the application of a rude form of
mathematics to the irrigation projects which were already in evi-
dence in the early centuries of the historic period. From each,
after the ice retreated, a human stream in due time flowed to
the more invigorating climate of the north and carried along
with it traces of the early mathematical lore which had already
begun to develop in the temples, and of the customs established
among the primitive traders of the more favored lands.

Antiquity of Mathematics. Although there is evidence show-
ing that^the human race has been on the earth for hundreds
of thousands of years, the earliest definite remains that have
come down to us are from the Early Stone Age, or Paleolithic
Age, which seems to have begun in the warm interval after the
third descent of the ice. This was not less than 50,000 years
before our era and may have been from 50,000 to 75,000 years
earlier still. In the remains of this period we find implements
which suggest the existence of barter and the need of numbers
for counting, although they date from a period thousands of
years before there had dawned upon human intelligence any
idea of written numerals. We may simply suppose that the
presence of such implements and the important discovery of
a means for making fire, which seems to have occurred about
50,000 years ago, are evidence of a degree of intelligence high
enough to assure some idea of number.

About 15,000 years before our era there is thought to have
begun the Middle Stone Age, the period of the fourth descent
of the ice. In this period we find the oldest known works of
art. These works show such an intellectual advance as to make
it quite certain that the world had reached a period when the
abstract notion of number must have been in evidence, a
judgment warranted by our knowledge of all primitive peoples
today who have reached this stage in art.

22 CHINA

The Late Stone Age is relatively a recent period, dating
from c. 5000 B.C., and by this time there had developed quite
elaborate number systems, and the observation of the stars
had become a fairly well organized science./ That this was
the case we know from various historical facts which will be
mentioned later.

Advent of Writing. Metal, as distinguished from natural
ore, was discovered c. 4000 B.C., possibly in the Sinai penin-
sula, and with this there came a new need for weighing and
measuring and a new impetus to a system of barter which was
doubtless very old even at that time. About 500 years later,
writing is known to have been in use, and the system of ruling
over masses of people had become so advanced as to render
possible the control of a population of several millions by one
government. The bearing of all this upon the development of
a number system and upon systematic taxation is apparent.
About 3000 B.C. the earliest stone masonry was laid and sea-
going ships began to cross the Mediterranean, and a little
later the pyramids of Egypt were erected, so that history now
enters a period in which mathematics reached out beyond mere
counting and into such fields as that of practical geometry,
including a primitive kind of leveling and surveying.

2. CHINA

Early Chinese Mathematics. We have no definite knowledge
as to where mathematics first developed into anything like a
science. Mesopotamia has several strong claims to priority,
and so has Egypt. As to China, we have little positive knowl-
edge of its earliest literature, the possibility of corruption of
its texts being such as to cast doubts upon its extreme claims.
Until native scholars develop a textual criticism commen-
surate with that which has been developed in the Occident, this
uncertainty will continue to exist. (The historical period begins
with the 8th century B.C., or, at the earliest, with the reign of
Wu Wang, the Martial Prince, in 1122 B.C. 1 In beginning with

1 A. J. Little, The Far East, p. 20. Oxford, 1905.

EARLY CHINESE MATHEMATICS 23

China, therefore, it must not be thought that we should recog-
nize the validity of all the claims that are often advanced for
the antiquity of her science.

Basing his opinion upon later historical descriptions of the
primitive astronomy of China, Professor Schlegel of The
Hague, as already remarked, asserts that the Chinese recog-
nized the constellations as early as 17000 B.C., which was about
the close of the Early Stone Age. There is nothing impossible
in such a supposition, although it is improbable. The race had
developed considerably by that time, and it may well have
extended its poetic fancy to the giving of forms to groups of
stars which it had looked upon for thousands of years. Pro-
fessor Schlegel also fixes upon 14700 B.C. as an approximate
date of the duodenary zodiac, other scholars asserting that
13000 B.C. is more probable and still others fixing upon 4000
B.C., a discrepancy that may well arouse skepticism as to the
validity of any of these hypotheses. Schlegel also believes
that there is evidence of the extended study of the celestial
sphere in China in or about 14600 B.C.

While such claims are generally doubted by competent Sinolo-
gists, it is quite likely that the Chinese developed some ac-
quaintance with descriptive astronomy at an early period, and
that this development necessitated such knowledge of mathe-
matics as the measure of time and angles and the use of fairly
large numbers. Reasonably well-founded tradition gives the
probable dates of Fuh-hi, 1 the reputed first emperor of China,
as 2852-2738 B.C., 2 and in his reign there were extensive as-
tronomical observations. In this general period the Chinese
are believed to have changed their zodiac into one of twenty-
eight animals.

1 In general, the transliteration of Chinese names is that of Y. Mikami, The
Development of Mathematics in China and Japan (Leipzig, 1913) (hereafter
referred to as Mikami, China}, and H. A, Giles, A Chinese Biographical Dic-
tionary (London, 1898). The transliteration varies greatly with different Sinolo-
gists, a title like I-king appearing as Yih-ching, Yi-ching, Ye King, Y-Ching,
and so on. In many cases I have been greatly assisted by my colleague, Pro-
fessor Friedrich Hirth, one of the greatest living Sinologists.

2 F. Hirth, The Ancient History of China, p. 7 (New York, 1908). Professor
Hirth follows Arendt's tables as being the most carefully considered.

24 CHINA

Reign of Huang-ti. In the year 2 704 B. c. 1 Huang-ti, the
Yellow Emperor, began his reign. Under his patronage it is
said that Li Shu wrote on astronomy and that Ta-nao estab-
lished the Chia-tsu, or sexagesimal system, both of these state-
ments being supported by copies (possibly altered) of ancient
records. 2 Even the emperor himself is said to have taken
such an interest in mathematics as to write upon astronomy
and arithmetic, and in his reign an eclipse of the sun was ob-
served and recorded. Tradition assigns to this period even
the decimal system of counting, although it is more likely that
some popular work on the subject was written at this time.
It was possibly during the reign of the emperor Yau 3 (c. 2357-
c. 2258 B.C.) that two brothers, Ho and Hi, made astronomical
observations. They are said to have suffered the displeasure
of the emperor through their failure to predict a solar eclipse, 4
an incident showing a state of mathematical advancement quite
equal to that in Greece in the time of Thales, some 1500 years
later. The story is told in the Shu-king (Canon of History),
an ancient record sometimes attributed to the pen of the em-
peror himself and sometimes to that of Confucius nearly two
thousand years later."'

1 According to Arendt and Hirth. Giles gives 2698 and others give 2697.
Huang-ti is said to have died at the age of in years.

2 As stated in Volume II, it is doubtful if the Chinese used anything like a
sexagesimal system at this time, although they may have learned from the
Sumerians that 60 is a convenient unit for subdivision.

3 Reputed to have lived nearly a full century. See A. T. de Lacouperie, The
Languages of China before the Chinese, p. 9 (London, 1887) ; Hirth, loc. tit.,
p. 29.

4 R. Wolf, Geschichte der Astronomic, p. 9 (Munich, 1877); Handbuch der
Mathematik, Physik, Geod'dsie und Astronomic, I, 7 (Zurich, 1872).

5 Hirth, he. cit., pp. 29, 33, 251, who believes it to be a late work, or at least,
if written in the time of Yau or of his immediate successors, to have been greatly
modified by later copyists. Some effort has been made to fix the date of the
eclipse as May 7, 2165 B.C. Other proposed dates are October n, 2154 B.C.,
October 12, 2127 B.C., October 24, 2006 B.C., October 22, 2155 B.C., and
October 21, 2135 B.C. The discrepancy between these later dates and those
tentatively assigned to Yau, as given above, has little significance in the
present state of knowledge as to Chinese chronology. The whole subject is
still in the conjectural stage and awaits extended research on the part of
capable Sinologists.

THE I-KING 25

It is this emperor Yau and his successor, the emperor Shun,
who, it is said, carried farther to the eastward the dominion
established by the Bak tribes which had come from western
Asia. These tribes had been under the civilizing influences of
the people of Susiana, who in turn had received their civilization
from Babylon. 1 If this theory proves to be correct, the simi-
larity between certain early forms of astronomy and mathe-
matics in the East and the West is more easily explained.

I-king. Of the "Five Canons" (Wu-king) of the Chinese
probably the third in point of antiquity is the I-king, or Book
of Permutations. 2 In this appear the Liang 7, or "two princi-
ples" (the male, yang, -; and the female, ying, ) and

from these were formed the Sz' Siang, or "four figures,"

and the Pa-kua (eight-kua) or eight trigrams, the eight permu-
tations of two forms taken three at a time, repetitions being al-
lowed. These Pa-kua had various virtues assigned to them and
have been used from a very early period until the present for
purposes of divination. It was probably Won-wang (1182
-i 135 B. c.) who wrote the I-king ; at any rate it was he who ex-
tended the Pa-kua into the sixty-four hexagrams now found in
this classic. 3

X A. T. de Lacouperie, loc. cit., pp. 9 seq.

2 It is often called the oldest of the Chinese classics, as in the edition by
J. Mohl, Y-King; Antigtiissimus Sinarum liber (Stuttgart, 1834-1839). In the
extensive literature on the I-king the following works may be consulted:
H. Cordier, Bibliotheca Sinica, II, cols. 1372 seq. (Paris, 1905-1906) ; A. T. de
Lacouperie, "The Oldest Book of the Chinese," Journal of the Royal Asiatic
Society, XIV (N. S.) (London, 1882), 781, reprinted in 1892, with an ex-
tensive bibliography; T. McCIatchie, "The Symbols of the Yih-King," The
China Review, I (Hongkong, 1872), 151; J. Edkins, "The Yi king of the
Chinese," Journal of the Royal Asiatic Society, XVI (N.S.; (London, 1884),
360; H. J. Allen, Early Chinese History, chap, viii (London, 1906). The first
European edition of the I -king appeared at Frankfort in 1724.

3 Hirth, loc. cit., p. 59. The Pa-kua are attributed to Fuh-hi byLiuHui, who
wrote c. 250 A.D. The Leibniz theory, set forth in his Philosophia Sinensi urn, 4,
that these symbols had some connection with binary numerals, has no historical
foundation in the I-king as originally written.

CHINA

It is hardly conceivable by the Western mind that such a set
of symbols should last for thousands of years, that it should
be the subject of such a large number of books and mono-
graphs as have appeared in explanation of its meaning, and that
it should be known today to everyone among the hundreds of
millions who have come under the influence of the Chinese
philosophy, not merely in China but all through the East.

.1-.

"""

....

Win

cJwn

sun

/'rtw

ton

^////

heaven

steam

fire

thunder

wind

water

mountain

earth

C/5 h-

< J4

H {*

c
J?

Q Q

w * &

*1 *

H ^

""?

w r>

5 ^

- O

,< w O

*-,

K GO

^ &

^ ^

S.E.

N.E.

s.w.

N.W.

THE PA-K.UA, OR EIGHT TRI GRAMS

From the 1-klng, or Book of Permutations. On the ordinary diviner's compass
these directions are reversed

An examination of the above interpretation of the Pa-kua,
the one commonly given by Oriental writers, suggests the Py-
thagorean doctrines with respect to numbers, and as we proceed
we shall find still more to strengthen the belief that the West
obtained much of its mysticism from the East.

Although there is no historical evidence that the Chinese
looked upon the Pa-kua as numerals, based upon the scale of
two, it is true that if we take for one and for zero, the
successive trigrams, beginning at the right, have values which
we may represent by our numerals as ooo, ooi, oio, on, 100,
101, no, and in. If these are considered as numbers written
on the scale of two, their respective values are o, i, 2, 3, 4, 5,
6 7 and 7.

THE PA-KUA

The Pa-kua are found today on the compasses used by the
diviners in every city and village of China. They are also

THIBETAN "WHEEL OF LIFE"

From a sheet of block printing done at Lhassa. This portion represents the
signs of the zodiac, the Pa-kua, and, in the center, a magic square

found on fans, vases, and many other objects of the home,
and on talismans of various kinds in common use in Thibet
and other parts of the Far East.

CHINA

The lo-shu and ho-t'u. The I-king also states that the Pa-
kua were footsteps of a dragon horse which appeared on a

MAGIC SQUARE

The rows, columns, and diagonals in this particular magic square have 15 as
their respective sums

river bank in the reign of the Emperor Fuh-hi, and that the
lo-shu, in reality the magic square here shown, was written

THE LO-SHU FROM THE I-KING

This is the world's oldest specimen of a magic square. The black circles are

used in representing feminine (even) numbers, the white ones in representing

masculine (odd) numbers

THE LO-SHU AND HO-T'U 29

upon the back of a tortoise which appeared to Emperor Yu
(c.2 200 B.C.) when he was embarking on the Yellow River.
The ko-t'u, also a highly honored mystic symbol, appears in
the same work.

It thus appears that the I-king is not a work on mathematics,
but that it contains the first evidence of an interest in permu-
tations and magic squares that has come down to us. It is

THE HO-T'U FROM THE I-KING

This was never considered so important as the lo-shu, lacking as it does the
interest of the magic square

reasonable to believe, however, that both these ideas were
already ancient when the book was written.

The Chou-pei. The oldest Chinese work that can be designated
as mathematical is the Chou-pe'i, or the Chdu-pei Suan-king?

^Suan-king, or Suan-ching, means "arithmetic classic." Also transliterated
in various other ways, such as Tcheou-pei-swan-king. See E. Biot, "Traduction
et examen d'un ancien ouvrage chinois intitule* Tcheou pet," Journal Asiatique
(1841), p. S9S, with a discussion of dates.

30 CHINA

a work relating chiefly to the calendar but containing informa-
tion referring to ancient mathematics, including some work on
shadow reckoning. The author and the date of the work are
both unknown, and there is some reason for believing that it
has undergone considerable change since it was first written.
The fact that Emperor Shi Huang-ti l of the Ch'in Dynasty, in
213 B.C., ordered all books burned and all scholars buried, would
seem at first thought to have given an opportunity for radically

CHOU-PEI SUAN-KING

A work written in the second millenium B.C. This illustration is from a very

early specimen of block printing. It shows the figure of the Pythagorean

Theorem, but gives no proof

altering all ancient treatises ; but such a sweeping decree could
not possibly have been executed, and even if every book had
been lost there would have been many who could have repeated
the ancient classics verbatim from memory. The probability is
that we have about as near the primitive form of these classics
as we have of the writings attributed to Boethius, Bede, or
Alcuin, or of certain Greek authors whose works we assume

transliterated Tsin Chi Hoang-ti and Tsin sch& huang ty (born 259 B.C.;
died 210 or 211 B.C.). The claims of such writers as Weber and J. B. Biot for a
high grade of mathematical learning in China before this time are contradicted
by L. Am. Sedillot, "De 1'astronomie et des mathematiques chez les Chinois,"
Boncompagni's Bullettino, I, 161.

THE CHOU-PEI 31

as known. In any case it is probable that we have in the
Chou-pe'i a very good record of the mathematics of about
1105 B.C., the year of the death of Chou-Kung, a party to
one of several dialogues which the book records. 1 One of
these dialogues is between the prince Chou-Kung and his
minister Shang Kao, and relates to number mysticism, men-
suration, and astronomy. Among the stories told of the energy
of Chou-Kung is one relating to his habit of rushing several
times from his bath, holding his long, wet hair in his hand, to
consult with his officials. Tradition also states that he had
a wrist like a swivel, on which his hand could turn completely
round, an odd fiction for those who are interested in stories
of mathematicians. A few extracts from the Chou-pei will
give some idea of the nature of the work :

The art of numbers is derived from the circle and the square.

Break the line and make the breadth 3, the length 4 ; then the
distance between the corners is 5. 2

Ah, mighty is the science of number.

Forms are round or pointed; numbers are odd or even. The
heaven moves in a circle whose subordinate numbers are odd ; the
earth rests on a square whose subordinate numbers are even.

One who knows the earth is intelligent, but one who knows the
heavens is a wise man. The knowledge comes from the shadow,
and the shadow comes from the gnomon. 3

The Nine Sections. Next in order of antiquity among the
mathematical works of China is the K'iu-ch'ang Suan-shu, or
Arithmetic in Nine Sections. 41 This is the greatest of the
Chinese classics in mathematics, and for many centuries has
been held in the highest esteem in the Orient. As to its author-
ship and the period in which it was written we are ignorant,

!Y. Mikami, China, p. 4; W. A. P. Martin, The Lore of Cathay, p. 30
(New York, 1901) ; A. Wylie, Chinese Researches, Part III, p. 159 (Shanghai,
1807).

2 This evidently refers to the right-angled triangle whose three sides are in th<-
ratio 3 14:5, a special case of the Pythagorean Theorem.

3 The gnomon was the index which cast the shadow on the sundial.

* In some editions, K'iu-ch'ang Suan-shu-ts'au-t'u-shuo.

32 CHINA

We know that not long after the burning of the books (213
B.C.) there appeared a mathematician by the name of Ch'ang
Ts'ang, that he collected the writings of the ancients, and that
he seems to have edited the K'iu-ch'ang Suan-shu. There is a
tradition, unsupported by positive proof, that the work was
originally prepared by direction of the Chou-Kung, who, as
already stated, died in 1105 B.C., and it has even been asserted
that it dates back to the reign of Huang-ti in the 2 yth century
B.C. 1 The evidence of tradition, therefore, places it very early,
and it seems probable that it existed, at least in great part, in
the period of which we are writing, that is, before 1000 B.C.

Topics in the Nine Sections. The work consists, as the title
says, of nine sections, books, or chapters. The titles and the
sequence of chapters vary somewhat in different editions, but
the following list is substantially correct as given in the revision
of the work in the 2d or 3d century B.C. :

1. Fang-t'ien (Squaring the farm), relating to surveying,
with correct rules for the area of the triangle, trapezium
(trapezoid), 2 and circle (\c*\d and \ cd), and with the
circle approximations f d 2 and ^ r 2 , where IT is taken as 3.

2. Su-mi (Calculating the cereals), relating to percentage
and proportion.

3. Shuai-fen (Calculating the shares), relating to partner-
ship and the Rule of Three. 3

4. Shao-Kuang (Finding length), relating to the finding of
the sides of figures, and including square and cube roots.

5. Shang-kung (Finding volumes), relating to volumes.

6. Chun-sfw, or Kin-shu (Alligation), relating to motion
problems (couriers, hare and hound) and alligation.

*A. Wylie, "Jottings of the Science of Chinese Arithmetic," North China
Herald, 1852, and the Shanghai Almanac for 1853; K. L. Biernatzki, "Die
Arithmetik der Chinesen," Crelle's Journal, Vol. LII (1856).

2 The meanings oi the words trapczoid and trapezium were curiously inter-
changed in England and America about a century ago, and the error still per-
sists in America. In this work the two words will be given as above, but the
meaning will always be the etymological one for trapezium, a quadrilateral with
two parallel sides.

3 A kind of proportion, discussed at length in Volume II.

THE NINE SECTIONS 33

7. Ying-pu-tsu, or Ying-nu (Excess and deficiency}, re-
lating to the Rule of False Position, 1 the terms "excess" and
"deficiency" relating to two concepts that are used in this rule.

8. Fang-ch'eng (Equation}, relating to simultaneous linear
equations, with some idea of determinants.

9. Kou-ku (Right triangle}, relating to the Pythagorean
Triangle.

These three works constitute the Chinese classics involving
mathematics which were probably written in whole or in part
before the year 1000 B.C. They show a degree of advance-
ment quite as high as that found in the other ancient countries,
and they prove that China was among the pioneers in the
establishing of the early science of mathematics.

3. INDIA

Early Hindu Mathematics. When we pass from a consid-
eration of Chinese mathematics to the mathematics of India,
Babylonia, and Egypt, we meet with the mental product of an
entirely different type of people, or rather of two different
types. There were two great branches of the human race af-
fecting the Western World on the one hand, and India, Meso-
potamia, and certain adjoining regions on the other hand. The
first of these branches is supposed to have wandered from
the Northern Grasslands, and constitutes what is known as the
Indo-Europeans. In the West its members appear as Celts,
Romans, and Greeks, and in Asia Minor it has several repre-
sentative groups. In the East this same stock is seen in the
Medes, Persians, and Hindus. The eastern branch is properly
designated as Aryan, from which we have the name "Iran" for
Persia. The people were generally highly imaginative, and
their work in mathematics developed along such lines as the
theory of numbers, geometry, and astronomy.

The second great branch is thought to have had its first
habitat in the Southern Grasslands of Arabia, and is represented

1 A primitive method of solving equations, considered at length in Volume II.

34 INDIA

by what is known as the Semitic peoples. These include
the inhabitants of Assyria, Babylonia, Phoenicia, and the
Phoenician colonies. They dwelt in the paths of trade from
East to West, and their work in mathematics developed chiefly
along the line of computation, a line which led to extensive
numerical work in the field of astronomy as well as in that of
commerce.

If the early mathematical achievements of the Chinese are
uncertain as to date and importance, much more so is the
early progress of the Hindus. Not only are we without any
satisfactory records of the remote past of these people, but we
are not infrequently confronted by claims that are preposter-
ous and that are so recognized by Hindu scholars themselves.
The first edition of the Surya Siddhdnta of the Swami Press
at Meerut, for example, says that the work was " Compiled
about 2,165,000 years ago," representing a period about four
times as long as it is thought the human race has been in
existence. With even more absurdity the Laws of Manu are
placed as far back as 6 x 71 x 4,320,000 years ago/ giving
almost an appearance of modesty to the ancient Chaldean
claims that their astronomical observations began more than
720,000 years ago. As a matter of fact this well-known work
on astronomy, the Surya Siddhdnta, was probably written
about the 4th or 5th century of our era. So little sympathy
had the early native scholars with those outside their own
caste that a general literature is wholly lacking, and it has
only been through the labors of those from other lands that an
all-round view of scientific progress has been attempted.
There is, however, sufficient evidence for the belief that pri-
mary schools existed very early in India, and that arithmetic
and writing were looked upon as the most important of the
seventy-two recognized branches of learning, at least in the
elementary stages of education. 2

1 On the extravagant ideas in the native Hindu chronology see J. C. Marsh-
man, Abridgment of the History of India, p. 2 (London, 1893). See also
W. Jones, "On the Chronology of the Hindus," in his Works, IV, i (London,
1807); M. Elphinstone, History of India, p. 136 (London, 1849 *

* A. Hillebrandt, Alt-Indien, p. in (Breslau, 1899)-

BABYLON 35

Lack of Authentic Records. As to authentic records, India
has none written before the first Mohammedan invasion,
c. 664 A.D. 1 All that we know of her earlier history is what
we can glean from her two great epics, the Mahabharata and
the Ramayana, and from coins and a few inscriptions. The
Mahabharata relates the skill in numerals possessed by the
ancient heroes, and the inscriptions tell us something of
the notation used by the Hindus two thousand years ago, but
neither gives us any knowledge of the period closing a thou-
sand ye?rs before our era. The Vedas, the sacred writings of
India, lead us to understand that in this period some attention
was given to astronomy, as was the case in contemporary
China, Mesopotamia, and Egypt.

All that we can say, therefore, about this period of Hindu
mathematics is that there is some evidence from ancient
literature that in very early times India paid attention to
astronomy and calculation, just as was the case with other
advanced peoples of that period. 2

4. BABYLON

Early Babylonian Mathematics. For our purposes Chaldea
and Babylonia are synonymous, each name referring to the
land extending from the delta of the Tigris and Euphrates
northward to Assyria, the hilly, forest-covered district origi-
nally surrounding the ancient capital of Assur (Asshur). In-
deed, it is convenient at present to consider as one large group
all those Semitic peoples descended from the wanderers from
the Southern Grasslands who settled in Assyria, in the region
about Nineveh, in Asia Minor, and along the Phoenician coast.
We shall also find it convenient to include a non-Semitic tribe,
the Sumerians, who dwelt in the land of Sumer at the head of
the Persian Gulf, directly in one of the chief paths of world

!The so-called Mohammedan period did not begin until 1001 A.D.

2 G. Oppert, On the Original Inhabitants of Bharatavarfa or India, p. i
(London, 1893); R. C. Dutt, A History of Civilization in Ancient India
(London. 1893).

36 BABYLON

commerce. These people, coming from the mountainous region
to the east, early developed a numeral system, and numerals
used by them in the 28th century B.C. are known to us through
certain inscriptions. Dwelling in a low country formed by
alluvial deposits, and thus deprived of stone for monumental
purposes, the primitive Sumerians resorted to the use of bricks

NUMERALS OF THE 28TH CENTURY B.C.

Sumerian tablet. The numerals at this time were made with the upper end

of the scribe's stylus and appear as curved symbols, and as such can easily be

recognized. From Breasted's Ancient Times

for the preservation of their records. Upon the surface of clay
tablets they pressed with a round and pointed stick, the result
being a circular, a semicircular, or a wedge-shaped (cuneiform)
character. These inscriptions were a mystery to the modern
world until the first half of the igth century, when Grotefend
'1802) suggested and Rawlinson (1847) perfected the key

EARLY CALENDARS 37

to the rich literature of ancient Mesopotamia. The clay tab-
lets, after being inscribed, were baked by fire or in the sun,
and thousands of them are now available for study in various
museums. These records of the Sumerians give us the infor-
mation that nearly 3000 years before Christ their merchants
were familiar with bills, receipts, notes, accounts, and systems
of measures. In no part of the world have we as clear evidence
of commercial mathematics at this early date as is revealed by
these Sumerian tablets. Here also we find evidence of an
approach to a scientific calendar, although of a later date than
similar evidence found in Egypt, and here is probably to be
found the first use of a kind of scale of 60 in counting.

Early Calendars. Some knowledge of mathematics must,
however, have long preceded the work recorded on these
Sumerian tablets. The old Babylonian year began with the
vernal equinox, and the first month was named after the Bull.
The calendar must, therefore, have been established at a
period in which the sun was in Taurus at this equinox, and
such a period began about 4700 B.C. A calendar of any kind
presupposes a system of numbers and some form of cal-
culation, so that we may safely say that some kind of arith-
metic existed in Babylonia in the 4th or 5th millennium B.C.
Indeed, so far as the calendar is concerned, it should be
said that the Sumerians celebrated the beginning of the year
at the vernal equinox as early as 5700 B.C., and possibly
even earlier. 1

Early Babylonia. What is commonly known as Early Baby-
lonia endured from about 3100 to about 2100 B.C. Sargon, the
first great ruler, flourished about 2750 B.C., his remarkable
career beginning in Akkad, the district just north of Sumer.
It was partly due to this proximity of territory that the people
of Akkad in particular and of Babylonia in general adopted
the business methods, the astronomy, 2 the calendar, the

!H. Radau, "Miscellaneous Sumerian Texts from . . . Nippur," in the
Hilprecht Anniversary Volume, pp. 408, 410. Chicago, 1909.

2 E. F. Weidner, Handbuch der Babylonischen Astronomic, Bd. I. Leipzig,
1015.

38 BABYLON

measures, and the numerals of the more highly cultivated
Sumerians. In Sargon's reign we find a record of eclipses,
so that the numeral system must have been well advanced/
and for him there was compiled the first great treatise on
astrology of which we possess any original fragments. 2

Among the tablets of about 2400 B.C. that have been de-
ciphered are various specimens dating from the reigns of kings
of the third dynasty of Ur 3 and recording the use of a kind of
draft or check, the measurement of land in shars, the weigh-
ing by talents (gur), the measurement of liquids by ka, the
taking of interest, the use of the fractions |, %,* and , and
the measurement of both liquids and solids by the qa (not
identical with the ka}.

In order to fix clearly in mind the period of which we are
speaking there should be mentioned not only the reign of
Sargon (c. 2750 B.C.) but the remarkable reign of Hammurabi
or Hammurapi (c. 2100 B.C.), in which the world's first great
code of laws, so far as we know, was written, and in which the
calendar was reformed. Among the other interesting relics of
the time of Hammurabi is the ruin of the oldest known school-
house. This was discovered by French archeologists in i894. r>
In the building were numerous tablets on which the pupils
had written their lessons, and it is from such tablets as these
that we have part of our knowledge of the arithmetic of the
Babylonians.

The general conclusion of archeologists, as will be elaborated
on page 40, is that these early Babylonians (in the thousand
years of their activity) developed a fair knowledge of computa-
tion, of mensuration, and of commercial practice, in spite of
an awkward numeral system by which they were handicapped.

1 See also F. Thureau-Dangin, in the Hilprecht Anniversary Volume, p. 156.

2 G. Bigourdan, L' Astronomic, 1920 ed., p. 27. Paris, 191 1.

3 G. A. Barton, Haver ford Library Collection of Cuneiform Tablets, Part I
(Philadelphia, n.d [1905]) ; ibid., Part II (1909).

4 These from Barton, loc. cit., Part I. On the taking of interest, the rates
running from 20% to 33^%, see E. Huber, "Die altbabylonischen Darlehns-
texte," in the Hilprecht Anniversary Volume, pp. 189, 217.

5 For a plan of the building see J. H. Breasted, Ancient Times, p. 136 (Bos-
ton, 1916) ; hereafter referred to as Breasted, Anc. Times.

ASSYRIA AND CHALDEA

ARAMEAN WEIGHT FOUND IN
ASSYRIA

The weight is of bronze and the inscrip-
tion is Aramaic. Fifteen of these lion
weights were found in Nineveh and tes-
tify to the common presence of Aramean
merchants in Assyria. From Breasted '5
Ancient Times

Early Assyria. As early as 3000 B.C. a Semitic tribe of no-
mads settled at Assur, and in due time it too adopted the
Sumerian calendar and such
of the mathematics of trade
as had been developed by
these people of the south.

Much later, and after 1200
B.C., the Arameans, or Syri-
ans, established kingdoms in
the region to the west of
Assyria. They were great
merchants, and the Sumerian
mathematics of trade, which
had worked slowly northward
through Babylonia and As-
syria, now found place in
the new territory. We have
bronze weights of this period,
showing that whole numbers,

fractions, measures, and elementary forms of computation
played a considerable part in the daily life of the people.

Early Chaldea. The desert tribe called the Kaldi came into
prominence long after the period now under discussion. It
gained a foothold in ancient Sumer and finally (606 B.C.)
conquered the Assyrians and established the Chaldean empire
in the region of Babylonia. Although their empire lasted only
to 539 B.C., they made great progress in science. In particular,
astrology was extensively cultivated, the equator was probably
divided into 360, the twelve signs of the zodiac definitely ap-
peared, and mathematics flourished as the handmaid of com-
merce and astronomy. Thus Babylonia became Chaldea, and
Chaldea became the patron of science and art.

Early Cuneiform Tablets. Our first important knowledge of
Babylonian arithmetic was derived from two tablets found in
1854 at Senkereh, the ancient Larsam or Larsa, on the Eu-
phrates, by a British geologist, W. K. Loftus. These tablets

40 BABYLON

contain the squares of numbers from i to 60 and the cubes of
numbers from i to 32.* Their date is uncertain, but the
evidence seems to show that they were of about the Hammurabi
period (c. 2100 B.C.).

Since the discovery of the Senkereh tablets there have been
unearthed some 50,000 tablets at Nippur, the modern Nuffar,
an ancient city lying to the south of Babylon, and among these
are many that relate to mathematics. 2 They are apparently
from a large library which seems to have been destroyed by
the Elamites about 2 1 50 B. c. or a little earlier, and again about
1990 B.C., and they constitute the most extensive mass of
ancient mathematical material ever brought to light. The
cylinders include multiplication and division tables, tables
of squares and square roots, geometric progressions, a few com-
putations, and some work on mensuration. Neugebauer's studies
(1935) of a large number of tablets show that the Sumerians
and Babylonians could solve special linear, quadratic, cubic,
and biquadratic equations and had some knowledge of nega-
tive numbers.

Babylonian Geometry. The tablets found at Nippur and else-
where also give us some knowledge of the Babylonian geometry.
From these it seems that as early as 1500 B.C. the Baby-
lonians could find the area of a rectangle, including that of a
square; the area of a right-angled triangle; the area of a
trapezium (trapezoid) ; and possibly the area of a circle, the
volume of a parallelepiped, and the volume of a cylinder.
There is ground for the belief that they knew the law of
expansion of (a + 6) 2 , although we have no knowledge as to
whether this was inferred from a geometric figure or from their
extensive study of square numbers. There is also some reason
to believe that they knew the abacus, since it has been sug-
gested that one of their signs (SID) may have been derived
from a pictograph of such an instrument.

1 Apparently from i to 60 originally, but part of the tablet is broken off.

2 H. V. Hilprecht, Mathematical, Metrological, and Chronological Tablets
from the Temple library of Nippur (hereafter referred to as Hilprecht, Tablets)-
Philadelphia, 1906.

SCALE OF SIXTY 41

Scale of Sixty. One peculiarity of Babylonian arithmetic is
the constant use of the number 60, a use which finally sug-
gested the development of sexagesimal fractions and which still
survives in our division of degrees, hours, and minutes into
sixty sub-units. It is generally thought that the Babylonians,
interested as they were in watching the stars, early came to
believe that the circle of the year consisted of 360 days. It is
also thought that they knew that the side of the regular in-
scribed hexagon is equal to the radius of the circle, this property
suggesting the division of 360 into six equal parts, and 60 being
thus looked upon as a kind of mystic number. This may y
indeed, be the origin of this use of 60, but we find other
nations using 40, 20, and even 15 in somewhat the same way,
with no apparent reason, so that all such customs may have
developed from racial notions which were started by some
leader or sect with no particular reason in mind. It is more
probable that 60 was chosen because of its integral divisors
2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, thus rendering work with
its fractional parts very simple.

Although the subject of fractions with the denominator 60 is
discussed in Volume II, a brief mention may be made at this
time of an important tablet first described in I92O. 1 It dates from
c. 2000 B.C. and illustrates the Babylonian custom of using either
360 or 60 for the denominator except in the cases of unit frac-
tions and of fractions in which the numerator is i less than
the denominator. For example, ^ 6 o may appear as J# or as |
fSUSSU), f| as $8- or f (SINIPU), and |{ft as gj or J
(PARAB).

5. EGYPT

Early Egyptian Mathematics. Whatever claims may properly
be made for the antiquity of mathematics in various other coun-
tries, claims of even greater validity can justly be made for the
science in Egypt. Civilization has generally developed along
great rivers; the Nile is one of the world's greatest arteries

iH. F. Lutz, "A mathematical cuneiform tablet," American Journal of
Semitic Languages, XXXVI, 240,

42 EGYPT

of commerce, and its fertile valley is one of the world's greatest
gardens. Egypt was a well-protected country, and civilization
had a more favorable opportunity for uninterrupted develop-
ment there than in such lands as Mesopotamia, Phoenicia,
India, and China. Furthermore, her art, as shown by wall
sculptures, was much farther advanced in the 4th millennium
B.C. than it was, say, with the Sumerians, and so there is every
reason to feel that her science was also in the lead of that in
other lands.

The earliest dated event in human history is the introduction
of the Egyptian calendar of twelve months of thirty days each,
plus five feast days, in the year 4241 B.C. 1 Such an achieve-
ment as the creation of this calendar, a better one than was
used in Europe from the time of the Romans until the reform
of Gregory XIII (1582), and in some respects better than the
one used at present, shows a high development of computation
as well as of astronomy. No authentic record of mathe-
matical progress in any other country dates back as far as
this ; it reaches back even into the Stone Age, more than a thou-
sand years before the earliest stone masonry and long before
any people had the slightest idea of an alphabet as we under-
stand the term. An event like this is a silent but powerful wit-
ness to the noteworthy arithmetic attainments of its sponsors
and to a long series of scientific observations by the temple as-
tronomers. Furthermore, our own calendar may be said to
be merely a poor adaptation of this ancient Egyptian one,
although containing the great improvement of having as
centennial leap years only those of which the numbers repre-
senting the hundreds are divisible by four.

Third Millennium B.C. in Egypt. When we approach the year
3000 B.C., toward the close of the Second Dynasty, we find our-
selves in a period of rapid development in practical engineer-
ing. We have no manuscripts of this period from which to
obtain direct information, but in the achievements of the
engineers it is possible to recognize a number of interesting

J Breasted, Anc. Times, p. 45.

THIRD MILLENNIUM B.C. 43

facts. Professor Breasted has characterized the development
of civilization in the 3Oth century B.C. in these words :

Hardly more than a generation before this 3oth century the first
example of hewn stone masonry was laid, and in the generation
after this 3oth century the Great Pyramid of Gizeh was built.
With amazingly accelerated development the Egyptian passed from
the earliest example of stone masonry just before 3000 B.C. to the
Great Pyramid just after 2900. The great-grandfathers built the first
stone masonry wall a generation or so before 3000 B.C., and the
great-grandsons erected the Great Pyramid of Gizeh, within a
generation after 2900. . . .

One finds it difficult to imagine the feelings of these earliest
architects ... as they paced off the preliminary plan and found an
elevation in the surface of the desert which prevented them from
sighting diagonally from corner to corner and applying directly a
well-known Egyptian method of erecting an accurate perpendicular
by means of measuring off a hypotenuse. . . .

The Egyptian engineers early learned to carry a straight line
over elevations of the earth's surface, or a plane around the bends
of the Nile.* In his endeavor to record the varying Nile levels in
all latitudes the Egyptian engineer was confronted by nice problems
in surveying, even more exacting than those which he met in the
Great Pyramid. A study of the surviving nilometers has disclosed
the fact 1 that their zero points, always well below lowest water,
are all in one plane. This plane inclines as does the flood slope
from- south to north. The Pharaohs' engineers succeeded in carry-
ing the line in the same sloping plane, around innumerable bends in
the river for some seven hundred miles from the sea to the First
Cataract. 2

Accuracy of Early Engineers. Such was the degree of ac-
curacy secured by these early surveyors that Petrie found the
maximum error in fixing the length of the sides of the Great
Pyramid to be only 0.63 of an inch, or less than Yi^inr of the
total length, and the angle error at the corners to be 12", or
only ^ 1 oinr of a right angle.

1 L. Borchardt, Nilmesser und Nilstandsmarken.

Z J. H. Breasted, " The Origins of Civilization," The Sckntific Monthly^ 8 7.

44 EGYPT

Speculations on the Great Pyramid. As to the speculations
relating to the Great Pyramid it is possible to make only
a brief statement in this work. That mathematics, and possi-
bly mathematical mysticism of some kind, played an important
part in the design of the structure is admitted by all scholars,
but precisely what the dominating principle was we do not
know. It has been suggested that four equilateral triangles
were put together for the pyramidal surface, but this theory
is not borne out by measurements, the base being considerably
longer than the sloping edge.

A second theory asserts that the ratio of the side to half the
height is the approximate value of TT, or that the ratio of the
perimeter to the height is 2 TT. It is true that this would give
the value of TT as about 3.14, an approximation that may have
been known to the pyramid builders ; but this was possibly a
mere matter of chance. If one searches in any building, or
indeed in any given object, for lines having this ratio, they
are not difficult to find. Nevertheless, there probably is some
mysticism of this kind in the proportions of the structure.

A third theory makes the claim that the angle of elevation of
the passage leading to the principal chamber determines the
latitude of the pyramid, approximately 30 N., or that the pas-
sage itself pointed to what was then the pole star; but even after
making all reasonable allowances in favor of this hypothesis, the
angular difference is too great to make out a very strong case.

It is also claimed that the pyramids have a constant angle
of slope, and it is true that the three at Gizeh vary but little,
being approximately 51 51', 52 20', and 51 ; but others have
slopes running from about 45 to 74 10'.

Recent measurements have been so accurately made that it is
probable that further study will reveal in the near future what-
ever mathematical principles actuated the architects. For the
present we may simply dismiss the speculations of such men
as Charles Piazzi Smyth 1 as interesting rather than scientific.

*An English astronomer; born at Naples, 1819; died 1900; astronomer royal
of Scotland (1845-1888). Our Inheritance in the Great Pyramid (1864), Life
and Work at the Great Pyramid, 3 vols. (1867).

REIGN OF AMENEMHAT III

COLLECTION OF TAXES, C. 3000 B.C.

Showing the clerks and scribes at the right,
with pen and papyrus, and the officials and
taxpayers at the left. From Breasted's An-
cient Times

Testimony of the Wall Reliefs. The wall reliefs of this gen-
eral period of the Pyramid builders testify to the collection of
taxes, probably in the form of grain, and the issuing of receipts
by the officials of the king. Nothing so tangible, showing the
applications of elementary arithmetic at this early period, has
been found in any other
region except Sumeria.

Reign of Amenemhat
III, or Moeris. About
1850 B.C., in the i2th
Dynasty, there came to
the throne one of the
most energetic of all the
kings of Egypt, Amen-
emhat III. 1 In his reign
there was carried out an

extensive system of irrigation, necessitating a knowledge of
leveling, surveying, and mensuration such as had probably
never beea. developed before this time in any other part of
the world, except perhaps in Mesopotamia. 2 There is good rea-
son to believe that in this reign, say about 1825 B.C., there
was written the original of the oldest elaborate manuscript on
mathematics now extant, the Ahmes treatise mentioned a little
later. If the conjecture is correct, the unit fraction was already
known in Egypt, as also seems to have been the case in Mesopo-
tamia, 3 and the simple equation with a fairly usable symbolism,

lf rhe name also appears as Ne-mat-re and as Amenemha. He is the Moeris
of Herodotus (II, 148-150), the Marros of Diodorus Siculus (I, 52), and the
Mares of Eratosthenes. He is also referred to by Strabo (XVII), Pliny (Hist.
Nat., V, 9, 50, and XXXVII, 12, 76), and Pomponius Mela (I, cap. 9). Recent
Egyptologists give the date of his reign as 1849-1801 B.C.; others place his
reign as c. 1986-^:. 1942.

2 On the Egyptian work in this line see A. Wiedemann, Aegyptische Ge-
schichte, I, 256 (Gotha, 1884) ; J. Lieblein, "L'Exode des Hebreux," Proceedings
of the Society of Biblical Archaeology, XXI (London, 1899) ,55. On the claims of
priority in the general development of mathematics, see E. Weyr, Ueber die
Geometric der alien Aegypter y p. 4 (Vienna, 1884).

8 See the author's review of Hilprecht's work in the Bulletin of the American
Mathematical Society , XIII (2), 392.

EGYPT

arithmetic and geometric series, and the elements of mensura-
tion were already familiar to the elite among the mathema-
ticians of the Nile Valley.

Mathematics of the Feudal Age.
Amenemhat III lived in the so-
called Feudal Age of Egypt, a
period which lasted for several
centuries, closing about 1800 B.C.
To this period belongs the oldest
astronomical instrument extant, a
forked stick used in sighting for the
purpose of obtaining the meridian.
Such a work presupposes some abil-
ity in calculation and in con-
structive geometry, and the exist-
ence of this ability is still further
proved by the Ahmes treatise.
There is another treatise, written
much earlier than this, 1 in which
we find mention of the civil calen-
dar of twelve months of thirty days
each, plus five extra days, as al-
ready mentioned.

About the close of the Feudal
Age a postal service existed in Asia
under Egyptian control, requiring
some means of payment on the
part of those whose convenience it
served. At the same time census
lists were prepared for use in the
taxation of the people, surveys for
irrigation projects were made, and
the Nilometer served to foretell the
beginning and the end of the rise
of the river, all of which involved,

lf That is, in the soth century B.C. See E. Mahler, "Der Kalender der
Babylonier," in the Hilprecht Anniversary Volume, pp. i, 9.

THE OLDEST ASTRONOMICAL
INSTRUMENT KNOWN

The original is in the Berlin
Museum. Part A was a plumb
line. By its aid the observer
could hold B over a given point
and sight along the slot to
some object like the North Star,
thus establishing a meridian
line. From Breasted's Ancient
Times

AHMES PAPYRUS 47

as already stated, the use of a considerable amount of mensura-
tion and computation, and adds to the evidence of an interest
in mathematics in this period of Egyptian history.

In the period of the Middle Kingdom (2160-1788 B.C.)
business arithmetic was such as to demand bills, accounts, and
tax lists. From this period we have various fragments of
papyrus rolls which were found in the remains of the libraries
of the feudal lords. These are the oldest libraries of papyrus
rolls thus far known. Among these remains were found such
evidences of the commercial activity above mentioned as are
seen in the fragments of papyri found at Kahun and now in
London and Berlin. 1

Ahmes Papyrus. About 1650 B.C. there lived in Egypt a scribe
named A'h-mose, commonly called by modern writers Ahmes. 2
He wrote a work on mathematics ; or rather he copied an older
treatise, for he says: "This book was copied in the year 33,
in the fourth month of the inundation season, under the maj-
esty of the king of Upper and Lower Egypt, 'A-user-Re',
endowed with life, in likeness to writings of old made in the
time of the king of Upper and Lower Egypt, Ne-ma f et-Re'.
It is the scribe A'h-mose who copies this writing. 77 Another
manuscript of the same period, containing a number of
lines on fractions, is in the British Museum. It was pub-
lished in 1927. The actual manuscript J of Ahmes has come
down to us, having been purchased in Egypt about the

ir The earliest date in the London fragments is in the reign of Amenemhat
III. See W. M. Flinders Petrie, Kahun, Gurob, and Hawara, chap, vi by F. L.
Griffith (London, 1890). This article places the date c. 1986-1(342 B.C., which
is somewhat later than that given by the earlier writers and a little earlier than
that given by some of the latest authorities.

2 "A'h-mose" was also the name of certain kings. Ahmes I, often known as
Amosis or Amasis, came to the throne at the beginning of the i8th Dynasty,
when Egypt entered upon a period of empire, and it was he who expelled the
Hyksos and pursued them into Palestine.

3 Since the first edition there have appeared editions of the Rhind Papyrus by
T. E. Peet (London, 1923) and A. B. Chace (Oberlin, O., 2 vols., 1927, 1928).
The Chace edition is the more elaborate, containing a facsimile of the papyrus, a
transcription into hieroglyphic and Latin characters, a complete translation, nu-
merous notes, and an extensive bibliography by R. C. Archibald. The preferred
form of the name is given as A'h-mose and the date as between 1750 and 1580.

48 EGYPT

middle of the igth century by the English Egyptologist,
A. Henry Rhind (whence the name "Rhind Papyrus"), and
having later been acquired by the British Museum. It is one
of the oldest mathematical manuscripts on papyrus extant.

A PAGE FROM THE AHMES PAPYRUS
Written c. 1550 B.C. The original is in the British Museum

The Ahmes manuscript is not a textbook, but is rather a
practical handbook. It contains material on linear equations
of such types as x + $x = ig] it treats extensively of unit
fractions ; it has a considerable amount of work on mensura-
tion, and it includes problems in elementary series. 1 The

iThe British Museum published an inexact facsimile of the papyrus in 1898
under the title Facsimile of the Rhind Mathematical Papyrus. The standard works
on the subject are those of Chace and Peet, mentioned on page 47, and (less valu-
able) that of A. Eisenlohr, Ein mathematisches Handbuch der alien Aegypter,
2d ed. (Leipzig, 1877). See also F, L. Griffith, Proceedings of the Society of Bibli-
cal Archaeology, 1891, 1894; A.Favaro, Atti delta R. Accad. . . . in Modena,
Vol. XIX.

COMMERCIAL MATHEMATICS 49

internal evidence shows the work to be a compendium of the
contributions of at least two or three authors. 1

Evidence of Commercial Mathematics, About 1500 B.C. there
was built by Queen Hatshepsut 2 the temple known at present
as Der al-Bahri. This is not far from Thebes 3 and in 1904
was uncovered and made known to modern scholars. On the
walls of this temple is pictured the receipt of tribute from the
land of Punt, probably on the Somali coast of Africa, 4 and
mention is made of "reckoning with numbers, summing up in
millions, hundreds of thousands, tens of thousands, thousands,
and hundreds," showing the extent to which numbers were
used in commercial matters even before coins were invented.
There are certain inscriptions of the same period in the tomb
of Rekhmire, at Thebes, giving the tax list of Upper Egypt,
and interesting because of the fact that the highest number is
1000 and that is the only fraction used. 5

Oldest Sundial. From this period or a little later, but from
about 1500 B.C., there dates the oldest sundial extant, an
Egyptian piece now in the Berlin Museum, showing that the
Egyptians had already developed, as we might have inferred
from the other mathematical and astronomical knowledge
possessed by them, a good system of timekeeping by means of
a primitive sun clock. On this clock the shadow shortened as
the forenoon advanced, and lengthened from noon to night.

1 This is seen particularly in the several rules which are evidently followed
in the formation of unit fractions.

2 Hat-shepset, Hatasu, or Hatshepsu, also known as Ramaka (Ma-ka-ra).
See E. A. W. Budge, The Mummy y p. 30 (Cambridge, 1893).

3 The No Amon (City of Ammon) of the Bible, also known to the Greeks as
Diospolis.

4 J. H. Breasted, Ancient Records, Egypt (Chicago, 1906, 1907), II, pp. 104,
114, 210, 211 ; IV, pp. 362 seq.; hereafter referred to as Breasted, Anc. Records
See also official accounts of the same period in facsimile in Golenischeff, Le$
Papyrus hieratiques nos. 1115, m6A et m6B de I'Ermitage imperial a St
Petersbourg, 1913.

5 Breasted, Anc. Records, II, 283. In later inscriptions, as of c. 600 B.C., the
fractions J and $ appear (ibid.,IV, p. 486). Still later, from 19 A.D. to 250 A.D.,
the papyri tell us of the periodic census introduced apparently by Augustus,
with taxes and the records of imports and exports. See A. S. Hunt, "Papyri
and Papyrology," Journal of Egyptian Archaeology, I (1914).

50 EGYPT

There were six hours in the forenoon and six in the afternoon,
from which division of the day came the system of twelve hours
later adopted in Europe. Such clocks, in various forms, were
afterwards used by the Greeks and gave rise to the sundials of

OLDEST SUNDIAL EXTANT

Egyptian specimen, restored after Borchardt, now in the Berlin Museum. Dates

from c. 1500 B.C. In the morning the crosspiece was turned to the east, and in

the afternoon to the west. From Breasted's Ancient Times

later times. The clock above shown bears the name of Egypt's
greatest general, Thutmose III, who has justly been called her
Napoleon.

Practical Problems. By the time of Seti I (c. 1350 B.C.)
business calculation had come to require larger numbers than
those needed in the time of Ahmes. This is seen from the
problems in the Rollin papyrus manuscript now in the Louvre, 1
one of which, line for line as in the text, is as follows :

1601 39 2 >325

together bread 107,893 makes in ten 364,371

bread 6121 loaves 1800 thcs makes in ten 2i ; 6oo

together 385,871

rest 6354

quantity of maize sacks 1601 makes in bread 112,090

makes in ten 392,306

brought to the magazine bread 114,064 makes in ten 385,971

The meaning is that here are two accounts of 1601 sacks oi
wheat each, the produce varying in the two cases. The weights

1 M. F. Chabas, Aegyptische Zeitschrift, 1869, p. 85. The manuscript wa:
published by W. Pleyte in 1868, and a new translation by Eisenlohr appearec
in 1897 in the Proceedings of the Society of Biblical Archaeology, XIX, 91, 115
M7, 252.

PRACTICAL PROBLEMS 51

are calculated in thes or ten, 12 ten making i thes, a ten being
about 316 grams. In the first case a loaf of unbaked bread
weighed 3.63 ten, 1.15 kg. ? or 2\ lb., and after it was baked it
weighed 3.37 ten, 1.06 kg., or 2^-lb. But the first case also
gives the weight of 6121 loaves as 21,600 ten, which is at the
rate of 3.52 ten per loaf when baked, so that the sizes evidently
varied. In the second case the bread weighed 3.55 ten per loaf,
possibly unbaked, and 3.38 ten per loaf when delivered. The
first account may be represented as follows :

107,893 loaves weigh 364,371 ten

6121 loaves weigh 21,600 ten

together they weigh 385,971 ten

there is left 6,354 ten

the total being 392,325 ten

The problem in itself is of little moment except as it shows
the practical use of large numbers in these early times.

Rameses II divides the Land. At the close of SetiWelatively
short reign his son, Rameses II (c. 1347 B.C.), known to the
Greeks as Sesostris, came to the throne. In his reign a re-
division of land took place among the people, and surveying
must have attracted much attention.

Herodotus (c. 484-*;. 425 B.C.), referring to information that
he had received from the priests, relates the following :

Sesostris also, they declared, made a division of the soil of
Egypt among the inhabitants, assigning square plots of ground of
equal size to all, and obtaining his chief revenue from the rent which
the holders were required to pay him every year. If the river carried
away any portion of a man's lot, he appeared before the king, and
related what had happened; upon which the king sent persons to
examine, and determine by measurement the exact extent of the loss ;
and thenceforth only such a rent was demanded of him as was pro-
portionate to the reduced size of his land. From this practice, I think,
geometry first came to be known in Egypt, whence it passed into

52 EGYPT

Greece. The sundial, however, and the gnomon, with the division of
the day into twelve parts, were received by the Greeks from the
Babylonians. 1

Harris Papyrus. Rameses IV came to the throne c. 1167
B. c. and immediately prepared a remarkable document setting
forth the great works of his father, Rameses III (1198-1167
B.C.), including a list of his extensive gifts to the gods. The
list shows the proportion of the wealth of ancient Egypt held
by the temples and is of value in giving the numerals of the
period. This document, known as the Harris Papyrus, is still
extant 2 and affords the best example of practical accounts that
has come down to us from the ancient world.

That surveying played a prominent part in the life of Egypt
is seen in an inscription on the tomb of Penno at Ibrim, in
Nubia, in the reign of Rameses VI (c. 1150 B.C.), in which the
boundaries and areas of five districts are given. 3

Evidence of Egypto-Cretan Relations. Thus we see that be-
fore 1000 B.C. Egypt had developed enough knowledge of as-
tronomy to devise an excellent calendar, and that she was in
possession of a commercial system requiring extensive work in
computation, of an elaborate scheme of leveling and survey-
ing, of a considerable knowledge of what we would now con-
sider as a kind of algebra, and of some ability in mensuration,
especially as it related to granaries and to the use of grain
products in the making of bread.

Recent excavations have shown the existence of a high de-
gree of civilization in Crete in this period of progress in Egypt,
and there is also evidence of amicable relations between these
two countries in early times. Our knowledge of the subject
is too limited, however, to determine whether it has any bearing
upon the history of mathematics. The deciphering of the Cretan
inscriptions still awaits the further efforts of scholars.

1 Herodotus, II, 109.

2 Breasted, Anc. Records, IV, 127 seq. See also S. Birch, Zeitschrijt fur
Aegyptische Sprache, pp. 119 seq. (1872).

3 Breasted, Anc. Records, IV, 233. For the Egyptian measures in common
use at this period, and thus far identified as to equivalents, see ibid., p. 88.

DISCUSSION 53

TOPICS FOR DISCUSSION

1. The countries in which mathematics flourished prior to 1000
B.C., and the reasons for this mathematical activity.

2. Reasons for supposing mathematics to have made some prog-
ress in the Late Stone Age, or even earlier.

3. Influences leading to an extension in the use of .mathematics
in the third millennium B.C.

4. Probable nature of the earliest mathematics of China, and
the influences which developed the study of this science.

5. General nature of the early written mathematical works in
China, with approximate dates.

6. The first traces of number mysticism in the East.

7. General period in which the Nine Sections was written. Nature
of the work.

8. Probable nature of the early Hindu mathematics.

9. Influences that developed Babylonian mathematics and the
method of recording the science.

10. General nature of Babylonian mathematics.

11. Evidence of early mathematics in Egypt. General nature of
the work in the earliest periods.

12. Mathematics of the Feudal Age in Egypt.

13. The Ahmes Papyrus, its origin and general nature.

14. Evidences of development of commercial arithmetic between
the time of Ahmes and 1000 B.C.

15. Types of problems in arithmetic, algebra, and mensuration
that interested the ancient Egyptians.

1 6. Comparison of the mathematical progress and interests of
China, India, Babylonia, and Egypt in early times.

17. A consideration of the reasons why this period was lacking in
power to advance its mathematics.

1 8. A study of the evidence of mathematics in Crete and Cyprus
before 1000 B.C., and the influence of this mathematics upon
Greek science.

19. The evidences of interrelation of mathematical ideas in Mes-
opotamia, Egypt, and the islands of the Mediterranean Sea.

20. Mathematical and astronomical instruments of this period.

21. The degree of accuracy apparently secured by engineers before
the year 1000 B.C.

CHAPTER III

THE PERIOD FROM 1000 B.C. TO 300 B.C.
i. THE OCCIDENT IN GENERAL

Geographical Limits. For our present purposes we may de-
fine the Occident of the period from 1000 B.C. to 300 B.C. as
practically identical with Greece and her colonies. Whatever
mathematics Rome had in this period was essentially Greek,
and most of the Mediterranean world, aside from the hinter-
land of Phoenicia and Egypt, may therefore be conveniently
classified as under the influence of the Hellenic civilization.
Phoenicia contributed little that was not commercial, and the
golden age of Egypt was already past.

Protected Regions. Philosophy, letters, mathematics, art,
and all the finer products of the mind require peaceful sur-
roundings for their development. It is for this reason that
mathematics at this time flourished best on the protected
islands of the ^Egean Sea, on the Greek peninsula, and in the
Greek towns of Southern Italy. In all these places invasion
was difficult and the rewards of the invader were few. Com-
mercial and intellectual communication with the rest of the
world was possible, so that peace without stagnation was, rela-
tively speaking, assured.

It would also be proper to include some mention of the
mathematics of Mesopotamia, since this was quite as occi-
dental as oriental ; but aside from its use in astronomy the
science was not sufficiently in evidence in Babylon at this time
to demand our attention.

Chronological Limits. The reason for taking the lower arbi-
trary limit of 300 B.C. is that a new era in the history of
mathematics begins with the founding of the Alexandrian
School at about that time. This event led to a reshaping of

THE GREEKS 55

mathematics either through the efforts of scholars connected
with the first great cosmopolitan university or through the
works written by those who came under their influence.

2. THE GREEKS

Birth of Greek Arithmetic. Commercial arithmetic was well
advanced in various neighboring states long before it was
known in Greece. The merchants of the Phoenician coast
(along and across which passed the routes of trade with the
Orient), receiving inspiration from Babylon, early developed
a fairly good business arithmetic, and in clue time became the
teachers of this art in Egypt, Asia Minor, and the ,/Egean
isles. The recent excavations in the ancient palace of Knossos
show that in early times the commercial arithmetic of Babylon
reached even as far west as Crete, and future studies are
likely to reveal much valuable information relating to this
island. Indeed, in what is called the Early Minoan Period of
Crete, Greece was still a forest, thinly peopled by a nomad
race, for this was long before the warlike Dorians, about a
thousand years before our era, made themselves masters of
Peloponnesus and changed the whole tenor of Greek life.
Thucydides describes the country at this early period as a
theater of frequent migrations, when "each man cultivated
his land only according to his immediate needs, with no
thought of amassing wealth." Under such conditions, before
the coining of money was known, only the most primitive
arithmetic was demanded. A little counting and a little rude
barter were all for which the ancient Greek civilization had
created a need. Even at a much later period than the one
we have described, Greece was little inclined to commerce.
Her older cities were not generally seaports, and what little
navigation she had was concerned with war and piracy rather
than with the development of trade.

External Influences. It was only when the Greeks began to
come into closer contact with other peoples that they showed
any interest in arithmetic. Indeed, contrary to the idea that
is commonly expressed, Greece always depended largely upon

56 THE GREEKS

external influences for her mathematics, and few who advanced
this science in her schools were born within her continental
area. But when we speak of the early efforts of Greece to put
herself in contact with the external world through colonization,
it must be understood that we are still ignorant as to how far
peninsular Greece was then a colonizer and how far she herself
was a colony, since her people lived as much along the Asiatic
as along the European coast.

Miletus. We are told by Herodotus (c. 484-*;. 425 B.C.) and
Strabo (c. 66 B.C.-C. 24 A.D.), however, that Miletus, the great-
est commercial town of the twelve forming the Ionian con-
federacy, was an Athenian colony,
although there are good reasons for
doubting the statement. Situated at a
strategic point on the coast of Asia
ANCIENT COINS Minor, it in turn became a great col-
_ . r ,.*.,. onizing center, and in the yth centurv

Coins found in Asia Minor. uv i j i A.

They are among the earliest B - c - established no less than ninety
known, dating from about towns along the shores of the Black
550 B.C. From Breasted's Sea and the Mediterranean, even open-

Ancient Times . -o * * i. i L.I

ing Egypt to her commercial settle-
ments. This fact had a bearing upon the early science of the
Greeks, since it was at Miletus that their mathematics had its
beginning; and it was here, doubtless, that their commercial
arithmetic first developed to any great extent. It was in Lydia,
just east of here, that coins were first struck in the West, in the
yth century B.C., and Miletus at once recognized and adopted
the new invention, anticipating Athens, indeed, by over half a
century. The influence of this movement, particularly in re-
lation to arithmetic, is evident. Without the aid of coins all
business calculation must have been very cumbersome, money
consisting of bars or ingots of metal that had to be weighed,
and small currency being practically nonexistent save in the
form of shells or trinkets. We can therefore determine fairly
well the time and place of the beginning of any noteworthy
business arithmetic among the Greeks, namely, about the yth
century B.C. and along the coast of Asia Minor.

LOGISTIC 57

Logistic. At the period of which we have been speaking, the
Greek science of numbers, the arithmetic proper, had not yet
been invented. Only the art of calculating had made any
appeal to these practical people. This branch of the subject
went by the name of " logistic," and its beginnings must be
sought in prehistoric times. Greek tradition states that it
came from the Phoenicians, whose trading instincts are well
known, and many comparatively recent writers have felt that
this tradition had a foundation in fact. It must not be thought,
however, that the interesting properties of numbers were en-
tirely unrecognized before this time. Various curious rela-
tions had been the subject of discussion in the Orient for many
centuries, and some knowledge of number mysticism had doubt
less been acquired by the priestly caste in Greece long before
logistic existed as a special subject of study.

Although by the tradesman in Miletus, and later in Corinth
and other seaport towns, logistic must have been looked upon
as important, it is probable that the ordinary Greek could
neither multiply one number by another nor perform any other
operation in what we now call arithmetic. There were doubt-
less schools at that time, for Herodotus (c. 450 B.C.) and
Diodorus Siculus (ist century B.C.) both speak of them as then
known, but logistic was looked upon as a technicality of trade,
just as we may today look upon the use of a slide rule or a
typewriter. A little later, however, it came more into favor,
for Plato refers to it rather than to the theoretical part of the
science when he says:

Very unlike a divine man would be he who is unable to count,
"one, two, three," or to distinguish odd and even numbers. . . .
All freemen, I conceive, should learn as much of these branches of
knowledge as every child in Egypt is taught when he learns his
alphabet. In that country arithmetic games have actually been
invented for the use of children, which they learn as a pleasure
and amusement.

Furthermore, Plato recommends the use of apples and other
objects in the presenting of the idea of number, quite as a
modern teacher would employ them.

$8 THE GREEKS

In spite of the extensive use of logistic among the Greek
merchants, not a single treatise upon the subject remains. A
Greek multiplication table, written on wax at about the begin-
ning of our era, and hence somewhat later than the close of
the period under discussion, is still preserved in the British
Museum ; and this, together with a few examples in addition,
subtraction, and multiplication, and an abacus, are all that
have come down to us that bears directly upon the practical

GREEK MULTIPLICATION TABLE ON A WAX TABLET

One of the few examples of the Greek logistic. This specimen is now in the
British Museum and dates from about the beginning of the Christian era

computations of the Greeks. To these examples reference will
be made when we come to consider the abacus and the various
operations.

Arithmetic. Although the precise nature of the Greek logis-
tic, the art of calculating, is very little known, fortunately
the same cannot be said of the Greek arithmetic, the theory of
numbers. As a subject for philosophers and by them com-
mitted to writing, it has come down to us as it was left by the
later Greeks, and probably with its details little changed from
the original form given to them in the earlier days. This topic,
relating to the remote ancestor of our present number theory,
will be considered, together with logistic, in Volume II.

GREEK GEOMETRY 59

Greek Geometry. Although both logistic and arithmetic de-
veloped in the Orient as well as in the Occident, geometry as a
logical science is purely a product of the western civilization.
On the other hand, intuitive geometry is universal, differing
as a matter of course in degree of accomplishment in the
various parts of the world. Egypt possibly knew the law of
the square on the hypotenuse of a right-angled triangle long
before Pythagoras, and there is reason to believe that China
and India were also familiar with it; but the first proof of
the theorem, and apparently the first idea of a geometric proof,
are both due entirely to the Greeks. Indeed, we may say that
all of our geometry, considered as a logical sequence of propo-
sitions, whether relating to two-dimensional or to three-
dimensional space and whether limited to circles and straight
lines or extended to include conic sections and higher plane
curves, had its origin solely in the Greek civilization. So com-
pletely was the Greek mathematics given over to geometry that
both arithmetic and the science that was much later known
as algebra were treated almost entirely from the geometric
standpoint. We shall therefore see that, although mathematics
among the Greeks included geometry, arithmetic, logistic,
music, and a kind of algebra, the central element was geometry.

Centers of Mathematical Activity. Mention has been made
of Miletus, and before proceeding farther it is desirable to lo-
cate the other centers of mathematical activity in Greece and
her colonies. The following cities and countries will be men-
tioned frequently, the numbers referring to the map on page 60 :

Abdera, 16. Clazomenae, 24. Jerusalem, 40. Rhodes, 21.

Alexandria, 41. Cnidus, 22. Laodicca, 32. Rome, 3.

Amisus, 31. Constantinople, 28. Larissa, 36. Samos, 18.

Antinoopolis, 43. Crete, 20. Medma, 7. Sicily, 4.

Apameia, 35. Crotona, 9. Mendes, 42. Smyrna, 25.

Aquitania, 2. Cyprus, 34. Miletus, 23. Stageira, 14.

Athens, 13. Cyrene, u. Naples and Pompeii, 6. Syene, 45.

Byzantium, 28. Cyzicus, 27. Nicaea and Bithynia, 30. Syracuse and

Cadiz, i. Elea, 8. Paros, 19. Messina, 5.

Chalccdon, 29. Elis, 12. Perga, 33. Tarentum, 10.

Chalcis, 37. Gades, i. Pergamum, 26. Thasos, 15.

Chios, 17. Gerasa, 39. Ptolemais, 44. Tyre, 38.

MATHEMATICAL-HISTORICAL MAP OF THE MEDITERRANEAN
COUNTRIES IN CLASSICAL TIMES

This map shows the location of places most frequently mentioned in relation
to Greek and Roman mathematics, with the names (page6i) of scholars con-
nected with them. The numbers are arranged on the map from left to right. The
dates following the names are merely approximate, and those before the Christian
era are indicated by the letters B.C., as in the text. France and Spain are not
shown, because they are mentioned only with respect to Cadiz (Gades) and
Aquitania (in southern France), and to include them would reduce in size the
more essential parts of the map. It will be observed that, as stated in the text,
mathematics flourished best in territory that was protected by sea, by desert, or
by mountainous regions

MATHEMATICAL CENTERS 6l

CITIES AND COUNTRIES BY MAP NUMBERS

1. Cadiz (Gades) : Columella, 25. In southwestern Spain, not shown.

2. Aquitania: Victorius (Victorinus) , 450. In southern France, not shown.

3. Rome: Varro, 60 B.C.; P. Nigidius Figulus, 60 B.C.; Vitruvius, 20 B.C.;

Frontinus, 100; Mcnelaus, 100; Hyginus, 120; Balbus, 100; Domitius
Ulpianus, ,200; Nipsus, 180; Epaphroditus, 200; Sextus Julius Africanus,
220; Censorinus, 235; Serenus, 200; Porphyrius, 275.

4. Sicily: Diodorus Siculus, ist century B.C.

5. Syracuse and Messina: Archimedes, 225 B.C.; Julius Firmicus Maternus,

340; Dicaarchus of Messina, 320 B.C.

6. Naples and Pompeii: Pliny, 75.

7. Medma: Philippus Medmaeus, 350 B.C.

8. Elea: Zeno, 450 B.C.; Parmenides, 460 B.C.

9. Crotona: Pythagoras, 540 B.C.; Philolaus, 425 B.C.

10. Tarentum: Pythagoras, 540 B.C.; Philolaus (?), 425 B.C.; Archytas, 400 B.C.

11. Cyrene: Theodorus, 425 B.C.; Nicoteles, 250 B.C.; Eratosthenes, 230 B.C.;

Synesius, 410.

12. Elis: Hippias, 425 B.C.

13. Athens: Solon, Ooo B.C.; Agatharchus, 470 B.C.; Socrates, 425 B.C.; Meton,

432 B.C.; Phaeinus, 432 B.C.; Euctemon, 432 B.C.; Thesetetus, 375 B.C.;
Plato, 380 B.C.; Spcusippus, 340 B.C.; Ptolemy, 150.

14. Stageira: Aristotle, 340 B.C.

15. Thasos: Leodamas, 380 B.C.

16. Abdera: Democritus, 400 B.C.

17. Chios: (Enopides, 465 B.C.; Hippocrates, 460 B.C.

18. Samos: Pythagoras, 540 B.C.; Conon, 260 B.C.; Aristarchus, 260 B.C.
IQ. Paros: Thymaridas, 380 B.C.

20. Crete: Early civilization, particularly at Knossos.

21. Rhodes: Eudemus, 335 B.C.; Geminus, 77 B.C.; Poseidonius, 100 B.C.

22. Cnidus: Eudoxus, 370 B.C.

23. Miletus: Thales, 600 B.C.; Anaximander, 575 B.C.; Anaximenes, 530 B.C.

24. Clazomence : Anaxagoras, 440 B.C.

25. Smyrna: Theon, 125.

26. Pergamum: Great library; parchment.

27. Cyzicus: Callippus, 325 B.C.

28. Byzantium (Constantinople): Proclus, 460; Psellus, 1075.

29. Chalcedon: Xenocrates, 350 B.C.; Proclus, 460.

30. Niccea and Bithynia: Hipparchus, 140 B.C.; Theodosius, 50 B.C.; Sporus, 275.

31. Amisus: Dionysodorus, 50 B.C. 36. Larhsa: Domninus, 450.

32. Laodicea: Anatolius, 280. 37. Chakis: lamblichus, 325.

33. Perga: Apollonius, 225 B.C. 38. Tyre: Marinus, 150; Porphyrius, 275.

34. Cyprus: Early civilization. 39. Gerasa: Nicomachus, 100.

35. Apameia: Poseidonius, 100 B.C. 40. Jerusalem: Religious.

41. Alexandria: Euclid, 300 B.C ; Eratosthenes, 230 B.C. ; Apollonius, 225 B.C.;

Aristarchus, 260 B.C.; Hypsicles, 180 B.C.; Heron, 50; Menelaus, 100;
Ptolemy, 150; Diophantus, 275; Pappus, 300; Theon, 300; Hvpatia, 400.

42. Mendes: Commercial. 44. Ptolemais: Synesius, 410.

43. Antinoopolis (Antinoe): Serenus, 50. 45. Syene: Eratosthenes, 230 B.C.

THE GREEKS

Early Greek Appreciation of Geometric Forms. Greece went
through the same stages of appreciation of geometric forms that
Egypt and Crete went through. This is seen in the use of
crude parallels, then of the less crude and more elaborate

forms, and finally
of the more delicate
forms found in the
period just preced-
ing the development
of the highest type
of Greek art. The
use of these geomet-
ric forms was es-
pecially noteworthy
just before the time
of Thales, what is
known as the geo-
metric style in the
decoration of vases
having reached its
climax in the 8th
century B.C.

Greek Algebra.
Algebra as a science
distinct from arith-
metic and geometry
was invented long
after Greece had
ceased to be a cen-
ter of civilization.
Certain identities
that we now express in algebraic form were well known to the
Greeks, however, and were demonstrated with even greater
rigor than is the case in our textbooks today, where the work
is practically limited to rational numbers. For example, the
Greeks proved that

GREEK GEOMETRIC FORMS JUST PRECEDING
THE TIME OF THALES

From a specimen of the 8th century B.C. in the
Metropolitan Museum, New York

GREEK ALGEBRA 63

although they had no algebraic shorthand by which to express
the fact, and although they considered only lines and rectangles
instead of numbers and products. In like
manner they knew such other identities as

(a + b) (a- b) = or- 6 2 ,
a ( x -I- y 4- z) = ax + ay -I- az,

and

a I

1/6

although they considered these also as geometric relations.
They could complete the square of the binomial expression

a 2

but, again, this was looked upon as the filling out of a geometric
figure that was made up of a square increased or decreased
by twice a rectangle. Greek algebra, as a form of arithmetic
distinct from geometry, was developed some time after the
period which we are now studying. When we come to consider
the life and works of Diophantus (c. 275), for example, we
shall see that the later Greeks made a remarkable advance in
the analytic treatment of this subject. They developed a fairly
good symbolism, and they considered algebra, under the name
" arithmetic/' as entirely distinct from the geometry of which
we have been speaking.

3. ORIGINS OF GREEK MATHEMATICS

The Makers of Greek Mathematics. There are three impor-
tant periods in the development of Greek mathematics, two
of them within the chronological limits now being considered
and one immediately following the later of these limits. The
periods may be characterized as, first, the one subject to the
influence of Pythagoras ; second, the one dominated by Plato
and his school ; third, the one in which the Alexandrian School
flourished in Grecian Egypt and extended its influence to
Sicily, the ^Egean Islands, and Palestine. We shall now con-
sider the names of some of those who made the mathematics

ORIGINS OF GREEK MATHEMATICS

of the first two of these important periods, and some of the in-
fluences which led them to undertake their epoch-making work. 1

Thales. The first of the Greeks to take any scientific in-
terest in mathematics in general, and in the union of astronomy,

geometry, and the theory
of numbers in particular^
was Tha'les. 2 Before his
time there had been the
usual interest of early
peoples in the mystery of
the heavens, as witness
the statement of the poet
Archirochus 3 that a solar
eclipse was observed some
time before Thales was
born. Not until the time
of Thales, however, did
the science of mathe-
matics begin in the Greek
civilization.

Miletus was then, as we
have seen, a trading and
colonizing center, a city
of wealth and influence.
Herodotus (c. 450 B.C.)
tells us that Thales was
of Phoenician descent;
but his mother, Cleobu-
line, bore a Greek name,
while the name of his
father, Examius, is Carian. The name of Thales himself was

THALES

Ancient bust in the Capitoline Museum at
Rome, not contemporary with Thales

1 The most recent and elaborate works on this period are A. Mieli, La
Scienza Greca, of which Volume I (Florence, 1016) deals with the history of
Greek science before Aristotle; and Sir T. L. Heath, A History oj Greek Mathe-
matics, 2 vols. (Cambridge, 1921), of which the first volume covers the period
from Thales to Euclid.

2 aXi}*. Born at Miletus, c. 640 B.C.; died c. 546.

s. Born c. 714 B.C.; died c. 676.

THALES

probably a common one. Indeed, we have in the Metropolitan
Museum in New York at present a Cypriote vase of his period
which bears the name in the form

evidently that of the owner. 1 Con-
sidering the general recognition of
the abilities of Thales, even during
his lifetime, it is not impossible that
a vase made in Cyprus may have
been intended for him, but there is
no further evidence that this was the
case.

Stories concerning Thales. Thales
was a merchant in his younger days,
a statesman in his middle life, and
a mathematician, astronomer, and
philosopher in his later years. In his
mercantile ventures he seems to have
been unusually successful, even in
dealing with the shrewdest of the
Greek trading races. Aristotle (c.
340 B.C.) tells us how he secured
control of all the oil presses in Miletus

and Chios in a year when olives promised to be plentiful,
subletting them at his own rental when the season came.
Plutarch (ist century) also testifies to his ingenuity in the
following anecdote: 2

Solon went, they say, to Thales at Miletus, and wondered that
Thales took no care to get him a wife and children. To this Thales
made no answer for the present ; but a few days after procured a
stranger to pretend that he had left Athens ten days earlier and

1 Reading from right to left, the characters have the values ta + le + se,
corresponding to the Greek Oa + X^ + s, Thales. The vase is part of the
Cesnola collection found in Cyprus.

2 In his life of Solon.

CYPRIOTE VASE WITH THE
NAME OF THALES

The name TA + LE + SE,
when viewed from above, has
the appearance shown in the
text. The vase is about con-
temporary with Thales of
Miletus. From the Metropoli-
tan Museum of Art, New York

66 ORIGINS OF GREEK MATHEMATICS

Solon inquiring what news there was, the man replied according as he
was told: "None but a young man's funeral, which the whole city
attended, for he was the son, they said, of an honorable man, the
most virtuous of the citizens, who was not then at home, but had
been traveling a long time." Solon replied, "What a miserable man
is he! But what was his name?" "I have heard it," said the
man, "but have now forgotten it, only there was great talk of his
wisdom and justice." [After Solon had been drawn on to pronounce
his own name and had learned that it was his own son,] Thales took
his hand and, with a smile, said, "These things, Solon, keep me
from marriage and rearing children, which are too great for even
your constancy to support ; however, be not concerned at the report,
for it is a fiction."

Solon (c. 639-559 B.C.), it should be observed, was inter-
ested in astronomy and was the one who introduced a "leap
month" into the Athenian calendar (594 B.C.).

Trade was then an honorable calling, and Thales seems to
have traveled in Egypt on his commercial ventures, and early
writers tell of his also visiting both Crete and Asia. He was
not the only mathematician to have thus turned trade to
profit, for Plutarch has this to say of him : "Some report that
Thales and Hippocrates the mathematician traded, and that
Plato defrayed the charges of his travels by selling oil in
Egypt." In this way Thales may have accumulated the wealth
that permitted him to indulge his taste for learning and to
found the Ionian School. It was through this indulgence that
he acquired such a reputation as to be enrolled as the first
among the Seven Wise Men of Greece, and that he was es-
teemed as the father of Greek astronomy, geometry, and
arithmetic.

Arithmetic of Thales. Of the nature of the arithmetic that
Thales brought back from Egypt we have little direct knowl-
edge, lamblichus of Chalcis (.325 A.D.) tells us that he de-
fined number as a system of units, and adds that this definition
and that of unity came from Egypt. This is not much, but it
is enough to show that Thales was interested in something
besides the merely practical. It is probable that he knew many

THALES 67

other number relations, for the Ahmes papyrus contains some
work in progressions, and such knowledge would hardly escape
so careful an observer as Thales. It is, however, in his work
in founding deductive geometry and in his capacity as a teacher
of Pythagoras rather than as a discoverer of facts that Thales
commands our attention.

Interest in Astronomy. He took much interest in astronomy,
and Herodotus (I, 74) tells us that he even succeeded in pre-
dicting an eclipse. Some authorities suppose this eclipse to
have occurred on May 28, 585 B.C., while others place it about
twenty-five years earlier. He could have obtained certain in-
formation on this subject from a study of the Chaldean records,
but whether this was his source of information we cannot say.
At the present time we have numerous cuneiform tablets of the
7th century B.C. which record such prognostications. One of
these reads: "To the king, my master, I have written that
there was about to be an eclipse. The eclipse has now taken
place. This is a sign of peace for the king, my master."

A man like Thales, possessed of an inquisitive mind, coming
in contact with scholars from other lands, either on his travels
or in the commercial center of Miletus, would lose no oppor-
tunity to secure information of this kind and to make use of it
in his teaching. Doubtless his scientific training led him to dis-
card the astrological notions of the Chaldeans but to retain
whatever of astronomy came to his attention.

Geometry of Thales. In geometry he is credited with a few
of the simplest propositions relating to plane figures. The list,
according to the most reliable ancient writers, is as follows:

1. Any circle is bisected by its diameter.

2. The angles at the base of an isosceles triangle are equal

3. When two lines intersect, the vertical angles are equal. 1

4. An angle in a semicircle is a right angled

Proclus, ed. Friedlein, pp. 157, 250, 299 (Leipzig, 1873).
9 There is a doubt about his knowing this. It is inferred from a statement
by Pam'phila (lla^/Xi/, a woman historian, ist century A.D.), but there is
no early authority for the statement.

68 ORIGINS OF GREEK MATHEMATICS

5. The sides of similar triangles are proportional. 1

6. Two triangles are congruent if they have two angles and
a side respectively equal (Euclid, I, 26)."

Importance of his Geometry. As propositions in geometry
these may seem trivial, since they are intuitive statements;
but their very simplicity leads us to believe that it was the
fact that Thales was the first to prove them that led Eudemus
(c. 335 B.C.) and other early writers to mention them. Up to
this time geometry had been confined almost exclusively to
the measurement of surfaces and solids, and the great contri-
bution of Thales lay in suggesting a geometry of lines and in
making the subject abstract. With him we first meet with the
idea of a logical proof as applied to geometry, and it is for this
reason that he is looked upon, and properly so, as one of the
great founders of mathematical science. In the history of
mathematics, as in the history of civilization in general, it is
the setting forth of a great idea that counts. Without Thales
there would not have been a Pythagoras or such a Pythag-
oras; and without Pythagoras there would not have been a
Plato or such a Plato.

Philosophy of Thales. In philosophy he is said to have as-
serted that water is the origin of all things, that everything is
filled with gods, that the soul is that which originates motion,
and that matter is infinitely divisible ; but his basis for belief in
these assertions is not very satisfactory. Like most of his con-
temporaries, he left no written works.

Anaximander. At the death of Thales the leadership of the
Ionian School passed over to Anaximan'der, 3 who is generally

iHe used this proposition in measuring the height of a pyramid by means
of the shadow of the pyramid and that of a staff. See Diogenes Laertius,
Vitas Philosophorum, ed. Cobet, p. 6 (Paris, 1878) ; Pliny, Hist. Nat., XXXVI,
17; Plutarch, Septem Sapientinm Convivium, ed. Didot, III, 174 (Paris, 1841).
Pliny's statement that he measured the shadow at the time of day when
the shadow "is equal in length to the body projecting it" is not very con-
vincing. This would be too simple and is quite contrary to Plutarch's version.

2 Eudemus (c. 335 B.C.), a disciple of Aristotle, refers this to Thales.

3 'Avat(ji,avdpos. Born c. 611; died c. 547 B.C. J. Neuhaeuser, Anaximander
Milesivs (Bonn, 1883).

THALES AND PYTHAGORAS 69

thought to have been his pupil. Anaximander or some con-
temporary of his first brought into use in Greece the gnomon,
an instrument resembling the sundial 1 and used for determin-
ing noon, the solstices, and the equinoxes. Aside from this,
Anaximander seems to have had no interest in mathematics.
It was about this time that the water clock (clep'sydra), 2 al-
ready known to the Assyrians, found its way into Greece, and
very likely Anaximander's gnomon came also to be used for
determining the time of day.

4. FROM PYTHAGORAS TO PLATO

Pythagoras. Of all the interesting figures in the history of
ancient mathematics Pythag'oras 3 ranks easily first, partly
from the mystery surrounding his life, partly from his own
mysticism, partly from the brotherhood which he established,
'and partly from the unquestioned ability of the man himself.

Early Life of Pythagoras. As with Euclid and Heron, of
whom we shall presently speak, so with Pythagoras, the date
and the place of his birth are both unknown. He seems to
have been born between the soth and $2d Olympiads, to use
the Greek system of chronology, or between 580 and 568 B.C.
of our calendar. Although called a Samian, we are not certain
that he was born on the island of Samos, for Suidas, a late
medieval writer (c. 1000), says that he was born in Italy, and

] Really, the pointer which casts the shadow on the dial.

2 KXc^i55pa, from ic\&rmi> (to hide) + vSwp (water).

. :i Born at Samos (?), c. 572 B.C.; died at Tarentum (?), c. 501. W. Schultz,
Pythagoras und Heraklit, Leipzig, 1905; A. Ed. Ohaignet, Pythagore et la
philosophic pythagoricienne, contenant les fragments de Philolaiis et d'Archytas,
Paris, 1873; W. Bauer, Der dltere Pythagoreismus, Bern, 1897; W. W. Rouse
Ball, "Pythagoras," in the Math. Gazette, London, January, 1915; G. J. Allman,
Greek Geometry from T hales to Euclid, Dublin, 1889 (hereafter referred to
as Allman, Greek Geom.) ; F. Cramer, Dissertatio de Pythagora, Prog., Sund,
l8 33 5 W. Lietzmann, Der pythagoreische Lehrsatz^ Leipzig, 1912 ; Armand
Delatte, "Etudes sur la litterature pythagoricienne," in the Bibliotheque de
I'Ecole des hautes Etudes, Vol. 217 (Paris, 1015). Of the early histories of
Pythagoras the one best known is that of lamblichus, c. 325. It first appeared
in print at Franeker, 1598. Better editions by Ludolph Kuster (Amsterdam,
1707), and A. Nauck (Petrograd, 1884).

70 FROM PYTHAGORAS TO PLATO

that as a child he migrated to Samos with his father/ Never-
theless the weight of authority favors his Samian birth, and a
number of coins of the island, struck
some centuries after his time, bear his
name and figure, and this would hardly
have been the case had he merely spent
his boyhood there. 2 Various stories are
told of his parentage, but we are quite
uncertain whether his father was an
FIGURE OF PYTHAGORAS engraver of seals or a merchant. At any

A coin of Samos of the rate he lived after Greece had enjoyed
reign of Trajan (98-117), two centuries of commercial activity,

and therefore much later and at the dawn of that golden age
than Pythagoras. It shows , . , , A , , , , , ., , ,

the honor in which he was whlch be S an m Athens in the 6th cen-
heid and the claim of Samos tury B.C. and closed for that city at the
as his native country end of the ^ cent ury B.C.

Period of Pythagoras. But in whatever land he was born,
and in whatever year, and of whatever parentage, Pythagoras
lived in stirring times and was himself one of the great makers
of the civilization of his period. Samos was just becoming a
center of Greek art and culture, Polyc'rates was just ascending
the throne, and Anac'reon was beginning to write his famous
lyrics in the Samian court. Pythagoras was therefore brought
up amid scenes that could hardly fail to stimulate a youth
of his native powers and urge him to a high intellectual life.
Moreover, the spirit of the times was active in great works.
Buddha was just promulgating his doctrine in India, and Con-
fucius and Lao-Tze were laying the foundation for their philo-
sophic cults in China; and, whether or not Pythagoras came
into personal touch with the Far East, he lived when the world
was ripe for great movements.

1 See also M. Barbieri, Notizie istoriche del Mattematici e Filosofi di
Regno di Napoli, cap. ii (Naples, 1778), who (p. 25) thought that Samos was
the modern Crepacore, a town in Southern Italy.

2 One of these coins is shown in the illustration. There are also a few gems,
of doubtful age, which are said to represent Pythagoras. See C. W. King.
Antique Gems and Rings, I, 212, and II, plate XXXVIII, No. i (London, 1872).

PYTHAGORAS 71

The fact that arithmetic and geometry took such a notable
step forward at this time was due in no small measure to the
introduction of Egyptian papyrus into Greece. This event oc-
curred about 650 B.C., 1 and the invention of printing in the istb
century did not more surely effect a revolution in thought than
did this introduction of writing material on the northern shores
of the Mediterranean Sea just before the time of Thales.

Studies and Travels of Pythagoras. Our knowledge of the
life of Pythagoras is very limited, the early writers having viec!
with each other in the invention of fables relating to his travels
his miraculous powers, and his teachings. He seems to have
sought out Thales and to have been his pupil. Tradition say?
that he was initiated by the master into the secrets of Zeus
on Mount Ida, and was then told that if he would have
further light he must seek it in Egypt. We now lose all definite
knowledge of Pythagoras for a considerable period. Appuleius. '
a Roman writer of about 150 A.D., asserts that he was cap-
tured by Cambyses the Persian, :i that he learned from the
Magi, and that he even sat at the feet of Zoroaster himself ;
but part of the story cannot be true, because Zoroaster probably
died about the time that Pythagoras was born, and possibly
much earlier than that, for the date is very uncertain. Isoc'-
rates, a writer of a century after Pythagoras, and Callim'achus.
librarian of the Alexandrian library, who lived in the 3d century
B.C., both assert that he spent some years in Egypt. Pliny,
writing in the ist century of our era, says that Pythagoras was
there in the time of Psammetichus, 4 and Strabo, about the be-
ginning of the Christian era, states that he studied in Babylon.
Others claim that he went as far east as India, but we have no
definite proof of any of these statements.

*In the reign of King Psammet'ichus (Psammitichus) I, soon after 660 B.C.,
sometimes given as c. 640-610 and sometimes as 671-617. See T. Gomperz, Les
Pcnseurs de la Grece, French ed., p. 13 (Lausanne, 1904). This is the Psemtek
of the monuments and the first king of the 26th Dynasty.

2 Usually so spelled in ancient texts, but occasionally Apuleius.

3 Reigned 520-522 B.C.

4 That is, Psammetichus III, who reigned 526-525 B.C., when Pythagoras was
about 46 years old.

72 FROM PYTHAGORAS TO PLATO

Contact with the East. In spite of the assertion of various
writers to the contrary, the evidence derived from the philoso-
phy of Pythagoras points to his contact with the Orient. The
mystery of the East appears in all his teachings. 1 His mysti-
cism of numbers is quite like that found earlier in Babylon,
and indeed his whole philosophy savors much more of the
Indian than of the Greek civilization in which he was born.
According to our best evidence the familiar proposition of
geometry that bears his name was known, as already stated, in
India, China, and Egypt ( ?) before his time, and all that can be
claimed for him in relation to it is that he may have given the
earliest general demonstration of its truth.

School of Crotona. When Pythagoras reappeared after his
years of wandering, he sought out a favorable place for a
school, and finally settled upon Crotona, a town on the south-
eastern coast of Italy, in a territory called by the Italic Greeks
of that time Great Greece. 2 This town was a wealthy seaport,
and it was to the young men of well-to-do families that Pythag-
oras made his appeal. Pretending to have the power of divi-
nation, given at all times to mysticism, and possessed in a
remarkable degree of personal magnetism, he gathered about
him some three hundred of the noble and wealthy young men
of Magna Graecia and established a brotherhood that has ever
since served as a model for all the secret societies in Europe
and America. He divided his disciples into two groups, the
hearers and the mathematicians, the latter having passed
through a probationary period as members of the former group.

Oral Teaching of Pythagoras. Pythagoras never embodied
his doctrines in any treatise. Like Thales, and also like those
Oriental teachers from whom he probably learned, he trans-
mitted his theories by word of mouth. This he did through the
elect of his brotherhood, thus making known his doctrines
freely to all who were deemed worthy to receive them. This

1 E. W. Hopkins, The Religions of India, p. 559 (Boston, 1902) ; L. von
Schroeder, Indiens Literatur und Cultur, pp. 718 seq.; Reden und Aufsatze,
p. 1 68 (Leipzig, 1913) ; Pythagoras und die Inder (Leipzig, 1884).

2 'H xxe'ydXi} 'EXXds, Magna Graecia.

SCHOOL OF CROTONA 73

method of imparting knowledge was not due merely to a spirit
of mysticism, but was quite as dependent upon a lack of good
writing material. Parchment had not as yet been invented,
the wax tablet was serviceable only for brief epistles, the clay
cylinders of Babylon were subject to similar limitations, and
the fragile papyrus of Egypt was probably somewhat rare in
Magna Grsecia. Pythagoras therefore followed the custom of
his time in passing his philosophy along by word of mouth,
just as the ancients had transmitted to his generation the songs
of Homer. Even in Plato's time there was no bookshop in all
Athens where worthy manuscripts could be purchased, nor was
there any when Euclid taught in Alexandria. Not until the
time of Augustus was the book trade established, making pos-
sible the easy and certain transmission of knowledge, and not
until fifteen hundred years later was printing known in Europe.
For the doctrines of Pythagoras we are indebted chiefly to
Eudemus of Rhodes (c. 335 B.C.), whose works, though lost,
are known to us through extracts preserved by later writers.
We also know of the doctrines of Pythagoras through passages
from a work by Philolaus of Crotona (who lived in the 5th
century B.C.), from a statement by Archytas of Tarentum
(c. 400 B.C.), a friend of Plato, and from a number of passages
in the works of later writers.

Philosophy of Pythagoras. Pythagoras based his philosophy
upon the postulate that number is the cause of the various
qualities of matter. This led him to exalt arithmetic, as dis-
tinguished from logistic, out of all proportion to its real impor-
tance. It also led him to dwell upon the mystic properties of
numbers and to consider arithmetic as one of the four de-
grees of wisdom, arithmetic, music, geometry, and spherics
(astronomy), these forming the quadrivium of the Middle Ages.
Aristotle (384-322 B.C.) tells us that Pythagoras related the
virtues to numbers, and Plutarch says that he believed that
earth was produced from the regular hexahedron, fire from the
pyramid, air from the octahedron, water from the icosahedron,
and the heavenly sphere from the dodecahedron, in all of which
the physical elements are related both to number and to form.

74 FROM PYTHAGORAS TO PLATO

Philolaus probably voiced the teaching of the master when he
asserted that five is the cause of color, six of cold, seven of
health, and eight of love.

The Chinese say that five represents wind, and two represents
earth, and these ideas are also claimed for the Pythagorean
system. 1 Here again the resemblance between the mysticism
of this school and that commonly found in the Far East leads
to the belief that Pythagoras must have come into relations with
the wise men of the Orient. Savoring of the East, too, is the
description given by Suidas, a late medieval Greek compiler,
of a ceremony called Pythagus, in which there is written some-
thing in blood on the face of a mirror, at the time of the full
moon, the words then being read in the reflection in the circle
of the moon; but there is no ancient authority for such a
statement. 2

Shakespeare refers to the acceptance by Pythagoras of the
Hindu belief in the transmigration of souls, in these words :

Thou almost mak'st me waver in my faith,
To hold opinion with Pythagoras,
That souls of animals infuse themselves
Into the trunks of men.

Merchant of Venice

Unity and Infinity. From various early writers we judge that
Pythagoras asserted that unity is the essence of number, the
origin of all things, the divine; that he had the idea of the
limited and the unlimited ; and that he held that from the latter
came the ideas of time, space, and motion. Diogenes Laertius
(2d century A.D.) says that he was interested in number, and
that the part of mathematics "to which Pythagoras applied
himself above all others was arithmetic"; * and Aristox'enus 4
says that he esteemed this science above all others.

1 J. Hager, An Explanation of the Elementary Characters of the Chinese,
p. xv. London, 1801.

2 J. C. Bulengerus, De Lvdis privates ac domestids veterum, p. 31. Lyons, 1627.

3 Diogenes Laertius, VIII, i, n, p. 207 ed. Cobet. Various references in
connection with Greek mathematics may be found in Allman, Greek Geom.

4'A/u<rr6ews, a philosopher. Born at Tarentum, c. 350 B.C.

ORIENTAL IDEAS 75

Geometry of Pythagoras. In the field of geometry Eudemus
(r. 335 B.C.) informs us that Pythagoras " investigated his
theorems from the immaterial and intellectual point of view,"
and that " he discovered the theory of irrational quantities and
the construction of the mundane figures." 1 Favori'nus, a phi-
losopher living in southern France c. 12 5, asserts that he em-
ployed definitions in his work in mathematics, this being the
first trace that we have of such use. 2 Tn particular, he defined
a point as "unity having position." 3 He or his school knew
that the plane space about a point may be filled by six equilat-
eral triangles, four squares, or three regular hexagons, a fact
which had doubtless been inferred as a result of the observa-
tion of mosaic pavements long before this time, but which no
doubt he was able to prove. It is probable that Pythagoras
proved the proposition relating to the sum of the angles of a
triangle, that he constructed a polygon equivalent to one given
polygon and similar to another, and that he could construct the
five regular polyhedrons ; and he may possibly have proved the
theorem relating to the square on the hypotenuse. It seems
likely that he taught that the earth is a sphere in space ; at
any rate, this theory was accepted by various later philosophers. 4

Pythagoras on Music. Pythagoras is said to have discovered
that the fifth and the octave of a note can be produced on the
same string by stopping at f and ,] of its length, respectively,
and it is thought that this harmony gave rise to the name of
"harmonic proportion," since

T . 1 T _ 2 . 2 _ 1
I -3- * 3-3 2

Although he seems to have derived some knowledge of music
from Egypt, 5 he is generally called the inventor of musical
science, or the harmonic canon (a mere tradition), but we

i/. p., of the five regular polyhedrons. Proclus (c. 412-485), ed. Friedlein, p. 65.

2Diogenes Laertius, VIII, i, 25, p. 215 ed. Cobet.

3 Proclus, ed. Friedlein, p. 95.

4 On doubts as to the Pythagorean Theorem see G. Junge, Bibl. Math., VIII
(3), 62, and H. Vogt, ibid., IX (3), 19- On the astronomical question see
P. Duhem, Le Systeme du Monde (Paris, 1913-1917).

r J. G. Wilkinson, Manners and Customs of the Ancient Egyptians, I, vi.
London, 1878.

7 6

FROM PYTHAGORAS TO PLATO

know nothing of the notes or of the system that he used. 1 With
his love for music and number it is natural to believe that he
must have taken great pride to himself for connecting the two
in the harmonic proportion. He seems to have held that the

P1TA.GORAS &a PYTACO.

PYTHAGORAS THE MUSICIAN

From F. Gafurius, Theorica Musice, Milan, 1402. One of the first crude at-
tempts to portray Pythagoras by means of a woodcut, and the first to portray
him as a musician. In the same work he is also shown as a bell ringer

intervals between the heavenly bodies were determined by the
laws of musical harmony, and hence arose the doctrine of
the harmony of the spheres.

The influence of Pythagoras became so great that the gov-
ernment caused his brotherhood to be dispersed, although the
members still spread the doctrines of the sect throughout
Greece. Pythagoras died an exile from Crotona, possibly at
Tarentum. Two centuries later, however, and during the
disasters of the First Samnite War, 343 B.C., the Senate
erected his statue in Rome, in response to an order of the Del-
phic Oracle to pay this honor to " the wisest and bravest of the
Greeks,' ' and the people learned to call him the preceptor of

3 A. Baumgartner, Geschickte der musikalischen Notation, p. u. Munich, 1856.

GEOMETRY AND MUSIC

King Numa, while even the great ^Emilian family was, in later
years, proud to claim him as one of their honored ancestors.
We shall now consider a few of the other Greek mathe-
maticians who attained prominence before the time of Plato.
All of them were influenced by the doctrines of Pythagoras, and
it is convenient and
proper to consider
them in close con-
nection with the
Pythagorean School,
including in the dis-
cussion the members
of the Eleatic School
mentioned below.

Lesser Waiters.
Among the contem-
poraries of Pythago-
ras was Anaxim'enes
of Miletus, 1 prob-
ably a pupil of An-
aximander. Dioge-
nes Laertius quotes
two letters from him
to Pythagoras, in
one of which he

MAP OF THE WORLD BY HECAT^EUS, 517 B. C.

Showing the primitive ideas held at the time of
Pythagoras. From Breasted's Ancient Times

speaks of Thales as his teacher ; but his tastes were in the direc-
tion of philosophy rather than of mathematics. In this period
there also flourished the geographer Hecatae'us, whose map of
the world serves to show how fragmentary was the knowledge
then possessed even by the best scientists.

About the time of Pythagoras there also flourished Ameris'-
tus, 2 a geometer of some prominence and brother of the poet
Stesich'orus. He is mentioned by Proclus (c. 412-485 A.D.),
but nothing is known of his work.

1 'Aj/a&M^s, born at Miletus, c. 585; died c. 528 B.C.

2 The Mamercus of J. Gow, History of Greek Mathematics, p. I4S (Cam-
bridge, 1884). Hereafter referred to as Gow, Greek Math.

78 FROM PYTHAGORAS TO PLATO

Zeno of Elea. 1 About the time of the death of Pythagoras
there was born at Elea 2 the philosopher Zeno, whose work on
motion represented a noteworthy advance in the science, even
though the mathematical feature is in evidence in only a single
instance. It was from Elea that the Eleatic School of philoso-
phers, one of the two great schools of southern Italy, derived
its name. Zeno asserted that on account of the infinite divisi-
bility of space through which an object must pass in moving,
motion could not begin; that Achilles could not pass a tor-
toise, even though he went faster than the tortoise; that a
moving object must be at once in motion and, because it occu-
pies space, at rest ; and that one space of time might, in differ-
ent relations, be both long and short, reminding us of certain
features of the modern doctrine of relativity. His argument
with respect to Achilles and the tortoise may be thus expressed
in modern units : If the tortoise has a mile the start and goes
one tenth as fast as Achilles, when Achilles reaches the point
where the tortoise was, the latter will be T V of a mile ahead ;
when Achilles has covered that distance, the tortoise will be
jJ-0- of a mile ahead ; and, similarly, whenever Achilles reaches
a spot where the tortoise was, the latter will still be ahead, and
so Achilles can never pass it. 3

Anaxagoras. Among the noteworthy contemporaries of Zeno
was Anaxag'oras, 4 the last of the celebrated philosophers of the
Ionian School. He was a friend and teacher of Euripides, Peri-
cles, and other great men of his time, but was condemned to
death 5 at the age of seventy-two for being favorable to the
Persian cause. Although his chief work was in philosophy,
where his prime postulate was that "reason rules the world," he
was interested in mathematics and wrote on the quadrature of

1 7,-fivwv (Zenon). Born at Elea, c. 496 B.C. He was living in the time of
Pericles (died 429). Heath, History, I, 271.

2 'EX&i or 'TCVXi; (Elea or Hyele) ; Latin, Velia\ in southern Italy.

SF. Cajori, "The History of Zeno's Arguments on Motion," Amer. Math.
Month., XXII, i, 39, 77, i9, H3, 179, 215, 253, 293; "The Purpose of Zeno's
Arguments on Motion," /sis, III, 7.

4 \vafay6pa9. Born at Clazomenae, Ionia, c. 499; died c. 427 B.C.

5 Ancient writers are not clear upon this point.

ZENO AND ANAXAGORAS 79

the circle and on perspective. 1 When banished from Athens he
remarked, "It is not I who have lost the Athenians, but the
Athenians who have lost me."

Agatharchus. About this period (470 B. c. ) an Athenian artist,
Agathar'chus, 2 applied stereometry to the theory of perspective.
He is said to have painted the scenery for a tragedy which
^Eschylus produced. In his work on drawing he showed how to
make use of the notion of projection upon a plane surface.

Socrates. 3 Although we do not commonly think of Soc'rates,
the Athenian statesman and philosopher, as a mathematician,
yet for his work on induction and for his insistence upon ac-
curate definition he should be mentioned in connection with
the early development of a logical geometry. As the teacher of
Plato he assisted in the development of that great maker of
philosophers and of those who based their mathematics upon
sound logic. Socrates has left us no writings of his own, but
\ve have the testimony of Plato, Euclid, and others that they
were greatly his debtors. Probably no more noble tribute
has been paid to him than that given by Dr. Jowett in his
paraphrase of the words of Plato : " And he, Socrates, is a mid-
wife, although this is a secret ; he has inherited the art from his
mother bold and bluff, and he ushers into light, not children,
but the thoughts of men." 4 Xenophon 5 and Diogenes Laertius 6
tell us, however, that he felt that geometry and astronomy
were useful merely for measuring fields and telling the time of
day, a view which, if really held by him, has been advanced
by men of far less mentality in every generation since that
time, and with the same empty results.

(Enopides of Chios. 7 Probably a Pythagorean and certainly
one of the leading astronomers of his time, (Enop'ides is

^Anaxagorae Fragmenta (Leipzig, 1827); better edition by Schorn (Bonn,
1829). See also F. Breier, Die Philosophic des Anaxagoras von Klazomend nach
Aristotetes (Berlin, 1840).

Born near Athens, 468 B.C.; died at Athens, 390.
4 Jowett's Plato, IV, 123. 6 Lives of the Philosophers, II, 32.

6 Memorabilia, IV, 7. 7 Otvoirtdv. Born in Chios; fl. c. 46$ B.C.

8o FROM PYTHAGORAS TO PLATO

thought to have learned the science of the stars and the
obliquity of the ecliptic from the priests and temple astrono-
mers of Egypt. He is said to have invented the cycle of 59
years for the return of the coincidence of the solar and lunar
years, giving the length of the solar year as 365 days and
somewhat less than 9 hours. Proclus (c. 460) attributes to
him the discovery of two problems of Euclid, one (I, 12)
referring to the drawing of a perpendicular to a given line
from an external point, and the other (I, 23) referring to
the making of an angle equal to a given angle. If this is
really the case, it shows how slight had been the advance
in demonstrative geometry, even in the century following the
death of Pythagoras.

Democritus. Democ'ritus, 1 known to later generations as the
Laughing Philosopher, inherited great wealth, spent his for-
tune in travel, met the learned men of many lands, was a man
of remarkable diligence in study, and died in poverty. His
works are lost, except for certain fragments. 2 One of his
teachers in philosophy is said to have been Leucip'pus, 3 the
founder of the atomic theory of the ancient philosophy which
asserted that the original characteristics of matter are func-
tions of quantity instead of quality, the primal elements being
particles homogeneous in quality but heterogeneous in form.
Archimedes tells us, in his work on Method, that Democritus
was the first to show the relation between the volume of a cone
and that of a cylinder of equal base and equal height, and
similarly for the pyramid and prism. In spite of the manifest
bearing of his work upon an infinitesimal calculus, it seems to
have had no influence in this direction among the Greeks. It is
said that Plato felt that all the writings of Democritus should
be burned. At any rate he had so slight an opinion of the latter
that he makes no mention of him in any of his works. Such

Born at Abdera, Thrace, c. 460 B.C.; died c. 357.
-F. W. A. Mullach (F. G. A. Mullachius), Democriti operum fragmenta.
Berlin, 1843.

3 Aftf/cifl-Tros. The date and place of his birth and of his death are unknown.

DEMOCRITUS Si

treatment at the hands of Plato was perhaps due to the boastful
nature of Democritus, who speaks of himself in these words:

I have wandered over a larger part of the earth than any other
man of my time, inquiring about things most remote; T have ob-
served very many climates and lands and have listened to very
many learned men ; but no one has ever yet surpassed me in the
construction of lines with demonstration ; no, not even the Egyptian
harpedonaptae ( e A/oTrcSoi/aTmu) , with whom I lived five years in all,
in a foreign land.

These harpedonaptse (literally, "rope stretchers") were the
surveyors of ancient Egypt, and the quotation suggests that the
logical demonstration of propositions was then practiced in that
country as well as in Greece.

Parmenides of Elea. Parmen'ides of Elea 1 taught at Athens
in the middle of the 5th century B.C., and among his theories
of the universe was the one that the earth is a sphere. His
work, however, was that of a philosopher rather than a mathe-
matician. It was in his time that Herodotus (c. 450 B.C.)
wrote his history, and it is in this work that the idea of a merid-
ian first appears in any literature now extant. From this
time on for several centuries the sphericity of the earth was
accepted as valid by many philosophers. The theory was re-
vived in the i2th century of our era and was strongly asserted
by Roger Bacon (c. 1250).

Philola'us, 2 a distinguished Pythagorean, was born at Cro-
tona, or possibly at Tarentum, and according to Plato was a
contemporary of Socrates. Although Pythagoras handed down
his doctrines by word of mouth, it is stated by Porphyr'ius
(fl. .275) that Ly'sis, who was a prominent philosopher, and an
obscure Pythagorean named Archip'pus, put into writing some
of the doctrines of the school and transmitted them to their
descendants as secret heirlooms. Philolaus, however, was the
first to write a treatise on the teachings of Pythagoras and to

1 Uappevld-r)*. Born in Elea; fl. c. 460 B.C.
-4>tXAXaos. Fl. c. 425 B.C.

82 FROM PYTHAGORAS TO PLATO

make it public. Judging from the fragments that have come
down to us, 1 his interest was in philosophy rather than in
mathematics, although he touches upon the latter field in his
description of a gnomon.

Hip'pias of Elis, 2 known both as a statesman and as a philos-
opher, belonged to the sophists, a class of teachers who traveled
from place to place and took money for their services, a
practice quite contrary to the ideas of earlier philosophers. He
accumulated wealth by teaching and public speaking, and
Plato speaks of him as a vain man, given to arrogance and boast-
ing. He seems to have been possessed of a wide but superfi-
cial knowledge. His contribution to mathematics was confined
to his invention of a simple device for trisecting any angle,
this device being known as the quadratrix. Since it was studied
and described at a later period by Deinostratus ( Dinostratus,
c. 350 B.C.), it generally bears the latter's name. 3

Hippocrates. 4 Various stories are told of Hippoc'rates,
among them one that he was an unsuccessful merchant, later

becoming a Pythagorean phi-
losopher with a special interest
in mathematics. Aristotle speaks
of him as skilled in geometry
but as otherwise stupid and
weak. He is mentioned by
ancient writers as the first to
arrange the propositions of geometry in a scientific fashion
and as having published the secrets of Pythagoras in the

X A. Bockh, Philolaos des Fythagorecrs Lchren, nebst den Bruchstuc ken seines
Werkes (Berlin, 1819); W. R. Newbold, " Philolaus," in Archiv fiir Geschichte
der Philosophic, XIX, 176.

2 'l7r7rfas. Born at Elis, on the west coast of Peloponnesus; fl. c. 425 B.C.

3 All man and Hankel do not believe that this is the Hippias to whom Proclus
refers as the inventor of the quadratrix. There is room for the doubt that these
eminent writers express, but Cantor, Montucla, and various other historians
feel that the evidence is in favor of Hippias of Elis. See also Gow, Greek
Matk. t p. 162. The name "quadratrix" is due to the fact that the curve can
also be used in the quadrature of the circle. On the application of this and
other curves to the problems of geometry, see Volume II, Chapter V.
Born in Chios; fl. c. 460 B.C.

HIPPOCRATES 83

field of geometry. In his attempts at squaring the circle
he discovered the first case of the quadrature of a curvi-
linear figure, 1 namely, the proof that the sum of the two shaded
lunes here shown is equal to the shaded triangle. The proposi-
tion holds equally for any right-angled triangle, isosceles or not,
although Hippocrates knew it only for the isosceles right-angled
triangle. Proclus (c. 460) ascribes to him the method of reduc-
tion (W^G^), the passing from one proposition to another
that seems more simple, proving the latter, and then reversing
the order. For example, Eratosthenes (c. 250 B.C.) tells us
+hat Hippocrates showed that the duplication of a cube can be
effected if two mean proportionals can be found between
two given lines."

Meton, Phaeinus, and Euctemon. That there was great in-
terest in mathematical astronomy in Athens between the time
of Pythagoras and that of Plato is seen in the work of the three
astronomers Me'ton, Phaei'nus, and Eucte'mon. 3 It is not
possible, however, to differentiate their contributions to the
subject. The philosopher Theophras'tus 4 says that Phaeinus
made astronomical observations on Lycabettus, at Athens, and
that from these Meton constructed the cycle of 19 years, since
known as the Metonic Cycle. The astronomer Ptolemy 5 says
that Meton and Euctemon made observations at Athens and in
other places. He adds that Meton made the length of the year
to be 365! days-f yV f a day, which is more than 30 minutes
too long. Whether the ig-year cycle is really due to Meton, or
was already known to (Enopides, or was obtained from Egyp-
tian or other sources is, and is likely to remain, unknown.

a W. Lietzmann, Dcr pythagoreische Lehrsatz (Leipzig, 1912). For a further
discussion see Volume II, Chapter V.

2 That is, if a : x = x : y y : 2 a, then x 2 ay, x* = a 2 ? 2 , y~ = 2 ax, and
hence x 4 2a'-*x, or x 3 2a 3 . F. Rudio, Der Bericht des Simplicius ilber die
Quadratures, des Antiphon und des Hippokrates, Leipzig, 1907, with Greek and
German text; P. Tannery, "Hippocrate de Chios et la quadrature des lunules,"
in the Memoires de Bordeaux (1878); La Geometric Grecque, p. 117 (Paris,
1887); "Le fragment d'Eudeme sur la quadrature des lunules," Memoires
scientifiques, II, 46, 339 (Paris, 1912); Gow, Greek Math., p. 164; M. Simon!
Archiv der Math., VIII (3), 269. 8 M<*Twv, Qacivfa, Eu/mfriwv. Fl. c. 432 B.C.

4 6e60pa<rT0y. Fl. c. 2So B.C 6 Born c. 85; died c. 165.

84 FROM PYTHAGORAS TO PLATO

The Method of Exhaustion. lamblichus (c. 325) mentions
Bry'son, 1 or Bryso, as one of the youths whom Pythagoras in-
structed in his old age. If this is true, Bryson must have been
born about 520 B.C., but it is commonly believed that he flour-
ished about 450 B.C. He was formerly thought to have con-
tributed to what is known as the method of exhaustion, a crude
approach to the integral calculus whereby the area between a
curvilinear figure (say a circle) and a rectilinear figure (say an
inscribed regular polygon) could be approximately exhausted
by increasing the number of sides of the latter. There is, how-
ever, no reliable ancient authority for connecting his name with
the theory." The method was effectively used by later writers,
notably by Eudoxus and Archimedes, and was extended to
include the mensuration of solids.

Antiphon and the Method of Exhaustion. Aristotle mentions
a Greek sophist named An'tiphon,' 1 or Antipho, whose attempts
at the quadrature of the circle led him into this phase of
geometry. Antiphon inscribed a regular polygon in a circle,
doubled the number of sides, and continued doubling until, as
he seems to have believed, the sides finally coincided with the
circle. Since he could construct a square equivalent to any poly-
gon, he could then, as he thought, construct a square equivalent
to the circle; that is, he could " square the circle," thus finding
its area. We have here another phase of the method of ex-
haustion, the area between the polygon and the circle being
exhausted as the process of doubling the number of sides pro-
ceeds. It is one of the first steps in the development of an infini-
tesimal calculus applied to integration, a type of mathematics
that had to wait two thousand years for serious consideration.

Archytas. In Plato's time Archy'tas, 4 a distinguished Pythag-
orean philosopher, achieved a high reputation as a mathema-
tician, a general, a statesman, a philanthropist, and an educator,
and Cicero (106-43 B - c -) speaks of him as a friend of the great

. ap. Rudio, Bibl. Math., VII (3) , 378.

:i \VTi<l>&v. Fl. C. 430 B. C.

4 'Apxtfras. Born at Tarentum, c. 428 B.C.; died c. 347. See Allman, Greek
Geom.j p. 102, for an excellent summary of his work.

ARCHYTAS 85

master himself. Horace (65-83.0.) refers to his death by
shipwreck in the Adriatic, speaking of him in these words :

The scanty present of a little dust

Near the Matinian shore confines thee, O Archytas,

Measurer of the sea, the earth, and the innumerable sand. 1

Archytas lived in Magna Gratia, then much more tranquil
than Greece itself, disturbed as the latter was by the Pelopon-
nesian War. It was because of these wars that many Pythag-
oreans returned to Crotona and Tarentum, the result being
that scholarship again flourished in this part of Italy. Vitru-
vius~ says that Archytas solved the problem of the duplication
of the cube by means of cylindric sections. He was the first
to apply mathematics in any noteworthy way to mechanics, and
he also applied the science to music and even to metaphysics. ''

Eudemus (c. 335 B.C.), speaking of his work in geometry,
tells us that he was one of those who "enriched the science with
original theorems and gave it a sound arrangement," and from
another statement we infer that he knew and doubtless proved
the following propositions :

1. If a perpendicular is drawn to the hypotenuse from the
vertex of the right angle of a right-angled triangle, each side is
the mean proportional between the hypotenuse and its adjacent
segment.

2. The perpendicular is the mean proportional between the
segments of the hypotenuse.

3. If the perpendicular from the vertex of a triangle is the
mean proportional between the segments of the opposite side,
the angle at the vertex is a right angle.

J Te maris et terrae numcroquc carentis harenae
Mensorem cohibent, Archyta,
Pulveris exigui prope litus parva Matinura
Munera.

Carmen i, 28

2 Praefatio to his De Architecture ix.

3 For a list of fragments attributed to him see J. A. Fabricius, Bibliotheca
Graeca, 14 vols., I, 833 (Hamburg, 1705-1728). There is a later edition, Ham-
burg, 1700-1809. See also Gow, Greek Math., p. 157.

86 FROM PYTHAGORAS TO PLATO

4. If two chords intersect, the rectangle of the segments of
one is equivalent to the rectangle of the segments of the other.

5. Angles in the same segment of a circle are equal.

6. If two planes are perpendicular to a third plane, their line
of intersection is perpendicular to that plane and also to their
lines of intersection with that plane. 1

Theodorus of Gyrene. 2 Among those who assisted in pre-
paring the way for scientific mathematics as distinguished from
the intuitive form, Theodo'rus of Cyrene deserves at least brief
mention. He was a Pythagorean philosopher, and Proclus
(c. 460) says that he was a little younger than Anaxagoras,
who was born c. 499 B.C. According to Appuleius (c. 150)
and Diogenes Laertius (2d century), Plato went to Cyrene
to study geometry under Theodorus, possibly learning from
him the theory of irrationality, which, as we know, had received
attention in the school of Pythagoras.

Theaete'tus : of Athens was a pupil of Theodorus and of
Socrates and is described by Plato as a man of unusual bril-
liancy. 1 Although his works are lost, there are references in
the writings of the ancient historians to show that he discovered
a considerable part of elementary geometry and wrote upon
solids. Euclid seems to have been indebted to him and to Eu-
doxus for some of the material used in writing the Elements.

With Thesetetus may be said to have closed the period which
began with Pythagoras and which prepared the way for Plato
and his school. Pythagoras made scientific study popular with
the leisure class, or at least he created an influential group
of scholars. Without the work of his school, supplemented by
the contributions of such schools as the one at Elea, the world
would not have been ready for Plato. The period just closing
supplied the raw material, and we shall find that Plato furnished
the tools for making good use of this supply.

1 Allman f Greek Geom. t p. 114.

2 9e65wpo$. Fl. c. 425 B.C. Cyrene (Kvp^io)) was a city on the north coast of
Africa.

s eeafrTjros. Fl. c. 375 B.C. ; died 368 B.C. H. Vogt, Bibl. Math., XIII (3) , 200.
4 Allman, Greek Geom* p. 206.

PLATO

5. INFLUENCE OF PLATO AND ARISTOTLE

Plato. To few men can the words of Carlyle be more
appropriately applied than to Plato: "In every epoch of the
world the great event, parent of all others, is it not the arrival
of a Thinker in the world ! " For never in all her early history
was Greece so des-
perately in need of
men of soul as she
was when Plato 1 be-
gan his life work.
While he was still
a young man (404
B.C.) Athens fell
before the Spartan
forces. The century
in which were first
produced the great
tragedies of ^Eschy-
lus, Sophocles, and
Euripides, which saw
the Acropolis adorned
with the masterpieces
of Ictinus, Phidias,
and Callicrates, and
which knew Athens
under the reign of

Pericles the Magnifi- A fanciful portrait. From a drawing by Raphael
, . 5 in the Accademia at Venice. Inserted to show

Cent, ttllS Century this artist , s concep tion of the philosopher

had passed away,

and with it had gone forever that glory of the city that
appealed to the masses, the glory of arms, of the drama,
of architecture, and of sculpture. The new century was
to see a new Athens, dead to the present but filled with
intellectual ambitions for the future. Three great names

PLATO

inxdrwi/ (Platon). Born at Athens, c. 430 B.C.; died c. 349.

88 INFLUENCE OF PLATO AND ARISTOTLE

of Athenian citizens has that future preserved, and by them was
it powerfully influenced, the names of Plato, Aristotle, and
Demosthenes.

Plato's Studies. Of the first of these three great leaders of
men Cicero has this to say :

It is reported of Plato that he came into Italy to make himself
acquainted with the Pythagoreans, and that when there he made the
acquaintance, among others, of Archytas and Timaeus J and learned
from them all the tenets of the Pythagoreans. 2

It is also said that Plato visited Egypt, partly, no doubt, for
purposes of trade, but chiefly that he might acquire knowledge.
It may have been from the priests along the Nile, but more
likely through the Pythagoreans, that he came to appreciate
so highly the value of geometry. At any rate, in later years
he is said to have placed above the entrance to his school of
philosophy (the Academy) the words, "Let no one ignorant
of geometry enter my doors," ;5 the oldest recorded entrance
requirement of a college, and to have spoken of God as the
great geometer. 4

Plato studied under Socrates and also under a certain
Eucleides (Euclid) of Meg'ara/' a philosopher who has often
been confused with Euclid the geometer (c. 300 B.C.). He
traveled extensively, visiting not only Egypt and lower Italy
but also Sicily and possibly Asia. He thus came in contact
with the mathematics and philosophy of these various countries
and returned to Athens filled with enthusiasm for an era of
splendid thought in place of the era of splendid action which
had characterized the century that had just closed.

j Not the historian, but a native of Locri. Probably no works of his are
extant, although there is one doubtfully attributed to him.
-Titsculan Disputations, I, 17
;{ MiySeis a-yew/n^rp-tyros tiff IT w /xoi TT)I> ffrtyyv.

4 "God eternally gcometrizes," 'Act 0e6s yeuperpeT. This is not in Plato's
works, but is stated by Plutarch as due to him. Plutarch, Convivalium Dis-
putationum libri novem, viii, 2, ed. Didot (Paris, 1841).

5 A Greek city.

PLATO 89

Of his philosophy it is unnecessary to speak, since this has
little bearing upon the problem in hand, but in the field of
mathematics his great contribution was to the underlying prin-
ciples of the science, including the method of analysis.

Plato's Interest in Arithmetic. In the study of numbers he,
like all the ancient philosophers, was interested in arithmetic
rather than logistic. In his Republic he says that the science
has a double use, military and philosophical.

For the man of war must learn the art of numbers or he will not
know how to array his troops ; 1 and the philosopher also, because he
has to arise out of the sea of change and lay hold of true being,
and therefore he must be an arithmetician. . . . Arithmetic has a
very great and elevating effect, compelling the mind to reason about
abstract number.

Mysticism of Numbers. One thing that particularly inter-
ested Plato was the mysticism of numbers. In his Republic
( Book VIII) he speaks in an obscure fashion of a certain mys-
tic number, but does not make it clear what this number is.
He calls it "the lord of better and worse births," and subse-
quent writers have often tried to find exactly what he meant.
One theory is that 60 4 , or 12,960,000, is the Platonic number.
This number played an important part in the mysticism of
the Hindus and the Babylonians, and it is possible that
Pythagoras found it on the banks of the Euphrates, if he
really studied there, and that he took it with him to Crotona,
passing it on to his disciples, who, in turn, told it to Plato
and his followers.

Although Plato esteemed the science of numbers highly, 2
he gives us no information concerning the way it was taught in
his school or what it included. We are about as ignorant of
the subject as presented by him, and of the ground it covered,
as we are of the ancient logistic/'

1 Evidently referring to the square, heteromecic, and triangular numbers
described in Volume II, Chapter I.

*Laivs, V.

3 P. Tannery, "L'education platonicienne : L'arithmetique," in the Revue
scientifique, XI (iSSi), 287.

go INFLUENCE OF PLATO AND ARISTOTLE

Plato on Geometry. More than any of his predecessors Plate
appreciated the scientific possibilities of geometry, of which
more will be said in Volume II of this work. By his teaching he
laid the foundations of the science, insisting upon accurate
definitions, clear assumptions, and logical proof. His opposi-
tion to the materialists, who saw in geometry only what was
immediately useful to the artisan and the mechanic, is made
clear by Plutarch (ist century) in his Life of Marcellus,
Speaking of the use of mechanical appliances in geometry,
Plutarch remarks upon " Plato's indignation at it and his invec-
tions against it as the mere corruption and annihilation of the
one good of geometry, which was thus shamefully turning its
back upon the unembodied objects of pure intelligence." Thai
Plato should hold the view here indicated is not a cause foi
surprise. The world's thinkers have always held it. No mar
ever created a mathematical theory for practical purposes
alone. The applications of mathematics have generally beer
an afterthought. 1

Immediate Followers of Plato. Among the followers of Plate
was his nephew, Speusip'pus, 2 who accompanied him on his
third journey to Syracuse and succeeded him as head of the
Academy (347-339 B.C.). He wrote upon Pythagorean num-
bers, integers like 3, 4, 5, which represent the sides and
hypotenuse of a right-angled triangle. He also wrote upon
proportion. We get some information concerning him from
an anonymous work of uncertain date called the Thcologu-
mena, which is also the title of a lost work by Nicomachus
(c. 100 A.D.). In this work it is related that Speusippus was the
son of Potone, the sister of Plato, and that he "ceased not tc
study with diligence" the lessons of the Pythagoreans, and
especially of Philolaus. The work also states that he treated
with rare elegance the subjects of linear, polygonal, plane, and
solid numbers. 3

x On the general subject of mathematics in Plato's time see B. Rothlauf
Die Mathematik zu Platons Zeit nnd Seine Beziehungen zu ihr (Munich, 1878)
Heath, History, I, 284.

8 See Volume II, Chapter I.

EUDOXUS 91

Of the minor followers of Plato mention should be made of
Leod'amas of Thasos, J who is referred to by Proclus (c. 460)
and Diogenes Laertius (2d century) and is said to have made
use of the analytic method of proof. There was also Philip'pns
Medmae'us, 2 an astronomer and geometer of Medma, ot
Mesma, in Magna Gnecia, who, under the guidance of Plato,
took up the study of mathematics. 1 Thymaridas, who devised
a rule for solving simultaneous linear equations, seems to have
lived about this time.

Eudox'us of Cnidus, 4 at one time a pupil of Plato, achieved
eminence in astronomy, geometry, medicine, and law/' It is
said that he introduced the study of spherics (mathematical
astronomy) into Greece and made known the length of the
year as he had found it given in Egypt. He was the first of
the Greeks, so far as is known, to give a description of the con-
stellations. Strabo asserts that the observatory of Eudoxus
still existed at Cnidus in his time, that is, about the beginning
of the Christian era. Seneca says that he brought from Egypt
to Greece the theory of the motions of the planets ; Aristotle
records that he made separate spheres for the stars, sun, moon,
and planets ; and Archimedes says that he found the diameter
of the sun to be nine times that of the earth and showed that a
pyramid is one third of a prism of the same base and the same
altitude, and similarly for a cone and cylinder. For the men-
suration of the cone and cylinder he probably developed the
method of exhaustion as a rigorous theory/ 5 Vitruvius gives
him credit for a new form of sundial called the spider's web, 7
which may, however, have been an astrolabe. Because of a
note, possibly due to Proclus, he is often credited with having
written a work on proportion which finally became Book V
of Euclid, but for this statement there is no definite historical

Fl. c. 3808.0. 2 <j>{x t7r7ros ' MeS/Acuos Born c. 375 B.C.

8 He is also known as Opuntius. H. Vogt, Bibl. Math., XIII (3), 193, 195.
4 E&5oos. Born c. 408 B.C.; died c. 355. See Allman, Greek Geom., p. 128;
Gow, Greek Math., p. 183; Heath, History, I, 322.

5 Diogenes Laertius, VIII, 86. 6 See Allman, Greek Geom., pp. 96, 139.

Part of the astrolabe resembles a web.

92 INFLUENCE OF PLATO AND ARISTOTLE

sanction. 1 Our principal knowledge of Eudoxus and his work
comes from an astronomical poem written by Ara'tus," and
from a commentary of Hipparchus upon it. 3

Menaech'mus 4 was a pupil of Eudoxus and a friend of Plato, 5
and possibly it is to him that we owe the first treatment of
conies. It is said that Alexander the Great was his pupil and
that he asked that geometry be made more simple for him ;
whereupon Menaechmus replied : "O King, through the country
there are private and royal roads, but in geometry there is only
one road for all."' J The conic sections which Proclus (c. 460)
says were considered by him were probably the "Menaechmian
Triads" of Eratosthenes (c. 230 B.C.). It is said that he ob-
tained them by cutting cones by planes perpendicular to an
element, the parabola from a right-angled cone, the hyperbola
from an obtuse-angled cone, and the ellipse from an acute-
angled cone. A friend of his, Theudius of Magnesia, wrote a
textbook on geometry.

Deinos'tratus, or Dinostratus, 7 was a brother of Mensechmus.
He is known chiefly for his study of the quadratix, a curve
already invented by one Hippias, very likely Hippias of Elis.
This curve enabled him to square a circle. 8

Xenoc'rates, 9 a native of Chalcedon, was a friend of Plato
and Aristotle and was prominent both as a philosopher and as
a diplomat. Besides various works on philosophy and govern-
ment he wrote on physics, geometry, arithmetic, and astrology.

1 For a discussion of the matter, see Allman, Greek Geom., p. 136 ; Sir T. L.
Heath, The Thirteen Books of Euclid's Elements, 3 vols., II, 112 (Cambridge,
iQo8) (hereafter referred to as Heath, Euclid).

2 "ApaTos. Fl. c. 270 B.C. The poem was the Phaenomena ($aiv6fj,va) , and
certain fragments were preserved by Hipparchus.

3 "brTrapxos. Fl. c. 150 B.C. J. B. J. Delambre, Histoire de I'astronomie ancienne,
I, 106 (Paris, 1817). This poem of Aratus was first printed at Venice in 1499.

*M&a<xAu>*. FI. 365-350 B.C. See Bibl Math., XIII (3), 194.

5 See AHman, Greek Geom., p. 153; Max C. P. Schmidt, "Die Fragmente
des Mathematikers Menaechmus," in Philologus, XLII (1884), p. 77; Gow,
Greek Math., p. 185; Heath, History, II, no.

fi The story is due to Stobaeus, a late Greek writer, c. 500. It is also related
of Euclid and King Ptolemy. 7 Aetj^Tparos. Fl. c. 350 B.C.

8 The details are considered in Volume II, Chapter V. See also Allman, Greek
Geom , p. 180; Gow, Greek Math., p. 187. BevoKpdTi}s. Born c. 396 B.C. ; died 314.

ARISTOTLE 93

Plutarch (ist century) tells us that he took the soul as a
"self-moving number," and deified unity and duality, 1 speaking
of the former as the first male existence, ruling in heaven, as
father and Zeus, as uneven number and spirit ; and duality as
the first female, the mother of the gods, and the soul of the
universe which reigns over the world, all of which theory
shows the Pythagorean influence. He also assumed the exist-
ence of indivisible lines and spoke of them as the elements of
certain Platonic triangles, perhaps with some intuition of an
infinitesimal calculus. He followed Speusippus as head of the
Academy and wrote a history of geometry in five books, which,
like his other works, is lost.

Ar'istotle 2 studied under Plato at Athens, and his diligence
and brilliancy led^ the Tatter" to call him the " intellect of the
school." 3 He became one of the instructors of Alexander the
Great, and later returned to Athens and founded the Peri-
patetic School of philosophy, probably so called from the place
where he taught. 4 He was a voluminous writer, but although
many of his works are extant the major part are lost. His
interest in the mathematical sciences lay chiefly in their appli-
cations to physics. He speaks of mathematics as standing half-
way between physics and metaphysics. He wrote two works of
a mathematical nature, one on indivisible lines and the other on
mechanical problems. Both have been edited and printed.
We know that, contrary to the doctrines of the Pythagoreans,
he advocated the separation of arithmetic and geometry. In his
systematizing of logic he contributed indirectly to the great
work of Euclid. To him, too, we owe the first known definition
of continuity: "A thing is continuous when of any two suc-
cessive parts the limits at which they touch are one and the
same and are, as the word implies, held together. 775 Aristotle
was also interested in the historical development of science,
and this seems to have influenced the work of his disciples in

and dvds.

Born at Stageira (Stagira), the present Stavro, 384 B.C.; died
at Athens, 322. 3 Nous r^s $iarpi/3?s.

4 '0 ircpliraros. 5 Gow, Greek Math.^ p. 188.

94 INFLUENCE OF PLATO AND ARISTOTLE

gathering materials for the history of mathematics. Among
those whose interests led them into this field was Theo-
phras'tus, 1 a pupil of Plato as well as of Aristotle. He wrote on
philosophy, oratory, poetry, botany, physics, politics, and
mathematics, but his works are known chiefly from fragments."

Eudemus. Eude'mus"' of Rhodes, another disciple of Aris-
totle, who flourished c. 335 B.C., was also much interested in
the history of mathematics. Most of his works are lost, but
certain fragments remain and serve to throw considerable light
upon the mathematics of the Aristotelian school. It seems to
have been to his care that we are indebted for the preservation
of certain works of Aristotle.

Dicaearchus. It is probable that Dicaear'chus 4 of Messina, a
city just north of Syracuse, in Sicily, was also a disciple of
Aristotle, although we know little of his life. He seems to have
flourished r. 320 B.C. and to have died c. 285 B.C. His work
in mathematics was connected chiefly with mensuration as ap-
plied to geography. There was another philosopher by the
same name, a Pythagorean, whom lamblichus (c. 325) quotes
as having contributed to the history of mathematics, but his
works are not extant.

Autolycus. Among the contemporaries of Aristotle should
probably be included, although we are uncertain as to the
date, the astronomer Autol'ycus. 5 Nothing is known of his
personal history except that he wrote two treatises on astron-
omy, both of which are extant. These are the most ancient
mathematical texts that have come down to us from the Greeks.
The first is on the motion of the sphere, and the second is on
the risings and settings of the fixed stars, and in each he shows
considerable skill in geometry.

Aristae'us, known as Aristaeus the Elder, is mentioned by
Pappus, a mathematician of the 4th century, as one of the three
geometers of the Greeks who were skilled in that branch of

1 6e60pa<rros. Fl. c. 350 B.C. 4 Ai/cafapxos.

2 There are various editions of his works. 6 AtirAXvicos. Fl. c. 330 B.C.

6 'A/>urTcuoj. Fl. C. 320 B.C.

MINOR WRITERS 95

geometry which treats of analysis, the other two being Euclid
and Apollonius. Pappus also relates that Aristseus wrote five
books on solid loci, 1 supplementing five others on the elements
of conies. Possibly these two works were the same. He also
wrote on the five regular solids, and the i3th book of Euclid
seems to have owed much to his skill. He was evidently one
of those mathematicians of the 4th century B.C. who, inspired
by Plato, helped to make possible the works of Euclid and
Apollonius.

Callip'pus or Calippus, 2 an astronomer of Cyzicus, was a
friend of both Eudoxus and Aristotle. Although not to be
looked upon as a geometer, his astronomical observations de-
serve brief mention, being frequently referred to by Geminus
and Ptolemy. The Callippic cycle of 76 years, 940 lunar
months, or 27,759 days was such an improvement on the
Metonic cycle of 19 years as to have been adopted by ancient
astronomers. We have the testimony of Simplicius (6th cen-
tury) that he was a pupil of Polemar'chus (4th century B.C.),
who taught at Cyzicus, and that he lived for a time with
Aristotle. Ptolemy tells us that he made astronomical observa-
tions on the shores of the Hellespont.

6. THE ORIENT

Orient and Occident. The rise of mathematics in Greece, its
remarkable development under the influence of such leaders as
Thales, Pythagoras, and Plato, and its distinct characteristics,
are such as to make it desirable to consider the Orient and the
Occident separately from the time of Euclid until the two were
joined by the new intellectual bond established by the Chris-
tian missions about the beginning of the i7th century. So little
was accomplished in the Orient from 1000 B.C. to 300 B.C.,
however, that we may properly mention that little in the present
chapter. Although each of these two great divisions of the
world always influenced the other in developing a system of

1 T$7roi (rrcpeot. 2KtXXnnros or KrfXunros. Fl. c. 325 B.C.

96 THE ORIENT

mathematics, the East has always been the East, and the West
has always been the West. They have had many points in
common, particularly in the application of mathematics to
astronomy; but the development of a logical geometry, with
all of its far-reaching results, is peculiar to the European
peoples, while the less rigid and somewhat more poetic
phases of mathematics have generally interested the Asi-
atic mind. Even the ancients recognized this difference,
for Quintilian (c. 35-^. 96) remarks: "From of old there
has been the famous division of Attic and Asiatic writers,
the former being reckoned succinct and vigorous, the latter
inflated and empty." 1

China. It is an interesting fact that Egypt developed a
worthy type of mathematics before 1000 B.C. and then stag-
nated, that Babylonia did the same, and that China followed
a similar course. Was it that the world's vigor was concen-
trated in Greece? Had the older civilizations burned out?
Or was there some subtle influence that subjected the orig-
inal seats of mathematical thought to canonical expression
instead of progressive action ? Whatever the answer, between
TOOO and 300 B.C. China produced no great classic in mathe-
matics, unless possibly the Nine Sections 2 already mentioned, or
the Wu-ts'ao Suan-king to be mentioned later, belongs to this
period. It was rather in the impetus given to commercial calcu-
lation through the introduction of coins in the 7th century B.C.,
at about the same time as they appeared in Asia Minor, that
China made her most noteworthy contribution to the progress
of arithmetic. Knife money and spade money appeared c. 670
B.C., the coins representing such common articles of value as
knives and spades. Circular coins were issued later and be-
came the standard forms in the 3d century B.C. As to the
methods used in calculating at this time, we are ignorant, but
some mechanical means were probably employed in China
as well as in other parts of the ancient world. About 542 B.C.
the Chinese are known to have used in their calculations bam-
boo rods, in size and appearance somewhat like a new lead

1 Institutes of Oratory. Bohn ed., XII, x, 16. 2 K'iu-ch'ang Suan-shu.

CHINA AND INDIA 97

pencil. About 375 B.C. there appeared the earliest Chinese
coins with weight or value inscribed upon them, and thus the
monetary material for commercial arithmetic became fairly
well perfected.

The Compass. As early as the 4th century B.C. there seems
to have existed some kind of instrument for indicating the
southern direction, probably the compass. In later literature
the ch'i-nan-ku^ (south-pointing chariot) is mentioned, but
what it was is unknown. 2 In works of the 4th century it is
ascribed to Huang-ti (2704-2595 B.C.) and is also mentioned as
being in use in the reigns of Ki-li (1230-1185 B.C.) and his
successor (1185-1135 B.C.) 1

India. As already stated, we have no authentic records of
India before the Mohammedan invasion (7th century), almost
our only sources of information being the Vedic literature, the
Buddhist sacred books, the heroic poems, such inscriptions as
remain on monuments, and the metal land grants. Of these,
the later Vedic literature, the heroic poems, and the Buddhist
writings are all that give us any knowledge of the mathematics
of the period from 1000 to 300 B.C. The Vedic writings prob-
ably extend down to about 800 B.C., although the Veddngas
(" Limbs for supporting the Veda") were written several cen-
turies later. The dates of the Sulvasutra period are unknown.
Taking the opinions of various scholars and forming a rough
estimate, we may put the ritualistic rules of the Sulvasutras in
the five centuries just preceding our era. The rules which have
any mathematical interest relate indirectly to the proportions
of altars in the temples. They include a statement about
Pythagorean numbers, that is, numbers satisfying the relation
x 2 + y 2 = 2 , and imply a statement of the Pythagorean
Theorem itself. There is no reason for believing, however, that
the Hindus had the slightest idea of the nature of a geometric
proof. There is also evidence of a knowledge of irrationals and

1 Chi means to point with the finger ; nan means south ; and kii means chariot.
2 F. Hirth, Ancient History of China, p. 129.
3 /6id., p. i^S-

9^ THE ORIENT

of an understanding of the uses of the gnomon. 1 The Sulvasii-
tras also state that the diagonal of a unit square is equal to

3 3-4 3-4-34

or 1.4142156. The area of the circle is asserted- to be
/ill i

\8 8T~29 ~~" 8 - 29~T(5 sTj;/". 6

Mathematics in the Sulvasutras. The Sulvasutras were
changed more or less by such commentators as Apastamba,
Baudhayana, and Katyayana. The following statements from
the Baudhayana edition show the style: 3

"The chord stretched across a square produces an area of
twice the size." 4

" The diagonal of an oblong produces by itself both the areas
which the two sides of the oblong produce separately." 5

The Lalitavistara, one of the sacred books of the Hindus,
speaks of the arithmetical prowess of the Buddha. 6 Sir Edwin
Arnold has put the statement in verse in his Light of Asia. 7

3 L. von Schroeder, Pythagoras und die Inder, Leipzig, 1884; and Indiens
Lileratur und Cidtur, Leipzig, 1887; H. Vogt, "Ilabcn die alten Inder den
Pythagoreischen Lehrsatz und das Irrationale gekannt?" Bibl. Math., VII (3),
6; A. Biirk, "Das Apastamba-Sulba-Sutra," Zeitschrift der deutschen Morgen-
Idndischen Gesellschaft, LV, 543; LVI, 327; M. Cantor, "Ueber die alteste
Indische Mathematik," Archiv der Math, und Physik, VIII (3), 63; B. Levi,
" Osservazioni e congetture sopra la geometria degli Indian!," Bibl. Math, IX

(3), 97; Smith and Karpinski, The Hindu-Arabic Numerals, p. 13 and biblio-
graphical notes throughout (hereafter referred to as Smith-Karpinski), Boston,
1911; G. R. Kaye, Indian Mathematics, Calcutta, 1915.

2 Thibaut in the Journal of the Royal Asiatic Soc. of Bexgjl, XLV (1875),
p. 227; Dutt, History of Civ. in Anc. India, I, 271.

s The translation is Dr. G. Thibaut's. See his memoirs "On the Sulvasutras."
Journ. Royal Asiat. Soc. Bengal, XLIV (1874) ; "The Baudhayana s'ulvasGtra,"
The Pandit, 1875; "The Katyayana Sulvasutra," The Pandit, 1882; G. Milhaud,
"La geometric d'Apastamba," Revue generate d. sci., XXI, 512.

4 That is, the square on a diagonal of a square is twice the original square.

5 That is, the square on a diagonal of a rectangle is equal to the sum of the
squares on the two sides; essentially the Pythagorean Theorem.

6 The date of the birth of Buddha is often placed at 543 B.C., but in Burma,
Siam, and Ceylon it is usually given as 80 years earlier, that is, 623 B.C.

7 It is given in Smith-Karpinski, p. 16. See also the translation of the
Mahabharata in . Arnold, Indian Poetry^ London, 6th ed., 1891.

INDIA

In all this there is nothing that is definite, but there is enough
to show that mathematics was not limited to the meager needs
of trade and that it was related, as with all thoughtful peoples,
to the higher life.

Mesopotamia. Just before the period of which we are speak-
ing the Arameans 1 established a flourishing dominion to the
north of the Hebrew territory of Palestine. Their mer-
cantile interests extended into the ancient cities of Assyria, as
is proved by such bronze weights as the one shown on page 39.

HEBREWS PAYING TRIBUTE TO SHALMANESER III, KING
OF ASSYRIA

This was about 850 B.C. The original is now in the British Museum. From
Breasted's Ancient Times

By 1000 B.C. they had developed a system of alphabetic writ-
ing, and their bills of exchange were known in Mesopotamia,
Persia, and India, as those of Babylon had been known before
them. All through this early period the records of taxes show
that this form of applied arithmetic was ever present.

In the 8th century B.C. the Assyrians subdued Mesopotamia
and much of the territory to the west and became the domi-
nating power in Western Asia. They maintained the first
great army equipped with weapons of iron and by this means
held a large territory in subjection. Militarism, however,
eventually proved a weakness, and they in turn succumbed to

iQr Syrians, as they are often called.

ioo THE ORIENT

the power of the Kaldi, Semitic nomads already mentioned,
who came from the South and who, known to us as the
Chaldeans, finally became the ruling power in Mesopotamia.

Contributions of Babylonia and Assyria. In the midst of all
these changes two steps in the history of mathematics deserve
special mention : ( i ) the Arameans brought the arithmetic of
commerce to a higher standard, and (2) the Babylonians and
Chaldeans extended the earlier work in astronomy. The
science of astrology had by this time developed as a potent
force in civilization, and astronomy had become recognized as
the science par excellence. Ptolemy the astronomer (c. 150)
refers to a Chaldean record of a lunar eclipse of 721 B.C. and to
the division of the circle into 360. The recognition of a zodiac
of twelve signs, and the study of the courses of the planets,
about 600 B.C., are further evidences of the interest of the Chal-
dean astronomers in this phase of applied mathematics.

As to astrology, that daughter of astronomy who nursed her
own mother, as Kepler writes of it, there are various tablets
of this period which show in what high esteem it was held. In
general they are reports of the following kind to the king:
"Two or three times of late we have searched for Mars but
have not been able to see him. If the king, my master, asks me
if this invisibility presages anything, I reply that it does not." 1

Science reached its highest point in the reign of Nebuchad-
nezzar, which closed in 561 B.C. It is true that we have lists
of the planets dating from 523 B.C. and from other years, 2
statements of the irregular insertion of intercalary days at
about the same period, and a definite recognition of the leap
year between 383 and 351 B.C., 3 but the mathematics of Meso-
potamia practically ceased to exist with the decay of the
Chaldean power.

ifiigourdan, L'Astronomie, p. 29; R. C. Thompson, The Reports of the
Magicians and Astrologers of Nineveh and Babylon in the British Museum t
London, 1900.

2 F. Hommel, "Die Babylonisch-Assyrischen Planetenlisten," in the Hilprecht
Anniversary Volume, p. 170.

3 F. H. Weissbach, "Zum Babylonischen Kalender," in the Hilprecht volume
above cited, p. 282.

DISCUSSION 101

TOPICS FOR DISCUSSION

1. Influences favorable to the development of mathematics among
the Greeks from 1000 B.C. to 300 B.C.

2. The nature of logistic and of arithmetic and the reasons for
their treatment as unrelated subjects.

3. The advantages of the Greek method of treating arithmetic
from the geometric standpoint, particularly in relation to the nature
of irrational numbers.

4. The influence of Thales upon the subsequent development
of mathematics in Greece.

5. The influences which contributed to the making of the char-
acter of Pythagoras.

6. The influence of Pythagoras upon mathematics in general, and
particularly upon geometry and the theory of irrationals.

7. Music as a branch of ancient mathematics.

8. Beginnings of a kind of infinitesimal calculus in Greece, par-
ticularly with respect to the method of exhaustion.

9. Types of geometric propositions that attracted special atten-
tion in this period, thus showing the nature of geometry before the
time of Euclid.

10. The influence of Plato upon mathematics in general and upon
geometry in particular.

IT. The influence of astronomy upon mathematics in Greece, par-
ticularly with reference to geometry and a primitive trigonometry.

12. The early steps in the invention of conic sections.

13. The study of higher plane curves among the Greeks in the
period under discussion.

14. The influence of Aristotle upon mathematics in general, and
particularly upon its applications.

15. Nature of mathematics in the Orient in this period.

1 6. General distinction between the mathematics of Greece and
that of the East.

17. Mysticism of numbers as found in the Orient, in Mesopotamia,
and in the West.

18. Early studies in the history of mathematics among the Greeks.

19. The recognition of the sphericity of the earth by various
leading Greek philosophers.

20. The nature of the mathematics of the Sulvasutras.

CHAPTER IV

THE PERIOD FROM 300 B.C. TO 500 A.D.
i. THE SCHOOL OF ALEXANDRIA

Chronological and Geographical Considerations. The reason
why the limitations of 300 B.C. and 500 A.D. are arbitrarily
chosen for this chapter is that these dates mark approximately
the period of influence of the greatest mathematical school of
ancient times, the School of Alexandria. Moreover, the first
of these dates is approximately that of Euclid, the world's
greatest textbook writer, and the second is that of Boethius,
whom Gibbon characterizes as " the last of the Romans whom
Cato or Tully could have acknowledged for their countryman."

Within this period Greek civilization passed away, Rome
rose and fell, and the ancient mathematics of the West de-
scended from its most exalted to its most debased estate. We
have, therefore, the most significant period of ancient mathe-
matical history, at least in the matter of actual production,
and we have the Mediterranean world, probably the most
interesting of all ancient civilizations.

The School ol Alexandria. The greatest mathematical center
of ancient times was neither Crotona nor Athens, but Alex-
andria. Here it was, on the site of the ancient town of Rhacotis,
in the Nile Delta, that Alexander the Great founded a city
worthy to bear his name. Upon the death of the great Mace-
donian conqueror (323 B.C.) the vast domain which he had
brought under his control was broken up. After the death of
Antig'onus, his ablest general, the empire fell into three parts.
Alexander's friend and counselor, and possibly his blood
relative, Ptol'emy 1 So'ter (Ptolemy the Preserver), came into

1 IlroXquaibs ; Latin, Ptolemaetis.
102

ALEXANDRIA 103

possession of Egypt, Antigonus the younger laid claim to
Macedonia, while Seleu'cus took for his part the provinces of
Asia. Under Ptolemy's benevolent reign (323-283 B.C.) Alex-
andria became the center not only of the world's commerce
but also of its literary and scientific activity. 1 Here was es-
tablished the greatest of the world's ancient libraries and its
first international university. Cardinal Newman, in speaking
of these two features, says with poetic feeling that "as the
first was the embalming of dead genius, so the second was the
endowment of living." Here were trained more great mathe-
maticians than in any other scientific center of the ancient
world. With Alexandria are connected the names of Eu'clid,
Archime'des, Apollo'nius, Eratos'thenes, Ptolemy the astron-
omer, He'ron, Menela'us, Pap'pus, The'on, Hypa'tia, Diophan -
tus, and, at least indirectly, Nicom'achus. Today, however, not
the slightest trace remains of the famous library and museum,
and even their exact locations are merely conjectural.

2. EUCLID

Euclid. 2 Of all the great names connected with Alexandria,
that of Euclid is the best known. He was the most successful
textbook writer that the world has ever known, over one
thousand editions of his geometry having appeared in print
since I482, 8 and manuscripts of this work having dominated

1 For a summary of the causes of its rise and a description of its library see
W. Kroll, Geschichte der klassischen Philologie, p. 12 (Leipzig, 1908) ; hereafter
referred to as Kroll, Geschichte.

2 EvK\ei'5ijs. Fl. r. 300 B.C. The leading work upon Euclid and his Elements is
that of Sir T. L. Heath, The Thirteen Books of Euclid's Elements, 3 vols., Cam-
bridge, 1908. The best Greek and Latin edition of Euclid's works is Heiber?
and Menge, Euclidis Elementa, Leipzig, 1883-1916. Of the many works ant-
articles on the life of Euclid the following may be consulted to advantage:
A. De Morgan, "Eucleides," in Smith's Diet, of Greek and Roman Biog.;
T. Smith, Euclid, his life and system, New York, 1902 ; W. B. Frankland, Th*
Story of Euclid, London, 1902; G. B. Biadego, "Euclide e il suo Secolo," in
Boncompagni's Bullettino, V, i ; P. Tannery, Pour I'histoire de la science hellene,
Appendix II (Paris, 1887); Gow, Greek Math., p. 195; M. C. P. Schmidt,
Realistische Chrestomathie aus der Litteratur des klass. Alter turns, I, i (Leipzig,
1900), hereafter referred to as Schmidt, Chrestomathie; Pauly-Wissowa's Real-
Encyclopadie, Vol. VI (Stuttgart, 1909), has an extensive article on Euclid.

3 P. Riccardi, Saggio di una Bibliografia Euclidea, p. 4. Bologna, 1887.

104 EUCLID

the teaching of the subject for eighteen hundred years pre-
ceding that time. He is the only man to whom there ever

* *

fr,

? r * , J mf"Dar^tmwf t^ISTltu^^/'flf^ 1 ,...,_

i .rrcmy ocfrm-ip&d
PAGE FROM A TRANSLATION OF EUCLID'S ELEMENTS

This manuscript was written c. 1294. The page relates to the propositions on

the theory of numbers as given in Book IX of the Elements. The nrst line

gives Proposition 28 as usually numbered in modern editions

came or ever can come again the glory of having successfully
incorporated in his own writings all the essential parts of the
accumulated mathematical knowledge of his time.

THE PERIOD

105

Of the life of Euclid nothing definite is known. A recent
writer expresses the belief that the evidence indicates that he
was born as early as 365 B.C. and that he wrote the Ele-
ments when he was about forty years old, 1 but we have no
precise information as to his birthplace, the dates of his birth
and death, or even his nationality. It was formerly asserted
that he was born at Meg'ara, a Greek city, but it is now known
that Euclid of Megara was a philosopher who
lived a century before Euclid of Alexandria.
The Euclid in whom we are interested may have
been a Greek or he may have been an Egyptian
who came to the Greek colonial city of Alexan-
dria to learn and to teach. There is some reason
for believing that he studied in Athens, but in
the way of exact information we have nothing
concerning him. Under any circumstances, the
period of his real influence upon mathematics
begins about 300 B.C.

The Books of Euclid. As was the custom in
the days when all treatises were written on long
strips of parchment or papyrus, the separate
parts of his work were rolled up and called
volumes, from a Latin root meaning to roll.
Because of the difficulty of handling large rolls,
they were cut into smaller rolls known as bibtia
(PifSKta, bibles), a word meaning books. Hence
we have the books of Homer, the books of geometry, the
books of the Bible, and so on. Euclid's greatest work is known
as the Elements? and in the books relating to geometry there
was arranged that mass of material treating of circles, recti-
linear figures, and ratios which had accumulated during the two
centuries following the death of Pythagoras. No doubt there
were also many propositions that were original with Euclid;
but the feature which made his treatise famous, and which

!H. Vogt, Bibl. Math., XIII (3), 193; but see Heath, History, I, 354.
2 Sroixcia. So great was Euclid's fame that he was known to the Greeks
as o <TToixwTjJs, "teacher of the Elements"

PARCHMENT
ROLL

Upon such a roll
a "book" of Eu-
clid was written

io6 EUCLID

accounts for the fact that it is the oldest scientific textbook
still in actual use, is found in its simple but logical sequence
of theorems and problems. It has been said of Shakespeare
that he "took the stillborn children of lesser men's brains and
breathed on them the breath of life," and so it was with
Euclid.

Contents of the Elements. The various books of the Elements
treated of the following topics, respectively: I, Congruence,
parallels, the Pythagorean Theorem; II, Identities which we
would now treat algebraically, like (a + 6) 2 =a j -f2 ab + 6 2 ,
but which were then treated geometrically; areas; the Golden
Section; III, Circles; IV, Inscribed and circumscribed poly-
gons; V, Proportion treated geometrically; in part, a geometric
way of solving fractional algebraic equations; VI, Similarity
of polygons; VII-IX, Arithmetic (the ancient theory of num-
bers) treated geometrically; X, Incommensurable magnitudes;
XI-XIII, Solid geometry. We have in this text the earliest
extant evidence of a systematic arrangement of definitions,
axioms, postulates, and propositions. Euclid differs from most
of our modern writers on geometry in his greater seriousness
of purpose, in his desire to be more rigorous, and in the follow-
ing details of treatment : He has no intuitive geometry as an
introduction to the logical; he uses no algebra as such; he
demonstrates the correctness of his constructions before using
them, whereas we commonly assume the possibility of con-
structing figures and postpone our proofs relating to construc-
tions until we have a fair body of theorems ; he does not fear
to treat of incommensurable magnitudes in a perfectly logical
manner ; and he has no exercises of any kind.

Euclid's Other Works. Euclid wrote a number of other works.
Among them are the Phenomena? dealing with the celestial
sphere and containing twenty-five geometric propositions ; the
Data\ possibly a treatise on music; 2 and works on optics, 3

or else theKaToro/^ Ka^vos, both doubtful. See Heath
Euclid, I, 17. 8 '09TTt/c(.

THE ELEMENTS 107

porisms, 1 and catoptrics. 2 He also wrote a work on divisions
of figures, touching upon questions which arise, for example,
in surveying.'

Immediate Effect of Euclid's Work. The natural effect of
Euclid's work on geometry was to give rise to the feeling that
elementary geometry had attained to perfection and that the
next step in the progress of mathematics must be in the direc-
tion of some kind of higher geometry or else in the field of
mensuration. As a result, mathematics pursued both courses,
at first with little effect, as was the case with the predecessors
of Euclid, and then, when another genius appeared, with great
rapidity.

Minor Writers. There were, for example, at first such minor
writers as Co'non 1 of Samos, who, influenced by his observa-
tions of the coiled basketry work of the Egyptians, may have
invented the spiral of which Archimedes developed the proper-
ties. He is also mentioned by Apollonius (c. 225 B.C.) as hav-
ing studied the number of points of intersection of two conies.
There was also Nicot'eles 5 of Cyrene, possibly a student in
Alexandria, of whom Apollonius speaks as his predecessor in
the study of conies. Still another writer of influence appeared
in the person of the astronomer Aristar'chus, a native of Samos
but a teacher at Alexandria. It was he who first showed how to
find, by means of the Pythagorean triangle, the relative dis-
tances of the sun and the moon from the earth, and for nearly
two thousand years no better plan was known. His instruments
of observation were such as to make his result far from being
even approximately correct. 7 His greatest glory, however, lies

i Relating to methods of solution. See M. Breton, in Journal de Math, pures
et uppliquees, XX (i), III (2).

3 llepl Aicup&rewv pipXlov. R. C. Archibald, Euclid's Book on Divisions of
Figures, Cambridge, 1915.

4 K6vwi/. Fl. c. 260 B.C.

6 NetKor^Xrjs or NtKOT^Xijs. Fl. C. 250 B.C.

6 'Apfcrrapxos. Born c. 310 B.C.; died c. 230. See Sir T. L. Heath, Aristarchus
of Samos, Oxford, 1913; The Copernicus of Antiquity, London, 1920.

7 Carl Snyder, The World Machine, chap, vii, "Aristarchus and the distance
and grandeur of the sun" (London, 1907) ; Bigourdan, L'Astronomie, 252.

io8 ERATOSTHENES AND ARCHIMEDES

in the fact that he was the first to place the sun in the center
of the universe, asserting that the earth and the other planets
revolved about it, thus anticipating Copernicus by seventeen
centuries. In the field of arithmetic he found \/2 , possibly by
a method analogous to that of continued fractions.'

There are also extant various papyri of the Ptolemaic period
containing information about the financial problems of Egypt.
These problems relate chiefly to taxes and the cost of various
commodities, but they add nothing to our information as to
methods of calculation in ancient times. 2

Men like Conon were merely the usual heralds calling out
the approach of genius. The three men whose advent they
heralded were Archimedes, Apollonius, and Heron. Before
speaking of Archimedes, however, reference should be made to
a scholar whose interests were so scattered as to make his con-
tributions to pure mathematics of relatively little importance.
This man was the noet, librarian, arithmetician, and geographer
Eratos'thenes.

3. ERATOSTHENES AND ARCHIMEDES

Eratosthenes 8 lived some years after Euclid, and was one of
the greatest scholars of Alexandria. 1 His admirers exaggerated
his attainments by calling him "the second Plato," 5 and some
have thought that his nickname "Beta" signified that he was
the second of the wise men of antiquity, the Greek letter beta
standing for two. Others have said that he was called by this
name because his room in the university bore the number two.
But whether or not his followers ranked him second among the
wise men of Greece, we are justified in calling him the first
prominent geographer of antiquity. He was educated at Athens

TP. Tannery, Memoires de Bordeaux, V (2), 237; IV (3), 79.
2 For a bibliography, a list of the papyri, and a summary of the information
available, see H. Maspero, Les Finances de I'Egypte sous les Lagides, Paris,

Born at Cyrene, c. 274 B.C.; died c. 194.
4 Schmidt, Chrestomathie, I, 29, 114. See also Bibl. Math., XIII (3), 193.
r 'B. Baldi, Cronica de Matematici, p. 29 (Urbino, 1707), hereafter referred to
as Baldi Cronica\ Heath, History, II, 104.

WORK OF ERATOSTHENES

tog

and is known to have taught at Alexandria after c. 240 B.C.,
to have been librarian of the university, and to have been a
poet of some merit. His contribution to arithmetic was his
sieve, 1 a method of sifting out the composite numbers in the
natural series, leaving only primes. This he did by writing all
the odd numbeis and then canceling the successive multiples of
each, one after the other, thus: 3, 5, 7, 0, n, 13, y$, 17, 19,
**, 23, ?$, #?, 29, 31, 3<$, 35, 37, #, . . . 2 Prime numbers

MAP OF THE WORLD ACCORDING TO ERATOSTHENES

This shows the knowledge of the geography of the world in the 3d century B.C.

and should be compared with the map of Hecatams (517 B.C.) shown on

page 77. From Breasted's Ancient Times

have been studied from that time until the present, but no gen-
eral formula is yet known for detecting all of them. For exam-
ple, we do not yet know whether there are an infinite number
of primes of the form x 2 + i, whether 2 =x y has an infinite
number of prime solutions, or whether a prime number can
always be found between ri 2 and (n + i) 2 .

Earth Measure. Of the mathematical achievements of the
Greek astronomers none is more interesting than the meas-
urement of the circumference and diameter of the earth by
Eratosthenes, the first noteworthy step in the science of

OV. It was called the Cribrum Arithmeticum by Latin writers
2 M. C. P. Schmidt, loc. dt. 9 1, 114 (Greek text).

no ERATOSTHENES AND ARCHIMEDES

geodesy. 1 Learning that the sun at noonday was exactly in the
zenith at Syene 2 when it was 7 12' south of the zenith at
Alexandria, he decided that Alexandria was 7 12' north of
Syene on the earth's surface. Since the distance was known
to be 5000 stadia, and since 7 12' = T> V of 360, he judged the
circumference of the earth to be 50 x 5000 stadia, or 250,000
stadia. This result he altered to 252,000 stadia so as to have

DIAGRAM SHOWING METHOD USED BY ERATOSTHENES IN MEASURING

THE EARTH

Eratosthenes found that, when the sun was directly over Syene, at the First
Cataract, it was 7 12' south of Alexandria. From this he computed the cir-
cumference of the earth. From Breasted's Ancient Times

700 stadia, a more convenient number, to a degree, and from
this he computed the diameter to be the equivalent of 7850
miles, in our system of measure, which is only 50 miles less than
the polar diameter as we know it. 3

Eratosthenes also stated that the distance between the tropics
is I \ of the circumference, which makes the obliquity of the

1 Carl Snydcr, The World Machine, chap, vi, "Eratosthenes and the earliest
measures of the earth" (London, IQO;) ; Schmidt, "Erdmessung des Eratos-
thenes, 1 ' Greek text, Chrestomathie, II, 105; see also I, 29.

2 Sy e'ne, 2u^, the modern Assouan (Arabic from al + Syene} at the first
cataract of the Nile.

3 The problem of earth measure is more fully treated in Volume II, Chapter V.

GEODESY in

ecliptic 1 23 51' 20". Plutarch tells us that he found the sun
to be 804,000,000 stadia from the earth, and the moon to be
780,000 stadia, results which are remarkably close when we
consider the instruments then in use. That the knowledge of
geography had increased in the preceding 250 years may be
seen by comparing his map with that of Hecataeus (517 B.C.)-"
In one of his letters Eratosthenes also discussed the problem
of the duplication of the cube.

Archime'des* 1 was a friend of Eratosthenes and, if the testi-
mony of Plutarch is accepted, was related to King Hiero.
Leibniz praised his genius by saying that those who knew his
works and those of Apollonius marveled less at the discoveries
of the greatest modern scholars. 4 These words are justified,
for Archimedes anticipated by nearly two thousand years some
of the ideas of Newton and his contemporaries, and in the
application of mathematics to mechanics he had no equal in
ancient times. One of the Italian historians of mathematics 5
uses the happy phrase that he had fr a genius more divine than
human/' and Pliny calls him "the god of mathematics," a
phrase which one of his French translators felicitously renders
as "the Homer of geometry."

It is related that Archimedes set fire to the besieging ships
in the harbor of Syracuse by the aid of burning mirrors, and
there is nothing improbable in the idea that he may have

ir This term seems first to have been used by Ambrosius Aurelius Theodosius
Macrobius, a grammarian of c. 400.

2 On the general subject of the history of mathematical geography consult
S. Giinther, Studien zur Geschichte der math, und phytik. Geographic, with
extensive series of bibliographies (Halle a. S., 1879).

8 *A/>x i MSi7s. Born at Syracuse, the modern Suacusa, Sicily, 287 B.C.; died
at Syracuse, 212 B.C. Sir T. L. Heath, Archimedes, Cambridge, 1897 (hereafter
referred to as Heath, Archimedes) ; German translation by Kliem, Berlin, 1914;
P. Midolo, Archimede e il suo tempo, Syracuse, 1912; C. Snyder, The World
Machine, chap, x, "Archimedes and the first ideas of gravitation," London,
1907; Schmidt, Chrestomathie, III, 64, TOO; Heath, History, II, 16.

*"Qui Archimedem et Apollonium intelligit, recentiorum summorum virorum
inventa parcius mirabitur," Archimedis Opera (Geneva, 1768), V, 460. The
definitive edition is that of Heiberg, Leipzig, 1880-1915.

5 Baldi, Cronica, p. 26: "Hebbe ingegno piu divino, che humano."

112

ERATOSTHENES AND ARCHIMEDES

made them at least untenable by the soldiers. 1 A glance at the
map, and particularly at the lesser harbor then in use, and a
consideration of the fact that the ships were then hardly
larger than our pleasure yachts of today, and that they were

all anchored close to
the rocky shore, will
show that the task was
not so great as one might
at first suppose. At a
time when the breeze was
blowing in from the sea,
escape would have been
difficult, even with oars.

Archimedes and Me-
chanics. Plutarch, in his
life of Marcellus, relates
this incident to illustrate
the genius of Archimedes
in mechanics:

Archimedes . . . had
stated that, given the force,
any given weight could be
moved; and even boasted
. . . that, if there were
another earth, by going into
it he could remove this one.
Hiero being struck with
amazement at this, . . .
[Archimedes] fixed accord-
ingly upon a ship of burden . . . which could not be drawn out
of the dock without great labor and many men ; and loading her
with many passengers and a full freight, sitting himself the while
far off with no great endeavor, but holding the head of the pulley
in his hand and drawing the cords by degrees, he drew the ship in
a straight line as smoothly and evenly as if she had been in the sea.

1 On the subject of burning mirrors in Greek literature see Sir T. L. Heath,
in Bibl. Math., VII (3), 225.

ARCHIMEDES

Conjectural portrait bas-relief in the Capitoline
Museum at Rome. Date uncertain

MECHANICS

The Sand Reckoner. Archimedes saw the defects of the
Greek number system, and in his Sand Reckoner 1 he suggested
an elaborate scheme of numeration, arranging the num-
bers in octads, or the eighth powers of ten. In this work he

Athenian Walls Finished
Athenian Walls Unfinished
Syracusan Wall.

MAP OF ANCIENT SYRACUSE

Showing the general situation at the time of the Third Peloponnesian War and

continuing until the time of Archimedes. On the river Anapus here shown

papyrus still grows luxuriantly. From Breasted's Ancient Times

recognized, in substance, that a m a n = tf" + M , a law that is the
basis of our present operations by logarithms.

Other Mathematical Activities. Among his many activities
was the summation Vw 2 , the first example of the systematic
treatment of higher series of any kind. By the intersection of

s (psammites) ; Latin, arenarius or harenarius.

H4 ERATOSTHENES AND ARCHIMEDES

conies he was able to solve cubic equations which we should now
write in the form x* T ax 2 b 2 c = o. He also succeeded in
squaring a parabola, 1 that is, in finding the area of a segment,
showing that it is two thirds of a circumscribed parallelogram.
In the measure of a circle he showed that 3.! > TT > 3-^. In his
work in mensuration Archimedes included the sphere, cylinder,
and cone, the rules concerning the two latter having already
been known to Mensechmus. He also studied ellipsoids and
paraboloids of rotation. In his treatise on the mensuration
of circles and round bodies he was aided by the method of
exhaustion which had been developed by Menaechmus and
others. In the study of specific gravity and the center of
gravity of planes and solids he was a pioneer, and in the study
of hydrostatics he was unequaled in the Greek period. He is
also known for his study of spirals, possibly led thereto by
his friend Conon. In general, he stands out as one of the
greatest mathematicians and physicists in all history.

Method of Archimedes. In 1906 Professor Heiberg, who had
already edited the works of Archimedes, discovered in Con-
stantinople a manuscript on certain geometric solutions derived
from mechanics. 2 This is especially interesting from the fact
that it sets forth the method taken by Archimedes in deriving
geometric truths from principles of mechanics. Some idea of
the working of his mind may be obtained from the following:

After I had thus perceived that a sphere is four times as large
as the cone whose base is the largest circle of the sphere and whose
altitude is equal to the radius, it occurred to me that the surface of
a sphere is four times as great as its largest circle, in which I pro-
ceeded from the idea that just as a circle is equal to a triangle whose
base is the periphery of the circle and whose altitude is equal to the
radius, so a sphere is equal to a cone whose base is the same as the
surface of the sphere and whose altitude is equal to the radius of
the sphere. 8

1 See Volume II, Chapter X.

2 Translated by Lydia G. Robinson (Chicago, 1909) and by Sir T. L. Heath
(Cambridge, 1912).

3 For his method with respect to the parabola, see Volume II, Chapter X.

DEATH OF ARCHIMEDES 115

Death of Archimedes. Of the death of Archimedes at the
siege of Syracuse under Marcellus (212 B.C.), Plutarch has this
interesting record :

Nothing afflicted Marcellus so much as the death of Archimedes
who was then ; as fate would have it, intent upon working out some
problem by a diagram, and having fixed both his mind and his eyes
upon the subject of his speculation, he did not notice the entry of
the Romans nor that the city was taken. In this transport of
study and contemplation a soldier unexpectedly came up to him and
commanded him to go to Marcellus. When he declined to do this
before he had completed his problem, the enraged soldier drew his
sword and ran him through. Others write that a Roman soldier ran
towards him with a drawn sword and threatened to kill him, where-
upon Archimedes . . . earnestly besought him to stay his hand that
he might not leave his work incomplete ; but the soldier, unmoved
by his entreaty, instantly slew him. Others again relate that Archi-
medes was carrying to Marcellus some mathematical instruments,
dials, spheres, and angles, by which the size of the sun might be
measured . . . , and some soldiers . . . thinking that he carried gold
in a vessel, slew him. Certain it is that his death brought great
affliction to Marcellus; that he ever after regarded the one who
killed him as a murderer ; and that he sought for the kindred of
Archimedes and honored them with signal favors.

Discovery of the Tomb of Archimedes. In his Tusculan Dis-
putations (V, 23) Cicero relates that he himself discovered the
tomb of Archimedes "when the Syracusans knew nothing of it
and even denied that there was any such thing remaining."
He relates the incident as follows :

I remembered some verses which I had been informed were en-
graved on his monument, and these set forth that on the top of the
tomb there was placed a sphere with a cylinder. When I had care-
fully examined all the monuments ... I observed a small column
standing out a little above the briers, with the figure of a sphere
and a cylinder upon it. ... When we could get at it and were
come near to the front of the pedestal, I found the inscription,
though the latter parts of all the verses were effaced almost half
away. Thus one of the noblest cities of Greece, and one which at

n6 APOLLONIUS AND HIS SUCCESSORS

one time likewise had been very celebrated for learning, had known
nothing of the monument of its greatest genius, if it had not been
discovered to them by a native of Arpinum. 1

Of his works that have come down to us, those which are of
chief interest in the history of mathematics are the ones on the
quadrature of the parabola, on the sphere and the cylinder, on
the measure of a circle, on spirals, conoids, and spheroids, and
on notation. Archimedes seems also to have been interested in
astronomy, 2 although no work of his upon this subject is extant.

4 APOLLONIUS AND HIS SUCCESSORS

Apollo'nius of Per'ga 3 was known as "the great geometer"
because of his work on conic sections. He was educated in
Alexandria, and since he died under Ptolemy IV (Philop'ator,
reigned 222-205 B.C.) he very likely knew Eratosthenes. He
improved on the numeration system of Archimedes by using
io 4 as the base. This number, the myriad, 4 had long been in
use in the Orient, and was the base of all great systems of
numeration in the East as well as in Europe for many centuries.
His chief work was on the conic sections, to which he gave the
names ellipse, parabola, and hyperbola. 5

This work consisted of eight books, the first four of which
have come down to us in Greek and the next three in Arabic,
the last book being lost. In the first book Apollonius shows
how the three conies are produced from the same cone. He
uses a kind of coordinate system, the diameter serving for what
we call the #-axis, and the perpendicular at the vertex serving
for the ;y-axis. Books I-IV probably contain little that was

iTranslated by C. D. Yonge. London, 1891.

2 Livy (XXIV, 34) speaks of him as "unicus spectator coeli siderumque."

3 Apollonius PergaeuB, 'An-oXXi6wos. Fl. c. 225 B.C.; born at Perga, in Pam-
phylia, on the south coast of Asia Minor; Heath, History, II, 126.

4 Mtfpux, ten thousand. 5 "EXXei^is, 7rapa/3oXiJ, urrep/foXiJ.

6 Sir T. L. Heath, Apollonius of Perga (Cambridge, 1896); J. L. Heiberg,
edition of his works (Leipzig, 1891) ; E. Halley, Apollonii Pergaei Conicorum
libri octo . . . (Oxford, 1710), and his De Sectione Rationis libri duo (Oxford,
1706) ; G. Enestrom, Bibl. Math., XI (3), 7.

WORKS OF CONICS 117

not already known, but he arranged the material anew, as
Euclid had arranged systematically many propositions that his
own predecessors had known. Books V-VII seem to contain
the discoveries which he himself had made. Book V treats of
normals to a curve ; Book VI, of the equality and similarity of
conies; and Book VII, of diameters and rectilinear figures
described upon these diameters. In general, his propositions
are those which we now treat by analytic geometry, his method
being synthetic and analogous to that of Euclid with respect
to the circle and rectilinear figures.

Apollonius wrote various other works on geometry, including
one on plane loci. 1 Ptolemy speaks of him as having been also
a contributor to astronomy, but he probably confused him with
another Apollonius who lived a little earlier.

In the works of Apollonius Greek mathematics reached its
culminating point. Without Euclid as a guide, Apollonius
could never have reached the summit ; together, they dominated
geometry for two thousand years.

Minor Writers, After the death of Apollonius no great writers
on mathematics appeared for about two centuries. Greek civ-
ilization was receding. War was taking its toll. Arithmetic
seemed for a time to sink into a comatose state after the
slight attempts of Eratosthenes, and elementary geometry
seemed to die with Euclid and Apollonius.

There was some indication at this time of the coming birth
of a geometry of higher plane curves, just as the centuries
immediately preceding Euclid and Apollonius foretold the
appearance of these masters. The Greek civilization, however,
had not strength to fulfill the momentary promise. Many gen-
erations had to come and go before another people, living far
to the north, speaking a new language and making use of new
symbols and of a new method, brought to light the theory.

1 For his geometric works see a convenient list in Gow, Greek Math., pp. 246,
261. See also various restorations of his lost works, such as Woepcke's in the
Memoires presentes par divers savants a VAcadimie des Sciences, XIV; reprint
(Paris, 1856) ; G. Enestrom, Bibl. Math., XI (3), 7, 8. Since we are concerned
at present with the elementary field, the reader who wishes to consider the gen-
eral history of conies should consult the Encyklopddie, III (ii), i.

n8 APOLLONIUS AND HIS SUCCESSORS

Perseus. Of those who treated of these curves, one of the
first was Per'seus, 1 who lived c. 150 B.C. He wrote on sections
of the anchor ring, 2 but his works are known only by references
of later writers.

Nicomedes. Among the minor geometers of this period one
of the best known was Nicome'des, 3 who flourished c. 180 B.C.
and who invented a curve called the conchoid (mussel-shaped),
by which the trisection of an angle is easily effected. 4

Diocles. A probable contemporary of his, Di'ocles/' invented
the cissoid/' by which the duplication of a cube can be accom-
plished. 7 He also studied 8 the problem of Archimedes, to cut
a sphere by a plane in such a way that the volumes of tbe
segments shall have a given ratio.

At about the same time, say 180 B.C., Zenodo'rus 9 wrote
upon isoperimetry, 10 but most of his writings are lost.

A little later, Poseido'nius 11 taught in Rhodes, acqHred a
high reputation as a cosmographer and geometer, and had the
honor of claiming Cicero and Pompey as his pupils. His meas-
urements of the distance to the sun and of the circumference
of the earth, known to us through the works of Cleome'des
(c. 40 B.C.), were far from being as accurate as those of Era-
tosthenes, but his results seem to have been more generaPy
accepted by ancient geographers.

Contributions of the Astronomers. Owing largely to the
influence of the Egyptian and Chaldean priest-astronomers,
whose achievements had attracted more and more attention
on the part of the Greeks as intercourse became more free,

I Ilc/xreus. Heath, History, II, 203.

' 2 'S l TreTpa, a torus, or ring-shaped solid of revolution (in a special case, an
anchor ring), a solid already studied by Eudoxus.

3 NIKO/^STJS. His birthplace is unknown. Heath, History, II, 109.

4 This method and the use of other important curves are considered in Vol-
ume II, Chapter V. 6 Ato/cX^s. " Ivy-shaped " curve.

7 Such cases arc more fully discussed in Volume II, Chapter V.

8 In his llcpl irvpclwv. 9 7,r)v6d<apos.

10 Fourteen of his propositions have been preserved by Pappus (V, Pt. I) and
Theon of Alexandria (Comment. Almagest.).

II IloflretSwwos, born at Apameia, in Syria, c. 135 B.C.; died c. 44. Sometimes
called the Apamean. The name is also spelled Posidonius.

THE ASTRONOMERS 119

the 2d century B.C. was noteworthy for its advance in the study
of the stars. In this century two names stand out prominently,
not merely for their work as observers but because of their
mathematical attainments.

Hypsicles. The first of these astronomers was Hyp'sicles 1 of
Alexandria, who may have written the so-called fourteenth
book of Euclid's Elements, containing seven propositions on
regular polyhedrons. He was also interested in polygonal
numbers, 2 in progressions, and in certain indeterminate equa-
tions. His prime interest, however, was in astronomy, and
about his time there begins, among the Greeks, the division of
the circle into 360 and the definite, scientific use of sexagesimal
fractions, which the Babylonians had already suggested.

Hipparchus. About this time Hippar'chus, 3 working chiefly
in Rhodes, wrote a famous work on astronomy in which were
set forth the basic principles of the science. For this work he
needed to measure angles and distances on a sphere, and hence
he developed a kind of spherical trigonometry. Plane trigo-
nometry had as yet taken only rudimentary form, and, so far
as we know, there were no tables of functions. Hipparchus
worked out a table of chords, that is, of double sines of half
the angle, and thus was definitely begun the science of trigo-
nometry. With him also began the theory of stereographic
projection, a phase of geometry which Agatharchus (470 B.C.)
had already put in practice. Hipparchus used it for the pur-
pose of representing the projection of the celestial sphere upon
the plane of the equator. He left a catalogue of 850 fixed
stars, a number which Ptolemy (c. 150) increased to 1022 and
which was not further materially increased until modern times. 4

1 "r\f/iK\f}*. Fl. c. 180 B.C.. De Morgan places him c. 160 A.D. on general rumor,
but asserts that he could not have lived before 550 A.D. and places Diophantus
even later ! The Arab writers say that he was born in Ascalon. See Smith's
Diet, of Greek and Roman Biog., II, 54*. 2 See Volume II, Chapter I.

**Iinrapxo*. Born at Nicaea, in Bithynia, Asia Minor, c. 180 B.C.; died c. 125.

4 F. Boll, " Die Sternkataloge des Hipparch und des Ptolemaios," Bibl. Math.
II (3), 185. This article disputes the usual assertion that Hipparchus listed 1022
stars, and asserts that he knew only about 850, the rest being catalogued by
Ptolemy. See also Heath, History, H, 255.

120 APOLLONIUS AND HIS SUCCESSORS

Mathematics of Rome. The next two or three centuries wit-
nessed the rise of the Roman military power and the conse-
quent suppression of intellectual ideals. Art, philosophy,
science, politics, ethics, and mathematics had all sunk to a low
level. In literature, however, Rome made progress, although
Vergil took Homer as his model, and Cicero followed in the
footsteps of Demosthenes. Even as early as the 6th century
B.C. Etruscan art had become wholly Greek in its technique
and in its use of Greek mythology and customs. Rome simply
followed in the same lines, not merely in art but in letters and
science as well. In mathematics she showed no originality and
possessed no high ideals. The science was worth to her pre-
cisely what it would fetch in the coin of the realm and no more.
Rome created a goddess Numeraria, but she favored the acquisi-
tion of wealth rather than the creation of men. Money is
everything only when man is nothing. 1 Whenever a genius
like Heron of Alexandria, for example, arose in Greco-Roman
territory, his interests were usually in the applications of the
science as already developed, not in extending the boundaries.
As to Rome herself, it is noteworthy how many of her scholars
and literary men were born outside of Italy. Spain furnished
the two Senecas, Lucanus, Martial, Quintilian, and probably
Hyginus ; France, Favorinus and Domitius Afer ; Palestine,
Josephus ; Egypt, Philo ; and Greece, Plutarch and Epictetus. 2
If Pythagoras and Archimedes may be ranked as dwellers in
Italy, they were essentially Greek, and after the death of the
latter, exact science may be said to have taken her departure.
Cicero lamented this attitude of the Latin mind, contrasting
the high honor in which geometry was held among the Greeks
with the lack of appreciation on the part of the Romans. 3

In this period there flourished Marcus Terentius Var'ro
(116-28 B.C.), whom Quintilian called the most learned of the

^'L'argent n'est tout que dans les siecles ou les hommes ne sont rien." Libri,
Histoire, I, p. xiv.

2 Libri, Histoire, I, p. 53.

3 "In summo honore apud Graecos geometria fuit; itaque nihil mathematids
illustrius; at nos ratiocinandi metiendique utilitate huius artis terminavimus
modum." Tusculanarum Disgutationum Libri F, I, 2.

MATHEMATICS OF ROME 121

Romans. St. Augustine said of him that he had read so much
that we wonder that he had time to write anything, and that
he had written so much that we can scarcely believe that any-
one could find time to read it all. Of such a dilettante nothing
very scientific could be expected, and his one extant work 1
certainly has no great merit. In his Disciplinarum Libri he
treated of arithmetic, and he wrote a Mensuralia or De Men-
suris which related to practical mensuration, but, so far as we
know, his works were mere compilations. 2 He is the one of the
few pre-Christian mathematicians of whom we have a contem-
porary portrait, his profile appearing on a coin struck when he
was the proqucestor of Pompey.

Geminus. Among those who showed any interest in the his-
tory of mathematics at this time the best known is Gem'inus, 3
who was a native of Rhodes but may have written in Rome. He
is said to have divided mathematics into two groups, the pure
group, including arithmetic ( in the ancient sense) and geometry,
and the applied group, including mechanics, astronomy, optics,
geodesy, canonics, and logistic. Froclus, who lived in the sth
century A.D., tells us that he wrote a geometry which treated
of spirals, conchoids, and cissoids. Only one of his works is
extant, the Phenomena* a treatise on astronomy. Proclus has
numerous historical notes based upon the works of Geminus,
these notes being found mostly in fragments that remain of the
latter's Arrangement of Mathematics.

Minor Sources. Another source of information in the history
of mathematics was written by a Sicilian who flourished a little
later than this, having apparently been living in the year 8 B.C.
Diodo'rus, usually called Diodorus Siculus, was born in Agyr-
ium on the island of Sicily. He wrote forty books on history,
and while his style is not good and his facts are ill-sorted, his

1 De Re Rustica Libri III.

2 Montucla, Histoire, I (2), 488, tells us that a MS. of his arithmetic was
extant as late as the close of the i6th century, but it is now lost.

8 Tcfuvos or IVeti/os. Fl. c. 77 B.C. M. C. P. Schmidt, "Was schrieb Geminos?"
Philologus, XLV, 63 ; Chrestomathie, I, 45 ; C. Tittel, De Gemini Studiis Mathe-
maticiSj Leipzig, 1805; Heath, History, II, 222.

4 Ei<raywy)j els rb, ^aiv6fj^va. It was first printed, in Greek and Latin, in 1590.

122 APOLLONIUS AND HIS SUCCESSORS

works give us considerable information on the nature of the
mathematics studied in the classical period, particularly in
the schools of Egypt.

Still another writer to whom we are indebted for numerous
bits of knowledge relating to the ancient mathematics is
Strabo 1 the geographer. His second book deals with mathe-
matical geography and passes certain criticisms on the map of
the world prepared by Eratosthenes.

A little later than Geminus there lived P. Nigid'ius Fig'ulus, 2
a Pythagorean philosopher, who was highly esteemed in his
time. He was known to his contemporaries as a philosopher,
statesman, mathematician, and astrologer, but his contributions
had little influence. In the field of mathematical astronomy he
wrote DC Sphacra Barbarica et Graccanica, but only fragments
of his works have come down to us.

Probably contemporary with Figulus (but we are not sure
of the dates) there was a certain geometer named Dionysodo'-
rus, 3 who lived in Asia Minor, probably in Ami'sus. 4 He is
known for a solution of the problem proposed by Archimedes
and already discussed by Diocles, to cut a sphere by a
plane in a given ratio. 5 He also invented a new type of
conic sundial.

Caesar the Mathematician. It is also proper to speak of the
contributions of Julius Caesar (100-44 B.C.) to the reform of
the calendar (46 B.C.), a work undertaken with the help of
Sosig'enes" of Alexandria, an astronomer of whom almost
nothing further is known. Caesar himself was well versed in
astronomy and wrote a poem on the subject and a work DC
Astris, neither of which is extant. He also planned extensive
surveys of the empire.

About 40 B.C. the Greek astronomer Cleome'des 7 seems to
have flourished and to have composed a treatise on the circular

ifiorn c. 66 B.C.; died c. 24 A.D. 3 Aioiw6$wpos. Fl. c. 50 B.C.

2 F1. c. 60 B.C.; died in exile 44 B.C. 4 Anurfa.

fi lt is preserved in Eutocius's commentary (c. 560) on II, 5, of the work of
Archimedes on the sphere and cylinder. The method employed is that of the
intersection of a parabola and a hyperbola.

CESAR THE MATHEMATICIAN 123

theory of the heavenly bodies. 1 It was said three centuries ago
that manuscripts of his treatises on arithmetic and the sphere
were still in existence, 2 but they have since been lost.

Vitruvius. Of the Romans who made extensive practical use
of mathematics, none is more prominent than Marcus Vitru'vius
Pollio, commonly known as Vitruvius. Although the dates are
uncertain, it is thought that his great work on architecture :i
was written between 20 and 14 B.C. In Book IX he treats of
various types of sundials, and throughout the work he show?
his early training as an engineer. He also has something to
say on perspective, the ancient science of optics.

Referring to the same general line of applied mathematics,
Lucius Junius Moderatus Columel'la (c. 25 A.D.) of Gades
(Cadiz) wrote on agriculture 4 and included in his treatise a.
certain amount of information on astronomy, the calendar, and
the art of surveying.

Although the name of Gaius Plin'ius Secundus/' commonly
known as Pliny, is connected chiefly with his Natural History,
a work in thirty-seven books, it should be recalled that he in-
corporated a certain amount of mathematics in his treatise.
Book II contains a brief account of astronomy and is particu-
larly valuable because of its historical information. Our knowl-
edge of the practical use of the Roman numerals is enriched by
his frequent reference to them in this work.

Frontinus. Next to Vitruvius, the most prominent of the
Roman writers who made any practical use of mathematics was
Sextus Julius Fronti'nus (c. 40-106), general, superintend-
ent of water supply, and author of a work on war and of one on

flecopfas pxTcApw ptp\la, 5i5o, first printed in Latin at Venice,
in Greek, at Paris, 1539.

2 BaMi, Cronica, p. 43. The date of Cleomedes is often given as a century
later, but since he mentions no writer later than Poscidonius (died c . 44 B.C.).
it is probable that he lived in the ist century B.C.

*De Architectura Libri X (first printed at Rome r. 1486) ; The Ten Books on
Architecture, translated by M. H. Morgan, Cambridge, Massachusetts, 1914-

*De Re Rustica.

5 Born at Como, 23 ; died at the destruction of Pompeii, 79- The first name
often appears as Caius.

6 Strategematicon Libri IV. There is an edition by Gundermann, Leipzig, 1888.

124

APOLLONIUS AND HIS SUCCESSORS

aqueducts. 1 Some appreciation of the engineering works of
this period may be formed from a consideration of the aqueduct
of Claudius, which was constructed in the ist century A.D.
There are also preserved certain other books, generally be-
lieved to have been written by Frontinus, setting forth the prin-
ciples of land surveying as commonly practiced by the Romans. 2

THE AQUEDUCT OF CLAUDIUS

Constructed about the time of Frontinus, or probably just before his period of
activity. From Breasted's Ancient Times

Hyginus. Among those who made use of mathematics in the
work of surveying, Hygi'nus 3 (c. 120), known as Gromaticu?
(the surveyor), is one of the most prominent. The gromatici
were those who used the groma* an instrument employed in
measuring and laying out land, and Hyginus was well known
as a writer on the subject, although the fragments of his works

*De Aquaeductibus urbis Romae Libri II. First printed in Rome c. 1490;
there is a recent edition in English by Herschel, London, 2d ed., 1913.

2 His writings are collected in the so-called Codex Arceriamti. See K. Lach-
mann and A. Rudorff, Gromatici Veteres, being Vol. I of F. Blume, K.
Lachmann, and A. Rudorff, Die Schrijten der Romischen Feldmesser, 2 vols.
(Berlin, 1848). 3 Also spelled Hygenus and Higinus.

4 Also spelled gruma. It is from yv&nwv (gnomon), the shaft set up for
the ancient shadow-reckoning, for sundials, and for general astronomical
purposes. See Lachmann and Rudorff, loc. cit., I, 108.

ROMAN SURVEYORS 125

extant show no mathematical contributions to the science.
There was an earlier Hyginus, 1 who wrote a work of no merit
on astronomy, 2 and who is sometimes confused with his more
prominent namesake, the surveyor.

The Roman surveyor Balbus (c. 100) was very likely con-
temporary with Hyginus Gromaticus, but his contributions 3
were unimportant.

Theodosius. There lived about this time, and certainly in the
reign of Trajan (98-117), the mathematician and astronomer
Theodo'sius. 4 He seems, on the testimony of Suidas, to have
been a native of Tripoli, on the Phoenician coast. He wrote
several works, the most important being his treatise on the
sphere/' While this work possessed but little merit, it was
translated into the Arabic along with most of the other Greek
works on astronomy, and its brevity gave it considerable stand-
ing in Arabian schools. He is often confused with a Theodosius
of Bithynia, who lived c. 50 B.C. and wrote on the sundial.

5. PERIOD OF MENELAUS

Heron, or He'ro, of Alexandria represented the applications
of mathematics more completely than any other writer of about
the beginning of our era. He seems to have been an Egyptian,

Julius Hyginus, a friend of Ovid and therefore living in the ist
century B.C. 2 Poeticon astronomicon Libri III I.

3 Expositio et ratio omnivm jormarvm. See Lachmann and Rudorff , loc. cit.,
I, QI. 4 Oeo56<rtos. FI. c. loo.

6 20ai/9tjr& tv /3i/3Xfois rpttrtv. It was first printed, in Latin, at Paris, in 1520.
See A. A. Bjornbo, "Wann lebte Theodosios ?" Abhandlungen, zur Geschichte
der Mathematik, hereafter referred to as Abhandlungen, Leipzig, v. d., XIV, 64.

Q"llpwv. Fl. c. 50 A.D. This date is based upon the careful researches of
Wilhelm Schmidt, Heronis Alexandrini Opera quae supersunt omnia, Leipzig.
1899-1914. He places Heron in the ist century A.D. It was formerly thought
that he lived under the Ptolemies Philadel'phus and Euer'getes (283-222 B.C.),
and it was also asserted that he flourished c. 100 B.C. See also R. Meier, De
Heronis actate, Leipzig, 1905; Abhandlungen, VIII, 195; T. H. Martin, "Re-
cherches sur la vie et les ouvrages d'Heron d'Alexandrie " in Memoires presentes
Par divers savants a I' Academic des Inscriptions, IV (i) (Paris, 1854) 5 F- Hultsch,
Heronis Alexandrini geometricorum et stereometricorum reliquiae, Berlin, 1864.
Heath, History, II, 298, states that the evidence at present favors the 3d cen-
tury A.D., but at best the date is very uncertain.

126 PERIOD OF MENELAUS

his style not being that of a Greek. He invented the pneumatic
device commonly known as Heron's Fountain, a simple form
of the steam engine, and various other machines, showing much
ingenuity in all his numerous activities. He wrote on pneu-
matics, dioptrics, and mechanics, but from the standpoint of
mathematics his work on mensuration is the most interesting.
In this he treats of land surveying, probably summarizing the
methods in use by the Egyptians. As is the case with many
of the Greek scholars, some of his works are lost. His formula
for the area of a triangle, A = ^/s(s a) (s b) (s c) , is well
known. It appears in the geodesy, 1 which is contained in his
metrics, 2 but the proof is given (possibly an interpolation) in
his dioptrics. 3 In his geometry may be found the first definite
use of the trigonometric rule which we express by the for-

mula c = - cot - , where n is the number of sides of a regular
4 ;/

polygon of area A and side s, and where cA/s 2 . He com-
puted c for n 3, 4, , 12, but his method is unknown.
He was able to solve the equation which we write in the
fo/m ax 2 -h bx = c, so that the general quadratic as we know it
today was thus fully mastered by the Greek mathematicians.

About this time there lived Sere'nus of Antinoop'olis. 4 He
was the author of a treatise on the Section of the Cylinder,
containing thirty-three propositions, and of one on the Section
of the Cone, with sixty-nine propositions. The latter has con-
siderable work on maxima and minima. He also employed the
principle of a harmonic pencil of rays.

Menelaus. Of those who, in the period of decay of Greek
mathematics, showed any evidence of genius, Menela'us 5 was one
of the most prominent. He was a native of Alexandria and wrote

8 Ilepl Si67T7y>a$. On the formula in the Middle Ages, see G. Enestrom, Bibl.
Math., V (3), 311.

4 Antin'oe, 'Aprtpfaa, a city on the eastern bank of the Nile. He is often
called Serenus of Antissa. See Cantor, Geschichte, I, chap. 20. The date of
Serenus is quite uncertain. J. L. Heiberg, who edited his Opuscula (Leipzig,
1895), is inclined (p. xvii) to place him in the 4th century.
Fl. c. 100. Heath, History, II, 260.

MENELAUS AND NICOMACHUS 127

a treatise on the sphere, 1 particularly with respect to the geo-
metric properties of spherical triangles. He is known to have
made astronomical observations in Rome in the year 98. Be-
sides his treatise on the sphere he also wrote six books on the
calculation of chords. One of his most important theorems
states that if the three lines forming a triangle are cut by a
transversal, the product of the lengths of three segments which
have no common extremity is equal to the products of the other
three. This appears as a lemma to a similar proposition relating
to spherical triangles, "the chords of three segments doubled"
replacing " three segments." The proposition was often known
in the Middle Ages as the regula sex quantitatum because of
the six segments involved. He also knew the invariant property
of the anharmonic ratio of the line segments formed by a trans-
versal cutting four concurrent lines, a property the discovery
of which was formerly attributed to Pappus,- who flourished
about two centuries later.

Nicomachus. The best known of the Greek writers on arith-
metic, although not the greatest arithmetician, was Nicom'a-
chus 3 of Gerasa, his birthplace being probably the modern
Jerash, a town situated about fifty-six miles northeast of
Jerusalem. Since he mentions Thrasyl'lus, 4 who lived under
Tiberius (reigned 14-37), but says nothing of the work of
Theon of Smyrna, who lived under Hadrian (reigned 117-
138), and since his work was translated from Greek into
Latin by Appuleius, who lived in the time of Antoninus Pius
(reigned 138-161), we are safe in asserting that he lived about
the close of the first century.

Nicomachus wrote a treatise on music and a work in two
books on arithmetic. 5 The arithmetic as it has come down to
us may be only a compendium of a larger work which has

1 The Latin title, by which it is best known, is Sphaericorum Libri III. There
are editions by Maurolycus (1558), Mersenne (1644), and later writers.

2 See the Abhandlungen, XIV, 96, 99.

3 Nuc6/iaxos repa<n?i>6s, or Fepaon^s. Fl. c. 100. Heath, History, I, 97.

4 Probably Thrasyllus of Rhodes, died c. 36.

5 This was first printed in 1538, at Paris. The best edition is that of Hoche,
Leipzig, 1866,

128 PERIOD OF MENELAUS

long since been lost. Some such work seems to have been known
to Boethius (c. 510) and to have been used by him in compiling
his own treatise on the subject.

The Works of Nicomachus. Nicomachus belonged to the
Neopythagoreans, a sect of philosophers then flourishing in
Alexandria and trying to revive the teachings of Pythagoras.
It is therefore quite possible that Nicomachus made the journey
from Gerasa to Alexandria to study their doctrines. 1 At any
rate there is a considerable amount of the Pythagorean theory
of numbers in the tiresome treatment that he accords to arith-
metic. The period was one of intellectual decadence, and had
he not happened to summarize the ancient teachings in a field
that had not been entered by writers of the first rank, we
should never have heard of him. 2 His arithmetic 3 was rather
an introduction to the philosophy of the subject than a schol-
arly treatment of the science itself. For lack of anything bet-
ter it was adopted as a textbook in the few remaining schools
of philosophy, and Boethius did much to perpetuate its in-
fluence. 4 In the Philop'atriSj 5 probably a spurious dialogue
inserted among the genuine works of Lucia'nus/ 5 perhaps as
late as the loth century, 7 it is said of a certain man that
"he reckons like Nicomachus of Gerasa." 8 The remark
is ludicrous, and very likely was so intended, because there
is no evidence that Nicomachus could reckon with any skill
whatever, his interest being rather in the theory of numbers,
which, as we have seen, was quite distinct from logistic.

a On the rise of other intellectual centers, however, as Alexandria began to
lose prestige, see Kroll, Geschichte, p. 32.

2 P. Tannery, Revue philosophique, XI, 280.

3 Introductions Arithmeticae Libri duo\ GTetk,'Api0fjir)TiKTjscl<raywyi]spip\tap.
There are various editions in Latin and Greek. For a summary of the work
in English see G. Johnson, The Arithmetical Philosophy of Nicomachus of
Gerasa (Lancaster, Pennsylvania, 1916) ; Heath, History, I, 97, and II, 238.

4 It was also known in Hebrew, at least in paraphrase, in 1317. See M.
Steinschneider, "Die Mathematik bei den Juden," Bibl. Math., XI (2), 79.

6 <f>iX67rarpts.

6 AovKMvk. A humorous Greek writer of the 2d century.

7 For discussion, see M. C. P. Schmidt, Chrestomathie, III, 19.

a> s Ni/c6/*axoj 6 Tcpa<riiv6s.

NICOMACHUS AND THEON 129

Nicomachus mentions the sieve of Eratosthenes and often
cites the Pythagorean doctrines. He gives an extended treat-
ment of figurate numbers, and in his work appears an early
form of the Greek multiplication table. Extensive multiplica-
tion tables are found in the Babylonian tablets, but no earlier
Greek example is known, unless it be the one on the ancient
wax tablet mentioned on page 58- The medieval name, mensa
Pythagorica, may mean that a certain form of the multiplication
table, mentioned in Volume II, came from the Neopythagoreans.

Another work of Nicomachus, the Theologumena? has been
lost, the extant work by that name being a later compilation.

The'on 2 of Smyrna, so called to distinguish him from Theon
of Alexandria, who is mentioned later, lived in the time of
Hadrian (reigned 117-138). He was interested in arithmetic
and astronomy, and was the author of a work 3 which is com-
monly known in the Latin translation as the Expositio. Of
this work, which set forth the mathematics necessary for the
reading of Plato, two books are extant, one on arithmetic and
one on astronomy, and very likely these are all that he wrote.
The former resembles the work of Nicomachus but is less
systematic. 4

Marinus of Tyre. Mari'nus 5 of Tyre, a Greek scientist, who
lived c. 150, may properly be called the founder of ancient
mathematical geography. Apparently with greater success than
Hipparchus (c. 150 B.C.) he definitely located places by refer-
ence to two coordinates, namely, latitude and longitude, and
his maps set a new standard which the astronomer Ptolemy
recognized a little later. The maps themselves, however, have
not come down to us. He established the prime meridian

1 Qco\oyotiimcva ap

' 2 etav. Fl. c. 125. Heath, History, 11,238.

8 TQv Kara rb /jia.O'^fjiaTLKbv ^prja'lfjuav cts TJJV rov IlXdrwi'OS dvdyvwcriv (/3t/3Xfa).
The best Greek edition is E. Killer, Theonis Smyrnaei Philosophi Platonki
Expositio . . . (Leipzig, 1878). There is a French translation by J. Dupuis
(Paris, 1892).

4 On his astronomy see the edition by T. H. Martin, Theonis Smyrnaei
Platonici Liber de Astronomia (Paris, 1840) ,* J. B. Biot, review in the Journal
des Savants (April, 1850).

130

PTOLEMY AND HIS SUCCESSORS

through the Fortunatae Insulae? and this meridian was adopted
by Ptolemy. At a later date the meridian was more definitely
located through Ferro, 2 one of the Canary Islands, and this
position was recognized until modern times.

6. PTOLEMY AND HIS SUCCESSORS

Ptol'cmy, or Claudius Ptolemaeus, 3 whose period of greatest
activity was c. 140-160, did for astronomy what Euclid did for
plane geometry, Apollonius for conies, and Nicomachus for

PTOLEMY'S MAP OF THE WORLD

This shows the great growth in the knowledge of geography from the time of
Eratosthenes. See page 109. From Breasted's Ancient Times

arithmetic. He brought together in a single treatise the dis-
coveries of his predecessors, arranging the material systemati-
cally, and, like the first two mentioned, was possessed of such
genius as to make his work a standard of excellence for many

1 Ai TUV MttKdpwv vijffoi, Islands of the Blessed, probably including the Canary,
Madeira, and Azores groups. It \vas here, in what Milton calls the "thrice
happy isles," that Hesiod and Pindar placed Elysium.

2 Ancient Pluvialia, the nXowrdXo of Ptolemy.

8 HroXcjixcuos KXatf&os. Born c. 85; died c. 165. Heath, History, II, 273.

THE ALMAGEST 131

centuries. As to his life we know only that he taught in
Athens and Alexandria. His greatest work, commonly known
as the Almagest* contains much information on the history
of ancient astronomy. He also wrote on the planisphere, on
music, and on applied mathematics. There is a question as
to the genuineness of a work on optics that is often attributed
to him. In the Almagest there is a summary of the computa^
tions of Eratosthenes, Poseidonius, and others as to the size
of the earth, the position of certain places, and the size of
islands and countries. In the application of mathematics to
astronomy and geography Ptolemy stands preeminent among
Greek scholars. He extended the use of sexagesimal fractions
and elaborated the table of chords already used by Hipparchus.
He also wrote a treatise on the postulate of parallels and a work
of an astrological nature which is generally known in English
as the Tetrabiblos*

Minor Writers. Among the minor writers who came after
Ptolemy there was the jurist Domi'tius Ulpia'nus (c. 170-228),
a prolific contributor to the law and the compiler of the first
table of mortality of which we have any knowledge.

Probably in the same period (c. 180) there lived the Roman
surveyor Marcus Junius Nip'sus, but his contributions to the
science relate chiefly to mensuration and are unimportant. 3 At

1 The original title is usually given as
but on this question see J. L. Heiberg, Ptolemaei Opera, II, p. cxl (Leipzig,
1898-1907). Since he wrote another o-iVra, the Arabs seem to have called the
greater work al ncyd^y, and afterwards al /ue-yhmy (Smith's Diet, of Greek and
Roman Biog., Ill, 570). From neylcmi, with the Arabic al (the), the Arabs
made the word which has come to us as Almagest, so that to speak of "the
almagest" is like speaking of "the the-greatest." The work was first printed,
in an abridged form prepared by Regiomontanus, at Venice in 1496; the first
complete edition appeared in Venice in 1515. For the latest work on the subject
see C. H. F. Peters and E. B. Knobel, Ptolemy's Catalogue of the Stars, a
Revision of the Almagest, Washington, 1915.

2 Ter/wtj&pXos (rtivraZis. The first printed edition appeared at Venice, 1484;
first Greek edition, Nurnberg, 1535. It is also known by the Latin name,
Quadripartitum.

3 They are given in the Codex Arcerianus under the following titles : fiuminis
uaratio, limitis repositio, uarationis repositio, lapides etc., podismus. See Lach-
mann and Rudorff, Gromatici Veteres, I, 285.

132 PTOLEMY AND HIS SUCCESSORS

about the same time (c. 200) there flourished another Roman
surveyor named Epaphrodi'tus, who wrote not only on survey-
ing but also on the theory of numbers. 1 He showed that if r is
the radius of the circle inscribed in a right-angled triangle of
sides 0, b y and hypotenuse c, then 2 r = a -}- b c. It is proper
to refer, chiefly for the sake of showing the low estate to
which learning had fallen, to the Chronicon of Sextus Julius
Africa'nus (c. 220), a considerable part of which work is lost,
but the extant portion of which contains information of value
on the history of the calendar, and also to another work attrib-
uted to him, in which some notes appear on the history of other
branches of mathematics.

Among the lesser Roman geometers and astronomers there
was Censori'nus (c. 235), who wrote a book (238) entitled
De die natali, a work primarily on astrology but containing a
limited treatment of chronology, astronomy, and computation.
It has been stated that he also wrote a geometry, although the
work, if it ever existed, is lost.

We are also told by early writers of the interest taken in
mathematics by the wealthy Roman dilettante Quintus Sam-
monicus Sere'nus (died 212). He was a prolific writer and his
works include medicine, mathematics, and other sciences, but
in general they merely show the debased state of learning. He
is not to be confused with Serenus of Antinoopolis, already
mentioned.

A little later (c. 275) Spo'rus 2 of Nicae'a wrote a work from
which we derive certain information relating to the history of
early mathematics, particularly with reference to duplicating
the cube and squaring the circle. He may have been the teacher
of Pappus, who is usually put a century later. 3

1 V. Mortet, "Un Nouveau Texte des Trails d'Arpentage et de Geometric
d'Epaphroditus et de Vitruvius Rufus," Notices et Extraits des Manuscrits de
la Bibl. Nat., XXXV (1896), p. 510.

2 Probably the same as Porus of Nicaea. The date is very uncertain; it is
often given as of the 2d century.

3 P. Tannery, Memoir es de Bordeaux y V (2), 211, and Memoires scientifiques,,
Paris, 1912, I, 178, thinks he was the teacher of Pappus, or possibly one of his
older pupils. The dates are so uncertain as to allow of either possibility.

MINOR WRITERS 133

It is possible that Metrodo'rus, 1 the compiler of the arithmeti-
cal epigrams in the Greek Anthology, 2 flourished about 325, but
the date c. 500 is more probable. These epigrams were puzzle
problems, like the one about the pipes filling the cistern, which
we should now solve by algebra. For a long time such problems
have interested students of arithmetic and algebra, and will
doubtless continue to do so for all time to come. Sir Thomas
Heath believes that their use dates back at least to the sth
century B.C.

7. DlOPHANTUS AND HIS SUCCESSORS

Diophan'tus { of Alexandria was one of the greatest mathe-
maticians of the Greek civilization. That he flourished about
the middle of the 3d century seems now fairly certain, al-
though various other dates have from time to time been given.
Psellus (nth century) says that Diophantus and Anato'lius 4
wrote on Egyptian computation and that "the very learned
Anatolius collected the most essential parts of the doctrine
. . ., dedicating his work to Diophantus." Very likely, there-
fore, Anatolius may have studied under Diophantus. Since he
became bishop of Laodicea c. 280, he doubtless wrote this
work some time before that date, and so Diophantus, who
seems to have been the elder, probably flourished c. 250-275^

1 Mi?rp65b>pos. 2 English translation by W. R. Paton, London, iqi8, p. 25.

:i Ai60aros. Also written Diophantes, Diophantis, and Diophantos. Fl. c. 250-
275. There were several writers by this name. Sir T. L. Heath, Diophantus
of Alexandria, 2d ed. (Cambridge, 1910). On the text see also Tannery's edition
of his Opera Omnia (Leipzig, 1893, 1895). The first Latin edition of his works
was that of Xylander (Wilhelm Holzmann), Basel, 1575; the second, that of
Bachet (Paris, 1621), contained the Greek text; the third was that of Bachet
with Fermat's notes, Toulouse, 1670. Stevin published a French translation of
the first four books in his Arithmttique, Leyden, 1585, with editions in 1625 and
1634.

4 " AVCLT&\IOS . Bibl. Math., IV (3), 396. Some fragments of his works arc
given in J. A. Fabricius, Bibliotheca Graeca, III, 275 (Hamburg, 1716). His
computus was published by J. P. Migne, Patrologia Graeca, Vol. X (Paris,

5 Heath, Diophantus^ 2d. ed., p. i. Tannery confirms this by an ingenious
study of the price of wine at this time, finding that it conforms to that which
Diophantus gives. See his Mtmoircs scicntifiques, I, 62 (Paris, 1912).

134 DIOPHANTUS AND HIS SUCCESSORS

All that is known of his life is given in a curious problem
in the Greek Anthology, probably dating from the 5th cen-
tury. The problem states that his boyhood lasted of his
life, his beard grew after y^ more, after f more he married,
5 years later his son was born, the son lived to half his father's
age, and the father died 4 years after his son. While the state-
ment is obscure at one point, it is generally thought to mean
that Diophantus married at 33 and died at 84.

Works of Diophantus. Diophantus wrote three works:
(i) Arithmctica^ originally in thirteen books, of which six are
extant 1 ; ( 2 ) a tract De polygonis numeris 2 of which a portion
is extant; (3) a number of propositions under the title of
porisms. Of these, the work of greatest importance is the
Arithmetic a. This work relates, as the title indicates, to the
theory of numbers as distinct from computation, and covers
much that is now included in algebra. The equations of the
first degree are determinate and are so framed as to give posi-
tive values for the unknowns. In solving determinate quad-
ratic equations Diophantus used only one root, even where
both are positive. He solved a single special case of a cubic
equation, but it is thought that further work on such equations
may have been given in the lost books. His indeterminate
quadratic equations are generally of the types Ax* + C=/ 2
and Bx+ Cy 2 . His simultaneous quadratics relate only to
special cases.' 5

Diophantus introduced a better algebraic symbolism than
had been known before his time. In general he anticipated
by several centuries the progress of algebra, as this progress
appears in the works of other writers ; and his work, while
known to the Arabs, was not really appreciated until its dis-
covery in Europe in the i6th century. He stands out in the
history of science as one of the great unexplained geniuses.
We do not know what teachers inspired him, we do not know

1 Heath, p. 16, lists altogether twenty-five MSS., each containing more or less
of the works of Diophantus. See also the Tannery edition, I, xxii.

2 Ilepl iro\vy&vwv dpi0/xwi>. See the Tannery edition, I, 450 ; Heath, Diophan-
tus, 2d ed., p. 247. 3 Heath, Diophantus, 2d ed., p. 93.

WORKS OF DIOPHANTUS 135

the books he read, and we cannot explain how it happened
that he appeared like a giant in a century of pigmies. Perhaps
Seneca's statement that "no age is shut against great genius" 1
is the only explanation to be expected.

Lesser Writers. Not far from this time there also flourished
the Neoplatonist Porphy'rius, 2 originally known as Malchus 5
the Tyrian and commonly spoken of as Porphyry. He wrote
on the life of Pythagoras 1 and a work on the music of Ptolemy.
He resided in Athens and Rome, spent some time in
Sicily, and is known chiefly for his philosophical works and
his antagonism to Christianity. His tomb, or one traditionally
designated as his, is still pointed out in Constantinople.

One of the pupils of Anatolius and Porphyrius was lam'bli-
chus, r> the author of several works, including one on arithmetic.
He wrote a commentary on Nicomachus, and we are indebted
to him for considerable information relating to the latter, to
Pythagoras, 7 and to other Greek writers. To him is due the
theorem that if a number equal to the sum of the three integers
372, 3^1, 373 2 is taken, and if the separate digits of this
number are added, and the digits of this result, and so on, the
final sum is 6.

About 340 Julius Fir'micus Mater'nus, a Sicilian, wrote a
work entitled Eight Books on Mathematics* but concerned

1 "Nullum saeculum magnis ingeniis clausum cst."

2 llop<f>6pios. Born in Syria, 232 or 233; died c. 300.

3 From Melekh, the Hebrew for "king"; in the Greek of that period,
3HXxos. The name was changed, according to tradition, to Porphyrius (wearer
of the purple).

4 TlvOay&pov jSfos, possibly a fragment of his history of the philosophers.

5 'l<fyi|8\ixos. Born at Chalcis, Coelesyria, c. 283 ; died c. 330.

{ 'It appeared in various editions in the i6th century. The title-page of the
1668 edition begins: Jamblichus Chalcidensis ex Code-Syria in Nicomachi
Geraseni Arithmeticam introductionem (Arnheim, 1668).

7 Ufpl UvBay6pov alpfoewt, of which four books are extant, the first containing
the life of Pythagoras. The latter was published in Greek and Latin, at Franeker,
in 1598. There have been other editions. See Bibl. Math., VIII (3), 309.

*lulii Firmici Materni Junioris Siculi V. C. Matheseos Libri VIII. It was
first printed at Venice in 1497. The definitive edition is that of Kroll and Skutsch,
Leipzig, 1897-1913. L. Thorndike, "A Roman Astrologer as a Historical
Source: Julius Firmicus Maternus," Classical Philology, VIII, 415.

136 DIOPHANTUS AND HIS SUCCESSORS

exclusively with judicial astrology according to the precepts
of the Babylonians and Egyptians. Such works have little
place in a history of mathematics except as they show from
time to time the tendencies of the devotees of the science.

There are also various other isolated cases of mathematical
interest in this period of general decay of scholarship, as in the
constructing of an astrolabe by Syne'sius of Cyrene (c. 378-
c. 430), the poet and orator, a pupil of Hypa'tia. He became
bishop of Ptolemais in 410.

About 390 The'on of Alexandria, known as Theon the
Younger, father of the learned Hypatia, edited Euclid's Ele-
ments and the great work of Ptolemy, wrote various scientific
treatises, and set forth a method for finding square roots by
the aid of sexagesimal fractions. Manuscripts of his edition
of Euclid have been helpful to modern writers in determining
the accurate text of the Elements.

A little later (c. 450) Domni'nus 1 of Larissa, in Syria, wrote
on arithmetic, philosophy, and optics. He followed the geo-
metric, deductive method of Euclid rather than the inductive
method of Nicomachus, and seems to have had access to some
important work that is now lost on the theory of numbers.

Pap'pus 2 of Alexandria, a late Greek geometer, flourished
probably in the 3d century, although the date is uncertain.
Suidas (c. loth century), not a very careful writer, however,
places him in the reign of Theodosius (379-395), but others
believe him to have lived two centuries earlier. Of his greatest
work, the Mathematical Collections, 3 only the last six of the
eight books that it originally contained have come down to us.
The third book treats of proportion, inscribed solids, and the
duplication of the cube; the fourth, of spirals and of such
other higher plane curves as the quadratrix ; the fifth, of maxi-
mum and isoperimetric figures; the sixth, of the sphere; the

1 Ao/mw>s. P. Tannery, Darboux Bulletin, VIII (2), 288.

2 Ildmrof . Fl. c. 300. Heath, History, II, 355.

8 Ma0i7/xariKwi/ <rwa.ywyu>v j3i/3Xa. The text of this work in Greek and Latin
was published with notes by Hultsch, Berlin, 3 vols., in 1876-1878. There was
a Latin edition published at Pesaro in 1588, reprinted without change at Venice
in 1589 and at Pesaro in 1602. See also Bibl. Math., XII (3), 252.

PAPPUS 137

seventh, of analysis and its history among the Greeks; and
the eighth, of mechanics. Two well-known theorems bear his
name, one on the generation of a solid by the revolution of a
plane figure about an axis, later known as Guldin's Theorem,
and the other a generalization of the Pythagorean Theorem. He
also knew the doctrine of the involution of points and the con-
stancy of anharmonic ratios in the case of a transversal cutting
a pencil, the latter having already been known to Menelaus.

Hypa'tia 1 of Alexandria was the first woman who took any
noteworthy position in mathematics, and on this account and
because of her martyrdom she has occupied an unduly exalted
place in history. She was the daughter and pupil of Theon,
and such were her attainments that she was called upon, so
tradition says, to preside over the Neoplatonic School at
Alexandria. Much that passes for history in her case seems to
be fiction, as the statement of Suidas (c. roth century) that
she married Isidorus of Gaza, the Neoplatonist. It seems cer-
tain, however, that she was slain in one of the city brawls
between followers of rival sects. Suidas says that she wrote a
commentary on an astronomical table of a certain Diophantus,
possibly the algebraist, and one on the conies of Apollonius.
Her works, however, are all lost. 2

Pro'clus, surnamed the Successor 3 because he was looked
upon as the successor of Plato in the field of philosophy, 4

Born at Alexandria, c. 370; died at Alexandria, 415.

2 For the romantic side of her life, see J. Toland, Hypatia, or the history
of a most beautiful, most vertuous, most learned . . . lady, London, 1720;
C. Kingsley, Hypatia, London, 1853; F. Mauthner, Hypatia, Roman aus dem
Alterlum, 2d ed., Stuttgart, 1892. For a critical study, see R. Hoche, "Hypatia,
die Tochter Theons," in Philologus, XV (1860), 435; S. Wolf, Hypatia, die
Philosophin von Alexandria, Vienna, 1870; W. A. Meyer, Hypatia von Alex-
andria, Heidelberg, 1886. See also Heath, History, II, 528.

3 llp6K\os AiASoxos. Born at Byzantium, c. 412; died 485. A certain Marinus,
not to be confused with Marinus of Tyre, gives his birth as February 8, 412.
The name also appears as Proculus. The best of the partial editions of his
works is that of Cousin, Prodi Opera, 6 vols., Paris, 1820-1827; 2d ed., 1864.
The best edition of his commentary on Euclid I is that of G. Friedlein, Prodi
Diadochi in primum Endidis Elementorum librum, Leipzig, 1873. His Institutio
Physica, edited by A. Ritzenstein, was published at Leipzig in 1012.

4 Or because he succeeded Syrianus, the philosopher, at Alexandria.

138 THE ORIENT

studied at Alexandria and taught at Athens. He was a prolific
writer and his works include a paraphrase of difficult passages
from Ptolemy, a work on astronomy, a commentary on Euclid I,
and a brief treatise on astrology. He also shows evidence of
a study of certain higher plane curves. His works are valu-
able sources of information on the history of Greek geometry.
For information concerning his life we are indebted to Ma-
ri'nus, 1 of Flavia Neapolis in Palestine (the old Sichem), who
succeeded him in 485." This Marinus, very likely a Jewish
scholar/ also wrote an introduction to the Data of Euclid.

At about this time Victo'rius 1 of Aquitania (457) wrote a
Canon Paschalis, one of the first of the Computi, books on the
finding of the date for Easter. He suggested beginning our era
at the time of the first full moon after the death of Christ. He
also wrote a calculus, that is, a practical arithmetic. In this
he gave considerable attention to fractions and to tables for
the multiplication of large numbers.

The name of Capella might, for chronological reasons, be
included in this chapter, but on account of the relation of his
work to that of writers of the 6th century it is considered in
Chapter V.

8. THE ORIENT

China. The period from 300 B.C. to 500 A.D. was one of
mathematical activity in China, and some slight but noteworthy
trace remains of an interest in numbers in Japan. 5 At the be-
ginning of this period the event of greatest concern in the his-
tory of Chinese mathematics was the burning of all books 6
(213 B.C.), as already mentioned in Chapter II, by order of
the emperor Shi' Huang-ti, 7 founder of the Ch'in (Ts'in)

As stated above, he must not be confused with the astronomer
already mentioned.

2 His life of Proclus was first printed at Zurich in 1559.
3 S. Krauss, Jewish Quarterly Review, 1897, p. 518.

4 Often written Victorinus. It is thought that he was born in Limoges.

5 Smith and Mikami, History of Japanese Mathematics, chap, i (Chicago,
1914) ; hereafter referred to as Smith-Mikami.

6 An exception was made of books on medicine, agriculture, and divination.

7 She Huang-ti, "the First Emperor,'* born 259 B.C.; died 210 or 211.

CHINA 139

Dynasty (221 B.C.), who wished to appear in the eyes of poster-
ity as the creator of a new era of learning. The penalty for not
burning the books was branding and four years' service on the
Great Wall. The records say that four hundred and sixty
scholars protested against this odious law and were buried
alive as an example to others. How many of the ancient clas-
sics survived, or how many were faithfully transmitted by
means of copies made from memory, we do not know, but it
is probable that Chinese scholars will in due time apply the
methods of textual criticism to the determination of this point.

About this time, and probably just after the burning of the
books, there lived the learned Ch'ang Ts'ang (c. 250-152 B.C.),
a statesman of highest rank, who wrote (176 B.C.) a new
K'iu-ch'ang Suan-shu (Arithmetic in Nine Sections) * basing it
upon fragments of the earlier work of the same name. The
nine chapters or sections have already been given (page 32).

Ch'ang Ts'ang gave the area of a segment of a circle as
\(c + a)a, where c is the chord and a is the altitude of the
segment. Among his problems is that of finding the height of
the trunk of a tree, the upper part of which was 10 feet high
but has fallen over and reaches the ground 3 feet from the
base. The rule for the area of the segment of a circle is later
found in the work of the Hindu Mahavira (r. 850), and the
problem about the tree is found in various Hindu mathematical
works after the time of Aryabhata (c. 510).

Minor Chinese Writers and Events. The period following the
burning of the books was, as might have been expected from
the need thus created, one of considerable intellectual activity.
In this respect, but from a wholly different cause, it was not
unlike the century following the impetus given to learning by
Plato. Ch'eng Kiang Chen (also known as Chun Shuen), who
died in 200 B.C., wrote on knotted cords which perhaps, like the
Peruvian quipu, were for keeping accounts.

iR. L. Biernatzki, u Die Arithmetik der Chinesen," Crelle's Journal, LIT, n ;
A. Wylie, Chinese Researches, Part III (Shanghai, 1897). These writers put the
date c. 100 B.C., but Ch'ang Ts'ang appears to have died in 152 B.C., upwards
of i oo years old, and to have written the work in 176 B.C. See Mikami, China, p. 9.

140 THE ORIENT

Then as always in Chinese history the regulation of the
calendar occupied the attention of scholars. Thus it is recorded
that c. 104 B.C. the emperor reestablished official astronomy
and a new calendar was devised. 1 It is also worthy of note,
as bearing upon the arithmetic of commerce, that about this
time (135 B.C.) coinage became a government prerogative. 2
The Chinese annals of this period also speak of the efforts of
the emperor 3 to open up communication with the region about
the river Oxus, all such efforts having relation to the unsolved
problem of the transmission of mathematical knowledge be-
tween the East and the West. The famous Chinese general
Ch'ang K'ien went to the countries of the Jaxartes and the
Oxus in the 2d century B.C., and about 100 B.C. an envoy was
sent as far west as Lake Baikal. 4 This intercourse between
the East and the West was maintained for several centuries.
For example, an Aramaic manuscript of the ist century
(c. 1-20), the earliest known specimen of rag paper, has been
found on the Chinese border. 5 That China had intercourse
with India is evident from the fact that the records show such
relations as early as 218 B.C. and that the name Sin-du appears
in the Chinese annals of about 120 B.C. It is also well estab-
lished that China was known in the West at this period.
Ptolemy the astronomer (c. 150) speaks of the country under
the name of Thin, and in 166 Marcus Aurelius sent an embassy
to the emperor's court.

1 J. B. Biot, Etudes sur Vastronomie Indienne et sur Vastronomie Chinoise,
p. 299. Paris, 1862.

2 H. B. Morse, " Currency in China," from the Journal of the North-China
Branch of the Royal Asiatic Society, XXXVIII; reprint, p. 2.

3 Wu-ti (140-87 B.C.). On the general subject of the relations of China with
the West see S. W. Williams, A History of China, p. 58 (New York, 1897) ;
F. Hirth, "The Story of Chang K'ien," Journal of the Amer. Oriental Soc.,
XXXVII, 89, 185, 186; T. W. Kingsmill, "The Intercourse of China with
Central and Western Asia in the 2d Century B.C.," Journal of the China Branch
of the Royal Asiat. Soc., XIV (N.S.), i; Hirth and Rockhill, Chau Ju-Kua:
His Work on the Chinese and Arab Trade in the twelfth and thirteenth cen-
turies (Petrograd, 1911), the preface to which considers the whole question from
earliest times to the i3th century.

4 E. Bretschneider, Mediaeval Researches, I, 32. London, 1910.

5 M. A. Stein, Ruins of the Desert of Cathay , II, 114. London, 1912.

CHINA 141

It is probable that this continued interchange of thought is
one of the causes of the frequent changes in the calendar and
of the study of the related geometric figure of the circle. About
25 A.D. there lived a well-known philosopher and astronomer
named Liu Hsiao, who was of the Imperial house of the Han
Dynasty. 1 He was one of the most prominent of the " circle
squarers" of his day. His son, Liu Hsing, 2 devised a new
calendar, 3 thus using his time to better advantage than the
father. A few years later (c. 75 A.D.) Pan Ku wrote a work 4
in which the use of the bamboo rods, a primitive form of abacus,
is mentioned. At about this time Ch'ang Hong (78-139), chief
astrologer and minister under the emperor An-ti, constructed
an armillary sphere and wrote on astronomy and geometry. He
gave VK> as the value of TT, this being one of the earliest uses of
this approximation. 5 Perhaps contemporary with him, although
we are uncertain, there lived Ch'ang ch'un-ch'ing, who wrote
a commentary on the Chdu-pei. About 1 90 there flourished Ts'ai
Yung, 6 one of the numerous experts on the calendar, but his
works are lost. He was sentenced to death for political reasons,
but the sentence was commuted to having his hair pulled out.
His convivial habits gave him the name of Drunken Dragon.

Wu-ts'ao Suan-king. Possibly about the beginning of the
Christian era, for the date is so uncertain 7 that we are
not safe in fixing the time even within the limits of several
centuries, there was written one of the best-known but
least worthy Chinese classics on mathematics, the Wu-ts'ao
Suan-king. 8 The author seems to have been Sun-tzi', 9 but

1 This dynasty lasted from 206 B.C. to 25 A.D.
2 Biot (p. 305) transliterates the name as Lieou-hin.

3 The San-t'ung calendar, devised in the year 66.

4 The Han Shu. Pan Ku died in 92.

5 On account of the unreliability of early Chinese texts, all such statements
are open to some doubt. Born 133; died 102.

*Mikami, loc. cit., p. 37, says in the former (beginning c. 206 B.C.) or later
(c. 25-220 A.D.) Han Dynasty. * Arithmetic Classic in Five Books.

Also given as Sun Tsze, Sun Tsu, Suentse, Sun Wu tsze, and Sun Tsu Yen
Ch'i-sun. The work is also known as the Sun-tzi Suan-king. Pere Vanhee
puts the date as probably the ist century A.D., while Biernatzki (p. 21) says that
Sun-tzi' may have lived 220 B.C.

I 4 2 THE ORIENT

even as to this we are uncertain. The work is obscurely written
and is not so accurate in its statements as the Nine Sections.
It relates chiefly to the mensuration of areas. A single prob-
lem will serve to show its nature :

" There is a quadrangular field of which the eastern
side is 35 paces, the western side 45 paces, the southern side
25 paces, and the northern side 15 paces. Required the area
of the field."

Evidently a solution is impossible through lack of sufficient
data; but the author assumes that he may take one fourth
the product of the sums of the pairs of opposite sides, 1 such
approximations as this being not uncommon all through the
East in these early times.

Liu Hui. The best-known Chinese mathematician of the 3d
century was Liu Hui. 2 In 263 he wrote the Sea Island Arith-
metic Classic? a work which probably took its name from
the first problem that it contains, this problem beginning with
the statement, "There is a sea island that is to be measured."
The work is concerned with the mensuration of heights and
distances, the rules seeming to show some familiarity with the
manipulation of algebraic formulas.

Liu Hui also wrote a commentary on the Nine Sections, and
it seems to have been in the performing of this task that he
accumulated the materials for his "Sea Island" work.

Minor Chinese Writers from 200 to 500. Of the minor writers
of the 3d century mention may properly be made of Wang Pi
(c. 225-249), the leading authority on the mysticism of the
I-king*; of Wang Fan (229-267), the astronomer, who as-
serted that 7r = - 1 j 4 5 2 -; of Siu Yo (c. 250), who wrote the
Omissions noted in the Art of Numbers 5 ; of Li Ping, the great

1 Mikami, China, p. 38.

2 Also transliterated Lew Hui, Lew Hwuy, and Lieou Hoei.

*Hai-tau Suan-king. Wylie says that this title first appeared in an edition
prepared in the 8th century. 4 See page 25.

5 Shu-shu-ki-yi, or Chou-chou-ki-yi. There are many commentaries on this
work. See A. Vissifcre, Recherches sur I'origine de I'abaque Chinois . . ., p. 22
(Paris, i8p2).

CHINA 143

irrigation engineer of the 3d century; l of L ii 2 (c. 289),

who is possibly the one who gave the so-called "Chih's value
of TT," that is, TT = 3 J ; and Hsu Yiieh, who wrote a commentary
on Siu Yo's work above mentioned.

The sth century is more interesting because of the evidence
that we have of intercourse between China and the rest of the
world than because of any definite contributions to mathe-
matics. A few names of mathematicians are known, 3 but it
was the visit of the Buddhist missionaries and pilgrims from
India that is significant. The result of this visit was the trans-
lation of an arithmetic and of various astronomical works
of the Brahmans, which stimulated the activity of Chinese
scholars in these fields. This interchange of thought was not
new, for Buddhism was transmitted from India to China at
least as early as the year 65. In 399 a Chinese Buddhist,
Fa-hien, went to India, and after his return in 414 he devoted
his life to the translation of Hindu works. Since religion was
closely related to astronomy, and astronomy to mathematics,
the influence of this interchange of religious thought must have
been stimulating to the science of China. Moreover, after
about the year 450 there are many references in the Chinese
annals to the people of Po-ssi (Persia), and thereafter many
embassies passed between the two countries.

Among the mathematicians of this period whose names have
come down to us is P'i Yen-tsung (c. 400-^. 450), who is said
to have computed a noteworthy value of TT which has since been
lost. There is also Tsu Ch'ung-chih (430-501), an expert in
mechanics, who revived the knowledge of the " south-pointing
vehicle" and constructed a motor boat, all details of which
are lost. He gave - 2 y 2 - as an " inaccurate value" of TT, and f-^|
as the "accurate value," and he also showed that Trlies between
our present decimal forms 3.1415926 and 3.1415927. About
the year 450 a new calendar was devised by Ho' Ch'eng-t'ien,

'H. K. Richardson, Asia, XIX, 441.

2 There was another mathematician of the same name (i3n-i375)> wno
devised a new official calendar.

3 For example, Tun Ch'uan (c. 425), who wrote the San-tong-shu, and Wang
Jong, an arithmetician.

144 THE ORIENT

and at about the same time one Wu, a geometer, gave the equiv-
alent of 3.1432 -f as the value of TT. These details have little
significance except as they show the nature of the scientific
interests of China during this long period.

Japan in Earliest Times. Prior to the year 500 Japan seems
to have made no progress either in literature or in science.
There is a tradition that Chinese ideograms made their way
through Korea and into Japan in the year 284. There is also
reference to the Jindai monji, or "letters of the era of the gods,"
in early times, possibly a kind of system of cabala with numer-
ical values assigned to the letters, but nothing is definitely
known upon the subject. A tradition also exists that in 660 B.C.
the Japanese had a system of numeration extending to very
high powers of ten. In this system the special name yorozu
was used for 10,000, corresponding to the Greek myriad already
mentioned, and this may possibly be some slight evidence of
the early interrelations between the East and the West. 1

Of the rest of Japanese mathematics in the early periods we
know only that there was a system of measures and that, as
among all other ancient peoples of any intellectual standing,
a calendar existed.

India. The noteworthy contribution of India in this period
was probably the Hindu numeral system, which will be dis-
cussed later. 2 A second event of importance in the history of
mathematics in India, and one which chronologically precedes
the writing of the numerals, was the invasion of this country
by the army of Alexander the Great (327 B.C.) and the sending
of Greek ambassadors to reside in Indian courts. How much
influence this event had upon the science and particularly upon
the astronomy of the Hindus it is difficult at present to say.
It is worthy of note, however, that the later Hindu writers used
such Greek adaptations as jdmitra (from the Greek
kendra (/eeVrpoz/), and dramma

1 For discussion and bibliography see Smith-Mikami, p. 4.
2 See Volume II, Chapter II.

3 G. R. Kaye, Indian Mathematics, p. 26 (Calcutta, 1915) (hereafter referred
to as Kaye, Indian Math.) ; H. T. Colebrooke, Algebra with Arithmetic and

INDIA 145

Just before the beginning of the Christian era there were
numerous invasions from the north that interfered seriously with
the spread of Greek science, and in the 4th century A.D. there
appeared at least one work which definitely sought to replace
the astronomy of Greece by the ancient science of India.

The first important work on astronomy produced in India,
so far as now known, was the Surya Siddhdnta, 1 probably written
about the beginning of the $th century, although known to us
only in later ^manuscripts. The ritualistic mathematical for-
mulas of the Sulvasutras now gave place to the mathematics of
the stars. This change was possibly due to the influence of
Greek scholars whose works might still have been appreciated
by the descendants of the ancient Greeks who settled in India
after Alexander's time. Varahamihira, who will be mentioned
later, speaks of five Siddhdntas, but places the Surya Siddhdnta
at the head. Among the five is the Paulisa Siddhanta, prob-
ably of about the same period. This contains an excellent
summary of early Hindu trigonometry, the rules, expressed in
modern symbolism, being as follows:

sin 30 =1, 7T=Vio,

- sin (90 - 2 <f>)
^~ -

\ 2
1

There is also included in this work a table of sines which
was apparently derived from Ptolemy's table of chords.

The absence of an authentic Hindu chronology and of a
careful study of the effect of the Greek civilization upon the
sciences in India renders difficult a satisfactory assessment of
her mathematical achievements in this period.

Mensuration, front the Sanscrit, p. Ixxx (London, 1817) (hereafter referred to as
Colebrooke, Aryabhata, or Brahma%upta, or Bhaskara, according to the part
of the work considered, and with the modern spellings as here).

!E. Burgess, "The Surya Siddhanta," in the Journ. of the Am. Oriental
Soc., VI (New Haven, 1860) ; G. R. Kaye, "Ancient Hindu Spherical As-
tronomy," Journ. and Proc. of the Asiatic Soc. of Bengal, Vol. XV; Bapu
Deva Sastri and L. Wilkinson, The Silrya Siddhdnta and the Siddhdnta Siro-
mani (Calcutta, 1861). Alberuni, the Arab writer on India (c. 1000), speaks
of the work as "the Siddhanta of the sun, composed by Lata."

146 THE ORIENT

Decay of Civilization in Mesopotamia. For about two thou-
sand five hundred years before the period now under considera-
tion Mesopotamia had maintained a high civilization. Assyria,
Sumeria, Babylonia, and Chaldea had contributed in a large
way to the world's commercial machinery, to its science, to its
laws, and to its art. Mathematics, medicine, religion, sculpture,
architecture, literature, and the science of government are all
indebted to the genius of those who dwelt in the lands border-
ing upon or in the vicinity of the Two Rivers.

With the close of the 6th century B.C., however, there came
a change that was disastrous to the native civilization of this
region. The Persian conquest of 539 B.C. and the subsequent
coming of the Parthians, the Greeks, and the Romans, each of
whom held in subjection some or all of the territory of Meso-
potamia, left little of her ancient glory. Trajan, hoping to
repeat the conquests of Alexander, visited Babylon early in the
2d century A.D., and "saw nothing worthy of such fame, but
only heaps of rubbish, stones, and ruins," and this was sym-
bolic of the decay of a civilization which had perhaps exerted a
greater influence upon the world than any that had existed
prior to the rise of Greece.

Astrology continued to retain its power over the mass of
people, as it does in a large part of Asia today. This is shown
by tablets of the 2d century B.C., in which reports are made to
the king with respect to predictions as to the positions of the
planets. If superstition affected the court, much more would it
have affected the people at large.

In all the records of this region only a single name stands
out that is worthy of mention in the history of the mathematics
of this period, and this only in connection with a sister science.
About 250 B.C. Berosus (probably Bar Oseas, that is, the son of
Oseas), a Chaldean, founded a school on the island of Cos, and
introduced into Greece the astronomy and the astrological be-
liefs of his people, constructing a sundial and probably other
instruments. 1

*A. Wittstein, "Bemerkung zu einer Stelle im Almagest," Zeitschrift fiir
Math., XXXII (HI. Abt.) (Leipzig, 1887), 201.

DISCUSSION 147

TOPICS FOR DISCUSSION

1. The School of Alexandria, its rise, its influence, the great
scholars connected with it, and its decay.

2. Euclid, his life, his works, and his influence.

3. The work of Eratosthenes, particularly with respect to geodesy.

4. 1'he life, the works, and the influence of Archimedes.

5. Apollonius and his contribution to the study of conies.

6. The mathematical contributions of the Greek astronomers.

7. Mathematics in the Roman civilization. Causes of the dis-
regard for the science.

8. The life and works of Heron. His influence upon the develop-
ment of applied mathematics as compared with that of Archimedes.

9. The work of Nicomachus compared with the works of Euclid
and Apollonius.

10. The work of Claudius Ptolemccus, or Ptolemy.

11. The life and works of Diophantus.

12. The decay of Greek geometry, with a special consideration of
the work of Menelaus, Hypatia, Proclus, and Pappus.

13. Causes and probable effects of the burning of the books in
China in 213 B.C.

14. The period in which the Nine Sections was written and the
general nature of this work.

15. The knotted cords of China and the general subject of knotted
cords in the keeping of records and in religious ceremonial.

1 6. Efforts at opening communications between the East and West
at this period, and the probable effect of these efforts on science in
general and mathematics in particular.

17. The periods and nature of the Arithmetic Classic in Five Books
and the Sea Island Classic.

18. Influx of Hindu learning into China in this period and the
probable effect of this intercourse on the mathematics of both China
and India.

19. The invasion of India by Alexander the Great and its effect
upon the mathematics of the East.

20. The nature of the Surya Siddhanta and the bearing of this
work upon the mathematics of India.

21. Causes of the decay of mathematics in Mesopotamia in the
five centuries after the time of Alexander,

CHAPTER V

THE PERIOD FROM 500 TO 1000
i. CHINA

Intercourse with India and the West. The five centuries
extending from 500 to 1000 saw the general trend of mathe-
matics to the West rather than in the opposite direction. Eu-
rope was intellectually dormant, drugged with a new narcotic,
while most of the East was, as always, superstitious but in-
quisitive. On this account it is proper to consider first the
work of this period as it appears in the Orient. Even in the
Dark Ages, however, the West influenced the East, passing
traces of the later Greek culture on to the intellectual centers
of China and probably to those of India.

In so far as this intercourse was commercial it influenced
the art of calculation, while the travel of pilgrims and the
movements of armies resulted in the exchange of a knowledge
of both astronomical and abstract mathematics. Moreover,
the priest, whose leisure allowed time for the study of mathe-
matics, was often an astronomer, and he or the professional
astrologer was looked upon as a natural attendant at court
or a necessary adjunct to the general's staff. Where the army
went, there went a knowledge of mathematics. Astrologers of
one country thus consulted with those of another. The itiner-
ant tradesman, the pilgrim, and the army were the means of
the exchange of ideas in all ancient times, just as books and
periodicals are the corresponding media in our day.

Evidences of this Intercourse. Of the many evidences of
intercourse that we have in this period, a few may be men-
tioned simply as typical. In 518 Hui-sing, a Buddhist pil-
grim, visited India; sometime in the yth century a Sanskrit

148

THE EAST AND THE WEST 149

calendar 3 was translated into Chinese ; in 615 an Arab 2 embassy
visited China; in 618 a Hindu astronomer 3 was employed by
the Chinese Bureau of Astronomy to devise a new calendar ; in
629 Hiian-tsang 4 went to India and after his return in 645
he devoted his life to the translation of Hindu works, of which
he had brought no less than 657 from India; in 636, so the
Chinese records assert, a Roman priest whom these records
speak of as A-lo-pen came to the capital of China ; and at the
end of the yth century Buddhist pilgrims sailed from Canton
to Java and Sumatra. In the 8th century Arab ambassadors
visited China several times, in particular in 713, 726, 756, and
later; in 719 an ambassador 1 "' was sent from Rome to the
Chinese court; between 713 and 825, foreign ships of large
tonnage visited Canton, and an important customhouse is
known to have existed there at that time; and about 775 the
geographer Kia Tan (730-805) wrote the itinerary of a
voyage by sea from Canton to Persia. About 800, when Bag-
dad was rapidly becoming the center of the mathematical
world, the Chinese received an embassy from A-lun (Harun
al-Rashid). In records of the Tang Dynasty (618-907) there
are numerous references to the Arabs ( Ta-shi ) , and until the
1 2th century the intercourse between the Chinese and these
people is frequently mentioned. Mas'udi (died at Cairo, 956),
the famous Arab geographer and historian, visited India, Cey-
lon, and China in 915, and his Meadows of Gold, in which he
mentions these countries, is well known. With such evidences
as these we have a simple answer to the question as to whether
it is probable that China knew of the status of Western mathe-
matics before her own period of remarkable activity, and
whether, on the other hand, the West could have known any-
thing of Oriental progress. The answer is that it would have
been very strange if each had not been the case.

1 The Chiu-cki-li, as it was called in Chinese. The translator was Chii-t'an
Ksi-ta.

2 That is, if the name Ta-shi is taken, as usual, to mean "Arab."
3 In Chinese, Chii-t'an Chuan.

4 Original name was Ch'on I. See Giles, Biog. Diet., No. 801.

5 Called in Chinese by the name T'u-huo-lo. 6 In the T'wg shu.

150 CHINA

The Sixth Century. The 6th century is an important one in
the history of Chinese mathematics, owing to the appearance of
several works of considerable merit. The earliest of the promi-
nent writers was probably the learned Buddhist Ch'on Luan, 1
who seems to have been living in 535, but who devised a calen-
dar in the second half of the century.- He wrote the Arithmetic
in the five classics? in which he included various problems of
the standard type that had appeared in earlier works. He also
wrote commentaries on several of the earlier treatises. 4

Probably about the same time as Ch'on Luan there lived
Ch'ang K'iu-kien 5 (c. 575), whose arithmetic' 5 in three books
is nearly all extant. The work is devoted chiefly to fractions,
and it seems quite clear that the author knew the modern rule
of division by multiplying by the reciprocal of the divisor. It
also treats of arithmetic progression, the Rule of Three,
mensuration, and indeterminate linear equations.

Another contemporary of Ch'on Luan was the arithmetician
Ksia-hou Yang 7 (c. 550), the author of a treatise that is
still extant. h This work includes, as was the custom in most
cases of the kind, some problems in mensuration as well as a
treatment of certain processes of arithmetic. The arithmetic
problems all relate to multiplication, division, and percentage.

In this century there also flourished a geometer by the
name of Men (r. 575), of whom little is known, but who is said
to have given 3.14 as the value of TT.

Seventh to the Tenth Century. The most prominent Chinese
mathematician of the 7th century was Wang Hs'iao-t'ung, 9

1 Given by Pere Vanhee as T^en Loan and by Biernatzki (p. 12), who puts
him early in the 7th century, as Tschin Lwan. On all these names Mikami's
work has been freely used.

2 In the reign of Wu-ti, of the Chou monarchy (557-581), in the Chin
Dynasty. 3 Wit-king Suan-shu.

4 For example, on the Chou-pei and the K'iu-ch'ang Suan-shu.

5 Biernatzki (p. 12) transliterates the name as Tschang Kiu Kihn and gives
the date as early in the 7th century. 6 Ch'ang K'iu-kien Snan-king.

7 Biernatzki gives the name as Hea Hau yang. The date is uncertain, but he
probably lived in the period from c . 550 to c. 600.

8 The Hsia-hou Yang Suan-king (Arithmetic Classic of Hsia-hou Yang).
9 Also written Wang Hiao-t'ong and Wang Heau tung.

SIXTH TO TENTH CENTURY

known to have been living in 623 and in 626. He was an
expert on the calendar and was one of the first of the Chinese
to write on cubic equations. His work, 1 most of which is ex-
tant, contains twenty problems on mensuration, and in some
of these problems the cubic equation enters. No method of
solving such equations, however, is given.
The 8th century saw no work of impor-
tance in mathematics. In 727 I-hsing de-
vised a new calendar, 2 and two centuries
later (c. 925) there appeared an astrologi-
cal treatise of some merit, 3 but neither con-
tained any mathematics beyond such as
was needed in the work on the calendar.
The Dark Ages of the West had spread
over the East as well.

2. JAPAN

Beginnings of Japanese Mathematics. 4
Although Chinese influence had begun to
show itself in the intellectual development
of Japan before 500, it was not until the
Buddhist missionaries began to appear, in
52 2, F) that any very pronounced results were
noticed. Indeed, it was not until 552 that
Buddhism was really introduced, and not
until two years later that two scholars,
learned in matters pertaining to the calen-
dar, 6 crossed over from Korea and brought
to Japan the Chinese system of chronology. Not far from the
year 600 a Korean priest. Kanroku, presented to the empress
a set of books on astrology and the calendar, and Prince
Shotoku Taishi showed so much interest in calculation that
tradition thereafter made him the father of Japanese arithmetic.

1 Ch'i-ku Suan-king. 3 The K'ai-yuan Chan-king.

2 The T'ai-yen calendar. 4 See Smith-Mikami.

6 The first to come was Szu-ma Ta, known in Japanese as Shiba Tatsu.
These were Wang Pao-san and Wang Pao-liang. See Smith-Mikami, p. 8.

SHOTOKU TAISHI,
C. 600

From a bronze of the
1 8th century, showing
the prince with a soro-
ban, 'd chronological
impossibility

152

INDIA

Chinese Influence in Japan. From now on for many genera-
tions Japan came completely under Chinese influence in all
her intellectual life. The Chinese system of measures was
adopted, a school of arithmetic was founded (c. 670), an
observatory was established at about the same time, and in

701 a university system was inau-
gurated. Nine Chinese works were
specified for students of mathe-
matics, 1 and these seem to have been
the classics which influenced the
Japanese study of mathematics for
several centuries.

Aside from ShStoku Taishi the
man whose name stands out most
prominently in the history of Japanese
mathematics in this period is Tenjin, 2
counselor and teacher at the imperial
court (c. 890) and a great patron of
science and letters.

Altogether the era was one of prep-
aration, contributing nothing new to
what China had already developed. Indeed it was not until
the 1 7th century that Japan really awoke to her possibilities
in the field of mathematics.

3. INDIA

General Nature of the Work. In the period from 500 to 1000
there were four or five mathematicians of prominence in India.
These were the two Aryabhatas, 3 Varahamihira the astronomer,
Brahmagupta, and Mahaviracarya. In the works of all these
writers there is such a mixture of the brilliant and the

1 These were (i) Chdu-Pei Suan-king, (2) Sun-tzt Suan-king, (3) Liu-chang,
(4) San-k'ai Chung-ch'a, (5) Wu-ts'ao Suan-shu, (6) Hai-tau Suan-shu, (7)
Kiu-szu, (8) Kiu-ch'ang, (9) Kiu-shu> of which the third, fourth, and seventh
are lost.

2 His name was Michizane, but after his death he was canonized as Tenjin,
"Heaven man."

3 For rules for pronouncing Hindu names, see page xxi.

TEN JIN, PATRON OF MATHE-
MATICS, C. 890

From a bronze. The portrait

is found also in early paintings

of the Japanese

ARYABHATA

'S3

commonplace as to make a judgment of their qualities depend
largely upon the personal sympathies of the student. Alberuni
(c . 1000), the Arab historian, speaks of this peculiarity of their
writings in these words.:

I can only compare their mathematical and astronomical litera-
ture ... to a mixture of pearl shells and sour dates, or of pearls
and dung, or of costly crystals and common pebbles. Both kinds
of things are equal in their eyes, since they cannot raise themselves
to the methods of a strictly scientific deduction. 1

Aryabhata. 2 The first of the great writers whose name has
come down to us is the elder Aryabhata, 1 ' born at Kusumapura
(Kousambhipura), the City of Flowers, 4 a small town on the
Jumma just above its confluence with the Ganges. 5 The place
is not far from the present Fatna (Patna), called by the
Mohammedans Azimabad, by the ancient Buddhists Pataliputra
(Patoliputra), and by Megasthenes, the Syrian ambassador,

iAlberuni's India, translated by E. C. Sachau, 2 vols., I, 25 (London,
IQIO) ; hereafter referred to as Alberuni's India. On the relation of Greek and
Hindu arithmetic see H. G. Zeuthen, Kibl. Math., V (3), 97. On the relation of
India to the West in general, see H. G. Rawlinson, Intercourse between India
and the Western World from the Earliest Times to the Fall of Rome (Cam-
bridge, 1916). For extreme Hindu claims, see Benoy Kumar Sarkar, Hindu
Achievements in Exact Science (New York, 1918).

2 Born 475 or 476; died c. 550.

;{ Sir M. Monier-Williams, Indian Wisdom, 4th ed., p. 175 (London, 1893)
(hereafter referred to as Monier-Williams, Indian Wisdom) ; j. Garrett, Classical
Dictionary of India, p. 767 (Madras, 1871) ; C. M. Whish, "On the Alphabetical
Notation of the Hindus," Trans, of the Literary Society of Madras (London,
1827) ; L. Rodet, "Lemons de Calcul d' Aryabhata," Journal Asiatigue, XIII (7),
393 ', L,. Rodet, "Sur la veritable signification de la notation numerique invcntee
par Aryabhata," ibid., XVI, p. 440. Certain fragments of his works were pub-
lished by H. Kern in the Journal of the Royal Asiatic Society, XX (1863), 371.
See also G. R. Kaye, Indian Math., p. n, and "Aryabhata," Journ. and Proc.
of the Asiatic Soc. of Bengal, IV (N. S.), p. in (hereafter referred to as Kaye,
Aryabhata)', an article on "Ancient Hindu Spherical Astronomy," ibid., XV;
and an article in Scientia, XXV, i, all claiming Greek origin for most of the
Hindu work.

4 The term is also applied to Pataliputra. If we_may trust to the rather
obscure statements of Alberuni, it was the younger Aryabhata, however, who
was born at Kusumapura. See the mention of him later.

5 Sir E. Clive Bayley, Journal of the Royal Asiatic Society, XV (N.S.), 21.

1 54

INDIA

Palibothra. 1 Because of this geographic proximity Aryabhata
is often said to have been born at Pataliputra. A tradi-
tion says that the city was originally called Pataliputraka,

being founded by Pu-
traka, the knight of
the magic cup and
staff and slippers, who
married the princess
Patali. 2 The tradition
further asserts that
Buddha, toward the
close of his life,
crossed the Ganges
at this point and
prophesied the future
greatness of the city. 3
By the beginning of
the sth century, and
nearly a century be-
fore the birth of Ar-
yabhata, it had lost
}f its ancient

MATHEMATICAL-HISTOK1CAL, MAP OF INDIA

At Delhi, Jaipur, and Benares are interesting relics
of native observatories ; Patna is approximately the
birthplace of Aryabhata (c. 475); Ujjain was the already mentioned,
leading mathematical center of ancient India and rlp<5rrihfc; ( r
is known particularly for Varabamihira (c. 505), UCbLOUtrb l<"
Brahmagupta (c. 628), and Bhaskara (c. 1150) ;
about 75 miles from Poona are the Nana Ghat
inscriptions with early numerals; it was at Mysore
that Mahavira (c. 850) lived

.
the Chinese

ruins of the royal pal-
ace which Asoka com-
missioned the genii

to build, although
he speaks of the remarkable hospitals and other institutions
still to be found there. 4 Aryabhata evidently wrote there or at

1 E. Reclus, 4'fl, American ed., Ill, 222. 2 J. Garrett, loc. cit., p. 770.

3 R. W. Fraser, A Literary History of India, p. 143 (N. Y., 1898); E.W.
Hopkins, Religions of India, pp. 5, 311 (Boston, i8q8).

4 Dutt, Hist, of Civ. in Anc. India, II, 58 (London, 1803). On the sojourn
of Megasthenes, 306-208 B.C., see Fraser, loc. cit., p. 175. On its importance
at about the time of Aryabhata, see the inscriptions of Chandragupta II in

ARYABHATA 155

Kusumapura, for he says in one of his works: "Having paid
homage to Brahma, to Earth, to the Moon, to Mercury, to
Venus, to the Sun, to Mars, to Jupiter, to Saturn, and to the
constellations, Aryabhata, in the City of Flowers, sets forth
the science venerable." 1

It was probably because Aryabhata lived so far from Ujjain,
the ancient center of mathematics and astronomy, that his
works were so little known among Hindu scholars of the cen-
turies immediately following.

Aryabhata's Work. His work, often called the Aryabhafiyd*
or Aryabhatiyam, consists of the Gltikd or Dasagitika, a col-
lection of astronomical tables, and the Aryastasata, which in-
cludes the Ganita, a note on arithmetic 3 ; the Kalakriyd, on
time and its measure ; and the Gola, on the sphere.

The arithmetic carries numeration by tens as far as 10%
treats of plane and solid numbers, and gives a rule for square
root. It contains a rule for summing an arithmetic series after
the />th term, which may be expressed in modern symbols thus :

It also has a rule which we express by the formula

The rest of the work shows a knowledge of the quadratic equa-
tion and of the indeterminate linear equation.

J. F. Fleet, Corpus Inscriptionnm Indicarum, III, pi. iv, A, B (London, 1888).
A century and a half later (629-645) the Chinese pilgrim Hiian-tsang remarked,
"Although it has long been deserted, its foundation walls still survive." Sec
Frascr, loc. cit., p. 248.

iRodet, loc. tit., p. 396. For a slightly different translation see Kaye, Arya-
bhata, p. 116.

2 Monier- Williams, Indian Wisdom, p. 175; Mrs. Manning, Ancient and
Mediaeval India, 2 vols. (London, 1869), largely from the Journal of the Royal
Asiatic Soc., I (N.S.), 392, and XX, 371.

3 Rodet, "Lemons," loc. cit., p. 395- He translates the second part, p. 396
See also Kaye, Aryabhata, p. in; Bibl. Math., XIII (3), 203.

156 INDIA

Among the rules relating to areas is one for the isosceles
triangle, and this will serve to show the imperfect form of state-
ment used by Aryabhata: "The area produced by a trilateral
is the product of the perpendicular that bisects the base, and
half the base." The formula_for the volume of a sphere is
very inaccurate, being irr*^/irr* 9 which would make TT equal
to -^f-, possibly an error for the (^-) 2 of Ahmes.

The rule for finding the value of TT is given as follows : "Add
four to one hundred, multiply by eight, and add again sixty-two
thousand ; the result is the approximate value of the circum-
ference when the diameter is twenty thousand." This makes TT
equal to S|SJ;j|, or 3.i4i6/ Aryabhata also gives a rule for
finding sines, and the Gltikd has a brief table of these functions.

His work is also noteworthy as containing one of the earliest
attempts at a general solution of a linear indeterminate equa-
tion by the method of continued fractions. 2

As stated above, the Aryabhata here mentioned is known as
the elder of the two mathematicians of the same name. This
fact appears in the work of Alberuni 3 and has been the subject
of comment by recent writers. 4 The date of the younger
Aryabhata is unknown, nor is it possible as yet to differen-
tiate clearly between the works of the two. He seems, from
the meager authorities now known, to have been born at
Kusumapura.

Varahamihira (c. 505). Among the astronomers of India 5
two appear with the name of Varahamihira, one living c. 200
and the other c. 505. 6 The latter of these scholars is the most
celebrated of all the writers on astronomy in early India.
He wrote several works, of which the Panca Siddhantika,
treating of astrology and astronomy, is the best known. It
includes the computation necessary for finding the position of

1 Kaye, in his Aryabhata^ questions whether this is from the works of the
elder Aryabhata, the one of whom we are speaking. He thinks it is due to the
younger one mentioned below. 2 Kaye, Indian Math., p. 12.

3 India, II, 305, 327. 4 See summary by Kaye, Aryabhata, p. 113.

5 A list with dates is given in Colebrooke, loc. cit., p. xxxiii.

6 The date is quite uncertain. Varahamihira is said by some Oriental author-
ities to have died c. 587.

BRAHMAGUPTA 157

a planet, shows an advanced state of mathematical astronomy,
but is chiefly valuable in the history of mathematics because
of the description that it gives of the five Siddhantas which had
been written just before this time. 1 He urged his people to
appreciate the work of the Greeks, saying: "The Greeks,
though impure, must be honored, since they were trained in the
sciences and therein excelled others. What then, are we to say
of a Brahman if he combines with his purity the height of
science?" 2

Varahamihira taught the sphericity of the earth, and in this
respect he was followed by most 'of the other Hindu astrono-
mers of the Middle Ages. 3 Two of his works were translated
into Arabic by Alberuni (c. iooo). 4

Brahmagupta. The most prominent of the Hindu mathema-
ticians of the yth century was Brahmagupta, 5 whose period of
activity has been fixed as c. 628, both from astronomical data
and from the testimony of various Hindu writers.' 5 He lived
and worked in the great astronomical center of Hindu science,
Ujjain or Ujjayim, a town in the state of Gwalior, Central
India, said to have been the viceregal seat of Asoka during his
father's reign at Patna. Varahamihira also carried on his work
at the observatory in Ujjain.

When he was only thirty years old, Brahmagupta wrote an
astronomical work in twenty-one chapters entitled Brahmasid-
dhanta? which includes as special chapters the Ganitad'haya*

*G. Thibaut and Sudharkar Dvivedi, The Pancha-siddhdntikd of Varaha
Mihira. Benares, 1889.

2Alberuni's India, I, 23. 3 Alberuni, loc. tit., I, 266.

4 For a list of his works, sec Alberuni, loc. tit., I, xxxix. For the influence of
the Greeks upon his work and upon Hindu astronomy in general, see Cole-
brooke, loc. tit., p. Ixxx.

5 Colebrook, loc. tit. Alberuni (c. iooo) speaks of him as "the son of
Jishnu, from the town of Bhillamala." Suryadasa, a commentator on Bhaskara,
also speaks of him as the son of Jishnu.

6 Colebrooke (loc. tit., p. xxxv) makes the date 581 or 582, from Brah-
magupta's reference to the position of the star Chitrd (Spica Virginis). The
Hindu astronomers make it c. 628. He seems to have been born c. 508.

7 Also called the Brdhma-sphuta-sidd'hanta, "Brahma correct system," pos-
sibly a revision. Alberuni (c. iooo) gives twenty-four chapters, with the title
of each. See his India, I, 154; II, 303- ^Lectures on Arithmetic.

i$8 INDIA

and the Kutakkddyaka. 1 The former begins by a definition
of a ganaca, that is, a calculator who is competent to study
astronomy : "He who distinctly and severally knows addition
and the rest of the twenty logistics and the eight determina-
tions, including measurement by shadow, is a ganaca" 2

Nature of Brahmagupta's Arithmetic. The arithmetic in-
cludes work with integers and fractions, progressions, barter,
Rule of Three, simple interest, the mensuration of plane figures,
and problems on volumes and on shadow reckoning (a primi-
tive plane trigonometry applied by him to the sundial). The
mensuration is often faulty, as where Brahmagupta states a
rule which would give the area of an equilateral triangle of
side 12 as 6 x 12, or 72 ; that of the isosceles triangle 10, 13,
13 as 5 x 13, or 65; and that of the triangle 13, 14, 15 as
7 x I x { 13 -f 15), or 98. He also states that the area of any
quadrilateral whose sides are a, b, c, d is

where s\(a + 6 + c + d), a formula that is true only for
cyclic quadrilaterals. His rule for the quadrilateral is as fol-
lows: "Half the sum of the sides set down four times, and
severally lessened by the sides, being multiplied together, the
square root of the product is the exact area."" He uses 3 as the
" practical value" of TT and Vlo as the "neat value."

Brahmagupta's Algebra. The Kutakhddyaka applies alge-
bra to astronomical calculations. For example, "One who tells,
when given positions of the planets, which occur on certain
lunar days or on days of other denomination of measure, will
recur on a given day of the week, is versed in the pulverizer." 4

1 Lectures on Indeterminate Equations. The kutaka (kuttaka, cutacd) is
defined by Colebrooke (loc. cit., p. vii) as "a problem subservient to the
general method of resolution of indeterminate problems of the first degree."
The word means "pulverizer" and is used as a name for algebra. Ibid., p. 325;
J. Taylor, Lilawati, p. 129 (Bombay, 1816) ; hereafter referred to as Taylor,
Lilawatt, with this spelling. The word khddyaka means "sweetmeat," such
fanciful names being common in the East.

2 Colebrooke, loc. cit., p. 277.

3 Ibid., p. 295. 4 Question 7 of the Colebrooke translation.

BRAHMAGUPTA 159

In his chapter on computation 1 Brahmagupta gives the
usual rules for negative numbers. He also has a chapter on
quadratic equations, the rule 2 for solving an equation of the
type x z + px q = o being substantially a statement of the
formula / ., ---

which evidently gives one root correctly.

In the case of simultaneous equations of the first degree the
unknowns are spoken of as "colors," and the problems are
chiefly astronomical. Indeed, Brahmagupta was the first In-
dian writer, so far as we know, who applied algebra to astron-
omy to any great extent. While the fanciful problems so
often found in Indian works are generally wanting, a com-
mentator has supplied various examples to illustrate certain
of his rules. Two such problems are as follows :

On the top of a certain hill live two ascetics. One of them, being
a wizard, travels through the air. Springing from the summit of the
mountain he ascends to a certain elevation and proceeds by an
oblique descent diagonally to a neighboring town. The other,
walking down the hill, goes by land to ^
the same town. Their journeys are
equal. I desire to know the distance of
the town from the hill, and how high
the wizard rose.

The commentator takes the case here shown, and finds
x to be 8.

A bamboo 18 cubits high was broken by the wind. Its tip touched
the ground 6 cubits from the root. Tell the lengths of the segments
of the bamboo. 3

Indeterminate Equations. It is indicative of the state of al-
gebra at this time that Brahmagupta was interested in the
solution of indeterminate equations. Aryabhata had already

iShat-trinsat-paricarman. 2 Page 346 of the Colebrooke translation.

3 From his arithmetic. For an early Chinese version, see page 139.

160 INDIA

considered the question of the integral solutions of ax by = c,
but Brahmagupta actually gave as the results

where t is zero or any integer and p/q is the penultimate con-
vergent of a/6. 1 He also considered the so-called Pell Equation
of the form ~ 2 _ /2

but the solution was first effected, so far as we know, by
Bhaskara in the i2th century.

For the sides of the right-angled triangle Brahmagupta gave
the two sets of values

2 mn, m 2 2 , m 2 + ri\

, / i Im \ i Im , \
and Vw, -( ;/l, ~( h),

values which he probably obtained from Greek sources.

Brahmagupta was accused of propagating falsehoods relat-
ing to science for the purpose of pleasing the bigoted priests
and ignorant rabble of his country, hoping thus to avoid the
fate that befell Socrates, 2 all of which shows that he was a man
of recognized importance in his day.

Progress retarded in India. From this time to the year 1000
learning seems to have made but little progress in northern
India. In the 8th century the Rajput dynasty succeeded the
high-minded Valabhis, and for two hundred years the history
of this part of India is a blank. Not a piece of literature of
any value remains, nor any work of art or of industry. 3 The
abode of mathematics now moved northward and is found for
two or three centuries in Persia and in the other lands which
had been brought under Moslem rule. In southern India, how-
ever, there must have been some encouragement of mathe-
matics, as will be seen from the great work of Mahavira.

c, Indian Math., p. 16.
2 See Sachau's note in his translation of Alberuni's India, II, 304.
3 Dutt, History of Civ. in Anc. India, II, 162.

MAHAVIRA 161

Mahavira. The third of the great Hindu writers of this
period is Mahavlracarya, Mahavira the Learned, who wrote
the Ganita-Sara-Sangraha} This writer probably lived at the
court of one of the old Rashtrakuta monarchs who ruled over
what is now the kingdom of Mysore, and whose name is gi^en
as Amoghavarsha Nirpatunga. This king ascended the throne
in the first half of the Qth century, so that we may roughly
fix the date of the treatise in question as c. 850, or between the
dates of Brahmagupta and Bhaskara, 2 though nearer to the
former.

The work begins, as is not unusual with Oriental treatises,
with a salutation of a religious nature. In this case the words
are addressed to the author's patron saint, the founder of the
religious sect of the Jainas (Jinas), a contemporary of Buddha :

Salutation to Mahavira, the Lord of the Jinas, the protector
[of the faithful], whose four infinite attributes, worthy to be es-
teemed in [all] the three worlds are unsurpassable [in excellence].

I bow to that highly glorious Lord of the Jinas, by whom, as form-
ing the shining lamp of the knowledge of numbers, the whole of the
universe has been made to shine.

Mahavira's Sources. In general it may be said that Mahavira
seems to have known the work of Brahmagupta. It would
have been strange if this had not been so, for the Brahma-
sphuta-siddhdnta was probably recognized in his time as one
of the standard authorities. Mahavira seems to have made the
effort to improve upon the work of his predecessor, and
certainly did so in his classification of the operations, in the
statement of rules, and in the nature and number of problems.
As a result his work became well known in southern India, al-
though there is no definite proof that Bhaskara (c. 1150),
living in Ujjain, far to the north, was familiar with it.

Mahavira's Work. The work itself consists of nine chapters.
The first is introductory and relates chiefly to the measures

*M. Rangacarya, TheGanita-Sdra-Sangraha of Mahavlracarya, Sanskrit and
English, Madras, iqi2; hereafter referred to as Mahavira. Ganita-Sdra means
"Compendium of Calculation." 2 He lived c. 1150. See page 275.

1 62 INDIA

used, the names of the operations, numeration, negatives, and
zero. Eight operations with numbers are given, addition (ex-
cept in series) and subtraction (even with fractions) being
omitted as if presupposed. One interesting feature is the law
relating to zero, which is stated thus : " A number multiplied
by zero is zero, and that [number] remains unchanged when
it is divided by, combined with, [or] diminished by zero."
That is, the law given by Bhaskara for dividing by zero is not
here recognized, division by zero being looked upon as of no
effect. The law of multiplication by negative numbers is
stated, and the imaginary number is thus disposed of : "As in
the nature of things a negative [quantity] is not a square
[quantity], it has therefore no square root."

In his arithmetic operations he first treats of multiplication.
He then considers in order the topics of division, squaring,
square root, cubing, cube root, and the summation of series.
In his work in series he includes some treatment of arithmetic
and geometric progressions and of Vyutkalita, that is, the sum-
mation of a series after a certain number of initial terms
(ista) have been cut off, a theory which, as we have seen
(P- T 55)> occupied the attention of Aryabhata.

The most noteworthy feature in his treatment of fractions
is that relating to the inverted divisor, the rule being set forth
as follows: "After making the denominator of the divisor its
numerator [and vice versa], the operation to be conducted then
is as in the multiplication [of fractions]." It is curious that
this device, which from another source we know to have been
used in the East, became a lost art until again adopted in Europe
in the i6th century.

His method of approach to the subject of quadratic and radi-
cal equations is through fanciful problems of which the follow-
ing is a type:

One fourth of a herd of camels was seen in the forest ; twice the
square root [of that herd] had gone on to mountain slopes; and
three times five camels [were] however, [found] to remain on the bank
of a river. What is the [numerical] measure of that herd of camels ?

MAHAVlRA 163

This evidently requires the finding of the positive root of
the equation \ x + 2 Vi- + 15^ -V, or, in general, the solution of
an equation of the type x (bx+c^/x+a) **, the rule for
which is given. The chapter also contains various other types
of equations involving some knowledge of radical quantities.

A single example will also suffice to show the nature of his
indeterminate problems :

Into the bright and refreshing outskirts of a forest, which were
full of numerous trees with their branches bent down with the
weight of flowers and fruits, trees such as jambu trees, lime trees,
plantains, areca palms, jack trees, date palms, hintala trees, palmy-
ras, punnaga trees, and mango trees [into the outskirts] , the various
quarters whereof were filled with the many sounds of crowds of
parrots and cuckoos found near springs containing lotuses with bees
roam'ng about them [into such forest outskirts] a number of
weary travelers entered with joy. [There were] sixty-three [numer-
ically equal] heaps of plantain fruits put together and combined
with seven [more] of those same fruits, and these were equally dis-
tributed among twenty-three travelers so as to have no remainder.
You tell me now the numerical measure of a heap of plantains*

Mahavira's Treatment of Areas. His work in the measure-
ment of areas is somewhat like the corresponding chapter in
Brahmagupta's treatise, although it is distinctly in advance of
the latter. Mahavlra makes the same mistake as Brahmagupta
with respect to the formula for the area of a trapezium (trape-
zoid) in that he does not limit it to a cyclic figure. The same
error enters into his formula for the diagonal of a quadrilateral,
which he gives as

I (ac + bd) (ab + cd) \(ac + bd)(ad+bc)

N ad+bc r N ab + cd

For the Pythagorean triangle Mahavlra gives rules similar
to those of Brahmagupta. For TT he uses Vio, a common value
all through the East and also in medieval Europe. He was the
only-Hindu scholar of the native school who made any sSious
attempt to treat of the ellipse, but his work was inaccurate.

1 64 PERSIA AND ARABIA

His rule for the sphere is interesting, the approximate value
being given as - (|d) 3 , and the accurate value as ^ - f (} 2 d) 3 ,
which means that ir must be taken as 3.03!.

All things considered, the work of Mahavlra is perhaps the
most noteworthy of the Hindu contributions to mathematics,
possibly excepting that of Bhaskara, who lived three centuries
later. Mahavlra may have known the works of Chinese
scholars, for the value that he gives for the area of the segment
of a circle, \ (c -f a) a, was given six centuries earlier by Ch'ang
Ts'ang, but in any case he was a man of scientific attainments.

Bakhshali Manuscript. Another work that stands out with
some prominence in this period is the Bakhshali manuscript. 1
This work, of uncertain origin and date, contains material re-
lating to both arithmetic and algebra. It was formerly referred
to the early part of our era and then to the 8th or gth century,
but it gives evidence of having been written even after the
latter period, and possibly it is not even of Hindu origin. The
nature of the work may be inferred from a single problem :

A merchant pays duty on certain goods at three different places.
At the first he gives J of the goods, at the second J f of the remainder]
and at the third J- [of the remainder]. The total duty is 24. What
was the original amount of the goods? 2

4. PERSIA AND ARABIA

Persia. We are apt to think that the rise of learning in the
lands conquered by the Mohammedans was due solely to Arab
influence, but this is not the case. In Persia, for example,
Khosru the Holy, 3 a generous patron of science, invited to his

!R. Hoernle, "The Bakhshali Manuscript," Indian Antiquary, Vol. XVIII
(1888) ; G. R. Kaye, "Notes on Indian Mathematics," in Journ. and Proc. of
the Asiatic Soc. of Bengal, III (2), 501 (hereafter referred to as Kaye, Notes) ;
and "The Bakhshali Manuscript," ibid., VIII (2), 349 (hereafter referred to as
Kaye, Bakhshali).

2 The answer is 40, which necessitates the bracketed words.

3 Khosru I, Anoschirvan. He was a contemporary of Justinian, who was
crowned emperor in Constantinople in 527. See W. S. W. Vaux, Persia, p. 169
(London, 1875); T. Noldeke, Aufsatze zur persischen Geschichte, p. 113
(Leipzig, 1887).

BAKHSHALI MANUSCRIPT

A portion of this manuscript of which the date is still unsettled. It may be of
the loth century. From Kaye's Indian Mathematics

1 66 PERSIA AND ARABIA

court scholars from Greece and encouraged the influx of West-
ern culture. In his reign Aristotle and Plato were translated
and doubtless the works of the Greek mathematicians were
made known.

Christian Scholars in Mesopotamia. At about the time of the
rise of the Mohammedan power there were various Christian
centers of learning in the regions over which the Arabs were
soon to hold sway. These were found in the monasteries which
were scattered throughout the Near East. Of the scholars
who taught in these retreats, the most learned one of the 7th
century was Severus Sebokht, 1 a titular bishop who lived in
the convent of Kenneshre on the Euphrates in the time of the
patriarch Athanasius Gammala (who died in 631) and his
successor John. He distinguished himself in the studies of
philosophy, mathematics, and theology, and in his time the
convent of Kenneshre became the chief seat of Greek learning
in western Syria. He wrote on astronomy, the astrolabe, and
geography. In one of the fragments of his works which have come
down to us, of date 662 , he directly refers to the Hindu numerals.
He seems to have been hurt by the arrogance of certain Greek
scholars who looked down on the Syrians, and in defending
the latter he claims for them the invention of astronomy. He
asserts the fact that the Greeks were merely the pupils of
the Chaldeans of Babylon, and he claims that these same
Chaldeans were the very Syrians whom his opponents con-
demn. He closes his argument by saying that science is
universal and is accessible to any nation or to any individual
who takes the pains to search for it. It is not, therefore, a
monopoly of the Greeks, but is international.

Sebokht on our Numerals. It is in this connection that he
mentions the Hindus by way of illustration, using the follow-
ing words:

*J. Ginsburp, "New Light on our Numerals," Bulletin of Ihe Am. Math.
Sac., XXIII (2), 366, from which extracts have been freely made. Attention of
English readers was first called to this writer's mathematical works by Professor
Karpinski, Science (U.S.), June, 1912. See also E. R. Turner, Popular Set.
Mo., December, IQ T I

BAGDAD 167

I will omit all discussion of the science of the Hindus, a people
not the same as the Syrians ; their subtle discoveries in this science
of astronomy, discoveries that are more ingenious than those of the
Greeks and the Babylonians ; their valuable methods of calculation ;
and their computing that surpasses description. I wish only to say
that this computation is done by means of nine signs. If those who
believe, because they speak Greek, that they have reached the
limits of science should know these things they would be convinced
that there are also others who know something.

Bagdad. It was at Bagdad, on the Tigris River, that mathe-
matics had its greatest encouragement under the Mohammedan
ascendancy. Built upon the ruins of an ancient town by the
caliph 1 al-Mansur (712-774/5), one of the Abbasides, 2 Bag-
dad 3 became the intellectual center of the Mohammedan world,
a second Alexandria in its fostering of learning. In al-
Mansur's reign (c. 766) a work mentioned as the Sindhind is
said to have been brought to his court by an Indian scholar
named Kankah (Mankah?), the Hindu astronomy and mathe-
matics being thus made known to the scholars of Bagdad.
This work may have been the Surya Siddhdnta or it may have
been some other work bearing the title Siddhdnta, this name
being nearer to Sindhind than any other Sanskrit word likely to
be meant. It is generally believed, however, that it was the
Brahmasiddhdnta of Brahmagupta, whose works are known
to have been brought to Bagdad at this time. 4

To the court of the Caliphs there also came, so the story
goes, a Persian by the name of Ya'qub ibn Tariq (died 796).
He is said to have written (775) on the sphere (mathematical
astronomy) and the calendar, and to have edited, and prob-
ably to have assisted in translating, the works of Brahmagupta

1 Calif, from Khaltfah, successor (of the Prophet).

2 Ab bas'ides or Ab'ba sides, the (at least pretended) descendants of Abbas,
uncle and adviser of Mohammed. Al-Mansur reigned from 753/4 to 774/5- For
rules for pronouncing Arabic names, see page xx.

3 Persian Bagadata, "God-given"; in Arabic, Dar al-Salam, "Abode of
Peace," Also spelled Baghdad.

4 Sachau's preface to his translation of Alberuni's India, I, xxxi. As to the ab-
sence of Arabic records to prove that any embassy came from India at this
time, see ibid., II, 313.

1 68 PERSIA AND ARABIA

above mentioned. To the same court there came (and the
records of this fact are somewhat more trustworthy) the astron-
omer Abu Yabya, 1 and there he translated the Tetrabiblos of
Ptolemy, thus assisting to begin the great movement that led
to the introduction of the classics of Greek mathematics into
the court of the Caliphs.

About the same time al-Fazari 2 (died 777), working also at
Bagdad, wrote on astrology and the calendar. He was the
first Moslem, so far as is known, to construct astrolabes and to
write on mathematical instruments. His famous con tempo-
rary, Jeber,* the greatest alchemist of the Arabs, also wrote
on the astrolabe and possibly on mathematics. 4

It was in this reign that another al-Fazari, c son of the one
already mentioned, a man of unusual scholarship, particularly
in the field of astronomy, was asked by the caliph to translate
the Siddhdnta brought to Bagdad by Kankah. It was on this
translation that Mohammed ibn Musa al-Khowarizmi (c. 825)
based his astronomical tables.

Harun al-Rashid. Harun al-Rashid, well known to us from
the Arabian Nights Talcs, was a great patron of learning.
Under his influence several of the Greek classics in science,
including part of Euclid's works, were translated into Arabic,
Indeed, it is to the Arabic versions that medieval Europe was
indebted for its first knowledge of Euclid's Elements. In his
reign there was a second influx of Hindu learning into Bag-
dad, especially in the line of medicine and astrology.

1 Abu Yahya al-Batriq, who died about 796-806.

2 Ibrahim ibn Habib ibn Sjoleiman ibn Samora ibn Jundab, Abu Ishaq al-Fazari.

3 Jabir ibn I.Iaiyan al-Sufi, Abu 'Abdallah (died c. 777), one of two promi-
nent scholars known by the name of Geber in the Middle Ages.

4 The matter is discussed briefly in H. Suter, "Die Mathematiker und
Astronomen der Araber und ihre Werke," in Volume X of the Abhandlungen.
The transliteration of Arabic names is taken from Suter's list, with the change of
el to al, of g to ;', of s to sh, of ch to kh, of w to v, and of ; to y as in
English. While this is not always desirable in the case of sh, as in Ishaq, it is
much simpler for the general reader. With respect to all these names the student
should consult Suter's work.

5 Mohammed ibn Ibrahim ibn Ilabib, Abu * Abdallah al-Fazari. He died
between 796 and 806.

6 Harun al-Rashid, Aaron the Just. He reigned from 786 to 808/9.

THE CALIPHS 160

Al-Mamun. Harun al-Rashid's son, al-Mamun (reigned
809-833), was also a great patron of learning; indeed, he was
more than a mere patron, for he erected an observatory at
Bagdad and himself took observations there. He is also
credited with supervising two geodetic surveys in Mesopo-
tamia for the purpose of determining the length of a degree of
the meridian. Under his direction the translation of the Greek
classics continued, the Almagest of Ptolemy being put into
Arabic and the translation of the Elements of Euclid being
completed. In order to show the great activity among the
Arabs in the field of mathematics, and the general nature of
the work accomplished, a brief list of names will be given,
with notes that are necessarily condensed, although it is appar-
ent that most of the names are unfamiliar and most of the
details will pass from the reader's mind.

It is evident that astronomy was the science of this period
that did most to bring mathematics into a favorable light at
court. Linked up with astrology on the one hand and with
mathematics on the other, it introduced just enough of super-
stition through the former to help establish the latter science

Writers in al-Mamun's Reign. Among those who, in al
Mamun's remarkable reign, wrote upon mathematical astron-
omy, thus assisting to advance the study of trigonometry, tht
following scholars deserve special mention, not so much foi
their genius as for their spirit: al-Tabari, 1 who wrote a com
mentary on Ptolemy's Tctrabiblos; al-Nehavendi, 2 who pre-
pared a set of astronomical tables ; al-Mervarrudi, 3 who made
astronomical observations at Damascus and Bagdad (c. 830) :
al-Astorlabi, 4 who lived in Bagdad (c. 830), wrote on astron
omy and geodesy, and was celebrated as a maker of astrolabes
and other astronomical instruments; Messahala, 5 a Jewisl

a Omar ibn al-Farrukhan, Abu IJafs al-Tabari, died c. 815.

2 Ahmed ibn Mohammed al-Nehavendi, died c. 835-845.

3 Khalid ibn 'Abdelmelik al-Mervarrudi. 4 'Ali ibn 'Isa al-Astorlabi

5 Ma-sha'-allah ibn Atari. The spelling in the text is the one commonly usec
in the West. The text of one of his MSS. was published by W. W. Skeat in hi
edition of Chaucer's Astrolabe, London, 1872. His chief work was done jus
before al-Mamun's reign.

170 PERSIA AND ARABIA

astrologer, who wrote (c. 800) a treatise on the astrolabe that
seems to have influenced the later works of Rabbi ben Ezra
(c. 1150) and Chaucer (c. 1400) ; and Alfraganus 1 (c. 833), to
use his European name, who wrote on sundials, astronomy,
and the Almagest.

Mohammed ibn Musa al-Khowarizmi. The greatest mathe-
matician at the court of al-Mamun was Mohammed ibn Musa
al-Khowarizmi, 2 Abu 'Abdallah (died between 835 and 845),
a native of Khwarezm, the country in which is now the city
of Khiva. Although an astronomer and the author of several
astronomical tables and of works on dials, the astrolabe, and
chronology, he is best known for having written the first work
bearing the name "algebra," a treatise based upon Greek
models. 3 He also wrote on arithmetic, this work being trans-
lated into Latin by Robert of Chester or by Adelard of Bath
under the title Algorltmi de numero Indorum, whence such
words as algorism and augrim? derived from al-Khowarizmi.
The title of the algebra was 'Urn al-jabr wa'l muqabalah, "the
science of reduction and cancellation." 5 After al-Mamun's
death mathematics continued to flourish in Bagdad for about
a century and a half, although, as might be expected, with some-
what less encouragement. 6

1 Mohammed ibn Ketir al-Fargani. The European translators also used such
forms as Alfergani and Alfragan. Johannes Hispalensis translated his version
of the Almagest into Latin, and it was printed at Ferrara in 1493 and again,
with a preface by Melanchthon, at Niirnberg in 1537.

2 This transliteration is more familiar to English readers than is Suter's el-
Chowarezmi or Chwarezmi. The name means Mohammed son of Moses, the
Khwarezmite. C. Huart, History of Arab Literature, pp. 131, 292, 297 (London,
1003), says that there were two others by the name of al-Khowarizmi, one
(the geographer) of 035-003 (or 1002), and the other of c. 1036, but Huart
seems to have been confused in this matter. The name also appears as al-
Khowarazmi and as al-Khowaruzmi.

8 L. C. Karpinski, Robert of Chester's Latin Translation of the Algebra of
al-Khoivarizmi, New York, 1915.

4 So Chaucer speaks, in the Canterbury Tales, of " augrim stones."

5 See Volume II, Chapter VI.

CJ Another Mohammed of Bagdad wrote a work on the division of surfaces.
On the relation of this work to Euclid's book on the divisions of figures, see the
careful study by Professor R. C. Archibald in his Euclid's Book on Divisions of
Figures^ pp. 1-8 (Cambridge, 1915).

AL-KHOWARIZMI 1 7 T

Other Scholars of Bagdad. Almahani 1 (c. 860), as he is
commonly called, an astronomer of high standing, is perhaps
best known for having written upon the familiar problem of
Archimedes relating to the cutting of a sphere into segments
having a given ratio of volume. In his stereometric solution
of the cubic equation involved in this problem he made use of
the sine of a trihedral angle. He also wrote commentaries
on Books V and X of Euclid's Elements and on the work of
Archimedes on the sphere and cylinder.

Alchindi" (c. 860), to use the name by which he was gen-
erally known in medieval Europe, was commonly called "the
philosopher of the Arabs." He wrote on a large variety of
topics, including astronomy, astrology, optics, and number.
Gherardo of Cremona (c. 1150) translated his work on optics
into Latin.

About 870 there lived in Bagdad three scholars known as
the Beni Musa (sons of Moses) or the Three Brothers. 5 They
were the sons of Musa ibn Shakir, a reformed robber who had
finally devoted himself to geometry and astronomy in al-
Mamun's court. Of these brothers, Mohammed, Ahmed, and
al-I.Iasan, the first-named was the most celebrated, but all
three gave attention to securing the best scientific works of the
Greeks and to having them translated. They wrote on medi-
cine, conies, geometry, mensuration, the trisection of an angle,
and other scientific subjects. They used the conchoid in the
trisection problem and the string fastened to the foci in the
construction of an ellipse.

At this period there worked for a time in Bagdad the cele-
brated Tabit ibn Qorra 4 (826-901), a physician of prominence,
but better known for his work in philosophy and mathematics,
and particularly for the claim that he was successful in applying

3 Mohammed ibn <Isa, Abu 'Abdaliah al-Mahani, of Bagdad, died probably
between 874 and 884.

2Ya f qub ibn Ishaq ibn al-$abbah al-Kindi, Abu Yusuf, died c. 873/4.

3 M. Curtze, "Liber Trium Fratrum de Geometria," in Nova Acta der K.
Leop.-Carol. Deutschen Akad. der Naturforscher, XLIX, No. 2 (Halle, 1885).

4 Tabit ibn Qorra ibn Mervan, Abu-Hasan, al-IJarrzlni, a native of Ijjarran
in Mesopotamia, where he also spent some of his later years.

172 PERSIA AND ARABIA

algebra to geometry. He revised the translation of Euclid's
Elements made by Ishaq ibn Ilonein, a renowned physician
(died 910), and the translation of the so-called "middle books,"
that is, of those books written between the time of Euclid and
that of Ptolemy. 1 He also wrote extensively on astronomy,
the Almagest, conies, elementary geometry, Euclid, magic
squares, amicable numbers, and astrology. Gherardo of Cre-
mona (c. 1150) and Johannes Hispalensis (c. 1140) trans-
lated certain of his works. He had a son, 2 a physician, who
also followed in his father's steps, writing on astronomy and
geometry, and revising one of the translations of Archimedes
from the Syriac into Arabic.

At about this time an Egyptian Ahmed ibn Yusuf 8 wrote
on proportion and astronomy and discussed the figura cata,
that is, the proposition of Menelaus relating to the segments of
the sides of a triangle cut by a transversal.

Christian and Jewish Scholars in Bagdad. To Bagdad there
also came at this time various Jewish and Christian writers,
their names being commonly given in Arabic form. Among
these were Sahl ibn Bishr, 4 an astrologer, who had already
gained considerable reputation in Khorasan. He wrote a work
on algebra. Part of his writings appeared in print in Venice
(1493) and part in Basel (1533). There was also Abu'l-
Taiyib, 5 who gave up his Jewish religion and adopted the
faith of Islam. He compiled a set of astronomical tables and
seems to have written on trigonometry. Among the Christians
there was Qosta ibn Luqa al-Ba'albeki 6 (died c. 912/3), a

1 L. M. L. Nix, Das fiinfte Buck der Conica des Apollonius von Perga, with
Arabic text and German translation. Leipzig, 1889.

2 Sinan ibn Tabit ibn Qorra, Aba Sa'id, died 943- See Suter's list, Abhand-
lungen, X, 51.

8 Ahmed ibn Yusuf ibn Ibrahim, Abu Ja'far, al-Misri (died c. 912/3). Al-
Misri means the Egyptian, and the name is applied to other writers as well.
There is some doubt as to his works. He was the son of Yusuf ibn Ibrahim
ibn al-Daya, who was known as "the Arithmetician" and lived in Damascus,
Bagdad, and Egypt.

4 Sahl ibn Bishr ibn IJabib ibn Hani (or Haya), Aba 'Otman (c. 850).

6 Sind ibn 'Alt, Aba'l-Taiyib (c.Sso).

6 Kosta, son of Luke, from Baalbek, known to early Europeans as Kustaben Luca.

PYTHAGOREAN THEOREM IN TABIT IBN QORRA^S TRANSLATION

OF EUCLID

The translation was made by Ishaq ibn JJonein (died 910) but was revised by
Tabit ibn Qorra, c. 890. This manuscript was written in 1350

I 7 4 PERSIA AND ARABIA

physician, who translated the Spherics of Theodosius and
parts of Aristarchus, Autolycus, 1 Hypsicles, Heron, and Dio-
phantus, and who wrote a geometry in catechism form. There
was also a Greek Christian, Nazif ibn Jumn (or Jemen), known
as al-Qass (the priest), who translated Euclid X; and another
of the same faith, al-Jorjani, 2 a physician, who wrote a com-
pendium of the Almagest.

It is possible that it was about this time and in this region
that the anonymous Hebrew work entitled Mishnath ha-Mid-
doth (Theory oj Measures} was written, but the place and
date are quite unknown. It is primarily on the measurement
of geometric solids, and some of its features recall the work of
al-Khowarizmi on mensuration. 3

Later Writers. After the reigns of the first three caliphs of
Bagdad the science of astronomy still continued to be the ante-
chamber of mathematics. Thus we find such writers in this
field as al-Mervazi, 4 who wrote extensively on astronomy and
astronomical instruments; Albumasar 5 (died 886), the most
celebrated of the Arab writers on astrology, who was led
by this science to the study of astronomy; Ahmed ibn al-
Taiyib 6 (c. 890), of Persian origin, a pupil of AlchindPs, who
wrote on algebra and arithmetic as well as on astrology and
music; and al-Dinavari, 7 who wrote on algebra, astronomy,
and the Hindu methods of computation. There was also the

1 A Greek astronomer who lived c. 360 B.C. The others have already been
mentioned.

*'fsa ibn Yahya al-Masiht, Abu Sahl, al-Jorjani, died c. 1009/10. AI-Masiln
means a believer in the Messiah, a Christian. The Suter list does not give the
place where either of the last two lived.

3 M. Steinschneider, Festschrift Zunz (Berlin, 1864); H. Shapiro, Abhand-
lungen, with translation and commentary, III, 3; F. Rosen, The Algebra of
Mohammed ben Musa, p. 70 (London, 1831).

4 Ahmed ibn 'Abdallah al-Mervazi, a native of Merv (probably died between
864 and 874), known as Habash al-yasib ("IJabash the computer").

B As he was commonly known in medieval Europe. His name was Ja'far ibn
Mohammed ibn 'Omar al-Balkhi (from Balkh, in Khorasan), Abu Ma'shar.

"Ahmed ibn Mohammed ibn Mervan, Abii'l-'Abbas, al-Sarakhsi, known as
Ahmed ibn al-Taiyib.

7 Ahmed ibn Da'ud, Abu Ilanifa, al-Dinavar! (died 895). He lived most of
the time in Dfnavar, his native place.

LATER WRITERS 175

well-known scholar Albategnius 1 (died 929), as he was called
in Europe, who was justly esteemed for his astronomical
writings"' and tables. Among the many other scholars of this
period there may be mentioned Rhases (died 93 2 ), 3 to use his
European name, a celebrated physician who wrote on geome-
try and astronomy; a grandson of Tabit ibn Qorra, 4 also a
physician, who wrote on conies, dialing, and elementary geom-
etry; al-Farrabi/' a native of Farab in Turkestan, who wrote
a commentary on Euclid and was a philosopher of high stand-
ing; Ibn Yunis, who, next to al-Battani, was the most cele-
brated astronomer among the Arabs; and al-l.larrani/ who
wrote a commentary on Euclid.

The loth century saw several writers of somewhat higher
attainments, among whom the best-known was Abu'1-Wefa 8
(940-998), celebrated for his improvements in trigonometry,
his introduction of the tangent (umbra versa), and his compu-
tation of tables of sines and tangents for every 10'; it is also
very likely that he is entitled to credit for the use of secants and
cosecants. He was also prominent as a writer on arithmetic,
algebra, geometry, and astronomy.

Among the other writers of this period who are worthy of
special mention were al-Haitam of Basra, who wrote on
algebra, astronomy, geometry, gnomonics, and optics; Abu
Ja'far al-Khazin (died between 961 and 971), who attempted

1 Mohammed ibn Jabir ibn Sinan, Abu 'Abdallah, al-Battani, a native of
Battan, in Mesopotamia. He is also known as al-Raqqi, from the fact that he
made his observations at Raqqa on the Euphrates.

2 Translated by Robert of Chester (c. 1140) or Robertus Retinensis, referred
to later. The work was printed in 1537.

3 Mohammed ibn Zakariya al-Razi, Abu Bekr.

4 Ibrahim ibn Sinan Tabit ibn Qorra, Ab& Isliaq, son of the Sinan already
mentioned. Born 908/9; died 946.

e Mohammed ibn Mohammed ibn Tarkhan ibn Auzlag, Abu Nasr, al-
Farrabi; died at Damascus, 950/1.

6t Ali ibn Abi Sa'id 'Abderrahman ibn Ahmed ibn Yunis (or Yunos), Abu'l-
IJasan, al-Sadafi; died 1009.

7 Ibrahim ibn Hilai ibn Ibrahim ibn Zahrun, Abu Ishaq, ai-FTarrani. Born
923; died at Bagdad, 9^4.

8 Mohammed ibn Mohammed ibn Yahya ibn Isma'il ibn al-' Abbas, AbQ'l-
Wefa al-Buzjanf.

9 Al-IJasan ibn al-IIasan ibn al-JJaitam, Abu 'All, c. 965-1039.

176 PERSIA AND ARABIA

the solution of the cubic equation by the aid of conies and who
wrote on Euclid and astronomy; and Kushyar ibn Lebban, 1
who wrote on arithmetic, trigonometry, and astronomy.

Al-Nairizi 2 (died c. 922/3 ) was one of the notable loth cen-
tury writers on Euclid., He was interested in astronomy and
geometry, writing commentaries on both Ptolemy and Euclid,
but it is the commentary on the Elements, translated into Latin
by Gherardo of Cremona, that is best known.

As a type of the lesser commentators on Euclid in the loth
century there may be mentioned al-l.Iasan ibn 'Obeidallah, 3 who
wrote a commentary on the difficult parts of the Elements.

Translators into Arabic. Among the noteworthy translators
of this period were al-Hajjaj, 4 who made two translations of
at least six books of Euclid's Elements, and also translated
Ptolemy's Almagest] al-Jauhari/' who made astronomical ob-
servations at Bagdad and Damascus (c. 830) and wrote a
commentary on the Elements of Euclid; Honein ibn Ishaq, 6
who translated various Greek works, possibly including Ptol-
emy's Tetrabiblos, and who wrote on astronomy, but was more
celebrated as a physician and a philosopher ; and his son
Ishaq, 7 who was a physician and translated Euclid's Elements
and Data, the Almagest, Archimedes on the sphere and cylinder,
and probably the Spherics of Menelaus. Somewhat less well
known, but worthy of mention, are al-Arjani, 8 who wrote a com-
mentary (c. 850) on Euclid X; al-Himsi, 9 who translated the
first four books of Apollonius; and Sa'id ibn Ya'qub, 10 a
physician, who translated parts of Euclid and of Pappus.

1 Kushyar ibn Lebban ibn Bashahri al-Jili, Abu'l-Ilasan, c. 97i-c. 1029.

2 Al-Fadl ibn liatim al-Nairizi, Abu V Abbas.

3 Al-IIasan ibn 'Obeidallah ibn Soleiman ibn Vahb, Abu Mohammed, c. 925.

4 Al-Hajjaj ibn Yusuf ibn Matar, c. 786-c. 835.
s Al-'Abbas ibn Sa'id al- Jauhari.

6 Honein ibn Ishaq, al-'Ibadi Abu Zeid; born 809/10; died at Bagdad, 873.

7 Ishaq ibn Honein ibn Ishaq al-'Ibadi, Abu Ya'qub, died 910.

8 Ibn Rahiweih al-Arjani, or Arrajani, according to Steinschneider the same
as Ishaq ibn Ibrahim ibn Makhlad at-Mervazi, who died at Nishapur in 852/3.

9 Hilal ibn Abi Hilal al-tfimsi, died 883/4. Al-JJimsi means "from Emessa,"
in Syria.

10 Sa'id ibn Ya'qub al-Dimishqi, Abu 'Otman. He was living in 915.

TRANSLATORS INTO ARABIC 177

Abu Kamil. Between 850 and 930 there lived in Egypt Abu
Kamil, 1 who is known for several works but especially for his
treatise on the pentagon and decagon- and for his arithmetic
and algebra. 3 No writer of his time showed more genius than
he in the treatment of equations and in their application to
the solution of geometric problems.

About the same time there lived Abu'l-Faradsh Mohammed
ibn Ishaq, known as Ibn Abi Ya'qub al-Nadim, whose Kitdb
al-Fihrist (Book of Lists), written c. 987, is a collection of brief
biographies of various prominent mathematicians, both Greek
and Mohammedan. 4

Close of the Golden Age of Bagdad. In a general way it
may be said that the Golden Age of Arabian mathematics was
confined largely to the gth and roth centuries ; that the world
owes a great debt to Arab scholars for preserving and trans-
mitting to posterity the classics of Greek mathematics; and
that their work was chiefly that of transmission, although they
developed considerable originality in algebra and showed some
genius in their work in trigonometry.

5. THE CHRISTIAN WEST

The Dark Ages. The period from 500 to 1000 extends from
about the time of the fall of Rome (4S5) 5 to the first reawaken-
ing of Europe under Pope Sylvester II (Gerbert). It includes
the so-called Dark Ages, the period of the slow civilizing of the
northern races, of the development of monastic schools, of the
work of Charlemagne, and of the contact with Oriental civiliza-
tion, chiefly through the Moors in Spain. In mathematics it
was the era of the development of the Christian calendar in the
West, and of little else. The barbarian had to be civilized, to
assimilate slowly the Roman culture which he would have
destroyed, and to receive a better religion. The Roman schools

1 Abu Kamil Shoja ibn Aslam ibn Mohammed ibn Shoja.
2H. Suter, Bibl. Math., X (3), 15, 33-

8 L. C. Karpinski, Amcr. Math. Month., XXI, 37, and Bibl. Math., XII (3),
40. See also H. Suter, Bibl. Math., XI (3), 100.

4 Suter's translation appeared in the Abhandlungen (VI, i) in 1892.

6 The barbarians entered the city first in 410. The final fall is often given as 476.

178 THE CHRISTIAN WEST

had to be supplanted by those of the cathedral and the monas-
tery, and all the mathematics required was limited to the needs
of trade, to the keeping of accounts, and to the fixing of dates
for Church festivals. In those parts of Europe less subject to
Northern influence, such as Marseilles, Aries, and Narbonne,
the needs of commerce were still such as to render necessary the
arithmetic of exchange in the training of the merchant's appren-
tice. These cities maintained in this period their trade with
Italy, Constantinople, and the Orient, sending dyes, cereals,
pottery, and salt to the East, and importing silk from China,
pearls from India, and even papyrus rolls from Egypt. 1

Boethius. Anicius Manlius Severinus Boe'thius,- a Roman
citizen, a member of the distinguished family of the Anicii,
statesman, philosopher, mathematician, man of letters, and
founder of the medieval scholasticism, lived at the opening of
the period now under discussion. Persecuted for his upright-
ness, executed for his fearlessness, accepted by the Church as a
martyr, his reputation and scholarship gave his books on
mathematics high standing in the monastic schools for many
centuries.

His greatest work, written while he was in prison, is the
Consolation of Philosophy. 3 His mathematical works are an
arithmetic, 4 a geometry, 5 and a work on music/ 5 a subject then

3 A. Rambaud, Histoire de la Civilisation Fran^aise, i2th ed., I, 115. Paris, IQII.

2 Born at Rome c. 475; died at Paviu, 524. The more nearly correct Latin
form is Boetius. 3 De consolatione philosophiae.

*Boetii de institutione arithmetica libr: duo, ed. Friedlein (Leipzig, 1867);
hereafter referred to as Boethius^ ed. Friedlein. The earliest manuscripts used in
this edition are three of the loth century, a fact worth noting in view of
questions as to interpolations discussed later. Readers of Boethius and other
Latin writers will find assistance in B. Veratti, "Sopra la Terminologia Mate-
matica degli Scrittori Latini," Memorie della R. Accad. . . . di Modena, Vol. V.

*Boetii quae jertur geometria, in the Friedlein edition cited above. The earli-
est manuscript used in this edition is one of the ioth century. There is serious
doubt as to whether Boethius wrote the Ars Geometriae attributed to him. See
Tannery, La Geometric Grecque, 128; H. Weissenborn, "Die Boetius-Frage," in
the Abhandlungen II, 185.

6 Boetii di institutione muska libri quinque, in the Friedlein edition cited
above. The earliest manuscript used in this edition was mostly of the gth cen-
tury, although Books IV and V were earlier and some parts were missing.

BOETHIUS

179

ranked as part of mathematics. The arithmetic was based on
the work of Nicomachus, and the geometry on the Elements
of Euclid. Neither showed any originality in the domain of
mathematics, but each was sufficiently successful in its presen-
tation of the subject treated to permit of the general use of

Cl PTOLEMAEO-AUX- 71'BOF.TIO

FROM A DRAWING BY RAPHAEL
Fanciful sketches of Ptolemy and Boethius, now in the Accademia in Venice

these books in those monastic schools that had advanced far
enough to demand courses in the theory of numbers and in
demonstrative geometry.

Minor Writers. It is natural to expect that among the first
Christian scholars few would be found with any interest in
mathematics or the natural sciences. Their religious faith
was too intense, their persecutions too real, and their lives too
precarious to permit of speculations in these fields. The names
of a few Christians have already been mentioned, but their
contributions to mathematics were insignificant. With the

180 THE CHRISTIAN WEST

close of the 5th century, however, Christianity had become
powerful enough to permit of the development of an intellec-
tual class with interests outside of religious faith, and in this
class we find the names of several scholars who showed some
knowledge of the mathematics of the classical period.

Among these writers was Magnus Aurelius Cassiodo'rus, 1 a
descendant of an ancient Roman family. 2 He was a statesman
of distinction and was honored both by the last of the Roman
rulers and by their Ostrogothic successors. He founded a mon-
astery at Vivarium, and passed his last years within its
walls. He insisted upon a high standard of scholarship for the
clergy, and his writings show that he himself possessed, within
the limits which conditions then imposed, that which he de-
manded for others. Cassiodorus wrote De artibus ac discipli-
nis liberalium literarum, a trivial sort of compendium of the
seven liberal arts, grammar, rhetoric, and dialectic composing
the trivium, and arithmetic, geometry, astronomy, and music
composing the quadrivium. 3 This work was widely used in the
schools of the Middle Ages, 4 and nothing could better show
the low state of learning than this feeble -attempt at scholar-
ship. There is also doubtfully assigned to him a Computus
Paschalis sive de indicationibns cyclis soils et lunae, written in
562, one of the first treatises on the Christian calendar. The
plan for the adoption of the Christian era, however, was worked
out by Dionysius Exiguus, a Roman abbot, c. 525.

a Born at Scylaceum (Squillace), c. 470; died c. 564, a date sometimes given as
585. The name is also spelled Cassiodorius.

2 For a popular but vivid account of his achievements see M. Crawford,
Rulers of the South, II, 9.

3 A common medieval verse reads:

Gram loquitur, Dia verba docet, Rhet verba colorat,
Mus canit, Ar numerat, Ge ponderat, As colit astra.

P^trus Pictaviensis, in a verse to Peter of Cluny, writes :

Musicus, astrologus, arithmeticus, et geometra,
Grammaticus, rhetor, et dialecticus est.

4 The first collected edition of his works was published at Paris in 1584 and
IS98.

MANUSCRIPT OF THE AR1THMETICA OF BOETHIUS

This MS., now in Mr. Plimpton's library, was written c. 1294. The scribe has

used modern instead of Roman numerals

1 82 THE CHRISTIAN WEST

A little before the time of Cassiodorus there flourished
Martianus Mineus Felix Capel'la, 1 author of an encyclopedia
known as the Nuptials oj Philology and Mercury? It is a
medley of prose and verse, one part of the work being on geom-
etry and another on arithmetic. In connection with the latter
Capella discusses various classes of numbers and the supposed
mysteries of the smaller numbers. The book is even more
arid than that of Cassiodorus, the only redeeming feature being
the statement that Mercury and Venus revolve about the sun
instead of the earth. 3

Before the close of the sth century there was born in
Damascus a Syrian who took his name, Damas'cius, 4 from his
birthplace. He was the last of the important Neoplatonists
and was a disciple of the Marinus who succeeded Proclus
(c. 485). In 510 he became director of the school at Athens.
When Justinian closed the heathen schools of philosophy in
that city (529), Damascius went to Persia, but returned five
years later (534). His works were mostly philosophical, but
his name has doubtfully been connected with a fifteenth book to
be added to Euclid's Elements.

Almost the last of the Greeks to show any appreciation of
mathematics before the medieval period fairly began was
Euto'cius a of Ascalon. He wrote commentaries on the first
four books of the conies of Apollonius. He also wrote on
certain works of Archimedes, the sphere and cylinder, the
quadrature of the circle, and the work on equilibrium ; and on
the Almagest of Ptolemy, this last commentary being lost.
These writings of Eutocius are of little value except as they
supply certain information relating to Greek mathematics.

ifiorn possibly at Carthage, c. 420; died c. 490. See E. Narducci, in Boncom-
pagni's Bullettino, XV, 50$, with biography and bibliography. The name might
properly have been given in Chapter IV, but Capella is more closely related to
Boethius and Cassiodorus than to the last of the Greeks.

2 The first edition appeared at Vincenza in 1499, Opus Martiani Capelle de
Kuptijs Philologie & Mercurij libri duo.

a ln the De Astronomia, the chapter entitled Tellus quod non sit centrum
omnibus planetis. Fol. 333 of the 1592 edition.

6 EVT&KIOS. Fl. C. 560.

ISIDORUS 183

In the 6th century there seems also to have been written the
Codex Arcerianus, 1 so called from the fact that it belonged at
one time (1566-1604) to one Johannes Arcerius in Gro-
ningen. While it relates largely to legal matters of a rural
nature, it contains considerable information concerning the
Roman, surveyors.

There is little else to say for the century. It represents the
lowest point on the curve of intellectual progress in Europe.
The ecclesiastical element was unable to overcome the general
ignorance of the masses, and aside from a faint light in the
Irish monasteries, Europe was in darkness.

Isido'rus. The centuries immediately following the death of
Boethius saw little interest in the literature and science of the
classical period. Even as eminent a man as St. Ouen (c. 609-
683) spoke of the works of Homer and Vergil as the trifling
songs of impious poets 2 and made two distinct personages of
Tullius and Cicero ; while Gregory of Tours (538-594 ) uttered
the lament: "Unhappy our days, for the study of letters is
dead in our midst, and there is to be found no man able to
record the history of these times." So debased was civiliza-
tion that the few who stood for even the remnants of the old
Latin cult resorted to doggerel verse, as Capella had done, or
diluted their learning in the form of encyclopedias.

Prominent among those who developed the latter plan was
Isidorus of Seville, 3 historian, grammarian, orator, theologian,
bishop, and general scholar, as well as one of the most re-
markable statesmen of the Middle Ages. St. Martin, in his
funeral oration, describes him as "generous in his giving,
affable in his entertaining, sober in his affections, free in his
sentiments, equitable in his judgments, indefatigable in his
ministrations," and celebrated for his integrity. A man of
fortunate birth, he was helped by his family connections to
begin a career of such remarkable success, relative to that of

iMommsen puts it 0.450, and Cantor (Die Romischen Agrimensoren, p. q$
(Leipzig, 1875) ; hereafter referred to as Cantor, Agrimensoren) thinks it not
later than the 7th century. See also Cantor, Gesckichte, I, chap. 26.

2 "Sceleratorum neniae poetarum." 3 Born at Seville, c. 570; died April 4, 636.

1 84 THE CHRISTIAN WEST

any contemporary, that the Council of Toledo (653), a few
years after his death, could truthfully speak of him as "the
extraordinary doctor, the latest ornament of the Catholic
Church, the most learned man of the latter ages, always to be
named with reverence." Since he was the most learned man
of his time, it would be expected that his encyclopedia of the
trivium and the quadrivium, the seven liberal arts, would con-
tain some mathematics of merit. This work, called by him the
Origines but often known as the Etymologies, consists of twenty
books, the third one being on mathematics. The treatment,
however, is trivial, the arithmetic being simply a brief conden-
sation of Boethius, and the rest of the work being of as little
scientific value.

Bede the Venerable. It was about a century after Isidorus
that there was born at Monkton in Northumberland one of the
greatest of the Church scholars of the Middle Ages, Baeda
(c. 673-735), commonly known as Beda Venerabilis, the Ven-
erable Bede, 1 and called by Burke "the father of English
learning."

Of him Hallam 2 remarked that he "surpasses every other
name of our ancient literary annals; and, though little more
than a diligent compiler from older writers, may perhaps be
reckoned superior to any man the world (so low had the East
sunk like the West) then possessed." Four years before his
death he prepared a list of the thirty-seven works which he had
written up to that time, and added these words : " I have spent
my whole life in the same monastery, and while attentive to
the rule of my order and the service of the Church, my

*G. F. Browne, The Venerable Bede (London, 1880). For a discussion of his
scientific works, see J. A. Giles, Miscellaneous Works of the Venerable Bede,
VI, pp. v, 123 (London, 1843) ; J. Mabillon, "Ven. Bedae elogium historicum,"
in the Opera Omnia of Bede (Paris, 1862) ; K. Werner, Beda der Ehrwiirdige
und seine Zeit (Vienna, 2d ed., 1881). Bede was buried at Jarrow, but his re-
mains were moved to Durham c. 1050 and his tomb may now be seen in the
Galilee Chapel of the Cathedral. A good setting for the study of the education
of the period may be found in F. P. Barnard, Companion to English History
(Middle Ages), p. 303 (Oxford, 1902).

2 Literature of Europe, Chapter I, 7 (London, n.d.).

BEDE AND ALCUIN 185

constant pleasure lay in learning, or teaching, or writing." 1
Taught by Aldhelm and by John of Beverley, two of the heirs
to the intellectual and spiritual treasure which Augustine be-
queathed to Canterbury, 2 he was also a disciple of Archbishop
Theodore of Tarsus and Abbot Adrian, two pioneers in bring-
ing a high grade of scholarship to the monasteries, and thus he
was well prepared to render service to the world and to lead
a life "consecrated in noiseless activity to God." 3

In mathematics his interests were in the ancient number
theory, the ecclesiastical calendar, and the finger symbolism of
number, and his writings include these and other mathematical
subjects. 4 To him we are indebted for the best work on the
calendar written during the Dark Ages, and for the best work
up to his time on digital notation. 5 Certain mathematical
recreations have also been attributed to him, but the evidence
concerning their authorship is not conclusive.

Alcuin of York. The next great European scholar in mathe-
matics was Al'cuin (735-804). Born in the year of Bede's
death, less of a scholar than the latter but more of a man of
action, he attained prominence in the State as well as in the
Church. He studied in Italy, taught at York, 7 was called
(782) by Charlemagne 8 to assist him in his ambitious project
for the education of his people, and became abbot of St. Martin

*" Semper aut discere aut docere aut scribere dulce habui," words worthy of
the one whom Green, the historian, speaks of as "the first great English scholar."

2 W. F. Hook, Lives of the Archbishops of Canterbury, I, 42 (London, 1860) ;
A. Neander, Church History, $th American ed., Ill, 12 (Boston, 1855) ; J. E. G.
de Montgomery, State Intervention in English Education, p. 6 (Cambridge,
1002). See page 187, note 2.

3 Neander. For a description of Bede's death see loc. cit., p. 153.

*De numeris, De temporum ratione, De numerorum divisione, De circuits
sphaerae et polo, De astrolabio.

5 His De temporibus comes down only to 701/2. His De temporum ratione
comes down to 726. This second work contains his De Indigitatione sive de
compute per gestum digitorum and his De ratione unciarum.

6 As shown by one of his letters (XV) ; see Libri, Histoire, I, 89.

7 On the nature of the schools, see W. W. Capes, The English Church in the
i4th and i$th centuries, p. 332 (London, 1900).

s Who addressed him as "Carissime in Christo praeceptor." Charlemagne
reigned as king or emperor from 768 to 814.

1 86 THE CHRISTIAN WEST

of Tours. He wrote on arithmetic, geometry, and astronomy, 1
and his name is connected with a certain collection of puzzle
problems" which has influenced the writers of textbooks for a
thousand years. It is uncertain how much he may have had to
do with this set of mathematical recreations, and considerable
doubt has been thrown upon his connection with them through
recent studies of a certain manuscript at Leyden.* This manu-
script dates from the first part of the nth century and is
thought to have been written by, or at least inspired by, a monk
named Ademar or Aymar, of the ancient house of Chabanais,
who was born in 988 and who died on his way to the Holy
Land in 1030. He had considerable reputation as a his-
torian and a controversialist 4 and seems to have collected a
large amount of material with no scientific care. These prob-
lems were very likely part of the medieval versions of
Fables, collections which, although probably begun by
in Samos, in the yth century B.C., were modified by Babrius
about the 3d century, and were still further corrupted in the
Middle Ages. While problems attributed to Alcuin are found
here, and probably interested Ademar as they did hundreds of
others, there seems to be no good reason to believe that Alcuin
may not have collected them from the medieval versions at-
tached to the Fables. Certain it is that letters of Alcuin show
that he wrote a set of puzzle problems, although there is no
direct evidence that this is the one. 5 It would have been in
keeping with his ideas to compile a book that should be amus-
ing enough to relieve education of the drudgery of the time. 6

1 For his life and works, see G. F. Browne, Alcuin of York (London, iqo8) ;
C. J. B. Gaskoin, Alcuin; his life and his work (London, 1004) ; R. B. Page,
The Letters of Alcuin (New York, 1909) ; A. F. West, Alcuin and the rise of
the Christian Schools (New York, 1912).

2 Propositions ad acuendos juvenes.

3 Cod. Vossianus Lat. oct. 15, edited by G. Thiele and published at Leyden
in 1905. 4 J. Lair, L'Histoire d'Ademar, Paris, 1899.

5 It has been published in the works of Bede as well as in those of Alcuin.
The oldest MS. of the work, written early in the nth century, is now in
Karlsruhe. In this are the words: " Dilectissimo fratri siguulfo presbytero al-
cuinus salutem," but naturally these words are not absolutely conclusive evidence.

6 When "sub virga degere" meant school life and "pueri subiugales" meant
pupils. T. Ziegler, Geschichte der Padagogik, p. 29 (Munich, 1895).

ALCUIN OF YORK 187

In the collection is to be found, for example, the problem of
the hare and hound, already ancient but made the more mys-
terious by the cipher title,

"De cursu cbnks be fugb lepprks,"
for "De cursu canis ac fuga leporis." 1

The continued private wars among petty lords in the loth
and nth centuries made France a poor field for mathematical
or other intellectual progress, and hence these two centuries
produced little that was noteworthy.

Decay of British Learning. After the death of Alcuin the
brilliant era that started in Great Britain with St. Au'gustine
of Canterbury 2 (died c. 604 or 613) closed as suddenly as it
began. The ravages of the Danes put an end to that feeling
of security which makes for intellectual development, and when
Alfred (848-900) came to the throne (871) he could only
lament, " There was a time when people came to this island
for instruction, but now we must obtain it abroad if we desire
it." When Aethelstan, 3 the grandson of Alfred, came to the
throne (925), however, he showed great interest in the foster-
ing of learning, and in a poem written in the i4th century
reference is made to the introduction of Euclid into England
in the reign of this powerful ruler :

Thys grete clerkys name wes clept Euclyde,
Hys name hyt spradde ful wondur wide. , . .
The clerk Euclyde on thys wyse hyt fonde,
Thys craf te of gemetry yn Egypte londe ;
Yn Egypte he tawjhte hyt ful wyde,
Yn dyvers londe on every syde . . .
Thys craft com ynto Englond as y $ow say
Yn tyme of good kynge Adelstonus day. 4

1 Cantor, Agrimensoren, 139, 142.

2 Not to be confused with the greater St. Augustine of Hippo (354-430).
SAthelston, Ethelstan, Adelstan, Adelston, Edelstan, and other spellings

Born c. 895; died 941.

4 The MS. is in the British Museum (Bib. Reg. iyA, I. p. 32), and was
published by J. O. Halliwell, The Early History of Freemasonry in England
(London, 1840).

1 88 THE CHRISTIAN WEST

Jewish Activity. Probably about the time of Alcuin a Jewish
mathematician, Jacob ben Nissim, wrote a work entitled Sefer
JeziraJ which, like various Hebrew writings on mathematics,
contains some material on the theory of numbers.

Hrabanus Maurus. Alcuin's most famous pupil was Magnen-
tius Hrabanus Maurus, 2 "Primus praeceptor Germaniae," ab-
bot of the monastery at Fulda ( 822 ), and archbishop of Mainz
(847). In his younger days he traveled extensively 1 ' and wrote
a worthy treatise on the calendar, based on Bede's work and
showing a commendable knowledge of astronomy, a science
which included most of the mathematics of his time. 4

One of his contemporaries, Walafried Strabus 5 (c. 806-
849), is known to have taught mathematics at Reichenau,
near Constance, but he left no works upon the subject.

Remigius of Auxerre. A second great pupil of Alcuin's and a
witness to the beneficent influence of the Church in France,
was Remi'gius' 5 of Auxerre, a Benedictine monk who did much
for the schools at Rheims and who founded a school at Paris
out of which the university is thought by some to have devel-
oped. 7 He wrote a commentary on the arithmetic of Capella, 8
not an important contribution to mathematics, but typical of a
period given to useless disputation and empty sophistry.

1 Book of Creation. See M. Steinschneider, "Miscellen zur Gesch. der Math.,"
in Bibl. Math., Ill (2), 35; IX (2), 23. An Arabic commentary is known to
have been written upon it in the loth century. The question of the authorship
of the Sefer Jezira is still unsettled. The work relates chiefly to number
mysticism.

-Born c. 776; died 856. The name appears also as Rabanus Maurus. The
date of his birth is also given as c. 784.

3 Ego quidem, cum in locis Sidonis aliquoties demoratus sim." See Neander,
Church History, III, 457.

4 He also wrote an encyclopedia, De universo libri XXII, sive etymologiarum
opus. On his life see J. N. Bach, Hrabanus Maurus der Schdpfer des deutschen
Schulwesens (Fulda, 1835) J D- Tiirnau, Rabanus Maurus, der praeceptor Ger-
maniae (Munich, 1900).

r 'Walafrid Strabo. Cantor, Geschichte, I (2), 792.
The name comes from Remy, Remi, i.e., Rheims. Died .908.
7 It is also said to have developed from a school of dialectics opened by
William of Champeaux, c. uoo to mo.

8 The Vatican codex was published in Boncompagni's Bullettino, XV, 572.

NINTH AND TENTH CENTURIES 189

Hrotsvitha. A certain amount of light is thrown upon the
barren field of monastic mathematics of this period by the story
of the learned nun Hrotsvitha 1 of the Benedictine abbey of
Gandersheim, in Saxony. She wrote several plays and in these
she shows a knowledge of the Greek language and of either
Greek or Boethian arithmetic. In the Sapicntia the emperor
Hadrian demands the ages of the three daughters of Wisdom
(Sapicntia), namely, of Faith, Hope, and Charity. Wisdom
then says that the age of Charity is a defective evenly
even number ; that of Hope a defective evenly odd one ; and
that of Faith an oddly even redundant one. Upon Hadrian's
remarking, "What a difficult and tangled question has been
raised about the mere ages of these girls ! " Wisdom replied,
"In this is to be praised the great wisdom of the Creator and the
marvelous knowledge of the Author of the universe." 2 Hrot-
svitha incidentally speaks of three perfect numbers besides 6,
namely, 28, 496, and 8i28. 3

Other Writers of the Tenth Century. In the loth century
there may also have been written a treatise on the abacus by

iBorn c. 932; died c. 1002. Hrosvithae Opera, edited by Winterfeld (Berlin,
iQ02) (in the Scriptores rerum Germanic arum) ; Hrotsvithae Opera, edited by
Strecker (Leipzig, 1906) ; all but one of her works, edited by Conrad Celtes, and
with engravings by DUrer, were published at Niirnberg in 1501 ; there was also
an edition by Schurzfleisch (Wittenberg, 1707), and a complete edition by
Barack (Niirnberg, 1858). See also Ch. Magnin, Theatre de Hrotsvitha re-
ligieuse Allemande du X e siecle (Paris, 1845) ; E. R. A. Kopke, Die alteste
deutsche Dichtcrin (Berlin, 1869), with a refutation of a charge made by Asch-
bach (1867) that Celtes had forged the works.

The old historian Henricus Bodo referred to her in saying, "Kara avis in
Saxonia visa est." The name appears in various other forms such as Roswitha
and Hrotsuit. In the Munich MS., apparently contemporary, it appears as
Hrotsvith and Hrotsuitha.

2 A few of the sentences will show the style of the original :

Sapientia. Placetne vobis, O filiae, ut hunc stultum arithmetica fatigem
disputatione ?

Fides. Placet, mater, . . .

Sapientia. O Imperator, si aetatem inquiris parvularum, Caritas imminutum
pariter parem mensurnorum [ = annorum] complevit numerum ; Spes autem
aeque imminutum, sed pariter imparem ; Fides vero superfluum imparitcr parem.

s "XXVIII, CCCCXCVI, VIII millia CXXVIII perfect! dicuntur." A
perfect number is one that is equal to the sum of its aliquot parts, that is, of its
factors and unity ; for example, 6=1 + 2 + 3.

THE CHRISTIAN EAST

Odo of Cluny (Syg-c. 942), although it may be the work of
a 1 2th century writer 1 ; but in general the period was a barren
one. Only one other writer is worthy of mention, Abbo of
Fleury (945-1003), a native of Orleans, who wrote on Easter
reckoning, 2 on astronomy, and on the arithmetic of Boethius.
His chief title to remembrance, however, is the fact that he was
a teacher of Gerbert, the most learned man of his time, whose
life and works are considered in the next chapter.

Another example of the ecclesiastical scholar is seen in the
case of Bernward, who became Bishop of Hildesheim in 993 3
and who wrote a work on mathematics which was devoted
chiefly to the Boethian theory of numbers. A manuscript of
this work, possibly the original, is still extant at Hildesheim.

6. THE CHRISTIAN EAST

Egypt and Constantinople. The eastern countries touching
upon the Mediterranean did little for mathematics for a period
of five centuries after the fall of Rome. Even, the brilliant
reign of Justinian (527-565), "the Lawgiver of Civilization,"
was not able to remove the fears of a barbarian invasion, nor
to suppress the disastrous feuds between the Blues and the
Greens in Byzantium. Add to this the great fire of 532 and
the terrible pestilence of ten years later, and it will be seen
that the banks of the Bosporus were not the place for an
intellectual revival.

Decay of Alexandria. In Alexandria the chance of progress
in the arts and sciences seemed to die out with the fall of Rome,
and with the rise of Mohammedanism as a world power the
last hope of any revival of the city's ancient glories definitely
disappeared. Eighty years after the death of Mohammed his
followers had conquered all of northern Africa and had estab-
lished themselves firmly in Spain. In 642 the great library

1 S. Gunther, Geschichte der Mathematik, I, 244 (Leipzig, 1908) (hereafter
r-ferred to as Gunther, Geschichte) ; Cantor, Geschichte, I (3), 843; Th. Martin,
"Les Signes Numfraux," Annali di Mat. pura ed applic., V, 50, and reprint,
Rome, p. 78 (1864). * Liber in calculum paschalem.

3 H. Duker, Der liber mathematicalis des HeUigen Bernward. Hildesheim, 1875.

EGYPT AND CONSTANTINOPLE 191

of Alexandria was destroyed by fire, probably the most serious
loss that ever befell any great institution of learning.

Nevertheless a few names appear in the Christian East.
Anthemius, 1 an assistant architect in the building of St. Sophia,
wrote on conies, and a century later (c. 610) Stephen of Alex-
andria wrote on mathematics and astronomy and taught in
Constantinople. In Alexandria, just before the Mohammedan
invasion, Asclepias of Tralles (c. 635") wrote a commentary
on Nicomachus, and Joannes Philop'onus (c. 640'), known
also as Joannes Grammat'icus, did the same and also wrote
upon the astrolabe. 4

Toward the close of the igth century there was found at
Akhmim, 5 in Upper Egypt, a Greek papyrus which seems to
have been written about the yth or 8th century. In this there
are tables of unit fractions similar to those found in the Ahmes
papyrus, but the work shows no advance over its predecessor
of more than two thousand years earlier. Science had long
been dead in Egypt except in that part which came under the
influence of Alexandria.

School of Cairo. In the early part of the loth century the
Fatimites, a branch of the Mohammedan ruling class, drove
their rivals for power out of the city which they thereupon
called al-Kahira, the Victrix, the modern Cairo. Here they
proceeded to establish a school which they ventured to hope
would rival that of ancient Alexandria, and which indeed be-
came a center of astronomical activity. With it were connected
the names of Ibn Yunis (p. 175) and al-I.Iaitam (p. 175),
but it was short-lived, the caliphate of Egypt being destroyed
by Saladin in 1171.

*Died at Constantinople, 534.

2 Possibly a century earlier.

3 The date is very uncertain, being possibly a century too late.

*De vsv astrolabii ejusque constructions libellus. It was published by H.
Hase, Rheinisches Museum fur Philologie, VI, 127 (Bonn, 1839). His work on
Nicomachus was edited by Hoche, Leipzig, 1864, and Wesel, 1867.

5 Or Ekhmim, the site of the ancient Chemmis or Panopolis. It became a
great religious center under the Christians of the early Middle Ages. Nestorius
(5th century), the patriarch of Constantinople, was deprived of his honors and
banished to Akhmim for heresy. See also Heath, History, II, 543.

192 SPAIN

It is probable that the Jewish scholar Sa'adia ben Joseph 1
studied at Cairo during this period. He wrote on the division
of inheritances and on the calendar. He taught in Babylon,
where he doubtless met with Isaac ben Salom, who wrote on the
Hindu arithmetic and on astronomy.

7. SPAIN

Oriental Civilization in the West. After the burning of the
Alexandrian library (642) the Mohammedans continued their
conquests, sweeping along the north coast of Africa and finally
entering Spain in 711, defeating the Visigothic king, and estab-
lishing themselves for a sojourn of eight hundred years.
Bringing with them the Oriental faith in astrology, their pri-
mary interest in mathematics was related chiefly to astronomy,
trigonometry, and the conies ; possessed of esoteric tastes, the
mysteries of numbers and of gematria 2 appealed to them;
coming into constant relations with the Jews, the cabala doubt-
less impressed them ; inspired by the intellectual brilliancy of
Bagdad, the classics of the Greeks found place in their schools.
By the time the intellectual supremacy of Bagdad was seriously
threatened in the East, Cordova was becoming the intellectual
center of Islam in the West. Alhakem II, who reigned from
961 to 976, established a considerable library there, and about
the close of the loth century al-Majriti, 3 a native scholar,
wrote on amicable numbers, astronomy, and geometry.

Even in the loth century the activity in the field of mathe-
latics was not great. The first writer of note was Muslim ibn
Ahmed al-Leiti, Abu 'Obeida, also called Sahib al-Qible (died
907/8), a native of Cordova and a writer on astronomy and
arithmetic. About the same time Cordova produced Salhab
ibn 'Abdessalam al-Faradi, Abu'l-' Abbas (died 922/3), an
arithmetician of some note.

*In Arabic, Sa'id ibn Yusuf al-Fayyumi. He died in 941. The Hebrew
name Sa'adia Gaon means Sa'adia the Genius (Great).

2 Largely concerned with the evaluating of names by the numerical value of
the letters.

3 Abu'l-Qasim Maslama ibn Ahmed al-Majriti, died 1007/8.

DISCUSSION 193

TOPICS FOR DISCUSSION

1. Intercourse between China and other countries, and its possi-
ble influence upon mathematics.

2. Progress of Chinese mathematics from 500 to 1000.

3. Nature and sources of early Japanese mathematics.

4. General nature of Hindu mathematics from 500 to 1000.

5. The work of the two Aryabhatas.

6. Brahmagupta and the School of Ujjain.

7. The work of Mahavira compared with that of Brahmagupta.

8. The Bagdad School, its rise and its relation to the Hindu and
Greek learning.

9. The nature of the contributions of the Persian and Arab
mathematicians of the ninth and tenth centuries.

10. The life and works of Mohammed ibn Musa al-Khowarizmi.

1 1 . Causes of the decay of eastern Arabic mathematics.

12. Indebtedness of medieval Europe to Oriental mathematics in
the Middle Ages.

13. Causes of the low state of mathematics in Europe during the
greater part of the Middle Ages.

14. Boethius as a mathematician.

15. The life and mathematical works of Bede.

1 6. The life, influence, and mathematical works of Alcuin.

17. Evidences of an interest in the Greek theory of numbers in
the Middle Ages.

18. The influence of mathematics in the Middle Ages upon the
science at present.

19. The mathematics of the quadrivium.

20. The nature of the encyclopedias produced by the Church
scholars of the Middle Ages.

21. Mathematical recreations in the Middle Ages.

22. Nature of the mathematics studied in the British Isles in the
early part of the Middle Ages.

23. The Church schools as preservers of mathematical knowledge
in the early part of the Middle Ages.

24. The mathematical contributions of the Mohammedans of the
ninth and tenth centuries in Spain.

25. The relation of medieval astrology to astronomy and also to
mathematics in general.

CHAPTER VI

THE OCCIDENT FROM 1000 TO 1500
i. CHRISTIAN EUROPE FROM 1000 TO 1200

Religious and Political Influences. Just how much influence
the passing of the first Christian millennium had upon the com-
mon people it is difficult to say. Historians pay much less
attention to the "terreur de Tan Mil" than was formerly the
case. It is not probable that many educated persons took
literally the biblical remark relating to the period of a thousand
years, but it is certain that it was so taken by some. At any
rate, the passing of this milestone saw the Christian world
aroused to new interests.

Then, too, there were the crusades (iog$-c. 1270), which
have been called "the first Renaissance," and which did for a
civilization that had long been dormant one thi^o; which the
World War did for the civilization of the 2oth century, it let
one part of the race know more of what other parts were doing
and thinking and hoping. It was war, but it was in general
beyond the boundaries of intellectual Europe.

There was also the potent influence in Europe of a foreign
and highly developed civilization in her midst, the Saracen
supremacy in Spain ; and it was the Saracen scholars who made
known to Latin scholars the best of the Greek and Oriental
civilizations.

Moreover, Europe was seeing the folly of her private wars,
the "Truce of God" was beginning to make its power felt,
and the blessings of peace were once more settling upon France
and her neighbors, rendering intellectual pursuits possible.

To these influences there should be added that of the Nor-
man Conquest, which, without prolonged warfare, awakened
and united England, and showed her what the Continent had
for her in the way of science and art.

194

GERBERT 195

As a result of such influences Europe entered upon a new era,
one in which cathedral building, 1 church reform, renewed atten-
tion to art, political experiment, and scientific achievement
played great parts.

Gerbert. Nevertheless, the period was still dominated by the
spirit of the earlier centuries of the Middle Ages, "when faith
overpowered intelligence" and " authority became the enemy
of investigation," when "scholars degenerated into schoolmen"
and "science lost itself in the morasses of alchemy or astrology
and became anathema to the faithful." 2 This is seen in the
attitude of the learned world toward that remarkable church-
man and scholar, Gerbert, 3 one of the greatest popes that ever
added lustre to the Church and to the city of Rome. Elevated
to the papal throne, he reigned under the name of Sylvester II
from 999 until 1003. He was born of humble parents, 4 but his
natural brilliancy led to his call to study under the monks at
Aurillac, and particularly under such a worthy scholar as Abbo
of Fleury, and to his being sent to Spain (967) to perfect his
education. 5 About 970 he went to Italy, where he was pre-
sented to the pope and by him to the emperor, returning to

1 " It was as though the world had arisen and tossed aside the worn-out gar-
mer'ts of ancient time, and wished to apparel itself in a white robe of churches."
Raoul Glaber (o85-c. 1046).

2 W. C. Abbott, The Expansion of Europe, I, chap, i, New York, 1918.

3 Born near Aurillac, in Auvergne, c. 950; died at Rome, May 12, 1003. The
name is pronounced zher-bar.

4 "Obscuro loco natum," as an old chronicle states.

R For bibliography and for a more elaborate sketch, see Smith -Karpinski,
p. no seq. See also Cantor, Geschichte, II, chap. 39; J. Havet, Lettres
de Gerbert (983-907), Paris, 1889; N. Bubnov, Gerberti posted Silvestri II
papae opera Mathematica, Berlin, 1899; A. Olleris, (Euvres de Gerbert, Paris,
1867; F. Picavet, Gerbert, un pape philosophe, d'aprh I'histoire et d'apres
la ttgende, Paris, 1897; H. Weissenborn, Gerbert. Beitrdge zur Kenntnis der
Math, des Mittelalters, Berlin, 1888; C. F. Hock, Gerberto o sia Silvestro
II Papa ed il suo secolo, Milan, 1846; A. Nagl, Gerbert und die Rechenkunst
des X. Jahrh., Vienna, 1888; G. Friedlein, "Die Entwickelung des Rechnens
mit Columnen," Zeitschrift fur Mathematik und Physik, X, HI. Abt, 241
(hereafter referred to as Zeitschrift (HI. Abt.)), and Gerbert^ die Geometrie
des Boethius, Erlangen, 1861; K. Werner, Gerbert von Aurillac, Vienna, 1878;
B. Carrara, Memorie dell' Accad. d. Nuovi Lincei, XXVI, 195; K. Schultess,
Papst Silvester II. (Gerbert) als Lehrer und Staatsmann t 8.1. a.

196 CHRISTIAN EUROPE FROM 1000 TO 1200

France in 972. He held various offices in the Church, and in
999 was elected to the papacy. He was a man of great learn-
ing, was "accused our learning's fate of wizardry," com-
bated error, aroused new interest in mathematics, acquired a
knowledge of the Hindu-Arabic numerals, gave some attention
to the study of astrology (a subject then looked upon as a
worthy science), and wrote on arithmetic, 1 geometry, 2 and other
mathematical subjects, and probably on the astrolabe. 3

Minor Church Writers. Contemporary with Gerbert, but liv-
ing a life as humble as Gerbert's was magnificent, was an English
monk of the abbey of Ramsey, Byrhtferth 4 by name. He trav-
eled in France and studied under Abbo of Fleury. Returning to
England he found waiting for him at Ramsey a group of stu-
dents to whom he proceeded to teach astronomy, the calendar,
and the principles of mathematics. 5 Times were not propitious
for study, however. For three centuries in England ( 1000-1300)
there was an average of a famine every fourteen years, and
life was hard. Perhaps the need for the conquest of mind over
matter, which such calamities set forth, was one of the in-
fluences that made possible the later thinkers of England.

On the Continent, St. Gall was one of the chief centers of
monastic learning at this time, and here the well-known scholar
Notker Labeo 6 (c. 950-1022) translated parts of the encyclo-
pedia of Capella and possibly some of the arithmetic of
Boethius, besides writing a computus. 7

1 Regulae de numerorum abaci rationibus; Scholium ad Boethii arithmeticam.

2 Gerberti Isagoge Geometriae. Some doubt has been expressed as to his
authorship, but he probably compiled the work.

3 Gerberti Liber de astrolabio t placed by Bubnov with other works among
the Opera Dubia.

* Or Bridferth. Fl. c. 1000.

5 De temporum ratione, De natura rerum, De indigitatione, De ratione un-
ciarum, De principiis mathematicis, the extant MSS. being merely notes of his
lectures. The Anglo-Saxon text of his Handboc was published by F. Kluge in
Anglia, VIII, 298. See also the Cologne edition (1612) of Bede's works. There
are two other works doubtfully attributed to him.

o" Notker the Thick-lipped," so called to distinguish him from earlier scholars
of the same family.

7 A. A. Bjornbo in the Reallexikon der Germanischen Alter turns kunde, IV,
465. Strasburg, 1916.

THE ELEVENTH CENTURY 197

Of the mathematical pupils of Gerbert the most prominent
was Bernelinus of Paris, who wrote an arithmetic 1 in which he
explained the use of Gerbert's counters, but concerning his life
nothing further is known. 2

A little later (c. 1028) Guido of Arezzo (Aretinus), a Bene-
dictine monk from Pomposa, near Ferrara, wrote on arithme-
tic, 3 and at about the same time (c. 1066) Franco of Liege
did the same and, what was not so common at this time, wrote
on the quadrature of the circle. 4 Among his contemporaries
was Wilhelm, abbot of Hirschau (1026-1091), who taught
mathematics and astronomy.

Hermannus Contractus. The most prominent of the succes-
sors of Gerbert in the nth century was Hermannus (1013-
1054), son of the Swabian Count Wolverad. His limbs having
been painfully contracted from childhood, he is known in his-
tory as Hermannus Contractus. 5 Educated in the monastic
school at Reichenau, he afterwards joined the Benedictine or-
der, became a lecturer on mathematics, and gathered about him
a large number of pupils. He wrote on the astrolabe, 6 the
abacus, and the number game of rithmomachia. 7

Psellus. The period of intellectual activity in the West had
very little counterpart in Constantinople. Life was still stag-
nant there. In the nth century only a single name stands out as
representing any interest whatever in mathematics in the eastern
capital, that of Michael Constantine Psellus 8 (1020-1110), a
Greek writer who studied at Athens, became a zealous Neopla-
tonist, and returned to Constantinople to teach philosophy.

i Liber Abaci. 2 Gerbert's (Euvres, ed. Olleris, p. 357 (Paris, 1867).

3 B. Baldi, Boncompagni's Bullettino, XIX, 590.

4 Abhandlungen, IV, 135.

5 Treutlein, Boncompagni's Bullettino, X, 589, where his Abacus is published;
Gunther, Math. Unterrichts, p. 47 ; Baldi, Cronica, p. 70. He is also known as
Hermann the Lame.

6 There is a beautifully written MS. of this work, I2th century, in the British
Museum (22,700), first published by Fez in Volume III of his Thesaurus
Anecdotorum and republished by Migne in Volume CXLIII of Patrologiae
cursus complete. 7 See page 198.

8 ^AXoj, called also Psellus the Younger, there having been another Psellus
who taught philosophy c. 8?o. Heath, History, II, 545,

1 98 CHRISTIAN EUROPE FROM 1000 TO 1200

He lived during the reigns of several rulers, consulted by
the emperors and honored by them with the title of Prince
of Philosophers. 1 An introduction to the study of Nicomachus
and Euclid is attributed to him, but the authorship is doubtful.
Partly because of the fact that he was almost the last of the
Greek writers on mathematics, partly because his works were
easily read, and partly because of his reputation for learning
in general, he is one of the few scholars of his time whose
mathematical contributions attracted any attention in the
Renaissance period. His leading works on mathematics 2 were
published at least thirteen times in the i6th century. The fact
that he takes \/8 as the value of TT shows how little he merited
his reputation as a scientist.

Rithmomachia. In speaking of the nth century mention
should be made of the number game of rithmomachia. 3 One of
the earliest treatises on the subject is due to Fortolfus, a monk,
who lived probably at the close of the nth century, 4 and the
indications are that it was not known before that century,
although it is occasionally attributed to Boethius and even to
Pythagoras. There is a manuscript in the Vatican library on
the subject, under the title "Ritmachya," written in 1077 by
a monk known as Benedictus Accolytus, and the game is also
referred to in a medieval poem De Vetula* Among the early

1 4>i\o<r60wi> vTraros.

2 Sapientissimi Pselli opus dilucidum in quattuor Mathematicas disciplinas,
Arithmeticam, Musicam, Geometriam, & Astronomiam, edited by Archbishop
Arscnius, Venice, 1532. This was the first edition, the text in Greek. The
Compendium Mathematicum, containing various works, appeared at Leyden
in 1647, but numerous others still remain unedited.

-Literally, "combat of numbers." The word is spelled in various ways,
rithmimachia, ritmachya, richomachie, and rhythmimachia being among the
most common forms.

A work on the subject by Boissiere, a French mathematician of the i6th
century, is entitled Nobilissimvs et antiqvissimus ludus Pythagoreus (qui
Rythmomachia nominatur), Paris, 1556. See Kara Arithmetica, pp. 12, 63,
271, 340.

*R. Peiper, "Fortolfi Rythmimachia," Abhandlungen, III, 167, 198.

B "O ut ; nam ludus sciretur Rythmimachiae !
ludus Arithmeticae folium, flos fructus et eius

gloria laus et honor." ,,, ,, TTT

Abhandlungen, III, 222

RITHMOMACHIA

199

writers who were interested in the subject were Hermannus
Contractus (1013-1054), as already stated, and both Jordanus
Nemorarius (died c. 1236) and Nicole Oresme (c. 1323-1382).

RITHMOMACHIA

From a work published at Paris in 1496. The middle portion of the board is

omitted. The part on Rithmomachia may be due to Bishop Shirwood of Durham

(died 1494), but is usually ascribed to Faber Stapulensis (1455-153^)

The game is based on the Greek theory of numbers as set
forth by Nicomachus. It was played upon a double chess-
board, rectangular in form, one side having eight squares and

200 CHRISTIAN EUROPE FROM 1000 TO 1200

the other sixteen. The pieces were triangles, squares, circles,
and pyramids, each possessing a certain value. These pieces
were arranged as shown in the illustration (p. 199) from a
work of 1496. The numbers were not taken at random, but
the plan on which they were arranged is too elaborate for
description in this work. Suffice it to say that when we form
the triangles we have 81 = 72 + \ of 72, 72 = 64 4- 1 of 64,
6 = 4 -h i- of 4, and 9 = 6 + \ of 6 ; that in the case of the square
pieces, 45 = 25 + 20 and 15 = 9 + 6; that the pyramids are
superposed squares such that 91 = 6 2 4- 5 2 + 4 2 + 3 2 -f- 2 2 -f i 2
and 190= 8 2 -f-; 2 + 6~ H-s 2 4-4 2 . In the case of the squares

there is a formula s = ( j /, in which the meaning of each

letter may be found by looking at the illustration, where
s 25, s' 15, n = 2 ; or s = 81, s' = 45, n = 4. The play is
very complicated, and for our purposes we may say that the
climax of the game was reached in the Victoria praestantissima,
in which it was necessary to get four numbers in a row, embody-
ing all three of the common progressions, arithmetic, geomet-
ric, and harmonic, the only possible solutions with these pieces
being six in number. It will be seen that the game requires
such familiarity with the Greek number theory as to make it
available only for the elite in mathematics in the Middle Ages.
Its popularity is attested by at least three manuscripts of the
nth century and three of the i2th and i3th centuries, besides
several printed treatises on the subject, all going to show that
there were more scholars in number theory than we should think
from the meager list of names that have come down to us/

A Century of Translators. The i2th century was to Christian
Europe what the 9th century was to the eastern Mohammedan
world, a period of translations. In the case of Bagdad, these
translations were from the Greek into Arabic; in the case of

1 For a description of the game, see D. E. Smith, "Number Games and
Number Rhymes," in Teachers College Record, XIII (New York), 385, together
with a history of "The Great Number Game of Dice." The article on rithmo-
machia may also be found in the Amer. Math. Month., April, 1911. See also
E. Wappler, Zeitschrift (HI. Abt.), XXXVII, i.

MEDIEVAL TRANSLATORS 201

Christian Europe, from the Arabic into Latin. The reasons for
this desire to know the science of the East are not difficult to
find. The causes already mentioned in connection with the
nth century were even more potent a hundred years later,
and the advancement of Moorish Spain in the arts and sciences
was already causing intellectual unrest in the higher class of
Church schools in France, Italy, and England. The result of
this unrest was an influx of students into Spain, an acquiring
of some knowledge of Arabic on the part of various scholars,
and a strong desire to know and to make known the science of
the East. Just as Bagdad never translated the Greek literature,
but sought diligently to know Greek science, so Europe gave
little attention to Arab letters, but devoted great care to those
works on astronomy, arithmetic, trigonometry, optics, astrology,
geometry, and medicine that had acquired reputation in the
capital of the caliphs. Even the Elements of Euclid became
known to the scholars of the Latin Church chiefly through its
Arabic translation instead of through the original Greek.

Italian and French Translators. In the i2th century Italy
and France produced two or three prominent scholars whose
knowledge of Arabic and taste for mathematics led them to
make known to the Latin world various classics of the Moham-
medan and Greek civilizations.

The first of these translators was Plato of Tivoli, or Plato
Tiburtinus, 1 who lived c. 1120. He translated the astronomy
of Albategnius (al-Battani), the Spherics of Theodosius, the
Liber Embadorum of Abraham bar Chiia (c. 1120), and
various works on astrology.

About this time Sicily was also active in the translation of
Greek and Arabic works. 2 Among the treatises thus brought
to the attention of scholars was Ptolemy's Almagest, which
was turned into Latin by an unknown translator, c. n6o, 3

1 B. Boncompagni, Delle versioni fatte da Platone Tiburtino, Rome, 1851.

2 C. H. Haskins and D. P. Lockwood, "The Sicilian Translators of the
Twelfth Century . . . ," in the Harvard Studies in Classical Philology, XXI, 75.

3 There is in the Vatican a MS. of this translation, written c. 1300. It is this
that was used by Professors Haskins and Lockwood in the work above cited.

202 CHRISTIAN EUROPE FROM 1000 TO 1200

from a Greek manuscript which had formerly been brought
from Constantinople to Palermo by a Sicilian scholar.

Some years later Gherardo Cremonense, or Gherardo of
Cremona (1114-1 187),* studied in Italy and then in Spain,
learning Arabic in Toledo. With him, as with many other
scientists in the Middle Ages and even later, astrology formed
a nexus joining medicine and mathematics, his interests there-
fore lying in all three lines. He translated various mathemati-
cal and astronomical works from the Arabic, including Euclid's
Elements and Data, the Spherics of Theodosius, a work by
Menelaus, and Ptolemy's Almagest? "for the love" of which
book he journeyed to Toledo. 3 In his translation is found one
of the early uses of the word sinus for a half chord, this being
the first of our modern names for the trigonometric functions. 4
There was a younger Gherardo of Cremona who lived in the
I3th century, called da Sabbionetta, who wrote on astronomy. 5

Among the Italian and French translators there may properly
be included Rudolph of Bruges, since most of his work was
done under French influence. About his time (c. 1 143) Hermann
of Carinthia translated Ptolemy's Planisphere. 6

1 Apparently a native of Cremona in Lombardy, although certain Spanish
writers have claimed him for Carmona in Andalusia. The name appears in
English as Gerard and in Latin as Girardus, with variants. B. Boncompagni,
Delia vita e delle opere di Gherardo Cremonense, Rome, 1851. A considerable
amount of information relating to such early Italian mathematicians is given
in B. Veratti, De' Matematici Italiani anteriori all' Invenzione della Stampa,
a pamphlet with bibliographical notes, Modena, 1860.

2 The translation was finished in 1175, as an old MS. asserts. This was
about fifteen years after the Sicilian translation, a work of which Gherardo was
apparently ignorant. See also Rose, in Hermes, VIII, 332. It was printed in
Venice in 1515. On the question of his translations see A. A. Bjornbo, Bibl.
Math., VI (3), 239-

8 Amore tamen almagesti, quern apud latinos minime reperiit,Toletum perrexit."

4 On the question of priority and of the use of the term by Plato of Tivoli,
see A. Braunmuhl, Geschichte der Trigonometric, I, 49 (Leipzig, 1900, 1903) ;
hereafter referred to as Braunmuhl, Geschichte. The term was probably first used
in Robert of Chester's revision of the tables of al-Khowarizml. See also Bibl.
Math., I (3), 521.

5 His Theorica planetarum was printed at Ferrara in 1472.

This was printed in 1507. See M. Chasles, Aperc.u historique sur I'origine
et developpement des methodes en geometric, Paris, 1837 ; 2d ed., 1875, hereafter
referred to as Chasles, Apergu; 3d ed., 1889. See also Bibl. Math., IV (3), 130.

ENGLISH TRANSLATORS 203

English Translators. England produced two or more trans-
lators of prominence in the i2th century, and Ireland seems
to have produced at least one. Of these the best known is
Adelard 1 of Bath (c. 1120), a British scholar who studied at
Toledo (1130), at Tours, at Laon, and also in the East, and
who journeyed through Greece, Asia Minor, Egypt, and possi-
bly Arabia, bringing back numerous mathematical works. 2 He
is credited with a knowledge of Greek and was one of the first
to translate Euclid into Latin, but he seems to have made this
translation from the Arabic. 3 Either he or Campanus seems to
have determined the sum of the angles of a stellar polygon,
a figure then attracting considerable interest, possibly because
of its use in astrology. He probably translated the astronomi-
cal tables of al-Khowarizmi, and he is said to have written a
commentary on the arithmetic of this author and to have com-
posed a work entitled Regulae abaci. 4 Adelard was by no
means the first to bring Euclid's name into England, for, as we
have seen (p. 187), it was probably known to British schola/s
in the loth century.

A few years after AdelarcPs sojourn in Toledo two other Eng-
lish scholars who were interested in mathematics went to Spain
to pursue their studies. The first of these was Robert of
Chester (c. ii4o), 5 who translated al-Khowarizmi J s algebra 6
into Latin and prepared several astronomical tables. He was
archdeacon of Pampeluna, in northern Spain, and seems also
to have traveled in Italy and Greece. He was the first to
translate the Koran into Latin (1143).

iThe older English form was Aethelhard. See C. H. Haskins, "Adelard of
Bath," English Historical Review, p. 401 (1911).
2 F. Woepcke, Journal Asiatique, I (6), 518.

3 From certain similarities in the different manuscripts there seems to have
been an unknown scholar whose version was consulted by Adelard and various
other early translators.

4 Boncompagni's Bullettino, XIV, i.

5 Robertus Retinensis, Robertus Ketensis, Robert de Ketene, Robert de
Retines, Robertus Cataneus, and other variants. He is known to have been in
Spain in 1141 and seems to have been studying at Barcelona with Plato of
Tivoli in 1136. As already stated, he translated the astronomy of Albategnius.

6 For reference to the translation see page 1 70, note 3.
i

204 CHRISTIAN EUROPE FROM 1000 TO 1200

The second of these English scholars was Daniel Morley, 1
who studied at Oxford in 1180. He went to Paris and thence
to Toledo, 2 and wrote on astronomy and mathematics, 3 quoting
freely from Arabic authors. That such men were compelled
to go abroad for their mathematics at this time is apparent
from the records of the work done in the schools of London,
this work being chiefly of the nature of grammar and disputa-
tion. 4 That they should go to Spain was quite natural, not
merely for linguistic reasons but because of the close ties that
existed between Castile and England, owing to the marriage
of Alfonso VIII (1158-1214) to Lenora, daughter of Henry II.

Other Scholars. One of Adelard's pupils, N. O'Creat, 5 wrote
a work on multiplication and division which shows his indebted-
ness to Arab writers on mathematics. Of O'Creat himself,
however, nothing further is known, but the name suggests the
country of his birth. The work contains a rule of Nicomachus for
squaring a number by using the formula a 2 = (a b)(a + b) + b' 2 ,
thus: log 2 = 100 118 + 81 = 11,881. He used the Roman
numerals, but with both o and a character like the Greek r
for zero.

1 Daniel of Merlai, Merlac, Marlach. In Latin, Morleius, Merlacus. A MS. in
the British Museum (J. O. Halliwell, Rara Mathematics London, 1838-1830, 2d
ed., 1841, p. 84; hereafter referred to as Halliwell, Rara Math.) begins, "Philo-
sophia magistri Danielis de Merlai ad Johannem Norwicansem episcopum."
See also C. Singer in Isis, III (1920), 263.

2 See A. a Wood, Historia et Antiquitates Vniversitatis Qxoniensis, I, 56
(Oxford, 1674) ; hereafter referred to as Wood, Historia Oxon.

3 Probably De principiis mathematicis.

4 See the Descriptio nobilissimae civitatis Londoniae written by Fitzstephen
(died c.i 190), prefixed to his Life of Becket, and published in John Leland's
Itinerary (London, 1770) ; J. Stow, Survey of London, p. 703 (London, 1633).

5 Probably the same as Joh. Ocreatus. See N. Bubnov, Gerberti . . . Opera
Mathematica, p. 174, n. 7 (Berlin, 1899) ; C. Henry, Abhandlungen, III, 129.
O'Creat begins his work with these words: "N. O. Creati liber de multi-
plicatione et divisione numerorum ad Adelardum Bathoniensem magistrum
suum." There is a i3th century MS. of the work in the Bibliot. nat. in Paris.
The prologue begins, "Prologus N. Ocreati in Helceph, ad Adelardum Baten-
sem magistrum suum." On the uncertain meaning of Helceph, see Henry,
loc. cit.

6 Possibly from the medieval rl<f>pa, from theca (teca), or from nfjcetv (to
come to naught).

MINOR SCHOLARS 205

About the year 1125 Radulph of Laon (died 1133) wrote on
arithmetic, and a little before this time (c. 1090) Gerland, prior
of St. Paul, of Besangon, 1 wrote a computus and a brief work
on the abacus. 2

Early in the i2th century there was an astrologer, geometri-
cian, and abacist by the name of Walcherus, a native of Lor-
raine, who attained considerable prominence in England and
wrote a work on astronomy. 3 Such names are of interest simply
as they bear witness to the nature of mathematics in the
Church schools of the time.

2. ORIENTAL CIVILIZATION IN THE WEST

Spain. After the year 1000 numerous Moorish scholars
appeared in Spain and contributed to the literature of arith-
metic and astronomy, and occasionally to that of algebra.
A list of a few of the most prominent of these scholars will
serve to show their range of interest and achievement in the
general field of mathematics.

Ibn al-$affar, 4 a native of Cordova, wrote on astronomical
tables and instruments. A little later (c. 1050) Ibn al-Zarqala, 5
probably a native of Cordova, wrote on astronomy and astrol-
ogy, and prepared a set of tables.

ancient record speaks of "Gerlandus vel Garlandus Prior S. Pauli, annis,
11311 1132." A document of 1134 records: "Huius praefatae concordiae testes
sunt . . . Garlandus magister . . . anno . . . M.C.XXXIIII." He is also men-
tioned in 1148: "Magistrum quoque Jarlandum Bisuntinum & magistrum
Theodericum Carnotensem [i.e., of Chartres], duoa fama & gloria doctores
nostri temporis excellentissimos." He is again mentioned in a letter written to
him in 1157: "Gerlando scientia trivii, quadriviique onerato & honorato."
See Boncompagni's Bullettino, X, 654.

2 See Boncompagni's Bullettino, X, 653; Cantor, GesMchte, II (2), 843.

3 In the south aisle of the chancel of the old priory at Great Malvern
may still be seen his tomb with this inscription, in part: Philosophvs dignvs
bonvs astrologvs Lothering vs | vir pivs ac hvmilis - monachvs prior hvjvs
... | ... geometricvs ac abacista : | Doctor . Walchervs . . . MCXXV ^
He should not be confused with the Walcherus of Lorraine who became Bishop
of Durham and was murdered in 1075.

4 Ahmed ibn 'Abdallah ibn ( Omar al-Ganqi, Abu'l-Qasim (died 1035).
'Ibrahim ibn Yahya al-Naqqash, Abu Ishaq.

e Schoner translated one of his works in 1534.

206 ORIENTAL CIVILIZATION IN THE WEST

In the latter part of the century Abu'1-Salt, 1 a Spanish physi-
cian from Denia, wrote on geometry and astronomy, and
Jabir ibn Aflah (died between 1140 and 1150), commonly
known as Geber, flourished at Seville and wrote on astron-
omy, spherical trigonometry, and the transversal theorem
of Menelaus. 2 He is often confounded with an alchemist of
similar name.

Jewish Scholars of the Eleventh Century. The most learned
scholars in Spain at the close of the nth century, however,
were not Mohammedans. The Jewish race, which may con-
veniently be mentioned in connection with the Oriental civiliza-
tion in Spain, was generally accorded better treatment under
Saracen than under Christian rule, although it had flourished
somewhat in Italy before this time. Through the encourage-
ment received from the Moors the Jews contributed in no small
degree to the advance of mathematics in Spain, and to them
the Christians were indebted for their first knowledge of the
Arabic works on the subject. 3 The first of their prominent
scholars in this century was Abraham bar Chiia 4 (Abraham
Judseus), commonly known as Savasor'da 5 (c. IOJQ-C. 1136),
a native of Barcelona. He wrote on astronomy, but is chiefly
known for an encyclopedia which included arithmetic, geome-
try, and mathematical geography. 6 Of this only fragments are
now extant. He also wrote a work entitled Liber Embadorwn?
treating of geometry but containing numerous definitions
used in the theory of numbers. In this he accuses the French

iQmeiya ibn 'Abdel'aziz ibn Abi'1-Salt, Abu'l Salt (1067/8-1133/4).

2 His astronomy was translated by Gherardo of Cremona and was printed

in 1534.

3 Libri, Histoire, I, 154 n. 4 Or Chijja, Chiya.

5 From Sahib al-Shorta, "Chief of the Guards." The transliteration to
Savasorda is due to Plato of Tivoli (c. 1120).

G Iesode ha-Tebuna u-Migdal ha-Emuna.

7 This is one of the sources of Fibonacci's geometry. See M. Curtze, Abhand-
lungen XII, where the Latin and German translations are given. The title is
medieval Latin from the Greek ifj.pa.86v, an area or surface. For his works and
for the contemporary Jewish writers see J. Bensaude, V Astronomic Nautique
au Portugal, p. 52 (Bern, 1912); hereafter referred to as Bensaude, Astron.
Portug. See BibL Math., 1896, p. 36.

JEWISH SCHOLARS 207

Jews of being ignorant of geometry and therefore weak in
their arithmetic. This work was translated from Hebrew into
Latin by Plato of Tivoli.

Rabbi ben Ezra. The second great Hebrew scholar of the
period was Abraham ben Ezra. 1 He wrote on the theory of
numbers, the calendar, magic squares, astronomy, and the
astrolabe, was much interested in the cabala, and is justly
ranked as the most learned Jew of his time. 2 He traveled ex-
tensively, going at least as far as Egypt to the east and as far
as London (1158) to the north. Besides his contributions to
astronomy, the calendar, and allied subjects, he wrote three or
four works on number: (i) Sefcr ha-Echad;* (2) Sefer ha-
Mispar* chiefly on arithmetic; (3) Liber augmenti ct diminu-
tionis vocatus numeratio divinationis, known only in Latin
translation and possibly not due to him; 5 (4) Ta f hbula, con-
taining the Josephus Problem, possibly a separate work, and
probably due to him. Of these the Scfcr ha-Mispar is the only
one of importance. It is based on the Hindu arithmetic but
uses Hebrew letters for the numerals, with a zero as in algorism.
He employed the check of casting out nines, as several of his
predecessors had done. The following is an example of his rules:
"Whoever would know how great the sum of the numbers is

1 Born at Toledo, between 1093 and 1096; died at Rome or Rouen, 1167.
This is the Rabbi ben Ezra of Browning's poem. He is sometimes confused
with Abraham ben Chiia, probably because each was called Abraham Judaeus.
On his life see M. Steinschneider, Bibl. Math., IX (2), 43; Abhandlungen, III,
57; Bensaude, loc. cit., p. 52.

2 Attention was called to him as a mathematician by O. Terquem, Journal
des mathematiques pures et appl., VI, 275. Since then his work has been studied
1 y Luzzato, Rodet, and Steinschneider. See also Smith and Ginsburg in the
Amer. Math. Month., XXV, 99.

*Book of Unity. This has twice been published: Bamberg, 1856; Odessa,
1867.

4 Book of Number. M. Silberberg published a German translation at Frank-
fort a. M. in 1895. See also Bibl. Math., IX (2), 91.

5 In favor of his authorship, Cantor, Geschichte, I (3), 730; against it, G.
Wertheim, Bibl. Math., II (3), 143. See also P. Tannery, Bibl. Math., II (3),
45. The Liber augmenti . . . divinationis was published by Libri. His Liber
de nativitatibus (Venice, 1485) was, however, the first of his works to appear
from the press.

208 ORIENTAL CIVILIZATION IN THE WEST

which follow one another in a series to a certain number,
multiply this by its half increased by J. The product is the
sum." 1

Although highly esteemed by Jews and Christians alike, hi
fate was not altogether a happy one, and in his struggle against
adversity he voices his lament in words like these :

Were candles my trade it would always be noon ;
Were I dealing in shrouds Death would leave us alone.

In connection with the Jewish activity of this period there
should also be recalled the name of one Hasan, a judge, who
may have written in the loth century, but whose country
is unknown, and of Yehuda ben Rakufial, who seems to
have been a physician in Spain. Both of these men wrote
on the Jewish calendar, and the former is referred to by
Rabbi ben Ezra.

Twelfth Century in Spain. The i2th century was even more
favorable than its predecessors to the study of mathematics in
Spain. The first of the Arab writers was Aver'roes (c. 1126-
1198/9), as he was commonly called in the Middle Ages, 2 who
wrote on astronomy and trigonometry. His most prominent
scientific contemporary was Avenpace, as he was called by the
Christians, 3 who lived at Seville and Granada c. 1140 and
wrote on geometry.

Jewish Writers of the Twelfth Century. In this as in the
preceding century, however, it was the Hebrew scholar who
made the greatest contributions to the advance of mathematics.
Aside from Rabbi ben Ezra, two of these scholars are deserving
of special mention: Maimonides 4 (1135-1204), a native of
Cordova, physician to the sultan, and an astronomer of

1 That is, s = n I - + -V Sefer ha-Mispar, ed. Silberberg, p. 24 (Frankfort

a. M., 1895). V2 2>

2 His name was Mohammed ibn Ahmed ibn Mohammed ibn Roshd, Abu
Velid.

3 His name was Mohammed ibn Yahya ibn al-Saig, Abu Bekr, also known
as Ibn Bajje and as Ibn Saig. The name "Avenpace" (also spelled Avempace)
is a Spanish form and, as such, is pronounced ah van pa'tha.

4 Rabbi Moses ben Maimun. He became rabbi of Cairo in 1177.

JEWISH ACTIVITY 209

prominence, 1 and Johannes Hispalensis 8 (fl. c. 1140), who
professed Christianity and wrote on arithmetic and astrology
(1142) and translated various Arabic works on mathematics
into Latin. 3

In the same century there were various Jewish scholars of
less prominence, such as Samuel ben Abbas, 4 who wrote on
arithmetic, 5 the Hindu numerals and their use, 6 algebra, and
geometry. There was also an unknown English Jew who wrote
a work called by English historians Mathematum Rudimenta
quaedam.

Jewish Writers of the Thirteenth Century. The i3th century
saw various translations made from the Arabic into Hebrew,
and several of the translators are known. Among these was
Moses ben Tibbon, 7 whose father and grandfather were cele-
brated as translators of philosophical and scientific works from
the Arabic into Hebrew. He was actively at work about the
middle of the century and translated (1259) the astron-
omy 8 of Alpetra'gius (c. 1200) and probably, as stated on
page 210, the arithmetic of al-l;Iassar.

The other Jewish scholars of this period also showed their
chief scientific interest in astronomy. Jehuda ben Salomon
Kohen of Toledo (died 1247), f r example, wrote upon Ptolemy's
Almagest, although he also prepared a brief extract from Euclid

1 There is a Jewish calendar of his among the manuscripts in the Bodleian
Library. Parts of his works on the calendar were printed at Paris in 1849, at
Leipzig in 1850, and at Berlin in 1881.

2 John of Seville, John of Luna. As in many such cases, the first name is
often written Joannes. The full name is also written Johannes Hispanerisis or
Johannes de Hispania. The date of his death may have been 1153.

3 At least some were translated into Spanish and were then put into Latin
by Domenico Gondisalvi. His Alghoarismi de Practica Arismetrice was pub-
lished by B. Boncompagni, Rome, 1857. It is based on Arab sources, but is
not a translation. His translations include works by Alfraganus (al-Fargani,
c. 833), Abu 'Ali al-Chaiyat (a prominent astrologer, died 835), and Tabit ibn
Qorra (c. 875). His works were published at Niirnberg in 1548.

4 M. Steinschneider, Bibl. Math., X (2), 81. He adopted the Mohammedan
faith, his Arabic name being Samu'il ibn Yahya ibn 'Abbas al-Magrebi al-
Andalusi. He died in 1174/5. 5 Al-Tab f sira.

6 Al-Qiwami, probably named after a patron, Qiwam ed-din Yahya.

7 Bibl. Math., X (2), 112. *Kitdb al-hei'a.

210 ORIENTAL CIVILIZATION IN THE WEST

and wrote a commentary upon it, and Isaac ben Sid, of Toledo
(died 1256), edited the Alfonsine Tables (see page 228) just
before his death.

About the middle of the i3th century there was born in
Cordova another descendant of the celebrated Tibbon family,
Jacob ben Machir, known as Prophatius. He lived in Mont-
pellier, wrote on a quadrant which he had invented (the quad-
rans Israelis or quadrans Judaicus}* translated from the Arabic
into Hebrew the Elements and Data of Euclid and the Sphere
of Menelaus, 2 and composed a work on the almanac.

Arab Writers of the Twelfth Century. Of the writers on arith-
metic among the western Arabs of the i2th century one of the
best known was Abu Bekr Mohammed ibn ' Abdallah, :i commonly
known as al-tlassar. 4 His work was so well received that, as
already stated, it was translated into Hebrew by Moses ben
Tibbon 5 (1259). The work is evidently Western, since it uses
the gobar numerals.

Arab Writers of the Thirteenth Century and Later. Early in
the i3th century Alpetragius, as the Christians called him,
lived in Spain, probably in Seville, and wrote on astronomy
(c. 1200). His theory of planetary motion, which gives him a
place in the list of mathematical writers, was translated into
Latin by Michael Scott.

Contemporary with Alpetragius there was a certain Ibn al-
Katib 7 (died 1210/11), who wrote two works which included a
little discussion of arithmetic, geometry, and architecture.

1 There is a good MS. of the work in the Columbia University Library, ap^
parently of the i$th century. The work has been several times translated into
Latin.

2 Boncompagni's Bullettino, IX, 595; Bibl. Math., XI (2), 35. The name also
appears as Propatius. He died c. 1308.

3 A Gotha MS. gives the name as Abu Zakariya Mohammed ibn 'Abdallah
ibn 'Aiyash. See Bibl. Math., XIII (2), 87.

4 That is, the Computer; but Suter thinks this a family name. See H. Suter,
"Das Rechenbuch des Abu Zakarija el-tfassar," in Bibl. Math., II, (3), 12, and
III (2), 109. See also ibid., XIII (2), 87.

5 Probably. The Vatican MS. has still to be studied critically.

N<ir ed-din al-Betrujt, Aba Ishaq.

7 Mohammed ibn 'Abderrahman, AM 'Abdallah.

ARAB WRITERS 211

Of the scholars born in northern Africa in the i3th century ,
and geographically closely related to the Spanish civilization,
the best known is Albanna,, or Ibn al-Banna. 1 From the fact
that he is also known as al-Marrakushi we infer that he was a
native of Morocco." He wrote on astronomy, mensuration,
algebra, the astrolabe, and proportion. His best-known work
is the TalchiSj a treatise on arithmetic. 3

There was also a Mohammedan scholar of Seville, known
as Ibn Bedr 4 or Abenbeder, who wrote a compendium of alge-
bra about this time. 5 The date is uncertain, but there is a
commentary upon it in verse which was written in 1311/12.

The last of the great Moorish arithmeticians of Spain was
al-Qalasadi, r> a native of Baza, a town near Granada. He
wrote extensively on arithmetic and seems to have had some
originality in the treatment of the theory of numbers. He
introduced a new radical sign and a sign of equality, and
proposed a system of ascending continued fractions. 7

3. CHRISTIAN EUROPE FROM 1200 TO 1300

General Activity of the Thirteenth Century. Whatever may
be thought of the mathematics of the i3th century, it is certain
that the century itself represents the real awakening of the
world after a long period of intellectual torpor. The centuries

1 That is, Son of the Architect. His full name is Ahmed ibn Mohammed
ibn 'Otman al-Azdi, AbuVAbbas. Born c. 1258; died in Morocco c. 1339.

2 A. Marre, Atti dell' Accademia Pontificia dei Njwvi Lincei, XIX (hereafter
referred to as Atti Pontiff ; M. Steinschneider, Boncompagni's Bullettino, X,
313. Suter's list omits the occasional name al-Marrakushi, and places the date
of his birth as c. 1258 or later, although it is sometimes given as early as 1252.
The father seems to have belonged to a Granada family.

3 Discussed in Cantor, Geschichte, I (3), 806.
4 Mohammed ibn 'Omar, Abu 'Abdallah.

5 Jos6 A. Sanchez Perez, Compendia de Algebra de Abenbeder, Arabic text
and Spanish translation, Madrid, 1916.

AH ibn Mohammed ibn Mohammed ibn 'Alt al-Qoreshi al-Basti, Abu'l-
IJasan. The name " al-Qalasadi " means the Upright, or Versed in the Law.
Suter gives the place and date of his death as Tunis, 1486.

7 Woepcke in the Journal Asiatique, 1854, II, 358, and 1863, I, 58. See also
the Atti Pontif., XII, 230, 399.

212 CHRISTIAN EUROPE FROM 1200 TO 1300

immediately preceding had produced writers on mathematics
in Europe, but they had produced no mathematicians. But now
a Spirit of the Times was abroad. The Far East felt its in-
fluence, and hence the remarkable revival and development of
algebra in China ; India felt it, and hence the appreciation of
the merit of Bhaskara, now a generation dead ; and all of in-
tellectual Europe felt it as never before. It was not a century
of great beacon lights, but it was one in which lanterns were
hung in all the thoroughfares of the West, promises of the
great illumination that was to come with the period of the
Renaissance. 1

Rise of the Universities. The most potent influence in the
development of the world's mathematical knowledge has, of
course, been the universities, and it is from the i3th century
that we trace the rise of these institutions in the modern sense
of the term. The earliest medieval universities grew out of
the cathedral or Church schools and hence their date of begin-
ning is necessarily obscure. In most cases the years in which
they received official privileges from some sovereign, civil or
ecclesiastical, are known, however, and are commonly taken
as the dates of foundation. In some cases there are two dates,
one of the receipt of the privilege from the State and the other
that from the Church, the latter giving to the holders of degrees
a right to teach. Thus Paris had a charter from the State in 1200
and its degrees were recognized by the pope in 1283. The cor-
responding dates for Oxford were 1214 and 1296; and for
Cambridge, 1231 and 1318. The University of Padua was
founded in 1222, and that of Naples in I224. 2 The i4th and
iSth centuries saw a number of other universities established,
but we may look upon the i3th century as the one which laid
the foundation for this type of higher education, although the
mathematics taught was still very meager. 3

1 J. J. Walsh, The Thirteenth, Greatest of Centuries, New York, 1907.

2 Some of these dates are uncertain, but they are approximately as stated.

3H. Suter, "Die Mathematik auf den Universitaten des Mittelalters," Fest-
schrift der Kantonschule in Zurich (Zurich, 1887), p. 39; hereafter referred to
as Suter, Univ. Mittelalt.

THE TOWER OF KNOWLEDGE

Illustrating the educational system of the Middle Ages. From the Margarita

phylosophka, 1503

214 CHRISTIAN EUROPE FROM 1200 TO 1300

Medieval Curriculum. The student began his study of gram-
mar with Donatus and Priscian, and took his logic from Aris-
totle and his rhetoric from Cicero. He then entered upon his

mathematical studies, such
as they were, arithmetic
according to Boethius, music
according to Pythagoras,
geometry according to Eu-
clid, and astronomy accord-
ing to Ptolemy. The goal
for those who were prepar-
ing for church activities was
the metaphysics and the-
ology of Peter Lombard
(c. 1150). This progress was
illustrated in a tower of
knowledge given by Gre-
gorius Reisch in his Mar-
garita phylosophica ( 1 503 ) ,
as shown on page 213.

Leonardo Fibonacci. The
first great mathematician of
the 1 3th century, and indeed the greatest and most productive
mathematician of all the Middle Ages, was Leonardo Fibonacci,
known also as Leonardo Pisano or Leonardo of Pisa. 1

1 Born at Pisa, c. 1170; died c. 1250. On his life and works see B. Boncompagni,
Scritti di Leonardo Pisano, 2 vols., Rome, 1857-1862 (hereafter referred to as
Boncompagni, Scritti Fibonacci); Delia vita e delle opere di Leonardo Pisano,
Rome, 1852; Intorno ad alcune opere di Leonardo Pisano, Rome, 1854; and Tre
scritti inediti di Leonardo Pisano , Florence, 1854 (hereafter referred to as
Boncompagni, Tre Scritti} ; Cantor, Geschichte, II, chaps, xli, xlii ; Libri, Histoire,
I, 156; E. Lucas, "Recherches sur plusieurs ouvrages de Leonard de Pise et sur
diverses questions d'arithm&ique superieure," Boncompagni's Bullettino, X,
129; G. Loria, "Leonardo Fibonacci," Gli Scienziati Italiani (Rome, 1919),
p. 4, with excellent bibliography; G. B. Guglielmini, Elogio di Lionardo Pisano,
Bologna, 1813; F. Bonaini, Memoria unica sincrona di Leonardo Fibonacci,
Pisa, 1858 (republished in 1867), and also in the Giornale Arcadico,Vo\. CXCVII
(N. S., LID, and in an article by G. Milanesi, Documento inedito e sconosciuto
intorno a Lionardo Fibonacci, Rome, 1867; V. A. Le Besgue, "Notes sur les
opuscules de Leonard de Pise," Boncompagni's Bullettino, IX, 583 ; O. Terquem,

LEONARDO FIBONACCI

Modern engraving. The portrait is not
based on authentic sources

FIBONACCI 215

At the time of Leonardo's birth, Pisa ranked with Venice
and Genoa as one of the greatest commercial centers of Italy.
These towns had large warehouses where goods could be stored
and duty paid in all important ports of the Mediterranean, the
head of such an establishment being a man of considerable promi-
nence. It was such a position that the father 1 of Leonardo held
at Bugia 2 on the northern coast of Africa, and in this town
Leonardo received his early education from a Moorish school-
master. 8 As a young man he traveled about the Mediterranean,
visiting Egypt, Syria, Greece, Sicily, and southern France, 4
meeting with scholars and becoming acquainted with the va-
rious arithmetic systems in use among the merchants of dif-
ferent lands. All the systems of computation he counted as
poor, however, compared with the one that used our modern
numerals. 5 He therefore wrote a work in 1202, Liber Abaci, 6

"Sur Leonard Bonacci de Pise et sur trois ecrits . . . ," Annali di Sri. Mat.,
Vol. VII (reprint, Rome, 1856); M. Lazzarini, "Leonardo Fibonacci," Bul-
lettino di Bibliogr. di. Sci. Mat., VI, 98, and VII, i ; P. Cossali, Scritti inediti, ed.
Boncompagni, p. 342 (Rome, 1857) ; Libri, Histoire, II, 21.

1 Guglielmo Bonaccio. But the name "Fibonacci" is thought by Boncom-
pagni and Milanesi to be a family name like Johnson, the form " filius Bonacci "
being merely a Latin translation. An ancient document of 1226 has "Leonardo
bigollo quondam Guilielmi," in which the Latin form " Bonaccius " does not ap-
pear, but in which the grandfather has this name. It seems more reasonable,
however, to think that when Leonardo himself wrote "filius Bonacci," "filius
Bonaccij," and "filius Bonacii," he knew what the words would mean to Latin
readers. Leonardo speaks of his father as being "in duan a bugee," in the
custom house at Bugia.

2 Modern Bougie, whence France imported her wax candles (bougies}.
Little of its ancient splendor remains except the Moorish gate (Bab-el- Bah r,
"sea-gate") in the old ramparts.

3 "Vbi ex mirabili magisterio in arte per nouem figuris indorum introductus."
Liber Abaci, p. i.

4 ". . . apud egyptum, syriam, greciam, siciliam et prouinciam."

5 "Sed hoc totum etiam et algorismum atque arcus pictagore quasi errorem
computavi respectu modi indorum," as it appears in the Florentine MS. pub-
lished by B. Boncompagni, Rome, 1857.

Early writers attributed to him the introduction of these numerals into Italy ;
thus ". . . e questi fu il primo, che port6 neir Italia i carrateri dei numeri
conforme testifica Luigi Colliado." Aritmetica di Onojrio Pvgliesi Sbernia Paler-
mitano, p. 12 (Palermo, 1670).

6 "Incipit liber Abaci Compositus a leonardo filio Bonacij Pisano In Anno
M cc ij." This is the title as it appears in the first line of Boncompagni,

216 CHRISTIAN EUROPE FROM 1200 TO 1300

in which he gave a satisfactory treatment of arithmetic and
elementary algebra. The work is divided into fifteen chapters,
and the following brief statement of the contents will serve to
show its general scope : i . Reading and writing of numbers in
the Hindu-Arabic system; 1 2. Multiplication of integers; 2
3. Addition of integers; 4. Subtraction of integers; 3 5. Divi-
sion of integers ; 6. Multiplication of integers by fractions ;
7. Further work with fractions; 8. Prices of goods; 9. Bar-
ter; 10. Partnership; n. Alligation; 12. Solutions of prob-
lems; 13. Rule of False Position; 4 14. Square and cube roots;
15. Geometry and algebra, the former being devoted to prob-
lems in mensuration.

Possibly it was his indulgence in travel that caused him to
write his name occasionally as Leonardo Bigollo, since in Tus-
cany bigollo meant a traveler. The word also means blockhead,
and it has been thought that he had been so called by the
professors of his day because he was not a product of their
schools, and that he retaliated by adopting the name simply to
show the learned world what a blockhead could do. It would
be human to hope that the latter explanation is the correct one,
just as it is human to rejoice that the son of a provincial
official became the greatest medieval mathematician. Such
a remarkable career as Fibonacci's warns us, as Froude so

Scritti Fiboracci, I, i, from the Codex Magliabechianus. The spelling of abacus
varies in this and other MSS., often appearing as abbacus. B. Boncompagni,
Intorno ad alcune opere di Leonardo Pisano, p. i (Rome, 1854).
1 "Nouem figure indorum he sunt

987654321

Cvm his itaque nouem figuris, et cum hoc signo O, quod arabice zephirum
appellatur, scribitur quilibet numerus." P. 2.

2 "Incipit capitulum secundum de multiplicatione integrorum numerorum."

3 "Incipit capitulum quartum de extractione minorum numerorum de
maioribus/'

4 "De regulis elchatayn . . . Elchataieym quidem arabice, latine duarum
falsarum posicionum regula interpretatur." See Volume II, Chapter VI.

5 "Incipit flos Leonardi bigolli pisani . . . ." See Boncompagni, Tre Scritti, i.
The word bigolli also appears as pigolli. See also F. Bonaini, Iscrizione . . . a
onore di Leonardo Fibonacci . . ., Pisa, 1858; 2d ed., 1867.

FIBONACCI 217

truly said, "that we should draw no horoscope; that we should
expect little, for what we expect will not come to pass." 1

In the same years and in the same regions in which
Leonardo was bringing new light into the science of mathe-
matics, St. Francis, humblest of the followers of Christ,
was bringing new light into the souls of men. Each was
one of the world's geniuses, and for a genius there is no
human explanation.

Fibonacci's Other Works. Leonardo also wrote three other
works, the Practica geometriae 2 (1220), the Liber quadrato-
rum* (1225), an d the Flos? besides which there is extant a let-
ter of his to Theodorus, philosopher to Frederick II, relating
to indeterminate analysis and to geometry. These works treat
of the theory of numbers in a way that shows that Leonardo
was a mathematician of remarkable ability, considering the
time in which he lived. His name attaches to the series
o, i, i, 2, 3, 5, 8, 13, . . ., in which u,=n n ^ + n n _^ where

So far as the schools were concerned, Leonardo's works were
like a voice crying in the wilderness. It is probably within
the bounds of truth to say that not a professor in the University
of Paris, to select what was soon to become the greatest intel-
lectual center of the world, could have made anything whatever
out of the fine reasoning of the Liber Quadratorum or could
have comprehended what the Flos was meant to convey to the

1 " Un brevet d'apothicaire n'empecha pas Dante d'etre le plus grand poete de
1'Italic, et ce fut un petit marchand de Pise qui donna 1'algebre aux Chretiens."
Libri, Histoire, I, xvi.

a On his knowledge of Euclid see G. Enestrom, Bibl. Math., V (3), 414.

3 R. B. McCIenon, "Leonardo of Pisa and his Liber Quadratorum," Amer.
Math. Month., XXVI, i. There is a question about the date 1225, although
it is given in the MS.

4 "Incipit pratica geometrie composita a Leonardo pisano de filiis bonaccij
anno M. cc. xx."

"Incipit liber quadratorum compositus a leonardo pisano Anni. M. CC.
XXV."

"Incipit flos Leonardi bigolli pisani super solutionibus quarumdam ques-
tionum ad numerum et ad geometriam uel ad utrumque pertinentium."

These titles are from the Boncompagni editions. Flos is a fanciful title,
blossom or flower.

2i8 CHRISTIAN EUROPE FROM 1200 TO 1300

mind. Since the course of study was concerned with little that
was scientific, 1 mathematics had no standing there or in the
schools of Italy.

Campanus. Roger Bacon speaks highly of a certain Master
Nicholas who lived about this time, but concerning whom we
know nothing further, and also of Master Campanus de No-
varia. 2 The latter is Johannes Campanus 3 (fl. c. 1260), some-
time chaplain to Urban IV, who reigned as pope from 1261 to
1264. It was he who prepared the translation of Euclid's
Elements that was used in most of the early printed editions,
but which seems to have depended upon at least three earlier
translations from the Arabic. Campanus also wrote a Tracta-
tus de Sphaera, a Theoria Planetarum, a Calendarium, a work
De Computo Ecclesiastico, a work on perspective, and a mem-
oir De Quadrature, Circuit which seems lost but which was
mentioned a century later by Albert of Saxony (c. 1370) . Con-
temporary writers have little to say of his life. He held rela-
tively minor positions in the Church, and it is probable that
in his later years he was a canon in Paris. 4

In the appendix to his translation of Euclid he showed how
to compute the sum of the angles of a stellar pentagon. It is
not improbable that the figures used by the astrologers of this
period account for the interest developed by various writers in
the study of stellar polygons in general. Campanus also con-
sidered the trisection problem, the irrationality of the Golden
Section (not yet known by this name), and the angle between
a circle and a tangent. He was, therefore, not merely a com-
piler of translated material but a man genuinely interested in
geometry. 5

1 The oldest statutes now extant (1215) record: "Non legant in festivis
diebus, nisi Philosophos et rhetoricas et quadrivalia et barbarismum et ethicam,
si placet." See Suter, Univ. Mittelalt., p. 56.

2 Of Novara, near Milan.

3 Giovanni Campano. Cantor, Geschichte, II (2), 90; C. S. Peirce, in
Science, XIII (N.S.) (New York), 809.

4 Pacioli (1509), in his De diuina proportione (I, 4) speaks of him as "el
gran philosopho Campan, nostro famosissimo mathematico."
6 B. Baldi, Boncompagni's Bullet tino, XIX, 591.

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FIRST PAGE OF THE CAMPANUS EUCLID

From Ihe MS. probably given by Campanus to Pope Urban IV, at that time
Jacques Pantaleon, Patriarch of Jerusalem. Now in the library of Mr. Plimpton

220 CHRISTIAN EUROPE FROM 1200 TO 1300

Other Italian Writers. In the i3th century Italian mathe-
matics consisted almost entirely of astronomy, and of the works
bearing upon this subject the most popular one produced was
the Tractatus Sphaerae of Bartolomeo da Parma, who is known
to have been teaching mathematics at Bologna in 1297. He
also wrote on geometry and astrology. 1

Whether Guglielmo de Lunis belonged to the i3th century
is uncertain, 2 but he translated an algebra from the Arabic/

Pietro d'Abano 4 (c. 1250-^. 1316), a professor of medicine 5
at Padua, wrote an Astrolabium planum? his interest in the as-
trolabe being doubtless due to its applications to astrology.

British Scholars. 7 Fibonacci was not a product of the uni-
versities, but he speaks of his master 8 as one who had studied
in the universities of Oxford and Paris, and no doubt, in his
mature years at least, he learned from him. This man was
"the wizard" Michael Scott, 9 who had not only studied at the

1 E. Narducci, Boncompagni's Bullettino, XVII, i, 43, 165.

2Bi. Math., XII (3), 270.

3 An algebra MS. of the i5th century treats of "la regola de Algebra amu-
cabale . . . secondo ghuglielmo de lunis." See Kara Arithmetica, p. 463; Bibl.
Math., IV (2), 96, and V (2), 32, 118.

*Petrus Aponensis. The dates are sometimes given as 1253-*;. 1319.

5 On the relations of mathematics to medicine two or three centuries later,
see the author's article on " Medicine and Mathematics in the Sixteenth Century,"
in Annals of Medical History, I, 125.

fl Published in Venice in 1502. 7 Cantor, Geschichte, II, chap. xlvi.

8 He dedicates to him the second edition of his Liber Abaci in these words :
"Scripsistis mihi domine mi magister Michael Scotte, summe philosopbe, ut
librum de numero, quern dudum composui, uobis transcriberem."

9 Spelled also Scot. Born possibly at Balwearie, Scotland, c. 1175; died
c. 1234. He was also called Michael Mathematicus.

"In these fair climes it was my lot
To meet the wondrous Michael Scott;
A wizard of such dreaded fame,
That when, in Salamanca's cave,
Him listed his magic wand to wave,
The bells would ring in Notre Dame ! "
Scott, Lay of the Last Minstrel^ II, xiii

"That other, round the loins
So slender of his shape, was Michael Scot,
Practised in every slight of magic wile."

Dante, Inferno, XX, Gary translation

BRITISH SCHOLARS 22 1

universities mentioned, but had learned Arabic and made
astronomical observations at Toledo. He was later appointed
astrologer to Frederick II and seems to have been employed
by this ruler to make known to scholars, through translations
from the Arabic, the newly discovered Greek texts.

Upon those who, unlike Michael Scott, studied chiefly in
England at this time, some influence may have been exerted
through the arrival at Oxford in 1224 of the first of the Fran-
ciscans. These men 1 were not learned in the sciences, but
they came from the intellectual centers of Southern Europe
and could not have been ignorant of what scholars were doing
beyond the Alps and the Pyrenees. Their presence in one of
the university centers was especially significant.

Sacrobosco. The second of the prominent British scholars
of this century was Sacrobosco, 2 who was educated at Oxford
and entered the University of Paris c. 1230. He afterwards
taught mathematics and philosophy in Paris and died there
c. I256. 3 He was buried in the Cloister Sodalium Mathurina-
lium, his astrolabe being placed on his tomb. 4

Sacrobosco wrote the most popular work on the sphere that
had appeared up to that time, and did much, through his

*A list is given by Wood, Historia Oxon., I, 67-77, and in A. G. Little,
The Grey Friars in Oxford, chap, i and p. 176 (Oxford, i8Q2).

2 Born at Halifax, Yorkshire, c. 1200; died at Paris, c. 1256. The name ap-
pears in various forms, such as Johannes de Sacrobosco, John of Halifax, John
of Holywood, Sacro Bosco, Sacrobusto. Sacrobosco is the Latin for Holywood
(Holy fax, Halifax). Widman (1489) writes the name lohane vo sacrobusto, and
Pacioli (1494) writes it Gioua de sacro busco. J. Aubrey, Brief Lives, ed.
Clark, I, 408 (Oxford, 1898) (hereafter referred to as Aubrey, Brief Lives}, says
that "Dr. [John] Pell is positive that his name was Holybushe."

3 The date of his death was formerly given definitely as 1256 on the author-
ity of G. J. Vossius, De Vniversae Mathesios Natvra & ConstUvtione Liber,
p. 179 (Amsterdam, 1650). P. Tannery has shown that the obscure verse from
which Vossius obtained this date refers to the completion of his Compotus, and
moreover that the date should be read 1244 instead of 1256. The verse is

"M Xristi bis C quarto deno quater anno." See Bibl. Math., XIII (2), 32.

4 J. C. Heilbronner, Historia Matheseos Vniversae, p. 471 (Leipzig, 1742);
hereafter referred to as Heilbronner, Historia. Wood (Historia Oxon., I, 85)
speaks of his teaching there: "Job. de Sacro bosco. Claruit apud Parisienses in
Mathesi & in Philosophia."

222 CHRISTIAN EUROPE FROM 1200 TO 1300

Tractatus de Arte Numerandi^ or Algorismus, to make the
Hindu- Arabic arithmetic known to European scholars. These
books were widely used for three hundred years, and continued
in use until the close of the i6th century."

The third of the prominent British scholars of this period
was Robert Grosseteste, or Greathead (died October 9, 1253), at
one time a student at Paris, later a student and teacher at
Oxford, and finally bishop of London. 3 His interest was chiefly
in the applications of mathematics to physics and astronomy, 4
but he also wrote a Praxis gcomctriae and a work on Euclid's
Optics:'

Among the Oxford men of this period was John of Basing-
stoke (died 1252), who learned Greek in Athens (1240) and
took back to England some knowledge of the numeral systems
and possibly of the mathematics of classical times.

Roger Bacon. The most prominent scholar in England in the
1 3th century, however, was Roger Bacon (1214-1294), a man
of erudition and of prophetic vision. His works show a knowl-
edge of Euclid's Elements and Optics, of Ptolemy's Almagest
and Optics, of Theodosius on the Sphere, of parts of the works
of Hipparchus, Apollonius, and Archimedes, and of the works
of various Arab writers. He was familiar with the writings of
Aristotle and with some of the commentaries upon them. Of

1 Printed in Halliwell, Kara Math., i. For early editions, see the Kara Arith-
metica. An edition by M. Curtze appeared in 1897.

-Suter, Univ. Mittelalt., p. 67; Bibl. Math., XI (2), 97; P. Riccardi, Bibl.
Math., VIII (2), 73.

3 The variants of his name, as given by Wood (Historia Oxon., I, 81), are
interesting. They include such forms as Grossum caput, Groshedius, Grouthede,
Grokede, and Groschede. He was also known as Robertus Lincolniensis and
Rupartus Lincolniensis.

4 Theorica planetarum, De astrolahio, De cometis, De sphaera coelesti, De
compute, Praxis geometriae, and a Calendarium.

6 See also L. Baur, " Der Einfluss des Robert Grosseteste auf die wissenschaft-
liche Richtung des Roger Bacon," in A. G. Little, Roger Bacon Essays, p. 33
(Oxford, 1914) (hereafter referred to as Little, Bacon), and Die phttosophischen
Werke des Robert Grosseteste, Munster, 1912.

6 Under this date Matthew Paris records: "Obiit magister Johannes de
Basingestokes, archidiaconus Legrecestriae, vir in trivio et quadrivio ad plenum
eruditus."

FROM A MANUSCRIPT OF SACROBOSCO

This MS. was written in Germany, c. 1442- shows distinctly the numerate
as they then appeared. Now in the library of Mr. Plimpton

224 CHRISTIAN EUROPE FROM 1200 TO 1300

algebra he knew little except the name. 1 Mathematics as then
understood was little more than astronomy, and for the work
of most of his contemporaries in this field he had a profound
contempt. This contempt was even more pronounced with re-
spect to teaching, in which he asserted that an enormous
amount of time was wasted. He stated that he had devoted
forty years to study, and that the entire ground could have
been covered in from three to six months." The teachers at
Paris he could only characterize by their "four defects, in-
finite and puerile vanity, ineffable falsity, voluminous super-
fluity, and the omission of all that is worthy." The charge was
untrue, as are most epigrams of the kind, for Bacon was given
to dipping his pen in vitriol. It is no wonder that his contem-
poraries generally hated him. Although spoken of in later
times as doctissimus mathematicus he contributed nothing to
pure mathematics, and his chief work in applied mathematics
was a calendar which the world was not ready to appreciate. 3

John Peckham. Of Bacon's influence upon his pupils, in the
direction of mathematics, we have little evidence. There is
some reason for thinking that he was possibly the one who in-
spired John Peckham 4 to take up the study of the science. At
any rate Peckham was a scientist of repute and his Perspective,
communis was looked upon as a classic for three hundred
years. 5 He became archbishop of Canterbury in 1279.

Un his Opus majus, he says: "Algebra quae est negotiatio, et almochabala
quae est census."

2 "Multum laboravi in scientiis et lingua, et posui jam quadriginta annos
postquam didici primo alphabetum . . . et tamen certus sum quod infra
quartam anni, aut dimidium anni, ego docerem ore meo hominem sollicitum et
confidentem, quicquid scio de potestate scientiarum et linguarum." Opus
Tertium, cap. xx.

3 Besides his published works by S. Jebb (1733, 1750), J. S. Brewer (1859),
J. H. Bridges (1897), and Robert Steele, consult E. Charles, Roger Bacon, sa
vie, ses ouvrages, ses doctrines . . . (Paris, 1861), and Little, Bacon.

4 Born in Kent, probably some time before 1240; died at Mortlake, Decem-
ber 8, 1292. The name also appears as Peachamus, Peccamus, and Pithsanus,
with various other modifications.

5 Facio Cardano (1444-1524) edited i't under the title Prospectiua cols d.
lohanis archiepiscopi Cdtauriesis, and it was printed s.l. a. (but 1482, at Milan).
There are various editions of this work.

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FIRST PAGE OF THE EARLIEST FRENCH ALGORISM
Written c. 1275, and now in the Library of Ste. Genevieve, in Paris

226 CHRISTIAN EUROPE FROM 1200 TO 1300

French Scholars. France produced no mathematicians of
importance in the i3th century. During a considerable part
of the time her great university was a place of rioting rather
than a seat of learning. In spite of this fact, however, several
respectable scholars appeared, one of the first being Alexandre
de Villedieu 1 (c. 1225), a Franciscan monk from Bretagne.
He wrote De Sphaera, De Computo Ecclesiastico, and De Arte
Numerandi, and taught in Paris. He is best known, however,
for his Carmen de algorismo, 2 a little arithmetic in Latin verse
that probably did more to make known the new Hindu- Arabic
numerals than any other work of the century. 3 A little later
(c. 1275) the first algorism in the French language was written.

Among the contemporaries of Alexandre de Villedieu there
was Vincent de Beauvais 4 (c. 1250), a Dominican, whose
encyclopedia, the Speculum Majus* written for Louis IX
("Saint Louis"), includes the quadrivium, the subject being
very poorly treated.

Roger Bacon mentions as one of the greatest mathematicians
of his time (c. 1265) a certain Petrus de Maharncuria, 6 but
all that is known of him is that he wrote a work on the magnet.

German Writers. Of the German mathematicians of the i3th
century only three deserve special mention. Of these the first
in order of time and of mathematical ability was Jordanus Nemo-
rarius, 7 who studied at Paris and wrote an Arithmetica decem
libris demonstrate,* and possibly an Algorismus demonstrates. 9
He also wrote a work on mathematical astronomy, Tractates
de sphaera] one on geometry, De triangulis] and one of
the leading books of the Middle Ages on algebra, Tractates

*De Villa Dei, or De Villa Dei Dolensis.

2 Song on Algorism, that is, on al-Khowarizmi's arithmetic methods.

3 It is printed in full in J. O. Halliwell, Rara Mathematica (London, 1838),
sd ed. (1841), p. 73.

4 Vincentius Bellovacensis. 5 His Opera appeared at Venice in 1494.

6 Also called Peter de Maharn-Curia and Petrus Peregrinus. Maharncuria
seems to have been Maricourt in Picardy. Boncompagni's Bullettino, I, i.

7 Also known as Jordanus de Saxonia and Jordan of Namur. He was born
at Borgentreich in the diocese of Paderborn and died in 1236 or 1237. Cantor,
Geschichte, II, chap, xliii. 8 Published at Paris in 1496.

As to the doubt upon this point see G. Enestrom, Bibl. Math., V (3), 9.

FRANCE AND GERMANY 227

de numeris datis. The Arithmetica 1 is on the theory of num-
bers as set forth in treatises like that of Boethius, and is the
least original of his works. The one noteworthy feature of the
book is the use of letters to represent general numbers. This
is already found to a certain extent in the works of earlier
writers, including Aristotle and Diophantus, but Jordanus uses
letters quite as they are used today, letting 6, for example,
represent any number whatsoever.' 2 The Tractatus de sphaera
was for a long time a classic and several editions were printed.
The De triangulis 3 is a work in four books containing seventy-
two propositions of the usual type, together with propositions
on such topics as the center of gravity of a triangle, curved
surfaces, and similar arcs.

The Tractatus de numeris datis is a system of algebraic rules.
The problems 4 generally relate to a numerus datus, a given
number, which has to be divided in some stated manner, as in
many of the problems in our current algebras. 6

He also wrote a work entitled De Ponderibus Propositiones
XIII, which was printed at Nlirnberg in 1533 and contains a
brief treatment of statics. He is the Jordanus de Saxonia who,
in 1222, became general of the Dominican order. 7

3 On the MSS. in the library ot the Royal Society and in Oxford, both of
which differ from the first printed edition, see J. O. Halliwell, A Catalogue of
Miscellaneous Manuscripts preserved in . . . the Royal Society (London, 1840),
and J. Wallis, Algebra, p. 13 (Oxford, 1693).

2 Cantor, Geschichte, II (2), 56; Enestrom, Bibl. Math., VII (3), 85.

3 "Jordani Nemorarii Geometria vel de Triangulis libri IV," in the Mil-
theilungen des Coppernicusvereins, Heft VI (Thorn, 1887).

4 See P. Treutlein, Abhandlungen, II, 135.

5 " Numerus datus est cuius quantitas nota est."

6 One of his first problems is practically this: To separate a given number
into two parts such that the sum of the squares of the parts shall be another
given number. E.g., x + y io, A 2 + y 2 = 58, whence # = 7, y=$. Abhand-
lungen, II, 136 (4).

7 One Oxford MS. distinctly calls him Jordanus de Saxonia. Nicolas Trivet,
an English chronicler of the i4th century, under the year 1222, states: "Hoc
anno in Capitulo Fratrum Praedicatorum generali tertio, quod Parisiis cele-
bratum est, successor beati Dominici in Magisterio Ordinis Fratrum Praedica-
torum factus est frater lordanus, natione Teutonicus, Dioecesis Moguntinae,
qui cum Parisiis in scientiis saecularibus et praecipue in Mathematicis magnus
haberetur. ..."

228 CHRISTIAN EUROPE FROM 1200 TO 1300

One of the greatest of the German scholars of this period
was Albertus Magnus, 1 Count of Bollstadt, a Dominican priest,
and Bishop of Regensburg. He studied at Padua and taught
at Bologna, Strasburg, Freiburg, Cologne, and Paris. So versa-
tile was he that he was called "Doctor Universalis." His in-
terests were chiefly in philosophy and physics, but his works
include material on astronomy and some reference to Pythag-
orean arithmetic. 2 Claude Fleury, who wrote an ecclesiastical
history in 1691, remarked that he could see nothing great in
him but his volumes.

At the close of the century (c. 1270) Witelo or Vitello, 3 prob-
ably from Thiiringen but possibly from Poland, 4 wrote on per-
spective (optics) and astronomy, a fact which shows the
interest in this phase of applied mathematics in Poland in the
J3th century.

Other Thirteenth-Century Writers. Of the other scholars of
this century whose works touched upon mathematics the most
prominent was Alfonso X, King of Castile (1223-1284), known
as el Sabio (the Wise). He was an astronomer of merit and his
name appears in the Alfonsine Tables, planetary tables which
improved upon the imperfect ones left by Ptolemy. Work upon
them began in 1248 and was completed in 1254. Tycho Brahe
is said to have deplored the waste of money involved in their
compilation, although they unquestionably stimulated the study
of mathematical astronomy.

ifiorn at Lauingen, Swabia, 1193 or 1205; died at Cologne, 1280. Albertus
Teutonicus, de Colonia, or Ratisbonensis. See Cantor, Geschichte y II (2), 86,
and Dixon's translation (London, 1876) of his biography by Sighart (Regens-
burg, 1857).

2 The first edition of his Opera Omnia appeared in Leyden in 1651. The best
edition is that of Paris, 1890.

3 In the oldest MSS. the name appears as Witelo. The forms Vitello and
Vitellius are later. There are many variants, such as Witilo, Witulo, Widilo,
Wito, and Vitellion.

4 He speaks of his country, saying: "In nostra terra, scilicet Poloniae habi-
tabili . . . ," and of himself as "Thuringo-polonus" and as "Filius Thu-
ringorum et Polonorum," so that possibly his mother was a Pole.

5 The date of completion is sometimes placed later than this. The tables
were first printed in Venice in 1483.

THIRTEENTH-CENTURY WRITERS 229

Another prominent writer of the period was Arnaldo de Villa
Nova 1 (c. 1235-c. 1313), who taught at Paris, Barcelona, and
Montpellier. While known principally as a physician and for
his twenty works on alchemy, he wrote a Computus Ecclcsias-
ticus & Astronomicus? probably being led to a study of the
subject through its relation to astrology. 1 *

Roger Bacon, in his condemnation of most of his contempo-
raries, speaks of "the notorious William Fleming who is now in
such reputation, whereas it is well known to all the literati at
Paris that he is ignorant of the sciences in the original Greek,
to which he makes such pretensions." This Flemish writer
was William of Moerbecke, 1 chaplain to Clement IV and
Gregory X. Among his translations were the catoptrics of
Heron and the writings of Archimedes on floating bodies. 5 It
is thought that Tartaglia took his translation of Archimedes
from this writer. 7 He also wrote on perspective.

Byzantine Writers. In the i3th century the only writer of
note in the Near East was Georgios Pachymeres," who may for
convenience be classified as European, although born in Asia
Minor. He wrote on the Four Mathematical Sciences? that is,
on arithmetic, music, geometry, and astronomy. The work is
important only as showing that interest in learning had not

1 Arnauld de Villcneuve, Arnald Bachuone, Arnoldus Villanovanus. He wa?
probably born at Villa Nova, Catalonia, but possibly at Villeneuve in Southern
France.

2 Printed at Venice in 1501. With respect to the edition see the Rara Arith-
metic a, p. 73.

3 On the geometric figures used in astrology the Ars Magna of Raymundus
Lullus (r. 1235-1315), a writer of this period, may be consulted.

4 Guilielmus Brabantinus or Flemingus. He died c. 1281 as archbishop of
Corinth.

5 De Us quae in humido vehuntur. See also J. L. Heiberg, Zeitschrift (HI.
Abt), XXXIV, 1-84; XXXV (HI. Abt.), 41-48, 48-58, 81-100, and later
volumes.

6 See page 297. 7 Cantor, Geschichte, II (2), 514.

8 Born at Nicaea, in Bithynia, about 60 miles from Constantinople, in 1242;
died c. 1316.

& Hepl TWV T(r<rdpw fj.a0v)^,drwv Haxvuepow jjuyaXov didafficdXov. There are vari-
ous MSS. extant. See E. Narducci, M Di un Codice Archetipo e Sconosciuto dell'
opera di Giorgio Pachimere," Rendkonti della R. Accad. dei Lincei, VII, 194,

230 CHRISTIAN EUROPE FROM 1300 TO 1400

wholly died out in the period between the capture of Nicsea by
the Crusaders in 1097 an( l its downfall before the Turkish
invaders in 1330.

Otherwise there is little known of the mathematics of Con-
stantinople in the i3th century. There is evidence, however,
to show that her scholars were using Greek numerical charac-
ters even as late as the isth century, augmented by a symbol
for zero resembling our inverted h. Their problems were trivial,
chiefly relating to mensuration. 1 Although they used the Greek
forms, they were acquainted with the numerical system of the
Arabs and spoke of it as Hindu in origin, but they were not
familiar with the numerals which we commonly call Arabic.

4. CHRISTIAN EUROPE FROM 1300 TO 1400

General Activity of the Fourteenth Century. After the bril-
liant beginning of a renaissance of learning in the i3th century,
it would naturally be expected that the i4th century would see
a notable revival of science and letters. To understand why
this expectation was not fully realized, it is necessary to con-
sider the peculiar conditions by which Europe was confronted.
As for Italy, this country was at last fully awakened to the
beauties of ancient literature, and so Dante (1265-1321), tak-
ing Vergil as his master, produced the Divina Commedia, the
"Epic of Medievalism"; Petrarch (1304-1374) made a no-
table collection of manuscripts of the ancient classics and
started a movement that resulted in a new appreciation of
the literature of Greece and Rome; and Boccaccio (1313-
I37S) showed great zeal in the attempt to collect and study
the works of the ancients. In this search Constantinople was
drawn upon for Greek manuscripts, with the result that Italian
scholars were interested anew in the study of science and let-
ters. Furthermore, the Florentine republic had just become
practically a government by the merchant class, owing to modi-
fications in the constitution between 1282 and 1292, a fact
that must have had much to do with the great prominence

!J. L. Heiberg, w Byzantiniscbe Analekten," Abhandlungen, IX, 161.

FOURTEENTH CENTURY 231

of Florentine arithmetic in the schools of the i4th century.
A general accumulation of wealth must also have followed,
which would naturally tend to foster the arts and sciences.
All this was promising, and the result would probably have
been the hastening of the period popularly known as the
Renaissance, had it not been for two deterring factors.

The first of these factors was the Hundred Years' War (say
1338-1453, although also given as 1328-1491), which over-
turned the economical and political systems of the two most
advanced countries of Europe north of the Alps. The battle
of Crecy (1346) struck at something besides feudalism.

The second deterring factor was the terrible ravaging of the
Black Death (1347-1349), by which from a third to a half of
the population of Europe is thought to have been swept away.

As to the universities in the i4th century, they did little
for mathematics. Those of Italy were behind their contem-
poraries in Paris, England, and Germany, the statutes of 1387
making no mention whatever of the subject. 1 In England,
Merton College, Oxford, was the mathematical center- and
made some pretense at work in this science, while Paris had
lectures on algorism, astronomy, and geometry, such as they
were. In the newly founded University of Erfurt (1392 ), which
may be taken as a German type, an elementary knowledge of
mathematics was offered but apparently was not required."

Italian Writers. No Italian writer of the i4th century stands
out as showing any real genius in mathematics, as a brief list
will bear witness. Cecco d' Ascoli (1257-1327), also known
as Francesco di Simone Stabili and as Francesco degli Stabili,
a native of Ascoli in Romagna, was professor of philosophy at
Bologna and Rome, wrote a commentary on the Sphaera of
Sacrobosco, 4 and did much to bring into high repute once more

*Suter, Univ. Mittelalt., p. 75- 2 Ibid., p. 83.

3 W. Hellmann, Ueber die Anfange des math. Unterrkhts an den Erfurter
. . . Schulen, I, 4. Erfurt, 1896.

4 On the early Bologna mathematicians in general, consult Silvestro Gherardi,
Di alcuni materiali per la Storia delta Facolta Matematica . . . di Bologna,
p. 17 (Bologna, 1846) ; hereafter referred to as Gherardi, Facoltb Mat. Bologna.

232 CHRISTIAN EUROPE FROM 1300 TO 1400

the ancient belief in astrology, a subject which perhaps reached
its greatest popularity in this century.

Andalo di Negro (c. i26o-c. 1340), a native of Genoa, had
considerable reputation as a mathematician and astronomer,
writing several works on the astrolabe, a book on the planets,
and a Tractatus de sphaera. There is also ascribed to him a
practical arithmetic. 1

Barlaam (c. 1290-^:. 1348), a native of Seminara in Calabria,
Italy, bishop of Geraci, studied in Constantinople and wrote on
computing, astronomy, 2 the science of numbers, 3 algebra, 4 and
Book II of Euclid.

Paolo Dagomari, 5 a native of Prato, in Tuscany, was promi-
nent in Florence as an arithmetician and astronomer. His
Trattato d'Abbaco, d'Astronomia, e di segrcti naturali c medi-
oinali contained a little commercial arithmetic and gave him a
reputation more extended than scientific. 6 That he wrote on
algebra is asserted by at least one later writer. 7 He may have
been the Paolo Pisano who is said to have lived about this time.

A more worthy writer on mathematics appeared in the per-
son of Rafaele Canacci (c. 1380) of Florence, author of an
algebra y with a number of historical notes.

1 C. de Simony, Boncompagni's Bullettino, VII, 313, 330.

2 Libri V logisticae astronomicae. See also B. Baldi, Cronica di Matematici,
p. 85 (Urbino, 1707), and Boncompagni's Bullettino , XIX, 598. Barlaam's
(Barlaamo's) given name may have been Bernardo, but this is uncertain. He
is occasionally known, from his birthplace, as Calabro.

3 Arithmetica demonstratio eorum quae in secundo libra elementorum
(Eudidis) sunt. It was printed at Strasburg in 1564.

4 Ao7i<rTtKTj, sive arithmetic ae, algebraicae libri VI. It was printed at Stras-
burg in 1572.

5 Born at Prato, c. 1281 ; died at Florence, 1365 or 1374. Known also r "is
Paolo dell* Abaco, Paolo Astrologico, Pagolo Astrologo, Paoli il Geometra, Paolo
Geometra, and Paolo Arismetra. F. Villani (fl. 1404), in Le Vite d' Uomini
illustri Fiorentini, 2d ed., Florence, 1826, speaks of him as "geometra grandis-
simo, e peritissimo aritmetico . . . diligentissimo osservatore delle Stelle e del
movimento de' cieli." For a resume of his work see D. Marlines, Origine e pro-
gressi dell' aritmetica, p. 59 (Messina, 1865); hereafter referred to as Martines,
Origine aritmet.

6 For a description of the work, see Rara Arithmetica, p. 435.

7 See Rara Arithmetica, p. 463, with reference to "w. paolo fiorj che circha
al. 1360. duro." 8 See Rara Arithm*tica t p. 459.

MATHEMATICS IN ITALY 233

There was also a Master Biagio 1 of Parma (died 1416) who
wrote an arithmetic and an algebra, but neither has been pub-
lished. He taught astrology and philosophy at Paris, Pavia,
Bologna, Padua, Venice, and Parma, wrote a commentary on
Oresme's DC latitudinibus jormarum* and wrote on statics and
perspective. The famous educator Vittorino da Feltre (1378-
1446, but the dates are doubtful), born in poverty, worked as

Vl-pt-
,*/>p*

,^fjjMU~-:-

-^iyio4Ljg*m*U

FROM DAGOMARI'S TRATTATO D J ABBACO

From an Italian MS. of c. 1339. Notice also the early per cent sign, p 100 lano
(per 100 the year), and the sign for Ib. The latter is possibly the origin of

the dollar sign

a scullery boy in Biagio's house so as to learn geometry from
him, and in turn became one of the best teachers of mathe-
matics of his time. 8

Toward the close of the century Antonio Biliotti 4 (c. 1383)
of Florence taught mathematics in Bologna, but he left no
works on the subject. Altogether the mathematical output of
Italy in this century was not encouraging.

Constantinople. Constantinople was at this time experienc-
'ng an intellectual revival similar to the one seen in Italy before
the coming of the Black Death. Prominent among her scholars
was Maximus Planudes (c. 1340), a Greek monk, at one time

1 Also Biagio da Parma and Pelacani. There was also a "m. biagio che
circha al. 1340. anj morj," as a MS. of c. 1440 asserts, although this 1340 may be
wrong and the two may be the same. See Rara Arithmetics, p. 463.

2 See page 199. The students of his day in Paris had a phrase "aut diabolus
est, aut Blasius Parmensis."

S W. H. Woodward, Vittorino da Feltre, Cambridge,

4 Also called Antonio dalT Abaco.

234 CHRISTIAN EUROPE FROM 1300 TO 1400

(1327) ambassador to Venice, who wrote on Diophantus and
who also wrote an arithmetic based upon the Hindu- Arabic
numerals. 1 He was a man of industry but of no genius, and
his arithmetic is of value chiefly as showing the influence of
Bagdad upon the mathematical thought of Constantinople.
It sets forth the system of notation by the "nine figures re-
ceived from the Hindus" together with the zero, and is the
first of the Greek works to give any attention to modern
methods of calculation. Planudes is also deserving of credit
as a translator of various Latin classics into Greek. 2

Among the minor contemporaries of Planudes there was
Joannes Pedias'imus (c. 1330), also called Galenus, who was
keeper of the seal of the patriarch of Constantinople. He wrote
a work on geometry in which he attempted to pattern after the
style of Heron of Alexandria, and also wrote upon the dupli-
cation of the cube, and upon arithmetic. { In general, however,
his work was literary and philosophical. Among his contempo-
raries he was known as the "Chief of Philosophers."

There lived in Constantinople a little later than Joannes
Pediasimus the celebrated grammarian Manuel Moschopou'-
lus, 4 a native of Crete. The dates are uncertain, but he seems
to have lived c. 1300. Although there were two men of the
same name, this one and his nephew, it seems from a manu-
script of a work by Nicholas Rhabdas, referred to below, that
this is the one who wrote a treatise on magic squares, the
earliest contribution to the subject in the Mediterranean
countries.

1 He called his work ^-n^o^opLa /car' 'Ii/5oi/s (Indian Arithmetic). There is
a Greek edition by C. I. Gerhardt (Eisleben, 1865), and a German translation by
H. Waschke, Das Rechenbnch des Maximus Planudes (Halle, 1878) (hereafter
referred to as Waschke, Planudes). See also Heath, Hhtory, II, 546.

2 Kroll, Geschichte, p. 70.

3 Boncompagni's Bullettino, III, 303.

4 Or Emanuel. In Greek the name appears both as Maw/rj\ and as 'E/uawi^X
Mo<rx6irov\oy. Heath, History, II, 549.

5 S. Gunther, Vermischte Untersuchungen zur Geschichte der math. Wissen-
schaften, p. 195 (Leipzig, 1876) (hereafter referred to as Gunther, Vermischte
Untersuch.) ; P. Tannery, "Manuel Moschopoulos et Nicolas Rhabdas," Bulletin
des Sciences math, et astr., VIII (2), September 2, 1884.

CONSTANTINOPLE 2 3 5

About this time Nicholas Rhabdas 1 (c. 1341), a Greek
"arithmetician and geometer" 2 from Smyrna, wrote from Con-
stantinople two letters on arithmetic, and particularly on finger
reckoning, 3 a subject first treated of with any completeness
by Bede. He also edited a work of Planudes on the Hindu
arithmetic, 4 possibly during the lifetime of the latter. With
him there flickered out what once had been a great beacon
light, the mathematics of the Greeks, and at the same time
any real appreciation of the language itself almost ceased to
exist. Petrarch began to study classical Greek in 1342, with
the aid of a monk who had lived in Constantinople, and a
learned scholar, Manuel Chrysoloras, lectured upon it in Flor-
ence from 1397 to 1400; but it was not until the i6th century
that mathematical works of the Greeks began to be known
again in the original tongue.

English Writers. 5 England produced several mathematicians
of more than ordinary ability in this century, all but one of
them doing his real work before the years of the pestilence.

Richard of Wallingford (born c. 1292; died 1336) lec-
tured on the liberal arts at Oxford and wrote on trigonometry*
and arithmetic. 7 He seems to have been one of the best-known
mathematicians of his time. 8 It was no doubt his influence

1 Nicholas Smyrnaeus, Rhabda, Artabasda, Artabasdes. In one rna-iuscript
the name appears as Nicolas Artavasdan. See P. Tannery, "Notice sur les
deux lettres arithmetiques de Nicolas Rhabdas," in Notices et extraits des
manuscrits de la Bibliotheque rationale, Paris, XXXII (1886), 121.

2 As he describes himself, w dpitf/^TiKou Kaiyeuntrpov"

3 "Efr0pa<rts r-v 5a.KTv\iKov nfopov. For a review of this arithmetic see Bibl.
Matk., I (2), 28. It has been printed several times, as by N. Caussinus.
Eloquentia sacra et humana, Paris, 1636, and by Morellus, Nic. Smyrnaei Arta-
basdae, graeci mathcmaticiSEK<t>paau numerorum notationis per gestum digitorum,
Paris, 1614. Heath, History, II, 550.

4 Published by Gerhardt, Eisleben, 1865. n Cantor, Geschichte, II, chap. xlvi.

G Quadripartitum de sinibus demonst r atis ; De sinibus et arcubus in circuit*
inveniendo, De chorda et arcu, and De chorda et versa. The word sinibus often
appears in the MSS. as sinubus, and arcibus commonly as arcubus. See Montucla,
Histoire, I (i), 529; Cantor, Geschichte , II (2), 101.

7 De rebus arithmeticis and De computo.

8 One of the medieval writers speaks of him as "in mathesi omnium sui
temporis primus."

236 CHRISTIAN EUROPE FROM 1300 TO 1400

that led John Manduith 1 (fl. c. 1320) to follow in his footsteps
and lecture on trigonometry 2 and astronomy at Oxford.

The most prominent of the English mathematicians of the
i4th century was Thomas Bradwardine, 3 known as the "Doc-
tor Profundus." He was professor of theology at Oxford, chan-
cellor of St. Paul's cathedral, and an upholder of liberalism,
and he died as archbishop of Canterbury. He wrote four
works on mathematics. In his Arithmetica Speculating he
followed the Boethian model, the work relating solely to the
theory of numbers. His other works were a Tractatus de pro-
portionibus, Geometria speculative^, and De quadratura circuit.
His geometry includes some work on stellar polygons, 5 isoperi-
metric figures, ratio and proportion, irrationals, and loci in space.

About this time there flourished in England a Cistercian
monk by the name of Richard Suiceth" (c. 1345), probably a
native of Glastonbury, in Somersetshire. He was educated at
Merton College, Oxford, and wrote an obscure work on mathe-
matics. 7 It treats of a subject just beginning to attract atten-
tion in England and in France, De latitudinibus formarum. 8

In this period there also lived a well-known writer, Walter
Burley, 9 whose work on the lives of the philosophers and

iMandwith, Manduit.

*De chorda et arcu recto et verso, et umbris, showing that he was acquainted
with the use of tangents.

3Born at Hertfield (Hartfield), Chichester, c. i2go; died at Lambeth Palace,
London, August 26, 1349. The name appears in such forms as Bragwardin,
Brandnardinus, Bredwardyn, Bradwardyn, de Bradwardina, and de Bredwardina.
Pacioli (Suma, 1494, fol. 68, r.) calls him Tomas beduardin.

4 Printed at Paris in 1495.

r> " . . . figuris angulorum egredientibus."

6 The first name may possibly have been Roger or Raymund, and the last
name appears in such forms as Suisset, Suicetus, Swincetus, Swineshead, and
Suineshevedus, a word derived from the Cistercian cloister, Vinshed, on the Holy
Island off the coast of Northumberland.

7 Opus aureum calculationum . . . Per . . . lohane de Cipro . . . emedat 9 et
explicit, s. 1. a., but Pavia, c. 1480; also Pavia, 1498. It may have been this
work, which also went by the name "calculator," that led to his being called
"calculator acutissimus" by one of the early writers.

8 See Volume II, Chapter V.

9 Born at Oxford, c. 1275; died c. 1357. The dates are very uncertain. Cantor,
following Prantl, gives his death as 1337. The Latin spelling, Gualterus Burlaeus,
and the late English Burleigh are also used.

BRITISH MATHEMATICIANS 237

poets 1 contains biographical notes on such prominent Greek
mathematicians as Pythagoras, Plato, and Ptolemy.

There also flourished in the latter part of the i4th century
a celebrated English mathematician and physician, Simon
Bredon. 2 He wrote on astronomy/' arithmetic, 4 the calcula-
tion of chords, 5 geometry/ 5 and other related subjects, and
various manuscripts of these works still exist. 7 He was one
of the earliest European scholars to pay much attention to
trigonometry.

Among the minor writers of the period there were William
Reade, 8 of Merton College, who had considerable reputation
as a mathematician, and who prepared some astronomical
tables, and Walter Bryte, 9 who is said to have written on
arithmetic, 10 astronomy, and surgery.

In the 1 4th century there was written an interesting but
anonymous manuscript on the mensuration of heights and
distances, 11 beginning: "Nowe sues here a Tretis of Geometri
wherby you may knowe the heghte, depnes, and the brede of
mostwhat erthely thynges." It is a practical work on shadow
reckoning and surveying, using the compass, staff, and quad-
rant. Like many such works it is divided into three parts, very
likely due to the Christian idea of the Trinity. 12

*De Vita et moribus Philosophorum ct Poetarum. The first printed edition
was s.l. a., but Cologne, c. 1467. There were at least fourteen editions printed
before 1501.

2 Born at Winchcomb; living in 1386. The name appears as Bridonus and

Biridanus.

3 /n demonstratione Almagesti.

*Arithmetica theorica.

* Calculations chordarum, and Tabulae chordarum.

"Quadratura circuit per Campanum et Simon Bredon.

*< See Suter, Univ. Mittelalt., 84.

SReede, Rede. He died in 1385 as bishop of Chichester.

9 Brithus, Brit, Brytte. Possibly identical with Walter Brute, a lay follower
of Wycliffe.

10 Tractatus algorismalis, De rebus mathematicis. There is much doubt as to
the authorship of each of the works assigned to him.

"Hafflwell, Rara Math., 56, where the complete text is given. The MS. is
Bib. Sloan. 213. xiv, fol. 120, in the British Museum.

12 "This tretis es departed in thre. pat es to say. hegh mesure. playne mesure.
and depe mesure."

238 CHRISTIAN EUROPE FROM 1300 TO 1400

Another anonymous manuscript of considerable interest,
The Crafte of Nombryng, was written c. 1300. It is one of the
first works on algorism to appear in the English language. 1

French Writers. 2 In spite of the calamities of war and
plague, France did some noteworthy work in mathematics in
the 1 4th century, not only through those born within her own
boundaries but through scholars from other lands who found
in Paris a more congenial intellectual atmosphere than they
could find elsewhere.

Among those whom France could claim by adoption was
Petrus Philomenus de Dacia/' a native of Denmark, 4 rector of
the University of Paris (1326 or 1327), author of works on
algorism 5 and the church calendar, and a compiler of cer-
tain tables. 7 Still another of the adopted sons of France
was Johannes Saxoniensis, or Johann Danck, who carried on
his astronomical work in Paris. He left various writings on
astronomy. 8

Of those born in France, the first in point of time was
Johannes de Lineriis or Jean de Ligneres (c. 1300-1350), pro-
fessor of mathematics at Paris, who adapted the Alfonsine
Tables to the meridian of that city. 9 Joannes de Muris,or Jean de
Meurs, 10 was a contemporary of his who studied at the Sorbonne
and taught there. He wrote (1321) on arithmetic, astronomy,

1 This manuscript is described more fully in Volume II.

2 Cantor, Geschichte, II, chap, xlvii.

3 There is an Easter computation of 1300 attributed to him.
4 Whence "de Dacia."

5 Commentum super algorismum prosaicum Johannis de Sacro Bosco. M
Curtze, Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacro-
bosco Commentarius, Copenhagen, 1897. *Computus ecdesiasticus.

7 H ; .s Tabula ad inveniendam propositionem cujusvis numeri contains a multi-
Plication table to 49 x 49. See Bibl. Math., IV (2), 32.

8 De astrolabio and on the Alfonsine Tables. See Boncompagni's Bullettino,
XII, 352.

9 His nationality is not certain, nor is it clear whether he is the same as
Johannes de Liveriis (or Liverius), whose work on fractions was printed at
Paris in 1483, and who may have been a Sicilian. See Kara Arithmetica, p. 13;
M. Steinschneider, Boncompagni's Bullet tino, XII, 345, 352, 420.

10 Born in Normandy 0.1290; died after 1360. The name also appears as
Johannes de Murs or de Muria. L. C. Karpinski, Bibl. Math., XIII (3), 99.

FRENCH MATHEMATICIANS 239

and music. Of his works on arithmetic, 1 the Canones tabula
proportionum, Arithmetic a communis ex diui Seuerini Boctij,
Tractatus de mensurandi ratione, De numeris eorumque divi-
sione, and Quadripartitum numerorum, the Quadripartitum
is the most noteworthy. It is partly in verse and contains
a certain amount of algebra. Among the algebraic equations
solved are x + i2 = Sx, with the roots 2 and 6 ; $x + 18 = x 2 ;
and one already given by al-Khowarizmi and by Fibonacci,
2\x 2 = 100. It also contains a close approach to a decimal
fraction.-

The greatest of the French writers of this period was Nicole
Oresme," a native of Normandy, sometime professor and "ma-
gister magnus" (1355) in the College de Navarre at Paris, pro-
tege of Charles V, dean of Rouen (1361), and finally (1377)
bishop of Lisieux, Normandy. He wrote Tractatus propor-
tionum, Algorismus proportionum, Tractatus dc latitudinibus
jormarum, Tractatus c?c unijormitate ct difformitatc inten-
sionum, and Traite de la sphere. He also translated Aristotle's
De coelo et mundo. In the Algorismus proportionum is the
first known use of fractional exponents, 2* being written 4
2 2*, and 9* appearing as ^.9^. He also wrote

and

1.2

for 4^, stating the value to be 8. r>

In the Tractatus dc uniformitatc there is set forth a sugges-
tion of coordinate geometry, by the locating of points by

!For first printed edition, see Kara Arithmetica, p. 117. See aLo A. Nagl,
Abhandlungen, V, 135; p. 139 for a list of his works.

-L. C. Karpinski, Science (N. Y.), XLV, 663.

3 Born probably at or near Caen, c. 1323; died at Lisieux, July IT, 1382.
Also known as Nicolaus Oresmus, Horem, Horin, and Oresmius. See M. Curtze,
Die Mathematischen Schriften des Nicole Oresnte, Berlin, 1870.

4 From the i$th century MS. in the University of Basel, used by Curtze in
the work above cited. Other MSS. have slightly different forms.

5 Cantor, Geschichte, II (2), 121.

240 CHRISTIAN EUROPE FROM 1300 TO 1400

means of two coordinates. 1 Oresme also stands out prominently
as a remarkably clear-thinking economist for his generation. 2
Of much less importance as a writer on mathematics, but of
greater reputation in his lifetime, is Petrus de Alliaco, 3 rector
of the University of Paris, bishop of Cambray, and cardinal.
His work on astronomy 4 throws considerable light on the early
computi.

Other Writers. The other contributors to mathematical lit-
erature in this century were in general possessed of less ability
than those of France and England.

Early in the century Hauk Erlendsson, 5 a Norwegian offi-
cial, wrote on algorism. This is the first trace that we have of
the Hindu-Arabic arithmetic in Scandinavia. There was also
a certain Swedish scholar, Master Sven, or Sunon, who lec-
tured on the sphere in 1340.

The leading Jewish mathematician of the I4th century was
Levi ben Gerson (i (1288-1344), who was also well known as a
theologian. His Work of the Computer" was written in 1321.
He also wrote a treatise on trigonometry 8 which was translated
into Latin under the title De numeris harmonicis, but neither
work showed any noteworthy power. 9

Isaac ben Joseph Israeli was apparently a contemporary of
Levi ben Gerson, but we are uncertain as to his dates. He

Volume II, Chapter V. For an early edition of the Tractatus de lati-
tudinibus see Kara Arithmetica, p. 117. See also Zeitschrijt (HI. Abt.), XIII;
Bibl. Math., XIII (3), 115, and XIV (3), 210.

2 Tractatus de origine, natura, jure, et mutationibus monetarum, edited by
Wolowski, Paris, 1864.

3 Born at Compiegne, 1350; died at Avignon, August 8, c. 1420. The name
also appears as Pierre d'Ailly, Alyaco, and Heliaco.

4 Cocorddtia astronomie cu theologia. First printed at Augsburg in 1490.

5 Born 0.1264; died 1334.

8 Also called Levi ben Gerschom and Gersonides, Leo Ebraeus, and Ralbag
(RLBG, for Rabbi Levi ben Gerson), but more commonly known as Leo
*le Balneolis or Master Leon de Bagnolo, having been born at Balnaolis or
Bagnolas, in Catalonia. See J. Carlebach, Lewi ben Gerson als Mathematiker f
Berlin, 1910.

7 Maassei Choscheb, edited by G. Lange, Frankfort a. M., 1909.

s De sinibus, chordis, et arcubus. See Bibl. Math., I (3), 372; IV (2), 73;
XII (2), 97. 'Bibl. Math., XI (2), 103.

JEWISH MATHEMATICIANS 241

wrote a work on astronomy 1 which contains a chapter on
geometry and also serves as a source of information on the
activity of Jewish and Arabic scholars in Spain.

Among the lesser Jewish scholars of the period were Joseph
ben Wakkar of Seville (died 1396), who worked out certain as-
tronomical tables for Toledo ; Jacob Poel of Perpignan (fl. c.
1360), who did the same for Perpignan; Imanuel Bonfils of
Tarascon (died c. 1377), whose astronomical tables were highly
appreciated and who wrote on the astrolabe; Jacob Carsono
(al-Carsi), who wrote both at Seville and Barcelona (c. 1375),
and whose tables were known to Tycho Brahe ; Isaac Zaddik
(al-Shadib), who wrote on the astrolabe and prepared various
tables of use to astronomers ; and Kalonymos ben Kalonymos,
a native of Aries (born 1286), known as Master Calo, whose
various translations include a paraphrase of Nicomachus.

Of the German writers, two or three are deserving of special
mention. The first of these was Heinrich von Langenstein,
Heinrich von Hessen, or Henricus Hessianus, 2 bishop of Hal-
berstadt, who taught mathematics at Vienna and had some
reputation as a mathematician and astronomer. 3 The second
was Chunrad von Megenberg (c. i^og-c. 1374), who wrote
(.1350) a work based on Sacrobosco's De sphaera. 4 About
this time there was another Chunrad (Conrad) who was inter-
ested in mathematics, Conrad von Jungingen (c. 1400). Ac-
cording to one of the manuscripts he seems to have been the
author of the Geometria Culmensis. 5 This work consists of five
parts, the first two relating to the mensuration of the triangle,
the third to the quadrilateral, the fourth to the polygon, and the
fifth to curvilinear figures.

1 Liber Jesod Olam sive Fundamentum Mundi, first published in Berlin in
1777, but with later editions in 1846 and 1848.

2 Born at Langenstein, near Marburg, 1325; died at Vienna, February n,
1397 (sometimes given as 1394).

*Quaestio de cometa. He also wrote on the circle.

4 F. Muller, Zeittafeln zur Geschichte der Mathematik, p. 80, with references
(Leipzig, 1802) ; hereafter referred to as Muller, Zeittafeln. See also O. Matthaei,
Konrads von Me