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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 Megenberg Deutsche Sphaera, Berlin, 1912.

6 Liber magnifici principis Conradi de Jungegen, magistri generalis Prusie,
geometric practice usualis manualis. See Cantor, Geschichte, II, chap, xlviii.

242 CHRISTIAN EUROPE FROM 1400 TO 1500

Contemporary with the first two of these German writers
there was Albert of Saxony, 1 or Albertus de Saxonia, who was
educated at Prag and Paris, taught at Paris 2 and Pavia, 3 was
the first rector of the University of Vienna (1365), and became
bishop of Halberstadt (1366-1390). He wrote several scien-
tific works, among them a theoretical treatment of proportion 4
after the mariner of Boethius, De latitudinibus jormarum? De
maxima et minima, and De quadratura circuit? Like most
medieval writers, he took 3^ for the value of ar, apparently
without considering it a mere approximation.

5. CHRISTIAN EUROPE FROM 1400 TO 1500

Influences leading to the Renaissance. Of the influences
leading to that revival of learning known as the Renaissance,
the two most potent were the transfer of Eastern scholarship to
Italy and the invention of printing. It is customary to speak
of the former as dating from the fall of Constantinople, that is,
from its capture by the Turks in i453. T When Mohammed II,
standing on the banks of the Bosporus, repeated the Persian
distich,

The spider has woven his web in the Imperial palace,

And the owl has sung her watch-song on the towers of Af rasiab,

he epitomized the situation of Greek culture in ancient Byzan-
tium. There was nothing left for the remnant of the Hellenic
civilization to do but to seek refuge in other lands.

Had Stephen Dusan not died when he did (1356), or had he
left a worthy successor to the Serbian throne, Constantinople
might have fallen to a western instead of an eastern conqueror.
It is interesting to speculate as to what would have been the

i Born at Riggcnsdorf, Saxony, c. 1325; died at Halberstadt, 1300.

2Du Boulay, Historia Universitatis Parisiemis, p. 362. Paris, ibbS.

sF. Jacoli, Boncompagni's Bullettino, IV, 495.

4 Tractatus proportionum, first printed c. 1478. See Rara Arithmetica, p. 9.

c Printed at Padua in 1505.

6 F. Jacoli, Boncompagni's Bullettino, IV, 493; H. Suter, Zeitschrift (HI.
Abt.),XXlX, 81.

7 For a vivid description see Gibbon, Decline and Fall of the Roman Empire,
Vol. VI, chap. Ixviii.

THE RENAISSANCE 243

result of such an event upon civilization in general and upon
mathematics in particular. 1 We have already seen, however,
that this transfer of Greek civilization began a century before
the fall of the city, and both Rome and Florence had begun
to acquire collections of Greek and Latin manuscripts that
were to be available when the printing press should be ready
to make their contents known.

About the opening of the isth century Niccolo de' Niccoli
(1363-1437) made a noteworthy collection of manuscripts at
Florence, and in 1414 Poggio Bracciolini, a secretary of the
Roman curia, began the copying and collecting of classical
works. During the century there were such munificent pa-
trons of learning as the Medici in Florence and Nicholas V in
Rome, and the results of their labors are still seen in the
Laurentian and Vatican libraries. There was also Federigo,
Count of Montefeltro (1422-1482), who had been filled with
enthusiasm by no less a teacher than Vittorino da Feltre him-
self, and who received from Sixtus IV the title of Due d'Urbino.
His library at Urbino, filled with manuscripts, was a rendez-
vous for the scholars of Italy. The humanist Giovanni Aurispa
(c. 1369-1460) also brought some 238 manuscripts of the an-
cient Greek writers from Constantinople to Venice.

With all this activity, however, Italy showed no native abil-
ity in mathematics in the isth century. What Symonds said
of Tuscan culture in general applies in particular to mathe-
matics: " Florence borrowed her light from Athens, as the
moon shines with rays reflected from the sun. The revival
was the silver age of that old golden age of Greece."

It is evident to one who studies the arithmetics of this cen-
tury that no national spirit had yet developed. It was the city,
not the state as we know it, to which men gave their allegiance.
Just as we have Venetian art, so we find Venetian and Floren-
tine and Roman arithmetics, quite as distinct from one another
as were the Pisan works from those of Nurnberg.

a See J. B. Bury, in the Cambridge Modern History, Vol. I, chap, iii (London.
1902). This important work, containing different essays, is hereafter referred
to as Cambridge Mod. Hist.

244 CHRISTIAN EUROPE FROM 1400 TO 1500

Origin of Printing. When we consider the effect of print-
ing upon the development of mathematics, we must recall the
fact that the art existed long before Gutenberg's time, about
1450. Not only had the Chinese printed from engraved blocks
many centuries earlier than this, but they had also made some
use of movable types. In Europe, too, block printing had
assumed considerable prominence before movable types were
invented.

It is also well to consider how mathematical knowledge was
disseminated before it was possible to send it abroad by means
of the printed page. There were three general periods in the
transmission of such knowledge in the Middle Ages :

1. From the founding of the monastery at Monte Cassino by
St. Benedict in 529, and continuing until c. 1200, the period in
which scholars went to the teacher in the monastery and heard
his lectures. This was the period of such men as the Vener-
able Bede and Alcuin of York.

2. From c. 1200 to c. 1400, when universities appeared in
considerable numbers and claimed control of the scribes and
booksellers. 1

3. From c. 1400 to c. 1460, when the manuscript trade be-
came more general, when large numbers of copies were made,
and when these were sold as we sell books today.

All this was, however, a crude way of disseminating knowl-
edge compared with the circulation of printed books. We
should therefore expect the isth century to begin to make
widely known the mathematical classics of the ancient civili-
zation, to meet the mathematical needs of a commerce to which
new worlds were opening, and to prepare popular summaries

ir rhe scribes had stands (stations) and the booksellers had shops near the
university. The bookshops near the Sorbonne in Paris today are relics of the
houses of the old librarii who, in the isth and i4th centuries, rented their
manuscripts. The stationarius was required to employ skilled copyists, who were
enjoined to perform their tasks "fideliter et correcte, tractim et distincte, assig-
nando paragraphos, capitales literas, virgulas et puncta, prout sententia requirat."
See G. H. Putnam, Books and their Makers during the Middle Ages, I, 200 (New
York, 1896, 1897) J hereafter referred to as Putnam, Books. On the whole sub-
ject of reproduction and sale see J. A. Symonds, Renaissance in Italy: The Age
of Despots, II, 120 (New York, 1883).

INFLUENCE OF PRINTING 245

of the mathematics already accumulated. It would hardly be
expected that this century would do more than receive these
accumulated treasures and transmit them, postponing any new
advance in science until the century following. It should be
added that the sack of Mainz by Adolf of Nassau (1462)
scattered the printers of that city all over Europe, an event
comparable in some respects to the results of the fall of
Constantinople.

The Universities. The universities were now beginning to
mention mathematics more commonly, but that was about all.
In the fragments of the Oxford statutes of 1408, only a little
arithmetic was required for the bachelor's examination, 1 and
in 1431 there was hardly any improvement. 2 In Paris the work
was reorganized by the Papal legate in 1452, but nothing was
done to better the condition of this science. The baccalau-
reate demanded only the reading of a little mathematics, 3 but
nothing definite was prescribed. In general, little was required
beyond arithmetic and a few pages of Euclid. 4

Italian Writers. The great commercial activity in Italy in
the 1 5th century gave rise to a large number of mercantile
arithmetics, and these set a standard for the treatment of the
subject that still influences, in some degree, the textbooks of
today. Among the best known of the commercial writers were
Matteo da Firenze 5 (c. 1400), his son Luca da Firenze
(c. 1400), Giovanni, the son of this Luca (c. 1422), and Andrea
di Giovanni Battista Lanfreducci (c. 1490), an officer of the

1 "Algorismus integrorum" and "computus ecclesiasticus."

2 For the licentiate, " Arithmeticam per terminum anni, videlicet Boethii,"
with a little more science and mathematics. Suter, Univ. Mittelalt., p. 90.

3 "Aliqui libri mathematici."

4 A MS. of 1515 in the Wolfenbiittler Bibliothek gives the following as repre-
senting the arithmetic: "Arithmetica communis ex divi Severini Boetii Arith-
metica per M. loannem de Muris compendiose excerpta; Tractatus brevis
proportioning abbreviatus ex libro de proportionibus D. Thomae Braguardini
Anglici, . . . Algorithmus M. Georgii Peurbachii de integris; Tractatus de
Minutiis phisicis compositus Viennae Austriae per M. loannem de Gmunden."
Monatsberichte der K. P. Akad. d. Wissensch. zu Berlin, 1867 (Berlin, 1868),

P- 43-

5 Matthew of Florence. See Rara Arithmetic^ p. 468.

246

CHRISTIAN EUROPE FROM 1400 TO 1500

Republic of Pisa in 1505. The works of these four writers
exist only in manuscript and contain little of value except

what they tell of the
commercial customs of
their time.

Among the illustrious
citizens of Florence one
of the 1 6th century
historians 1 mentions a
mathematician named
Benedetto, commonly
known as Benedetto da
Firenze. His arithmetic 2
was written c. 1460, and
relates to the mercantile
needs of his native city.

It is one of the most

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complete works of this
kind that appeared in
the i sth century, but it
has never been printed.
Of the Italian writers
of this period who had
some knowledge of
mathematics beyond the
field of commercial arith-
metic the earliest was
Prosdocimo de' Belda-
mandi, 3 a native of
Padua and a student and professor in the university in that city
(1422-1428). He wrote on arithmetic, music, and astronomy,

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FIRENZE'S WORK, 1422

This portion relates to the calendar. The MS.

contains the statement: "questo d dottobre

i|22" and the further date "ad 28 dottobre

1422"

1 U. Verino, De illustrations urbis Florentiae libri tres. Paris, 1583.

2 " Inchomincia el trattato darismetricha," as in the MS. in the library of
Mr. Plimpton. Sec Kara Arithmetica, p. 464.

3 Born at Padua, c. 1370-1380; died at Padua, 1428. The name also appears
as Beldomandi, Boldomondo, Beldemandi, and Prosdocimo Padvano. Ber-
nardino Scardeone (De Antiqvitaie Vtbis Patavii, Basel, 1560) speaks of him as
"& nobiii familia Patauina ortus: egregius Musicus, & eximius philosophus, &
clarus astrologus."

ITALIAN WRITERS 247

and also studied medicine. 1 The arithmetic was limited to the
ordinary operations and contained no commercial problems.-

About 1425 one Leonardo of Cremona, or Leonardo de'
Antonii (born c. 1380), wrote a brief practical geometry of
which three manuscripts are known. 3 A little later (c. 1449) a
fellow townsman of his, Jacob of Cremona, a teacher at Mantua
and Rome, translated some of the works of Archimedes. To-
wards the close of the century Georgius Valla (1430-1499),
a native of Piacenza, lectured on physics and medicine at
Pavia and also at Venice, and in his magnum opus, which was
merely a compendium of knowledge, treated of Boethian arith-
metic, Euclidean geometry, optics, and the astrolabe, as well
as a variety of other subjects. The work 4 was printed in
1501, but is notable chiefly for its size.

About 1475 an Italian painter, Pietro Franceschi, also called
Pietro della Francesca, 5 wrote a work, DC corporibus regulari-
bus, in which he treated of the mensuration of regular poly-
gons, the sphere, the five regular polyhedrons inscribed in a
sphere, and solid figures in general. His problems may be
illustrated by the cases of finding the area of a regular octa-
gon circumscribed by a circle of diameter 7, and the finding
of the surface of a cube circumscribed about a sphere also
of diameter 7. 7

J This on authority of two early MSS.: "Exanicn medicinae magistri Pros-
docimi de padua M iiij undecimo," that is, 1411; "Padovani dottori delle
arti e Medicina," with the assertion that he "fu esaminato e dottorato nelle
arte . . . 1409."

2 For description, see Ram Arithmetka, p. 13; Boncompagni's BulleUino,
XII, i.

"Leonard* Cremonensis artis metrice practice compilatio. Primus tractatus.
The date is doubtful. See Kara Arithmetics, p. 474 ; Bibl. Math., IX (3), 280.

4 Georgii Vallae Placentini viri clariss. de e.\petendis, et fvgiendis rebus opus.

5 Born 0.1410-1420; died 1402. See E. Harzen, "Ueber den Makr Pietro
degli Franceschi und seinen vermeintlichen Plagiarius, den Franziskanermonch
Luca Pacioli," Archiv /. d. zeichn. Kunste, II, 231; W. G. Waters, Piero della
Francesca, London, 1901 ; and Mancini's monograph mentioned below.

(J " Diameter circuli qui circumscribit octagonum est 7. Quanta igitur sit
octagon! superficies invenire." Tract. I, Problem XL. He uses- 2 /- for?r.

7 Tract. II, Problem XVI. The work is printed in full in G. Mancini,
"L' opera 'De corporibus regularibus' di Pietro Franceschi," in the Atti d. R.
Accademia del Lincei, XIV (5), 488.

248 CHRISTIAN EUROPE FROM 1400 TO 1500

Printing was introduced into Italy in 1464 by Juan Turre-
cremata, 1 abbot of the monastery of Subiaco, near Rome, the

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FIRST PAGE OF A MANUSCRIPT OF LUCA DA FIRENZE'S WORK (C. 1400)
This MS. dates from c. 1475. Now in the library of Mr. Plimpton

first book appearing in the following year. 2 Fourteen years
later (1478) the first printed arithmetic appeared at Treviso,

1 The Latin form of the family name Torquemada. He was a native of
Valladolid, in Spain. 2 Putnam, Books> I, 405.

FIRST ITALIAN PRINTED BOOKS 249

then an important commercial town a day's travel from Venice.
Three presses had already been established there, and it was
from the one of Manzolo, or Manzolino, 1 that this work ap-
peared. That it was anonymous was merely in keeping with
the custom of a land where the glory of the individual was
absorbed in the glory of the state ~ Boncompagni learned of
only eight copies of this arithmetic in various degrees of
preservation, 3 so that the book is rare. It is quite commercial
and never had any noticeable influence on the arithmetics of
Italy. 4 It was, however, the first step in a remarkable move-
ment, for up to the close of the isth century there were printed
in Italy at least 214 mathematical works, the number rising to
1527 in the following century. 5

Three years after the Treviso book appeared, the first purely
commercial arithmetic was published. The author seems to
have been one Giorgio Chiarino, of whom nothing further is
known. The work is a mere compilation of measures and of
such customs of exchange as were needed by Florentine mer-
chants, 7 but there was enough in it to lead Pacioli (1494) to
borrow freely from it.

Three years later Piero Borghi, N a Venetian, published the
most noteworthy Italian commercial arithmetic of the century, 9

iDomenico Maria Federici, Memorie Trevigiane, p. 73. Venice, 1805.

2 "When Byron swept with superficial yet brilliant eyes the rolls of Venetian
history, what did he find for the uses of his verse ? Nothing but two old men,
one condemned for his own fault, the other for his son's, remarkable chiefly
for their misfortunes/' Mrs. Oliphant, in The Makers of Venice.

s Atti Ponttf., Vol. XVI. The only copy of this arithmetic that ever reached
America is in the library of Mr. Plimpton.

4 For a description, see Kara Arithmetica, p. 3.

5 P. Riccardi, Biblioteca Matematica Italiana, Parte seconda, XI, XV, seq.
(Modena, 1880), hereafter referred to as Riccardi, BibL Mat. Hal.

G Qvesto e ellibro che tracta di Mercatantie et vsanze de paesi (Florence,
1481). See Rara Arithmetica, p. 10.

7 "Finito ellibro de tvcti ichostvmi: cambi: monete: pesi: misvre: & vsanze
di lectere di cambi : & termini di decte lectere che nepaesi si costvma et in diverse
terre."

8 Piero Borgi, as the name appears in the first edition. The name is also
given as Pietro Borghi and Pietro Borgo. He seems to have died after 1494.

*Qui comenza la nobel opera de arithmethica, the earliest printed books
generally having no title pages. In later editions this was sometimes called the
Libro de Abacho. The first edition was Venice, 1484.

250 CHRISTIAN EUROPE FROM 1400 TO 1500

a work intended for the mercantile class of Venice. One
of the features of the book is that, like certain others of the

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cto acDt"flppcUaf > .<Lcrniin*c qo vmulcutulq 3 nnis c G^tgura
c qcniuio vl rrrmio ?nncr Oircul A c ngura puna vna qdem U/
nca ptctai q ctrcu fcrcnaa noiaf :m cufmedio pucr'c : a quo'ocs
Imeerccte adcircufcrcfii ejrcii tco libiiutcc^ lut cqualcc. c btc
quide puctcctru circuit o:.G/Diamct er circuli c Imca recra quo
liiEcrccnEp tralicnsejrtremitatelqjUias circu teretic applicant
circulu i ono media omidit G^emiarculue c ngnra plana oia/
metro circuit i mcdictate circiifcrennc picnta.Glfbomo circu /
It e figura plana recta Imca 1 pane circu ferctte ptcra: lemictrcu/
lo quide aut maio: aut mtnp:. Gi&ccnlmce figure liit q recns h/
ucto connenf quarti qucda mlaterc q mb'Vectuj Unas: qucd j
quadnlatcrcq qtuojrecnslmeiD.qctamlhlaterc quu plunbus
qsqnatuo: rcctis Itneia continent. G ^tgurarii tnlateraru-.alta
dltiianguluobristrialatcracqnalia '^llia tnangnluooaobne
eqltalatcra ^lliamanguln&n'iuinequaliniii btcru. UOay itcru
alia cfto2tbogoiuu:vnu.l rectum anijuliiiii babeno.aiia cam'
bltgomum aliqucm obtulumangulutn babcne.ahaeft opgom
urn :in qua treeaiiguh Innt acutt G^iguraru auto quadrilateral:
3llia eft tidrarum quod eft cqiulateru atqs rectangulu . alia eit
tctragon^long 9 -q eft figura rcctaiii3ula : led equilarcra non eft.
2llla eft bdmuavm. que eit equilatcra : led rcctangula non eit.

DC pnncipii* pfc note:rpmo oe oiffint/

Crcnlu*

.nJIcqbuU^

FIRST EDITION OF EUCLID, 1482
First page, reduced, from the Venice edition of 1482

period, it begins with multiplication, putting addition and
subtraction after division and therefore presupposing some
ability in computation. The work with integers is followed by

LUC A PACIOLI 251

fractions, the Rule of Three, and the usual applications of the
time, partnership, profit and loss, barter, and alligation.

These works, however, were mere advance couriers. Arith-
metics had been published in Germany and Italy, Euclid's
Elements had appeared from the famous press of Ratdolt, in
Venice, 1 and the psychological moment had arrived for a gen-
eral treatise summarizing the mathematical knowledge of the
time. In such a period of unrest as that centering about the
year 1494, when France was at war with most of Italy, when
Florence was in arms against her sister towns, when Charles
VIII was invading the peninsula and Savonarola was proceeding
as an ambassador to Pisa to allay his wrath, in such a period
it would be surmised that a book like this could be prepared
only in the peaceful atmosphere of a cloister.

Luca Pacioli. Such was the case, and Luca Pacioli, 2 known
from his birthplace as Luca di Borgo, was the author. As a
boy he may have come under the influence of his townsman, the
artist Pietro Franceschi, already mentioned (p. 247), from
whose work he freely took considerable material. At about
the age of twenty he went to Venice (1464) and became a tutor
to the three sons of a wealthy merchant, 3 and some six years

iPreclarissimus lilcr elementorum Euclidis, Venice, 1482. See the facsimile.

2 Born at Borpo Sun Sepolcro, Tuscany, c. 1445 ; died probably after 1509. The
name is spelled in various other ways, such as Paciolo, Paciolus, and Paciuolo.
It does not appear at all in his Sfima, and in the De diuina proportione (Venice,
1500) it is given only in the Latin genitive, "Lucae pacioli ex Burgo sancti
Sepulchri . . . epistola." In his "Supplica . . . al Doge di Venezia" of Decem-
ber 29, 1508, in which he asks for permission to print the De diuina proportione^
the name appears as " Luca de pacioli dal borgo sa sepulchre." It is from such
contemporary evidence that Boncompagni (Bullettino, XII, 420) was led to
speak of him as Pacioli, although Bernardino Baldi, writing in 1589, says "Fu de
la famiglia de Paciuoli ignobile per quanto mi credo e di poco splendore," which
led Cantor to adopt this later spelling. For the contemporary documents, in which
he is generally spoken of as Luca or Lucas di Borgo, see Boncompagni, he. at.

The best sketch of his life is that by H. Staigmuller, " Lucas Paciuolo. Erne
biographische Skizze," in the Zeitschrift (HI. Abt.), XXXIV, 81, 121. See also
the "Elogio di Fra Luca Pacioli" in B. Boncompagni, Scritti inediti del P. Z>.
Pietro Cossali (Rome, 1857), p. 63; B. Boncompagni, in his Bullettino, XII, 377-

" . . . nostri releuati discipuli ser Bart , e francesco e paulo fratelli
deropiasi da la gudeca: degni mercatanti in vinegia: figliuoli gia de ser An-
tonio." Suma, 1494 ed., fol. 67, v. ? 1. 4.

252 CHRISTIAN EUROPE FROM 1400 TO 1500

later he wrote an algebra which he never published. 1 In 1471
he went to Rome and, possibly influenced by his two brothers
who had already entered the brotherhood of St. Francis, joined
the Minorite order. In 1476 we find him teaching in Perugia
and writing a little book for his pupils. 2 Five years later,
at Zara, he wrote still another work, "more subtile and rigid," 3
but neither of these works was printed. He traveled exten-
sively in Italy 4 and possibly in the Orient, 5 and even after he
became a Franciscan he was a wanderer. We find him back
in Perugia in 1487 and working on his Suma, in Naples in
1494, in Milan in 1496, in Florence and Rome in 1500, and in
Venice in 1508.

The great work of Pacioli, summing up not only his previous
and unpublished works but also the general mathematical
knowledge of the time, appeared in Venice in I494, 7 having
been written seven years earlier, when he was at Perugia. 8
It is a remarkable compilation with almost no originality. 9
He borrowed freely from various sources, often without
giving the slightest credit, but in this he merely followed

iAs to his MSS., now in the Vatican, see G. Enestrbm, Bibl. Math., XIII
(2), 53; B. Boncompagni, in his Bullettino, XII, 381, 428.

2 " ... alo giouani de peroscia . . . nel. 1476," as he says in the Suma of
1494, fol. 67, v.

3 "E anche in quello die a gara nel. 1481. de casi piu sutili e forti com-
pongmo." Ibid.

4 "Ma da poi che labito indegnamente del seraphyco san francesco ex voto
pigliamo: p. diuersi paesi ce conuenuto andare peregrinando." Ibid.

5 So Cossali, writing in the i8th century, says that he "per desio di scienza
viaggiasse in Oriente, ed in Arabia precipuamente, dove a que' tempi erano le
matematiche dottrine in gran fiore " ; but there is no contemporary authority for
the statement. Certainly the expression "in gran fiore" is without foundation.
On the contrary, there is internal evidence from the Suma that he was never
in the East.

6 See also P. Treutlein, Abhandhmgen, I, 10.

7 For the title-page, see page 253. The form Suma is a contraction for
Summa.

8 "E al presente q i peroscia . . . correndo glianni del nostro segnore Jesu
Christo. 1487." Suma, 1494 ed., fol. 67, v.

9 G. Mancini, "L'opera 'De corporibus rcgularibus' di Pietro Franceschi,"
Atti d. R. Accademia del Lincei, XIV (5), 488. See also Chiarino's work, from
which he did not hesitate to take what he wished, and Baldi's article in Bon-
compagni's Bullet tino, XII, 426.

LUCA PACIOLI

253

the custom of the age.
His use of material
from Euclid, Ptolemy,
Boethius, Fibonacci,
Jordanus, Sacrobosco,
Biagio of Parma, Pros-
docimo, and Suiceth
shows that he was at
any rate rather well-
read. He was, however,
a careless writer, so
much so that Cardan 1
has a chapter devoted
to errors in his book. 2
The work includes such
algebra as was then
known, a large amount
of business arithme-
tic, a poor summary of
Euclid, and a treatment
of double-entry book-
keeping. 3

In 1497, while at
Milan, he wrote his
De diuina proportione,
publishing it at Venice
in i sop. 4 This is a
work of more merit
from the standpoint of

pouionalita*

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^oponionic.ppomoaliraanotuia aclj? ocucU

dt e re turti (i almToi COxi*.
biauiouero cuutenrie numero.i ;.D fe$nra ronti'

nue,ppo2tioaK0el.6?c*7! oeudiaecjratte.
ffutte le rfreblao:fliiK>:cfoe rckuare . JKM% multfr

plicarffimarc^fotrarccoturrcfuc^udJanicrot'

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IfcamV.mulnplicar.fummanc fotrar oelep:opo^cu>

nicoetutteforn'radt'cf*

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fmdmriegcmralioucrcondufioni n?^abfolucre

ogntcafodxperrego[e^;dinrien6 fi podcflh

dxefondamcntt
ompagntei tuttimodi'.eTo:pflitilr&
&odde oe belttami*. e lo: parcire

ambt

PART OF THE TITLE-PAGE OF PACIOLl'S

WORK OF 1494

This shows the general nature of the treatise,
forming a kind of table of contents

1 In his Arithmetica of 1539.

2 "De erroribus F. Lucae quos vel transferendo non diligenter examinavit,
vel describendo per incuriam praeteriit, vel inveniendo deceptus est." See also
Cossali, "Elogio di Fra Luca Pacioli," in the Scritti inediti, p. 63 (published by
B. Boncompagni, Rome, 1857). On the variations in each of the two editions
(1494 and 1523) see E. Narducci, Intorno a due edizioni delta Summa de
Arithmetica, Rome, 1863.

3 On this phase of the work see V. Gitti, Gli Scrittori Classici della Partita Dop-
pia, a reprint from the Annali del R. Istituto Indust. e Professionale di Torino,
Vol. V. (Turin, 1877). ^German translation by C. Winterberg, Vienna, 1896.

254

CHRISTIAN EUROPE FROM 1400 TO 1500

geometry than the Suma, but one that could not, in the nature
of the case, be as popular. His figures of the regular solids,

* s f ven in th u is
book, were the

best that had as
yet appeared in
print, 1 and have
been attributed to
Leonardo da Vinci.
He also published
an edition of Eu-
clid in 1509, but

the work is of lit-
tle merit.

Of the Italian
astronomers of the
1 5th century who
gave considerable
attention to math-
ematics the most
eminent were Gio-
vanni Bianchini
(c. 1450) , profes-
sor of astronomy
in the University
of Padua, Frances-
co Capuano, Gio-
vanni Batista de
Manfredonia (c.
1450-1490), and
Domenico Maria
Novara da Fer-
rara( 1454-1 504).

Italian science, however, had hardly yet awakened to the need

for more advanced mathematics.

TITLE-PAGE OF CALANDRl's ARITHMETIC, 1491

An example of the fanciful portraits of the Greek
mathematicians. Such portraits became common dur-
ing the Renaissance. The anachronisms in this one
are evident

ipacioli here takes his material freely from Franceschi's work, already men-
tioned.

AUSTRIA AND GERMANY

Of the arithmeticians of this time one of the most noteworthy
was Filippo Calandri, 1 whose arithmetic appeared in Florence
in 1491 and contained the first printed example in long division
by our modern method and the first illustrated problems pub-
lished in Italy. Nothing further is known of his life, but
Sfortunati," writing in 1534,
speaks of him as a learned

eglicuna torrccbcc

pic wparia lino finmc
cfocclargbo ?o brae
cia. uo fapcrc qaamo
fara tiwglxi ima func
cbc Ha ^ppicvua alia 11
iu del hnmc i alia ci
maiclLuorre

far* tonga

rooj >oo

I Uoo

rimafcnro \C brae
da z ? 4 bracaa fcnc
ruppc

Francesco Tellos or Pel-
lizzati, a native of Nice,
published a commercial
arithmetic at Turin in 1492,
in which, as will be shown
in Volume II, use is made
of a decimal point to denote
the division of a number by
a power of ten.

Austria and Germany. Be-
fore considering the names
of those who advanced the
cause of mathematics in the
German states in the i$th
century it is necessary to
say a word concerning the
influence and work of the
Rechenmeisters. In the i3th
century there had been a
great revival of trade in

Germany. The Hanseatic League, a union of commercial towns
in the Teutonic countries, had shown that the demands of trade
must be recognized. It employed force against the pirates,

i Italian writers of the period give this form of the name. Latin writers use
Fhilippus Calender and Philippus Calandrus. Calandri himself gives only the
Latin genitive form, Philippi Calandri. See Rara Arithmetica, p. 47-

2 Giovanni Sfortunati (born at Siena, c. 1500), Nvovo Lvme Libra di
Arithmetic^ Venice, 1534. See page 306.

3 "Filippo Caladri Cittadino Fiorentino, huomo certamete in tale disciplina
erudito." From the 1545 edition, fol. 3, r.

Eglicunalbcroitifufa
nua dun fiumc clqua
tccalro? o br&a* cl
fmmc c largbo ? o bra
cu 7 per forttina di uc
to firuppc tural Uiogo
cbcUtcimadclUlbcro
roccaita Uriua del fiu
me. do fapcrc qiuntu
bracad fcnc ruppc 2
qnanronenmafcnno

fo -- Jo

FIRST ILLUSTRATED PROBLEMS
From Calandri's work, Florence, 1401

256 CHRISTIAN EUROPE FROM 1400 TO 1500

purchased settlements in foreign cities, and even made war upon
Denmark and England in order to protect the interests of its
members. The Church schools having failed to prepare boys
for business, the League undertook this work, offering not only
reading and writing, in which the Church schools gave instruc-
tion, but also business arithmetic, in which they gave none.
Out of this kind of teaching of apprentices there arose a type
of commercial school known as the Rechenschule, presided
over by a Rechenmeister. In due time the Rechenmeisters
formed guilds, claimed a monopoly in their vocation, and
finally came to be looked upon as regular officials of the town.
Not infrequently their duties included the sealing of measures,
the city gaging of casks, and occasionally the minting of money ?
in those times a function of the city instead of the state. Occa-
sionally they were also the writing masters, and they finally
came to be looked upon as among the dignitaries of the town. 1
In the larger cities the Rechenmeisters' guilds admitted appren-
tices who served for six years, then becoming Schreibers
(writers) with the privilege of becoming assistant teachers.
When a locus opened, the eldest of these cadets was subjected
to an examination 2 and, if successful, was given the position
with the rank of Meister. The guild of Schreib- und Rechen-
meisters continued in Liibeck, for example, until 1813.

In the 1 5th century the power of the Rechenmeisters was
first shown to any considerable extent, and for the next three
hundred years they clung more or less tenaciously to their privi-
leges, demanding that arithmetic should be taught by them in-
stead of by the common schoolmasters. Their influence extended
to the commercial towns of Holland, and the Rechenmeister is

iThus, the Burgermeister and Council of Rostock, in 1627, sent an official
call, couched in dignified language, as follows : " Wir Burgermeister und Rat zu
Rostock urkunden hiermit, dass wir den chrenfesten und wohlgelahrten Jeremias
Bernstertz zu unserm und gemeiner Stadt Schreib- und Rechenmeister be-
stellt und angcnommen haben." For the entire document see F. Unger, Die
Methodik der praktischen Arithmetik in historischer Entwickelung, p. 26 (Leip-
zig, 1888) ; hereafter referred to as Unger, Die Methodik.

2 One of the questions from an old paper reads: "Wie wird besagte Chilio-
heptacosioheptacontatetragonal-Zahl, deren Latus 6 formirt und aus solcher
gefundencn Polygonal-Zahl die Wurzel wieder extrahirt?"

RISE OF GERMAN MATHEMATICS 257

frequently mentioned in the Dutch arithmetics of the 1 7th cen-
tury. 1 The excellent commercial arithmetics which began to
appear in Germany in the isth century and which continued
there and in Holland until the igth century were the work
of the Rechenmeisters or of those whom they influenced. 2

Owing possibly to the early advance in printing in Germany,
possibly to the trend of the classical influence from Constan-
tinople through the Balkan states to Vienna, or possibly to
the influence of the leisure which wealth afforded, the Teutonic
countries forged ahead in the isth century, taking rank with
Italy in their men of mathematical ability. Four of these men
were scholars whose standing was recognized abroad ; the rest
were mediocrities.

Late in the century Johann Widman (born c. 1460), a native
of Eger in Bohemia, wrote on arithmetic and algebra. He was
a student at Leipzig in 1480, B.A. in 1482, M.B. in 1485,
and M.A. in 1486. He may have received a doctor's degree in
medicine, for a medical work by a Johann Widman appeared
in 1497, but this was probably another person. 3 That he gave
lectures on algebra, possibly the first given in Leipzig, is
shown by a contemporary manuscript. 4 He was probably the

1 Thus, Coutereels (1631) addresses a problem to one Ghileyn Pietersz
"Schepen [sheriff] deser Stadt, goet Reken-meester " ; Cardinael (1650) speaks
of himself as " Reecken-meester tot Amsterdam " ; the printer of Vander Schuere's
arithmetic (1634 ed.) speaks of the book "die ik door de correctie van een
goet Reken-Mr. het verbetert"; Eversdyck, who revised Coutereels's arithmetic
in 1658, is described as the "Reken-meester ter Reken-Kamer van Zeelandt";
and even as late as 1792 an edition of Bartjens is described as due to "den
Wel-ervaren Reekenmeester Klaas Bosch."

2 The Italian " master of the abacus " was probably suggested by the German
title, or vice versa. The expression is found in Italian manuscripts of this period
but does not seem to have been used in England. Indeed, even the word "abacus"
never had as extensive use there as on the Continent.

B Tractatus clarissimi medicinal doctoris Johdnis widman. In other works
the spelling Widmann is sometimes used. Boncompagni's BulleMino, IX, 210.

4 See Kara Arithmetica, p. 36. J. G. Bajerus (J. W. Bayer), De Mathe-
matvm . . . introdvctione (1704) (hereafter referred to as Bajerus, De Math.),
says (p. 9): "Joh. Widemann, natione Noricus, patria Egrensis, disciplina
Lipzensis, vir in Mathematicis abunde eruditus. Qui capessis in Philosophia insig-
niis, cum multa admodum in mathematica, & potissime in spcciebus in studio
Lipzensi, non sine auditorum summo applausu, aliquot annis volvisset . . ."
On his life and works see Cantor, Geschkhte, II, chap. Iv.

258 CHRISTIAN EUROPE FROM 1400 TO 1500

author of an Algorithmus Lincalis (Leipzig, post 1489), the
first printed treatise on calculation by the aid of counters. He
wrote the first important German textbook on commercial
arithmetic, 1 and in this appear for the first time in print the
signs + and , not as symbols of operation but to express
excess and deficiency in packages of merchandise.

More capable as a mathematician but less known by the
populace, Johann von Gmiinden 2 was educated at Vienna,
taught there, and was the first Austrian to occupy a chair de-
voted wholly to mathematics. He wrote a treatise on sexagesi-
mal fractions, 3 one on trigonometry, 1 and one on the computus. 5

A few years 'later, Nicholas Cusa," son of a fisherman, gave
proof of what industry and genius will do even for those born
in humble estate. He rose rapidly in the Church and held
various positions of honor, including the bishopric of Brescia.
He was made a cardinal and became the governor of Rome in
1448. He wrote several tractates on mathematics, including

l Behede vnd hubsche Rechnung auf alien kauff manse ha fit, Leipzig, 1480.

2 Born at Gmunden on the Traunsce, Gemiind in Lower Austria, or Gi'imind
in Swabia, c. 1380 ; died c. 1442. The name also appears in such forms as Johannes
de Gmunden, Johann von Gcmunden, Johannes de Gamundia. He has been
identified with the Johann Schindel or Sczindel, called Johannes de Praga, a
native of Koniggratz, and possibly the identification is correct. His name has
also been given as Wissbier and Nyden, but on doubtful authority. Sec Bibl.
Math , X (2), 4. Bajerus, De Math, has this note (p. 6): " Eo vix A. C.
MCCCXCVII. fatis furicto, Vindobonam ornare coepit Johannes dc Gmunden
kidem natione Germanus, (Ricciolus Johannem de Egmunda vocat) Theo-
logus & Astronomus Celebris, anno MCCCCXL1I. humanis excmtus." See also
Bibl. Math., Ill (3), 140.

s Tractatus de Minucijs phisicis, first printed at Vienna in 1515.

4 De sinibuSj chordis, et arcubus.

c It exists in manuscript, closing with the words : tf Explicit kalendariu mgri
Joh'is gmiind." See Kara Arithmetic^ p. 440.

6 Born at Kues (Cues) on the Mosel, 1401; died at Todi, Umbria, 1464. The
name appears in such forms as Nicolaus von Cues, Nicholas Cusanus, Nicholas
von Cusa, Nicolaus Chrypffs, Nicolaus Cancer, Nicolaus Krebs. His father's name
was Johann Chrypffs or Krebs. "Huomo di mostruoso ingegno impatronissi
delle tre lingue megliori, e diede opera all' arti liberali, & alle scienze," Baldi,
Cronka, 95. Rossi, in his Niccolo di Cusa (p. n), describes him as "originate
pensatore in molteplici discipline." There is a good biography by Dr. Schanz,
Prog., Rottweil, 1872. The best biographies are those of J. M. Dux, Der
deutsche Cardinal Nicolaus von Cusa, 2 vols., Regensburg, 1847, and E. Van-
steenberghe, Le Cardinal Nicolas de Cues, Paris, 1921.

GERMAN MATHEMATICS 259

in the subjects treated the quadrature of the circle, the reform
of the calendar, the improvement of the Alfonsine Tables, the
heliocentric theory of the universe (a theory which was looked
upon as a paradox rather than a scientific probability), and the
theory of numbers. Wallis 1 asserted that he was the first
writer known to have worked on the cycloid, but this is not
supported by the evidence. 2 His Opuscula appeared c. 1490,
and his Opera appeared in Paris in 151 1. 3

Much better known as a mathematician, Georg von Peur-
bach 4 studied under Nicholas Cusa and other great teacher?,
learned Greek from Cardinal Bessarion in order to be able to
read Ptolemy, lectured at Ferrara, Bologna, and Padua, and
became professor of mathematics at Vienna, making this uni-
versity the mathematical center of his generation. Although
interested primarily in astronomy and trigonometry, 5 he wrote
an arithmetic/' but this was merely for the use of students in
these branches of science. Melanchthon considered the work so
excellent that he wrote a preface for the edition of 1534. Peur-
bach compiled a table of sines, which was extended after his
death by his pupil Regiomontanus, and he also wrote various
works on astronomy.

The most influential and the best known of the German mathe-
maticians of the isth century was Johann Miiller, 7 generally
known, from the Latin name of Konigsberg, as Regiomon-
tanus. 8 At the age of twelve he was a student at Leipzig. He

1 Philosophical Transactions of the Royal Society of London, p. 561 (1697) ;
hereafter referred to as Phil. Trans.

2 See Bibl. Math., I (2), 8, 13. *Rara Arithmetica, p. 42.

4 Born at Peurbach, Upper Austria, May 30, 1423; died at Vienna, April 8,
1461. The name is also spelled Peuerbach and Purbach.

5 Tractates Georgii Purbachii super Propositions Ptolemaei de sinubus et
chordis, Nurnberg, 1541; Theoricae novae planetarvm, Venice, 1405, with an-
other edition s. 1. a. (Nurnberg) and one at Venice, 1490.

*Elementa Arithmetices Algorithmvs de numeris integris. It went by various
other titles. The first printed edition appeared in 1492.

7 Born at Unfied, near Konigsberg, Lower Franconia, June 6, 1436; died
July 6, 1476. Cantor, Geschichte, II, chap. Iv. In the British Museum there is
an interesting block-book almanac, not later than 1474, prepared by him, in
which his name appears as Magister Johann van Kunsperck.

8 Also as Joannes de Monteregio.

260 CHRISTIAN EUROPE FROM 1400 TO 1500

afterwards studied under Peurbach, lectured at Venice, Rome,
Ferrara, and Padua, and lived for a time in Niirnberg. 1 In
1475 he was called to Rome by Pope Sixtus IV on account of
one of the frequent attempts to consider a reform of the
calendar, and was made titular bishop of Ratisbon. He studied
the mathematics of the Greeks in the original, and was "the
first who made humanism the handmaid of science." He wrote
De triangulis omnimodis libri V (c. 1464), the first work that
may be said to have been devoted solely to trigonometry." He
also wrote an Introductio in Elementa Euclidis with some sup-
plementary work on stellar polygons, and had certain definite
ideas as to the circumnavigation problem. ?

The earliest known German algorism was compiled in 1445,
more than two centuries after Fibonacci prepared a work on
the subject, and about seventy years after the first French
manuscript relating to it is known to have been written. The
earliest example of a German algebra is found in a Munich
manuscript of 1461, with text also in Latin. The first printed
German arithmetic appeared at Bamberg in 1482.

France. France received the Italian humanism in the spirit
of a sister country speaking a kindred language, but for a num-
ber of decades after Gregory Tifernas (1458) went to Paris
to teach Greek the taste of the learned class was for letters
rather than science. 4

It is perhaps because of this fact that France produced fewer
noteworthy mathematicians than Germany or Austria in the
1 5th century, but it is more probably due to such causes as
the constant turmoil in which the University of Paris found her-
self at this time. With a continual warfare between Church

speaks of the great glory he brought to Niirnberg: "Noriberga
turn Regiomontano f ruebatur : mathematici inde & studii & operis gloriam tantam
adepta, ut Tarentum Archyta, Syracusae Archimede, Byzantium Proclo, Alex-
andria Ctesibio non justius quam Noriberga Regiomontano gloriari possit."
Scholarum Mathematicarum libri units et triginta, p. 62 (Paris, 1569) ; hereafter
referred to as Ramus, Schol. Math.

2 First printed edition, Niirnberg, 1533.

3 A. Ziegler, Regiomontanus . . . Vorlaufer des Columbus, Dresden, 1874.

4 See also R. C. Jebb, Cambridge Mod. Hist., Vol. I, chap. xv\.

FRENCH MATHEMATICIANS 261

and State over the control of her work, and with continual pro-
tests from within her walls, she was in no mood to foster either
science or letters.

A type of the best product of the educational system of
France at this time is seen in the person of a very mediocre
and almost unknown scholar, one Rollandus (c. 1424), who, al-
though probably a native of Lisbon, spent his life in Paris.
He was a physician and a minor canon of the Royal Chapel 1
and may have been the Rolland who was rector of the university
in 1410. He evidently was acquainted with the general field
of pure arithmetic and algebra, as is shown by a manuscript
of about 1424 still extant." He also wrote on physiognomy and
surgery.

The most brilliant of the French mathematicians of the
period, however, was Nicolas Chuquet, 3 who, although a native
of Paris, lived in Lyons. 4 He wrote (1484) the Triparty en la
Science des Nombres, a work touching upon the three fields of
arithmetic. 5 The first part relates to computation with rational
numbers, the second to irrationals, and the third to the theory
of equations. Nothing is known about Chuquet except his
statement that he was a bachelor of medicine and that he
wrote his work at Lyons. 6

Great Britain. Great Britain produced relatively few writers
on mathematics in the isth century, and none of any special
prominence. John Killingworth may be taken as a type of

iHe speaks of himself as "prebenda capelle palacij regalis parisiensis."

*Scientia de numero ac virtute numeri, now in the library of Mr. Plimpton.
See Kara Arithmetic^ p. 446.

3 Born at Paris; died c. 1500.

4 Estienne de la Roche (1520), speaking of the "plusieurs maistres expertz
en cest art," mentions "maistre nicolas chuquet parisien."

5 It was printed in Boncompagni's Bullettino> XIII, 555. See Ch. Lambo,
"Une algebre Franchise de 1484. Nicolas Chuquet," Revw les Questions
Scientifiques (Brussels, October, 1902).

6 " Et aussi pour cause quil a este fait par Nicolas chuquet parisien Bacheliet
en medicine. Je le nomme le triparty de Nicolas en la science des nombres
Lequel fut commance medie et finy a lyon sus les rosne de salut. 1484." From
the original copy of his MS. made for Boncompagni by A. Marre, and now in
the author's library.

262 CHRISTIAN EUROPE FROM 1400 TO 1500

the best scholars of the period. We know little about him, 1
but the records show that he became a fellow of Merton Col-
lege, Oxford, in 1432 and that he died on May 15, 1445. He
seems to have been chiefly interested in astronomy and to have
prepared a set of tables for the use of students in this science.
There is extant in the Cambridge University Library an algo-
rism written by him in 1444." In this he refers to the use of a
slate for purposes of computation, but as to the operations
themselves there is no evidence of any originality.

Other Countries. The early mathematics of Russia was
largely devoted to questions relating to the calendar and to
number puzzles. In some of the medieval manuscripts there
are the results of very complicated computations, but we have
no knowledge of the methods by which these computations
were performed. 3 Several such cases are found in the Russkaya
Pravda* At the end of the isth century, work on the calendar
is known to have been done by the Metropolitan, Zosima, and
by Gennadi, bishop of Novgorod, but the results are not extant.
In all this work the numerals seem to have been alphabetic,
the system being similar to that of the later Greeks. Even as
early as the i4th century, the clergy placed geometry and
astronomy under the ban, and not until the i7th century was
there any opportunity for the study of mathematics. 5 Even
when, in the i8th century, the attempt was made to inaugurate
scientific work, all the leaders in mathematics were brought
in from abroad.

a L. C. Karpinski, "The Algorism of John Killingworth," English Historical
Review (1914), p. 707.

2 "Incipit prohemium in Algorismum Magistri lohannis Kyllyng Worth."
The work itself begins: "Obliuioni raro traduntur que certo conuertuntur
ordine. Regulas igitur et tabulas ad breua computationem operis calculandi
vtiles in formam certam secundum ordinem specierum Algorismi curabo
redigere." Ibid., p. 713.

3 A. N. Peepin, History of Literature (in Russian) (Petrograd, 1911 ed.) y

If 253.

4 A work of which there are three versions, one of the nth century, one of
the 1 2th, and one of the 13th.

5 V. Bobynin, "De 1'etude sur 1'Histoire des math, en Russie," in Bibl. Math..

II (2), 103.

MINOR COUNTRIES 263

In the isth century (c. 1450) a Cretan writer commonly
known as George of Trebizond 1 (1396-1486) made a new
translation of the Almagest into Latin and also translated some
of Theon's commentaries upon it. He was a quarrelsome man,
of little honor to science, to letters, or to manners.

In Hungary there was a certain Georgius de Hungaria who
wrote an Arithmeticae summa tripartita in I499, 2 and this per-
haps shows the high mark of mathematics in that country in
the isth century.

The Jewish activities in this century were very slight, being
chiefly manifest in translations from the Latin. Jacob Caphan-
ton (died by 1439), probably a native of Castile, a physician
and a teacher, wrote an arithmetic; 3 and Jehuda Verga (c.
1450), known for his compilation on the calamities of the Jews,
was also the author of a compendium of the same type. 4 There
were various writers who left works on astrology and the
calendar, contributions unimportant in themselves but requir-
ing the computation of tables and offering some encouragement
to the astronomer.

Sporadic efforts were made here and there in the isth cen-
tury to advance the study of mathematics in other countries,
but with no effect beyond the establishing of an interest in
the science. Thus Joao (John) II, who was with great diffi-
culty placed (1481) upon the throne of Portugal, sought to
elevate scholarship, and particularly astronomy and navigation.
by establishing at Lisbon his Junta dos Mathematicos,"' but the
result was not noticeable in pure mathematics. A few scholars,
such as the bishops Calsadilha and Don Diogo Ortiz, sought
to advance the applications of mathematics in such fields as
map drawing, for the purpose of aiding the Portuguese navi-
gators, but this was about all that was done. A little later
there were such astrologers as Diogo Mendes Vizinho and

1 Georgios Trapczuntios. -It was reprinted at Budapest in 1804.

3 Bar Noten Ta'am le-Chacham (a Talmudic phrase). See Bibl. Math, XIII
(2), QQ. 4 ?'>/. Math., II (3), 62.

5 A. Marre, Boncompagni's Bullettino, XIII, 560; Bensaude, Astron. Pnrtn%., t
104; R. Guimaraes, Les Mathtmatiques en Portugal, 2d ed., p. n (Coimbra,
1009) (hereafter referred to as Guimaraes, Math. Portug.).

264 CHRISTIAN EUROPE FROM 1400 TO 1500

Thomaz Torres, but the former knew little beyond the elements
of cartography, and the latter seems to have been interested
only in the drawing of horoscopes. It was not until the i6th
century that mathematics commanded any noteworthy atten-
tion in the Portuguese universities.

Some idea of the new Italian spirit in mathematics and let-
ters succeeded in reaching Spain at about the same time. Men
like Barbosa lectured on Greek at Salamanca, and Lebrixa
(Nebrissensis) returned from Italy in 1474 and lectured at
Seville, Salamanca, and Alcala, but the interests of the learned
were in medieval theology and little attention was given to the
advancement of the sciences.

The nature of the Spanish works of the isth century may be
inferred from the Visio delectable de la philosophia & artes
liberates, a kind of general encyclopedia, written by Alonso
Delatore and published at Tolosa in 1489 and again at Seville
in 1538. The fourth chapter of this work is entitled Dela
arismethica y dc sus inuetores, but it has no merit and is little
better than the treatment of arithmetic given by Capella and
Isidorus several centuries earlier. Spain was at this time
occupied in suppressing the bandit nobles, in recodifying her
laws, and in banishing the last of the Moors, and it was not
until 1492 that the political unification necessary to future
peace was effected. Nevertheless the foundations seemed in
process of being laid for a new type of commercial mathematics
and for the rapid development of the science of navigation.
Such voyages as those of Columbus and, in the following
century, of Ponce de Leon, led to an expansion of the colonial
empire and to a rapid increase in wealth. Unfortunately, how-
ever, the Spanish leaders were unable to grasp their opportu-
nity, and they squandered their newly acquired possessions
in ambitious schemes that led to profitless wars. The chance
which the Medici so happily improved was cast aside by the
rulers of Spain, and neither art nor science was fostered. Dur-
ing most of the century, therefore, the situation was such as to
dampen any scientific zeal, and not a single mathematical work
of any consequence was printed in Spain in the isth century.

DISCUSSION ^3

TOPICS FOR DISCUSSION

1. Influences at work in the nth century to improve the intel-
lectual status of Europe.

2. The life and works of Gerbert, who became Pope Sylvester II.

3. A study of the number game of Rithmomachia.

4. The 1 2th century as a period of translations.

5. Oriental civilization in Spain in the nth and i2th centuries.

6. Jewish activity in mathematics in the nth and i2th centuries.

7. Influences tending to foster mathematics in the i3th century.

8. The life and works of Leonardo Fibonacci.

9. The translations of Euclid and their influence on mathematics
in the Middle Ages.

10. The life and works of Johannes Sacrobosco and his influence
on mathematics in general and on algorism in particular.

11. Roger Bacon; his contemporaries and his projected reforms.

12. The life of Jordanus Nemorarius and his influence on medieval
mathematics in general and on algebra in particular.

13. Influences at work in the i4th century to improve the intel-
lectual status of Europe, particularly in the field of mathematics.

14. The mathematics of England in the i4th century. In what
ways did the science meet the needs of the time ?

15. A comparison of the mathematics of England, France, and
Italy in the i4th century.

1 6. The various methods of diffusing knowledge among scholars
during the Middle Ages.

17. The effect upon mathematics of the influx of Greek manuscripts
into Italy in the i5th century.

1 8. Influences leading to the Renaissance, and their bearing upon
the development of mathematics.

19. The general nature of the mathematics of Italy in the i5th
century, and its effect upon modern education.

20. Influence of the universities in the i5th century, particularly
on the development of mathematics.

21. The leading printed mathematical works of the i$th century.

22. Some of the probable reasons why mathematics made such a
slight advance in the i5th century.

23. Reasons for the prominence of commercial arithmetics in the
latter half of the 1 5th century.

CHAPTER VII

THE ORIENT FROM 1000 TO 1500
i. CHINA

The General Period. The most interesting period in the his-
tory of Chinese mathematics, and indeed one of the most in-
teresting periods in the history of the world, is that of the
five centuries from the year 1000 to the year 1500. It is a
period not unlike the contemporary epoch in Europe, a time of
awakening from sleep. Europe had seen the glories of the
classical period die out; she had passed through a season of
darkness ; she was awakened by the crusades ; she arose with
a great feeling of refreshment in the i3th century, and in the
iSth century she opened a new world of thought through the
invention of printing from movable types, and a new world
of commercial activity through the discovery of the Western
Hemisphere.

China passed through a similar experience, making great
advances in algebra in the i3th century and learning of a new
civilization through the work of the Jesuit missionaries three
hundred years later.

The Eleventh Century. The nth century saw little progress
made in mathematics in China. Indeed, the only name of any
importance to be found in her annals is that of Ch'on Huo
(1011-1075), president of the Bureau of Astronomy, a min-
ister of state, and the author of a work in which there is found,
perhaps for the first time in China, the summation of a series
of any difficulty. This summation appears in the solution of
a problem on the number of wine kegs in a pile whose form
is a truncated pyramid, the upper row containing 2* kegs, the
lower one i2 2 , and there being n rows. The author considers

.266

RELATION OF OCCIDENT TO ORIENT 267

also the question of finding the length of a circular arc in terms
of the radius, and speaks of the difficulty of the problem.

In 1083 the government showed its interest in mathematics by
printing Liu Hui's (c. 250) Sea Island Classic? and a year later
(1084) it printed the arithmetic of Ch'ang K'iu-kien 2 (c. 575).

The Twelfth Century. The i2th century was about as barren
as its predecessor. The Huang-ti K'iu-ch'ang A seems to have
been printed (c. 1115), and Ts'ai Yuan-ting 4 wrote on the
I-king, but these events simply emphasize the fact that the
century was barren of achievement. What is much more sig-
nificant is the fact that the East was again coming in contact
with the West; commerce was exchanging the problems of
business, and the astrologer was doing the same for the higher
and more mysterious strata of mathematical knowledge. We
know that in this century metals were transported from Arabia
to China, and in 1 1 78 a Chinese work 5 was written in which a
great export trade is described and a list of the merchandise is
given. In 1128 an immense army from China invaded Turke-
stan and found there many Chinese residents. 6 The stimulus
of world intercourse was becoming stronger, and the effect of
this stimulus was to appear in the century following.

Exchange of Thought in the Thirteenth Century. The in-
crease of opportunity for the exchange of thought in the i2th
century was carried over into the i3th century , quite as the
peripatetic scholars in Europe increased in number and in activ-
ity in the same period. Metal was imported from Persia
(Po-sz') and from the country of the Arabs (Ta-shi). 7 In 1219

ir The Hai-tau Suan-king (see page 142). This was, of course, from blocks,
not from movable type.

2 The Ch'ang K'iu-kien Suan-king (see page 150).

3 " Yellow Emperor's Nine Sections." One of the versions of the K'iu-
ch'ang Suan-shu (see page 31).

*Born 1135; died 1198; student and historian (see Giles, loc. cit., No. 1985).

6 The Ling-wai-tai-ta of Chou K'u-fei. Upon this work Chau Ju-kua's great
treatise was based. See the edition by Hirth and Rockhill, Petrograd, 1911.

6 E. Bretschneider, Mediaeval Researches, 2 vols., I, 232 (London, 1910);
hereafter referred to as Bretschneider, Mediaeval Res.

7 This is related in the Si Shi ki (travels) of Ch'ang-ti. Mention is made of
pin t'ie, probably ste*,l. See Bretschneider, loc. cit., I, 146.

268 CHINA

the conquering Chinghiz Khan 1 carried out his great expedi-
tion to Western Asia, established roads connecting Eastern Mon-
golia with Persia and Russia, sent armies even to Eastern
Europe, and opened the way to trade between the Orient and
all parts of the Occident. He captured Bokhara, Samarkand,
Herat, Merv, Nishapur, Kiev, and probably Moscow (Moscoss).
He invaded Poland, Galicia, Silesia, and Hungary, and his
astrologers must have mingled freely with their guild in the
capitals of Eastern Europe. In 1221 the Chinese traveler
K'iu Ch'ang ch'un reached Samarkand 2 and, what is especially
significant, records the fact that "Chinese workmen are living
everywhere" in the city, and that he saw " peacocks and great
elephants which had come from Yin-du" (India). He also
speaks of meeting and conversing with an astronomer of Samar-
kand, as we should naturally expect of a man of his learning.
In 1236 the Mongols invaded Bulgaria; in 1238 France re-
ceived a Mussulman ambassador asking aid against the Oriental
forces; in 1241 the Mongols invaded Galicia, and in 1259
they burned Cracow for the second time.

The fact is also significant that in 1266 the king of Ceylon
had Chinese soldiers in his service, while about this time the
Chinese traveler Chau Ju-kua speaks of the Hindus from his
personal knowledge as being "good astronomers and calcula-
tors of the calendar." 3

Europeans in the East. Moreover, Europeans were fre-
quently seen in the East, and the Grand Duke Yaroslav, in
1246, met one Fra Piano Carpini, a Franciscan, at the Mongol
court. Only a few years later Rubrouck, another of the friars
minor, visited the same court, and both he and Carpini have
left accounts of their travels. There were also Eastern trav-
elers who went to the West, such as the Taoist monk Ch'ang
ch'un, from 1220 to 1224; Ch'ang-ti, who was sent out in 1259
by Mangu Khan, and who visited Bagdad ; and Ye-lii Hi-liang,
who traveled in Central Asia from 1260 to 1262.

1 Jenghiz Khan, Genghes Khan, Jinghis Khan; originally Temuchin.
2 The Semiscant of medieval writers, known to the Chinese as Sie-mi-sz'-kan
and Sun-sz'-kan. 3 Hirth and Rockhill, loc. cit., p. in.

EUROPEAN INFLUENCE 269

The second half of the century saw other travelers from the
West in the courts of Mongolia and China. Haithon (Hethum),
king of Little Armenia, visited Mongolia in 1254, and the
account of his journey is well known. Marco Polo, who set
out from Venice in 1271, spent seventeen years in China and
traveled in a leisurely way through Arabia, Persia, and India.
Rashid ed-din, vizier of the Persian Empire in 1298, wrote a
history of the Mongols showing thorough familiarity with their
country. All these events, unimportant in detail, are very
significant taken as a whole, because they afford evidence of
the free interchange of ideas between the East and the West.
Thus they serve to clear up many questions as to whether the
algebra of China, for example, could have found its way to
Italy in the i2th century. We repeat that it would be a cause
for wonder if it had failed to do so. Moreover, the i3th
century was a period of wealth, of luxury, and also of oppor-
tunity. Chinghiz Khan, in a letter to Ch'ang ch'un the Taoist
traveler, asserted that "Heaven had abandoned China owing
to its haughtiness and extravagant luxury." 1 This may have
been true ; a conqueror would be apt to say so ; but it is
certain that it was a century in which mathematics ought to
have flourished in both continents. We shall see that this
was the case.

Mathematical Activity in the Thirteenth Century. The I3th
century, in China as in Europe, was a period of awakening.
Indeed, it may be said that it was the period of the highest
development of native mathematics in the East. Whether as
a result of an interchange of thought with the West, or of the
leisure which wealth brought to the country before the invasion
of Chinghiz Khan, or of the growth of idealism which war is
said by its advocates to foster, or of various other causes,
China made a noteworthy advance in algebra at this time.

Perhaps the foremost scholar in this movement was Ch'in
Kiu-shao, a man whose intimate history is quite unknown, but
who was a soldier in his early days, was in government service

1 Bretschneider, Mediaeval Res., I, 37

270 CHINA

in 1244, was governor of two provinces, and wrote the Nine
Sections of Mathematics in I247. 1 The work relates chiefly to
numerical higher equations, in which the author anticipates to
some extent Horner's Method (1819); but it also considers
indeterminate equations and the application of algebra to trigo-
nometry. The "nine sections" of his work are not the same
as those of the one mentioned on page 31. For the value of
TT the author gives 3, $-*-, and VTo. In his work, too, the sym-
bol O is used for zero and the place value is used in the
writing of all numbers. The author shows little interest in
applying his knowledge of algebra to the solution of practical
problems, preferring to look upon it as a pure science.

Li Yeh's Work. The next noteworthy step in this direction
was taken by Li Yeh (1178-1265). He wrote the Sea Mirror
of the Circle Measurement 2 in 1249, the I-ku Yen-tuan in
1259, and various other works. In his early life he was en-
gaged in public service, and in 1232 was governor of Chim
Chou. He was later held in high esteem by Kublai Khan,
whose reign began in 1260.* Li Yeh directed his son to burn
all his works except the Sea Mirror. The I-ku Ycn-tuan was
also preserved, however, and each has since been looked upon
as among the great works of China. While Ch'in Kiu-shao
had given his attention chiefly to the solution of abstract equa-
tions, Li Yeh devoted himself to the forming of equations rep-
resenting various complicated problems, the solution being
neglected.

1 Chang Ch'i-mei (1616) says that the author called the work Su-shu or
Su-hsiao, but in the Chinese bibliographical works the title is Su-shu Kiu-
ch'ang. Biernatzki (pp. 13, 28) transliterates the title as Su schu kiu tschang
and the author as Tsin Kiu tschaou. Vanhe"e gives the date as 1257 and the
author as Ts'in K'ieou-Chao. For details as to his methods see Mikami, loc.
cit., p. 63.

-Ts'o-yuan Hai-king. Vanhee (Toung-pao, XIV, 537) gives the date as
1248 and the title as Ts'e yuen hai king. Biernatzki gives the name of the
author as Le Yay and Le yay Jin king, and the title as Tsih yuen ha king.
Li Yeh also used the nom de plume Ching-chai.

3 The Yuen Dynasty, founded by the Mongols, began in 1280, however. In
1260 the Mongols issued paper money, an event bearing somewhat on the
history of arithmetic problems.

THE THIRTEENTH CENTURY 271

About this time another scholar, Liu Ju-hsieh, wrote a
treatise on algebra, but the work is not extant. 1

Chinese Astronomers. That astronomy was the mathematical
subject of chief interest at this time, as indeed at all times in
early Chinese history, is shown by the considerable list of
names of scholars devoted to the science. Among the leaders
at this particular period was Ye-lii Ch'u-ts'ai, 2 who lived c.
1230. He established a great school at Peking (then Yen-
king) and accompanied Chinghiz Khan to Persia, occupying
himself with the calculation of eclipses and coming into con-
tact with the Persian astronomers.

Yang HuPs Work. In 1261 Yang Hui wrote The Analysis
of the Arithmetic Rules in Nine Sections," a work in which
he explained some parts of the original Nine Sections (see
page 31). In this work he gives a graphic representation of
the summation of an arithmetic series. In another work 4 he
gives rules for summing the series

I + (I + 2) + (I +2 + 3)+-.. +(1+2 + 3 +... + //)

and r+2 2 +3 2 + + ;r,

but offers no explanations. He wrote several other works, 5 and
among his problems are such as that of the hare and hound
and several involving simple and compound proportion.

Yang Hui's teacher, Liu I, a native of Chung-shan, wrote
(c. 1250) a work which is known only by name/ 5 but it doubt-
less related to numerical higher equations.

3 He is mentioned in a work of 1303 as a prominent contemporary. His
work is said to have been written before 1300. Since a commentary upon it
was written by Yuen Hao-wen, who was a friend of Li Yeh (1178-1265), the
date may be taken as c. 1260.

2 Remusat, in his Nouveaux Melanges Asiatiques, II, 64, does not speak of
his mathematical attainments, but in the Yuen-Shih (Historical Records of the
Yuen Dynasty) these are discussed.

*Hsiang-kieh K'iu-ch'ang Suan-fa. Yang Hui is also known in European
works as Yang Hwuy, Yan Hui, Yang Houei, Yang Hwang, and Kien-kouang.

4 Suan-fa T'ung-pien Pen-mo.

5 His six works on arithmetic, thought to be lost, were discovered in
Shanghai in 1842. *l-ku Kon-yu^n.

272 CHINA

Kou Shou-king. In 1267 the Mongols are known to have
employed various Arab artillery officers, 1 so that contact with
Arabia existed through the army. But this contact is the
more apparent in the scientific achievements of Kou Shou-
king 2 (1231-1316), a man well versed in the astronomy of the
Arabs. He was a native of Hsing-t'ai 3 and was remarkable
for his attainments even in early childhood, seeming to have in-
herited the scholarly qualities of his grandfather, Kou Yung,
a mathematician of repute. In early manhood Kou Shou-king
developed into one of the greatest engineers of China. He was
appointed by Kublai Khan to reform the calendar, 4 and for
this purpose he replaced the armillary sphere which had been
made about 1050, and which was calculated for Peenking, the
former capital, differing about 4 in latitude from Peking, by
the earliest of the great bronze instruments now on the wall of
the latter city. The instruments constructed by him included
several that were adapted to observations made in the day-
time as well as at night, 5 and show that he had consider-
able knowledge of spherical trigonometry. Only two of his
instruments seem to be extant, and these were found and
described by Matteo Ricci when he visited Peking early
in the i7th century. Ricci also speaks of having seen similar
instruments at Nanking.

With Kou Shou-king may be said to have begun the study
of spherical trigonometry in China, a subject already far
advanced in the Arab schools.

Further contact with the West was made at this time through
the sojourn of Friar Odoric in Canton from 1286 to 1331.

1 Two are called by the names I-se-ma-yin (probably Ismael) and La-pu-tan
in the Chinese records.

2 Also transliterated as Ko-cheou-king, Kouo Cheou-kin, and Kou Shou-ching.
Since his cognomen is given as J6-sze, it is possible that he is identical with
the Liu Ju-hsieh already mentioned.

3 Hing-tae, with other transliterations, a district in the prefecture of Shun-tlh.

4 He devised this calendar, the Shou-sh'i-li, in 1280 and it was adopted in
1281.

5 For a list and for the general subject of Kou Shou-king see A. Wylie,
"The Mongol Astronomical Instruments in Peking," Travaux de la 3* session
du Congres internal. *des Orientalistes, Vol. II. 6 See page 303.

THE THIRTEENTH CENTURY 273

Chu Shi-kie. The i3th century in China closed with the re-
markable work of Chu Shi'-kie, 1 a native of Yen-shan. As to
his private life, we know only that for more than twenty years
he was a wandering teacher. He wrote two works, the Introduc-
tion to Mathematical Studies 2 in 1299 and The Precious Mirror
of the Four Elements'' in 1303. With him the old abacus alge-
bra, in which the coefficients were represented by sticks placed
on a checkered board, 4 reached its highest mark. In the first
of his works there appears the algebraic rule of signs and an
introduction to algebraic processes in general. In the second
treatise, however, he considers a variety of new questions in
higher algebra. He begins with what is called at present Pas-
cal's Triangle, giving the values of the binomial coefficients and
referring to the scheme as an old one. He considers higher
equations with more than one unknown quantity, his treatment
showing some knowledge of elimination by a determinant no-
tation. He shows much ingenuity in his solution of numerical
higher equations by the method already used by Ch'in Kiu-shao
and which resembles Horner's Method.

2. JAPAN

Close of the Dark Ages. For Japan, as well as for Europe,
there was a period which may properly be spoken of as the
Dark Ages. For a thousand years after Buddhism was in-
troduced into Japan there were few other events of intellectual
significance to record. Japan was awaiting the world's rebirth,
and this period of rebirth came to the East at about the same
time as to the West.

Only two men stand out as worthy of mention in the history
of Japanese mathematics between the year 1000 and the year
1500. The first of these is Fujiwara Michinori, a daimyo or

1 Tchou Che-kie, Tchou Che-Ki6, Tschu Schi kih, Choo Che-kie, Chu-Shih-
Chieh, Tchou Che-kie, and various other transliterations. He is also called Chu
Sung ting.

2 Suan-hio-ki-mong, or Suan-hsiao Chi-meng.

*Szu-yuen Yu-kien, or Szu-yuen Yu-chien. The introduction was written
by T:U Yi Chi Hsien Fa. 4 See Volumd II, Chapter VI.

274 INDIA

feudal lord in the province Hyuga, who wrote a work on permu-
tations 1 between the years 1156 and 1159. The work is now
lost, but it was thought important enougn to be considered by
the leaders of mathematics in Japan in the ryth century.

The second name worthy of note is that of the Buddhist
priest Gensho, who lived in the first part of the i3th century.
No trace remains of any of his writings, but tradition says
that he was possessed of remarkable arithmetical ability.

3. INDIA

Sridhara. The first of the Hindu writers of this period seems
to have been Sridhara, commonly known as Sridharacarya,
Sridhara the Learned, who was probably born in 991. 2 His
work is known as the Ganita-Sara (Compendium of calcula-
tion} but is more commonly designated by the subtitle Trisatika,
a name referring to its three hundred couplets.' The subjects
considered are numeration, measures, rules, and problems, and
the order bears very close resemblance to the one followed
about a century later by Bhaskara in his Lildvati. The latter
writer was acquainted with Srldhara's work, as he himself
testifies in his Bija-Ganita.

Under the general topics above mentioned are included series
of natural numbers, multiplication, division, zero, squares,
cubes, roots, fractions, Rule of Three, interest, alloys, partner-
ship, mensuration, and shadow reckoning. The statement re-
lating to zero is noteworthy as the clearest one to be found
among the Hindus : "If zero is added to a number, the sum is
that number itself; if zero is subtracted, the number remains
unchanged ; if zero is multiplied, the result is zero ; and if _a
number is multiplied by zero, the product is zero only." The
question of division by zero is not considered.

1 The theory known as Keishizan.

2 The date is uncertain, being placed by one writer three centuries earlier.
The question is discussed by N. Ramanujacharia and G. R. Kaye in the Bibl.
Math.,XUl (3), 203.

3 The Sanskrit text was published in 1899. The English text is given in the
article mentioned above. The name may also have come from the fact that
it originally had 103 couplets.

SRIDHARA AND BHASKARA 275

For dividing by a fraction Sridhara gives the rule of multi-
plication by the inverted divisor, a rule already known to
Mahavlra (c. 850). Like the latter, too, he uses \/io for TT .

Bhaskara ( 1 1 1 4-0. 1185). There is only one other writer who
stands out prominently in the history of Hindu mathematics
from 1000 to 1500, and that is Bhaskara, commonly known

*JjT

^TO^T^irH^{^^^*^f^^^^f^^^
jf.V^^RT^fif3^qr?^^^|^^4jai59^^

^w^wrt Q^^|i^LJc?T^$^f^3

SRIDHARA'S TRISATIKA, r. 1025

Two pages from the copy used by Colebrooke in his works on Hindu mathe-
matics. From Kaye's Indian Mathematics

as Bhaskara the Learned (Bhaskaracarya), 1 a native of Bid-
dur 2 in the Deccan, but working at Ujjain. An ancient temple
inscription refers to him in the following terms : " Triumphant
is the illustrious Bhaskaracarya whose feet are revered by the
wise, eminently learned, . . ., a poet, . . . endowed with good
fame and religious merit. . . ." 3

1 The name is variously transliterated. Thus, we have Bhascara Acharya,
Bhaskaracharya, and other forms. See J. Garrett, loc. cit., p. 92; Taylor,
Lilawati, i; T. W. Beale, The Oriental Biographical Dictionary, Calcutta, 1881;
Colebrooke, Bhaskara. 2 Probably the modern Bidar.

3 Kaye, Indian Math., 37.

2 76 INDIA

Bhaskara's Lilavati. Bhaskara wrote chiefly on astronomy,
arithmetic, mensuration, and algebra. His most celebrated work
is the Lildvati, a treatise based upon Sridhara's Trisatika and
relating to arithmetic and mensuration. 1 This work was trans-
lated into Persian by Fyzr in 1587 by direction of the em-
peror Akbar, a great patron of letters. Fyzi states, though it
does not appear upon what authority, that Lilavati was the
name of Bhaskara's daughter and that the astrologers pre-
dicted that she should never wed. Bhaskara, however, divined

PALM-LEAF MANUSCRIPT OF THE LILAVATI

Showing the form in which the Hindu manuscripts appeared before paper
became a common medium. This manuscript was copied c. 1400. From the

author's collection

a lucky moment for her marriage and left an hour cup floating
on the vessel of water. This cup had a small hole in the bot-
tom and was so arranged that the water would trickle in and
sink it at the end of the hour. Lilavati, however, with a natural
curiosity, looked to see the water rising in the cup, when a
pearl dropping from her garments chanced to stop the influx.

1F The first translation into English is that of Taylor (1816), already men-
tioned.

2 Also spelled Faizi and Feizi. He was a brother of Akbar's secretary, Abu
Fazil. The work was printed at Calcutta in 1827.

BHASKARA

277

So the hour passed without the sinking of the cup, and Lila-
vati was thus fated never to marry. To console her, Bhaskara
wrote a book in her honor, saying : "I will write a book of your
name which shall remain to the latest times ; for a good name
is a second life and the groundwork of eternal existence."

The work begins, as is the custom in the East, with an ad-
dress to the Deity : "Salutation to the elephant-headed Being

FROM BHASKARA'S LILAVATI

From a manuscript of c. 1600. The original work was written c. 1150. The illus-
tration shows the form of Hindu manuscripts just following the use of palm-leaf
sheets. This page has the following statement: "Assuming two right triangles
[as shown], multiply the upright and side of one by the hypotenuse of the other:
the greatest of the products is taken for the base ; the least for the summit ; and the
other two for the flanks. See" [the trapezoid]. Colebrooke's translation, page 82

who infuses joy into the minds of his worshipers, who delivers
from every difficulty those who call upon him, and whose feet
are reverenced by the gods." The book includes notation, the
operations with integers and fractions, the Rule of Three, the
most common commercial rules, interest, series, alligation, per-
mutations, mensuration, and a little algebra. The rules relating
to zero are also given, to the effect that + = 0, powers of o
are o, and a o = o. The statement that a -f- o == o (corrected
by his commentators) was evidently not clear to him, for his
statement is "A definite quantity divided by cipher is the

2 7 8

INDIA

PAGE FROM THE FIRST PRINTED

SANSKRIT EDITION OF BHASKARA*S

LILAVATI

Printed at Calcutta, 1832. This is a con-
tinuation of the portion shown in manu-
script on page 277. The statement, as
translated by Colebrooke, is as follows:
"Length of the base, 300. Summit, 125.
Flanks, 260 and 195. Perpendiculars, 189
and 224." From these the other parts
are found

sub-multiple of nought,"
while his illustrations are
that 10-5-0=-^- and
3 ^. o=g, 1 the latter be-
ing accompanied by the
statement that "this frac-
tion, of which the denomi-
nator is cipher, is termed
an infinite quantity."

The Bija Ganita. Bhas-
kara also wrote the Bija
Ganita? a work on alge-
bra. 3 In this he discusses
directed numbers, the neg-
atives being designated in
Sanskrit as "debt" or
"loss" 4 and being indi-
cated by a dot over each
number, as in the case of
3 for 3, and the usual
rules being stated cor-
rectly. The imaginary is
dismissed with the state-
ment, "There is no square
root of a negative quan-
tity : for it is not a square."
Where several unknown
quantities are used, they
are mentioned as colors:
"'so much as' and the

1 Colebrooke, he. tit., pp. 19, 20, 137.

2 Variously transliterated as Vljaganita, Vija-Ganita, and the like. The term
literally means "seed counting" or "seed arithmetic." The work was translated
by Colebrooke, Bhdskara, p. 129, with the subtitle Avyacta-Gahita.

3 Translated into Persian by Ata Allah Rusheedee in 1634. The English
version of this translation was published by Edward Strachey, Bija Ganita^
London, n.d. (c. 1812), apparently with the aid of a translation made, with the
help of a pundit, by S. Davis, c. 1790. *Rina or cshs.ya.

BHASKARA

279

colors 'black, blue, yellow, and red/ and others besides these,
have been selected by venerable teachers for names of values

FROM A MANUSCRIPT OF FYZl'S TRANSLATION OF THE LILAVATI

Fyzi, counselor of Akbar, made the translation into Persian in 1587. This manu-
script is dated 1143 A.H., or 1731 A.D. The Persian manuscripts were written in
our ordinary book form. The above is page 32 of Taylor's translation of 1816.
The problem is: "A number is multiplied by 5; from the product is sub-
tracted one- third of itself, and the remainder is divided by 10 ; to the quotient
is added of \ of of the assumed number, and the result is 68. What is

the number ? w

of unknown quantities." 1 Surds are treated extensively, as in
many medieval works on algebra, the difficulty of handling

1 The initial Sanskrit syllables of the names of the colors are used.

280 INDIA

irrationals of all kinds being particularly great in the era of
poor symbolism. As with Aryabhata and Brahmagupta, the
" pulveriser" is given extensive treatment. Simple and quad-
ratic equations receive more attention and are more clearly
discussed than is the case with other Hindu writers. Besides
numerous problems relating to geometric figures there are the
usual poetic types, of which the following may serve as an
illustration :

The son of Prlt'ha, 1 exasperated in combat, shot a quiver of ar-
rows to slay Carria. With half his arrows he parried those of
his antagonist ; with four times the square root of the quiverful
he killed his horse ; with six arrows he slew Salya 2 ; with three he
demolished the umbrella, standard, and bow; and with one he
cut off the head of the foe. How many were the arrows which
Arjuna let fly ?

Sridhara's Rule for the Quadratic. The rule used for solving
the quadratic is given as Srldhara's. 3 The method of writ-
ing an equation has some evident advantages, the equation
i&x 2 = i6:r -f- goc 4- 18 being written

yav 18 yao ruo

ya v 16 ya 9 ru 18

which is then transformed into 2 x 2 gx = 18, thus :

yav 2 yap ruo
yav o yao rui8 4

Bhaskara's Siddhanta Siromani. A third work of importance
written by Bhaskara is the Siddhanta Siromani (Head jewel
of accuracy}* in which, in the book Goladhia (Theory of the

1 The son's name was Arjuna, mentioned later in the problem.

2 The charioteer of Carria. *See page 274.

4 Fa v is for ydvat-tdvat, "as many of" (the unknown), and is used for the
highest power. Ya is a color, green, and is used here for the first power. Ru
is for rupa, the known number. See Volume II and Colebrooke, Bhaskara, p. 130.

5 Various notes on this work will be found in H. T. Colebrooke, Miscel-
laneous Essays, 2 vols., 2d ed., Vol. II (Madras, 1872). It should be stated
that the Siddhanta fhromani is thought by various scholars to include the
other works mentioned, but this is merely a question of division of material.

BHASKARA

281

Sphere), he treats of astronomy and asserts, as various ancient
Greek philosophers have done, the sphericity of the earth.
The ancient inscription referred to on page 275 relates that
Bhaskara's grandson, Changadwa, was chief astrologer to King
Sirhghana, and that in his time a college was founded to ex-
pound the doctrines of Bhaskara. 1

From a manuscript of Bhaskara's work on astronomy, being the fourth and

last chapter of the Siddhdnta Siromani. This work was written c. 1150.

The reproduction is greatly reduced

In the forming of Pythagorean triangles Bhaskara follows
Brahmagupta in stating the relations

i (m \ i Im

*, -( ;/), -I

2\u I 2\n

and adds the two further relations

and

Other Parts of South Asia. In Eastern Sumatra (the Chinese
San-fo-ts'i) it is related by Chinese travelers of the 13* cen-
tury that the people were able mathematicians and could cal-
culate future eclipses of the sun and moon. 2 No doubt the

!G. R. Kaye, Indian Mathematics, p. 37.
2 Hirth and Rockhill, lot. cit., p. 64.

282

INDIA

same could have been said at that time of other parts of Asia
whose records have not come down to us, such calculations

FROM THE FIRST PRINTED EDITION OF BHASKARA'S GOLADHIA

This shows in print the same page shown in manuscript on page 281. The work
was printed at Calcutta in 1842

having been a part of the general stock in trade of the astrolo-
gers in various countries for many centuries preceding this
particular period.

DECAY OF BAGDAD 283

4. PERSIA AND ARABIA

Decay of Bagdad. For two centuries after the golden age of
the first three caliphs Bagdad continued to be a center of scien-
tific activity, in spite of the fact that it began to lose political
prestige after the death of al-Mamun (833) . By the year 1000,
however, the spiritual supremacy of the city had passed and
the seats of learning of the Western Arabs had begun to take
the place of the capital of the caliphs in Mesopotamia. The
Seljuk Turks, an intolerant Tartar tribe, overran much of the
territory formerly so well governed by the caliphs, captured
the holy cities of Palestine, and by their ruthless behavior gave
excuse for the crusades. In 1258 the Mongols took Bagdad,
and thenceforth it was little more than a name. Brute force
had put an end to the idealism that had been so noticeable in
the eastern Mohammedan empire.

Al-Karkhi. Among the last of the real contributors to mathe-
matics in the city of the caliphs was al-Karkhi, 1 who died
c. 1029. His first work of note was an arithmetic, the Kdji fit
Hisdb, 2 probably written between 1010 and 1016, and drawn
largely if not exclusively from Hindu sources : ' It not only
contains the elements of arithmetic as set forth by many writers
of the time, but gives the rule of quarter squares,

a rule probably due to the Hindus. It also gives such methods
of multiplication as are expressed by the formulas

(10 a -f a)(iob -h b) = [(io a -f a)b + ab] 10 -f at
and (10 a -f 6) (10 a + c) = (10 a + b -f c)a 10 -f be.

iMohammed Abu Bekr ibn al-Hasan (or al-Iiosein), al-Karkhi.

*Book of Satisfactions. A. Hochheim, Kail fll Hisdb des Abu Bekr Mu*
hammed Ben Alhusein Alkarkhi, 3 parts, Halle a. S., 1878-1880; hereafter re-
ferred to as Hochheim, Kdji fit Hisdb.

8 H. Weissenborn, Gerbert, p. 196 seq. (Berlin, 1888). But see Cantor's earlier
opinion that it came irom Greek sources; Geschichte, I (i), 655.

284 PERSIA AND ARABIA

In the approximations for roots al-Karkhi gives, among
others, _

for m = a 2 + r, Vw = a + r/(2 a + i) ;
for r = a, ^/ m a

He also considers the mensuration of plane figures, particu-
larly as it involves^surdjiumbers, and includes the Heron
formula Vj (s a) (s b) (s c).

The work closes with a treatment of algebra, including quad-
ratic equations and the usual Arabic explanation of the terms
al-jabr and al-muqabala, discussed in Volume II.

Al-Karkhi's Fakhri. Al-Karkhi is best known, however, for
his algebra, the Fakhri,? which includes the usual operations on
algebraic quantities, roots, equations of the first and second
degrees, indeterminate analysis, and the solution of problems.
The quadratic equations include such forms as x* -f 5 x~ =126,
and the solution of quadratics in general depends upon rules
such as that represented by the equation

a;r 2 -f- hx = c

and the formula

The rules are explained geometrically, as in the works of the
earlier Arab writers. Various problems given by him are ap-
parently suggested by al-Khowarizmi and Diophantus, and
these include such cases as the finding of integral solutions for

x = 1 Aj(-) + &c --- 1 ' <*

and

and the finding of fractional solutions for

1-2 _ ^ ,3 _ r.2
,,1 y <w

and .r 3 + f = A

The work ranks as the most scholarly algebra of the Arabs.

*F. Woepcke, Extrait du Fakhri . . . par . . . Alkarchi (Paris, 1853). For
an explanation of the term Fakhri see Volume II, Chapter VI.

AL-KARCHI 285

Minor Writers of the Eleventh Century. Of the minor writers
of the nth century the following deserve brief mention:

Mohammed ibn al-Leit 1 lived about 1000, was interested in
the trisection problem, and wrote on the construction of regu-
lar polygons of seven and nine sides.

tlamid ibn al-Khidr 2 wrote on the astrolabe and asserted
that the equation x s + y 3 = z* cannot be solved.

Mansur ibn 'Ali 3 wrote on astronomical instruments, trigo-
nometry, spherical sines, and Ptolemy's Almagest.

Al-Nasavi wrote on Hindu arithmetic and on the works
of Archimedes. 4

One of the most brilliant writers on the contemporary his-
tory of mathematics at the opening of the nth century was
Alberuni/' He was one of the munajjimin, or astrologer-
astronomers of the Arabs. He visited India and made a careful
study of that country and of its work in mathematics and the
other sciences. He summarized the debased state of knowledge
of his day in the words, "What we have of sciences is nothing
but the scanty remains of bygone better times." In later life
he wrote his work on India, and to this we are indebted for the
best summary of Hindu mathematics that the Middle Ages
produced.

Avicenna. Among the contemporaries of Alberuni there was
the famous physician and philosopher known in Christian
Europe as Avicen'na (980-1037). He was born in Safar, near

1 Mohammed ibn al-Leit, Abu'l Jud.

2 Hamid ibn al-Khidr, Abu Mahmud, al-Khojendi, died c. 1000.

3 Mansur ibn 'Ali ibn 'Iraq, Abu Nasr, c. 1000.

4 'Ali ibn Ahmed, Abu'l- Hasan, al-Nasavi, c. 1025. Woepcke in the Journal
Asiatique, 1863, p. 406. He was born at Nasa, in Khorasan. On his arithmetic
see Suter in Bibl. Math , VII (3), 113.

5 Mohammed ibn Ahmed, Abu'l Rih an (or Raman), al-Beruni. Born probably
in Khwarezm, 073 ; died 1048. He may have been born at By run in the valley
of the Indus, his name also appearing as al-Biruni. On his life and works, see
his Athar-el-Bakiya, Chronology of Ancient Nations (London, 1879), and his
India ; Boncompasmi's Bullettino, II, 153; S. Giinther, Zeitschrift (HI. Abt.),
XXI, 57; E. C. Sachau, Zeitschrift d. deutsch. Morgenl Gesellschaft, XXIX.

Al-Hosein ibn 'Abdallah ibn al-IJosein (or tjasan) ibn 'AH, Abu 'AM, al-
Sheich al-Ra'is, ibn Sina. See K. Lokotsch, Avicenna als Mathematiker
(Erfurt, 1912).

286 PERSIA AND ARABIA

Kharmitan, not far from Bokhara. He wrote on Aristotle,
Euclid, astronomy, music, medicine, and arithmetic, his treat-
ment of numbers being based upon Greek models.

Ibn al-^alah, 1 who died in 1153/54, was one of the later
generation of the Persian scholars who made Bagdad so famous.
Like so many mathematicians of the East he was also learned
in philosophy and medicine, the latter partly on account of
the supposed connection between the healing an and astrology.
He was born in Bagdad and finally went to Damascus and died
there. He wrote on geometry, and manuscripts of his works,
apparently fragmentary, are still extant.

Omar Khayyam. The i2th century saw less attention to
mathematics in the ancient Arab seats of learning, and more
attention to the science in Persia. Of those whose names added
luster to Persian mathematics and letters the most prominent
was the poet who is generally known to English writers as
Omar Khayyam 2 (c. noo). While he is known to the Western
world chiefly as the author of the Rubaiyat* he wrote on Euclid
and on astronomy, 4 and contributed a noteworthy treatise on
algebra. 5

Minor Writers. A little later than Omar Khayyam another
Persian, a native of Khorasan, made for himself a great name.
This writer was al-Razi, known as one of the leading philoso-
phers, physicians, and mathematicians of the Persians. His
contributions to mathematics were chiefly in the domain of
geometry.

iAlimed ibn Mohammed ibn al-Sur3l Nejm ed-din, Abu'l-Futuh.

2 'Omar ibn Ibrahim al-Khayyami, Giyat ed-din, Abu'1-Fath. He was born
at Nishapur (Nishapur, Nishabur) c. 1044 and died there in 1123/24.

8 Largely through the remarkable but not very exact translation of Edward
Fitzgerald, London, 1859.

4 "Ah but my computations people say
Reduced the year to better reckoning."

*L'Algebre d'Omar Alkhayyami, Arabic and French texts, by F. Woepcke,
Paris, 1851. On his life, see various editions of the Rubaiyat, and also J. K. M.
Shirazi, Life of Omar al-Khayydm, Edinburgh, 1905.

6 Mohammed ibn 'Omar ibn al-IJosein, Abu 'Abdallah, Fahr ed-din al-Razi,
ibn al-Khatib, 1149/50-1210.

DECAY OF LEARNING 287

Of the Arab scholars of the i2th century one of the best
known was Kemal ed-din ibn Yunis, or ibn Man'a, 1 who was
born at Mosul on the Tigris river. His works on the theory of
numbers and conic sections were highly esteemed by his Arab
contemporaries.

Contemporary with the last-named writer was Ta'asif," a
native of upper Egypt, a jurist, an engineer, and a mathemati-
cian. He showed his interest in the foundations of mathematics

by writing upon Euclid's postulates.

The Mongol Scourge. If the Seljuk Turks had been intol-
erant in the nth and i2th centuries, rendering difficult the
leading of an intellectual life, the great Mongol scourge, led
by Chlnghiz Khan between 1206 and 1227, rendered such a life
well-nigh impossible. His conquests and his son's included a
considerable part of the civilized world from Northern China
through Turkestan and Persia, and down to the banks of the
Indus. The son, Oktai (died 1241 ), quite as brutal as his father,
ravaged nearly half of Europe, and the result of the total con-
quest was the impoverishment of all the intellectual centers
that were once the glory of central and western Asia. To be
sure, two of the successors of these tyrants, Kublai Khan
(1216-1294) and Timur, or Tamerlane (1336-1405), contrib-
uted to the better things of life, but in general the record of the
Mongol invasions for two centuries is one of the blackest in
all history.

Decay of Learning. In the i3th century only one Persian
writer deserves special mention, and even he spent his closing
years in Bagdad. This writer was Nasir ed-din, 3 a native of
TUS, in Khorasan. He was an all-round scholar, writing upon
trigonometry, astronomy, computation, geometry, and the
construction and use of the astrolabe.

iMusa ibn Yunis ibn Mohammed ibn Man'a, Abu'1-Fath, Kemal ed-din.
Born at Mosul, 1156; died at Mosul, 1242.

2 Qaisar ibn Abi'l-Qasim ibn 'Abdelgani ibn Musafir, 'Alam ed-din, known
under the name of Ta'asif. Born 1170 (possibly 1169) ; died 1251.

3 Mohammed ibn Mohammed ibn al-IJasan, Abu Ja'far, Nasir ed-din al-Jusi,
1201-1274.

288 PERSIA AND ARABIA

Of the Arab writers, Ibn al-Yasimin, 1 who lived in Morocco,
is known chiefly for the influence of a poem which he wrote
on algebra, the Arjuza, Several manuscripts still exist, and
it seems to have had some such influence in popularizing alge-
bra as the Carmen de Algorismo (p. 226) had with respect
to algorism. Ibn al-Lubudi, 2 a native of tialeb (Aleppo), was
known in his century for works on arithmetic, algebra, and
Euclid. The Arab interest in learning was rapidly waning,
however, and only one other name deserves mention in the
record of the i3th century, that of al-Jusi, 3 another promi-
nent native of Tus. He wrote on geometry and algebra and
invented one form of astrolabe known as "Tusi's staff."
Islam had lost its hold upon mathematics ; the mathematical
world was becoming a hyperbola having one focus in China
and the other in Christian Europe, with nothing between
the branches.

The most notable of the Christian writers in the Near East
at this time was Bar Hebrscus, 4 whose father, a Jew named
Aaron, had entered the Christian church. When the son was
twenty years old (1246) he was made Jacobean bishop of
Gubos, near Malatia, and later he occupied other positions of
ecclesiastical importance. He wrote on astronomy and lectured
on Euclid and Ptolemy. 5

In the 1 4th century only three Mohammedans of any con-
siderable prominence appear among the world's mathemati-
cians, and no one of these was a genius. Two lived at least
part of the time in Egypt, Ibn al-Ha'im, a writer on

1 *Abdallah ibn Mohammed ibn Ilajjaj, Abu Mohammed. He died c. 1203-
1205.

2 Yahya ibn Mohammed ibn 'Abdan ibn 'Abdelvahid, Abu Zakariya Nejm
ed-din, 1210/11-1267/68.

3 Al-Mozaffar ibn Mohammed ibn al-Mozaffar Sharaf ed-din al-Tusi. He
died c. 1213.

4 That is, Son of the Jew. His Arabic name was Juhanna Abu'I-Faraj Bar-
Hebraius. Born at Malatia, in Eastern Asia Minor, 1226; died at Mosul, July,
1286.

9 F. Nau, in No. 121 of the Biblhtheque de VEcole des Hautes Etudes.
Paris, 1809.

c Ahmed ibn Mohammed ibn 'Im&d, Abu'l-' Abbas Shihab ed-din. Born at
Cairo, 1352 or 1355; died at Jerusalem, 1412.

ARAB ACHIEVEMENTS 289

arithmetic, and Ibn al-Mejdi, 1 who wrote on astronomy, trigo-
nometry, arithmetic, the calendar, and mathematical tables.
A third Arab writer of the period, commonly known as Ibn
al-Shatir, 2 left works on trigonometry, the astrolabe, and
astronomy, and prepared a few mathematical tables.

Ulugh Beg, the Royal Astronomer. Of the representatives
of the Arab-Persian interest in mathematics who lived in the
1 5th century the only one who seems to have possessed any
genius was Ulugh Beg* (1393-1449). and even this genius was
rather perseverance than any unusual endowment of intellect.
He was a Persian prince, born at Sultanieh, and his interest in
astronomy and astronomical tables was shown in the observa-
tory which he founded at Samarkand. The tables which were
worked out under his direction were highly esteemed in Europe
as well as in the East. 1 His assistant, al-Kashi, 5 wrote a short
treatise in Persian on arithmetic and geometry."

Summary of Arab Achievements. With these names the
achievements of the Mohammedan writers practically close.
As we sum up these achievements we are struck by the interest
of the Arabs in science but by their lack of originality. They
received their astronomy first from the Hindus and then from
the Greeks, their geometry solely and their algebra chiefly
from the Greeks, and their trigonometry largely from the
Hindus in connection with astronomy. As already stated
(p. 177), they originated nothing of importance either in arith-
metic or in geometry, they systematized algebra to some extent,

ibn Rajcb ibn Tiboga, Shihab ed-din AbuV Abbas. Born 1359;
died in Egypt, 1447.

2 'Ali ibn Ibrahim ibn Mohammed al-Mot'im al-Ansari, Abu'l-Hasan. Born
1304; died 1375/76 or 1379/80. 3 Ulug Beg.

4 See L. P. E. A. Sedillot, ProUgomtnes des Tables Astronomlques d'Ouloug
Beg, Paris, 1847; T. Hyde, Tabulae Longitudinis et Latitudinis Stellarum
Fixarum ex Observatione Ulugbeighi, Oxford, 1665; E. B. Knobel, Ulugh Beg's
Catalogue of Stars, Washington, 1017.

6 Jemshid ibn Mes'ud ibn Mahmud, Giyat ed-dm al-Kashi. He is also
known as Kazi Zadeh al Rumi and Ali Kushi. Died c . 1436.

H. Hankel, Geschichtc der Mathematik, 289 (Leipzig, 1874) (hereafter
referred to as Hankel, Geschichte) ; Taylor, Lilawati, Introd., p. 14. The intro-
duction to his Miftdh al-hisab (Key of arithmetic) was translated by Woepcke.

290 PERSIA AND ARABIA

they improved upon the astronomy of their predecessors, and
they made some real contributions to trigonometry. All these
matters will be discussed in the appropriate chapters. But on
the whole the Arabs of this period were still transmitters of
learning rather than creators, and to them Europe is chiefly
indebted for preserving in their translations many of the
important works of the Greeks. To this rather sweeping asser-
tion, however, one noteworthy exception may justly be taken,
for it seems quite certain that it is to an Arab, or rather to a
Turkish, scholar that we owe the first actual use of a decimal
fraction. This step was taken independently at a later period
by European arithmeticians, but the decimal fraction seems
certainly to have been in use in Samarkand early in the isth
century. In a work by al-Kashi, or Jemshid (p. 289), the ratio
of the circumference to the radius of a circle is given, in part,
as follows:

Integer

6 28318

the full result being correct to sixteen decimal places.

Justice also requires that the Arabs of the four centuries
beginning with the year 800 should be judged not with respect-
to the great achievements of the golden age of Greece but rather
in comparison with the very meager results secured by their
contemporaries in Europe and the Far East. If we consider
Europe during the same period, we shall find the names of few
original scholars in the domain of mathematics. The number
of Arab, Persian, and Turkish scholars in this period exceeds,
so far as we yet know, that of their European contemporaries,
and their achievements were more significant. It was only dur-
ing the period from 1200 to 1400 that European mathematics
forged ahead, and even then the Arab influence was one of
the prominent moving causes.

Justice further requires the admission that for lucidity of
statement the scholars of Bagdad surpassed their contempo-
raries both in the East and in the West during a period of
about six centuries, and were quite their equals in originality.

DISCUSSION 291

TOPICS FOR DISCUSSION

1. A comparison of the general nature of Chinese mathematics
in the Middle Ages with that of the mathematics of Europe.

2. Evidences of the interchange of thought between the East and
the West in the Middle Ages, and the possible effect of this inter-
change upon the mathematics of Europe and Asia.

3. Influence of astrology upon astronomy and upon pure mathe-
matics, both in the East and in the West.

4. Nature of astronomy in the Middle Ages in the East and the
border line between this science and astrology.

5. Nature of the mathematics of China in the period of its
greatest development, the i3th century

6. The reliability of Chinese texts in the Middle Ages, based upon
a general study of Chinese literature.

7. The justice of the claims oi China in the field of numerical
higher equations.

8. The nature of the problems that interested Chinese scholars
in the Middle Ages.

9. Early history of mathematics in Japan. Reasons for the
failure of the science to advance.

10. The nature of the mathematics of India in the Middle Ages. A
comparison of this mathematics with that of China.

it. The nature of the problems that interested Hindu scholars in
the Middle Ages.

12. The works of Bhaskara; their nature and influence.

13. Causes of the decay of mathematics in the Mohammedan coun-
tries, beginning with the nth century.

14. Nature of algebra as developed by Mohammedan writers
after the Golden Age of Bagdad.

15. The contributions to mathematics made by the Persian poet
Omar Khayyam.

1 6. The life and works of the prince-astronomer Ulugh Beg and
the reasons why his influence was not more powerful in Moham-
medan lands.

17. A summary of the contributions of the Arabs to the science
of mathematics.

1 8. General position of Hebrew mathematics in the Near East
in the Middle Ages.

CHAPTER VIII

THE SIXTEENTH CENTURY

i. GENERAL CONDITIONS

The Sixteenth Century in General. Until about the year
1500 the mathematics of the world was, so far as any records
tell us, limited to a small number of individuals in each cen-
tury. Printing having only just been invented, there was no
simple way for comparatively obscure workers, even in the isth
century, to make their contributions or their interest known,
and so they left no record of their achievements. In the i6th
century, however, the printed page began to perpetuate names,
and it now becomes impossible to do more than select a few out
of the many for such comment as space may allow. 1

Science and Letters. Furthermore, historical events now be-
gan to be recorded more freely, the world moved more rapidly,
and the influences that bear upon the development of mathe-
matics become more difficult to trace. That the opening of a
new world would greatly increase the interest in commercial
mathematics is evident, and the fact is abundantly proved
by the printed books of the i6th century; but the influence of
the great literary movement illustrated by such writers as
Shakespeare, Cervantes, and Camoens, or of such world
events as the defeat of the Spanish Armada, is not so apparent
in the scientific field. Indeed, we may say that it was the
influence of such scientists as Leonardo da Vinci, Copernicus,
Palissy, and Tycho Brahe that stimulated the literary renais-
sance of the period, rather than the reverse.

1 Thc reader who has access to the seventh edition of the Encyclopaedia
Britannica will find that the "Dissertation Third," by John Playfair, on "The
Progress of Mathematical and Physical Science since the Revival of Letters in
Europe" forms a very good background for his studies of this period.

292

MATHEMATICAL CONDITIONS

293

Mathematical Conditions. The conditions in the field of
mathematics were such as to mark out the course of progress.
Euclid's Elements had appeared in print in 1482 and the
Conies of Apollonius was known in manuscript, 1 and hence the
most promising field in pure mathematics was in the domain
of analysis. The
quadratic equa-
tion had been
fully solved, and
so the next step
was to attempt
the solution of the
cubic equation,
with which the
Greeks and Arabs
had been success-
ful only in special
cases and by hav-
ing recourse to
the intersection of
conies. It was
here, then, that

mathematics would naturally be expected to advance. Along
with this there would be expected to develop a better symbol-
ism, and one that was also suited to the needs of typography.
After the cubic was solved, the mathematical world would be
expected to try the equations of the fourth and higher degrees.
We should anticipate, for geographic reasons, that the lead
would be taken in Italy, and we should also expect that the
demands of astronomy and navigation would require a more
rapid development of trigonometry. All these suggested ex-
pectations, as we shall see, were fulfilled. Indeed, in the space
of a single century mathematics made more advance than had
been achieved since the days when the Alexandrian School
dominated the scholastic world.

MATHEMATICS IN THE 16TH CENTURY

Concept of the range of the science. From the title-
page of Coignet's Arithmetic^ Antwerp, 1580

*It was first printed in Venice in 1537. The Commandinus edition appeared
at Bologna in 1566, and the Hailey edition at Oxford in 1710.

294 ITALY

2. ITALY

Leonardo da Vinci. No list of the Italian mathematicians of
the 1 6th century would be complete without some mention of
Leonardo da Vinci 1 (1452-1519), and such were the remark-
able attainments of this gifted man that his name may properly
be given in its chronological order, thus standing first on the
record. He was born at Vinci, near Florence, resided in Flor-
ence, Milan, and Rome, went to France at the invitation of
the king in the year 1516, and died near Amboise in 1519.
Famous as a painter, sculptor, goldsmith, investigator of the
circulation of the blood, general scientist, architect, and writer
on mechanics, optics, and perspective, he would have ranked
as a worthy mathematician had not his talents in this direction
been obscured by his unusual gifts in these other lines. In
applied mathematics he may be looked upon as one of the
founders of the modern theory of optics. In geometry he dis-
tinguished between curves of single and double curvature, gave
much attention to the subject of stellar polygons, was interested
in constructions with a single opening of the compasses, and
gave various correct or approximate constructions of regular
polygons. In physics he knew the theory of the inclined plane,
found the center of gravity of a pyramid, worked in the field
of capillarity and diffraction, knew the camera obscura without
a lens, and studied the resistance of the air and the effect of
friction. The world has rarely produced such an all-round
genius.

Early Workers in the Field of Equations. Since the solution
of the equations of the third and fourth degrees was the chief
mathematical achievement in Italian mathematics of the i6th
century, it is proper to group together those scholars who were
most prominent in this work. Although the details of their
chief contributions will be reserved for Volume II, a brief
statement of their achievements will now be given.

*P. Duhem, fitudes sur Leonard de Vinci, 2 vols., Paris, 1906-1013. For
a popular sketch see D. Merejkowski, The Romance of Leonardo da Vinci,
New York [iQO2l.

CUBIC EQUATIONS 295

Scipione del Ferro, 1 a native of Bologna, whom Cardan calls
by his Latin name of Scipio Ferreus, was professor of mathe-
matics in the city of his birth. In geometry he was interested
in constructions depending on a single opening of the com-
passes. In algebra he found a method of solving the cubic
equation for the special case of x l -f ax = b.

In 1506 he revealed this method to his pupil Antonio Maria
Fior/ J a Venetian, who proceeded to turn the information to
account in the popular mathematical contests that were then
in vogue. Of the life of Fior little is known, but he is said by
Tartaglia 3 to have been living in 1536.

Zuanne de Tonini da Coi 4 (c. 1530) was a teacher in Bre-
scia and was interested in mathematics from the standpoint of
problem solving/' In 1530 he sent as a kind of challenge to
Tartaglia the two equations

3 + 3* 2 = 5
and r* + 6* 2 + 8# = 1000.

For some time Tartaglia was unable to solve them, but, as we
shall see in Volume II, he finally succeeded in doing so, this being
an important step in the general problem of the cubic equation.

Cardan. The first of the two prime movers in the solution of
the cubic was Giro'lamo Carda'no.' 5 He was the illegitimate

3 Born r. 1465; died at Bologna between October 2Q and November 16, 1526.
L. Frati, Bollett. di bibliogr. d. sci. matem., XII, i. The name also appears as
Ferri and as Ferreo.

-This on the statement of Tartaglia (1546). Cardan (1545) says it was
about 30 years earlier than the time of his writing, which would make it c. 1514
or 1515. The name appears with various spellings, particularly in the Latin
form of Antonius Maria Floridus and in the form of Antoniomaria Fior.

"Pronounced tar tii'lya. See page 297.

4 Also Zuane, Giovanni, Giovanno, John, with his last name sometimes given
as Colle or in the Latin form of Colla.

r 'In a letter written in 1540, Tartaglia speaks of "that devil" having returned:
"Eglie ritornato qui quel diauolo de Messer Zuanne Colle."

Born at Pavia, 1501 ; died at Rome, September 21, 1576. The name appears
as Hieronymus Cardanus and Jerome Cardan. He is commonly called Cardan
by writers of English. H. Morley, The Life of Girolamo Cardano, 2 vols.,
London, 1854 (hereafter referred to as Morley, Cardan) ; Cantor, Geschichte^ II,
chaps. 64, 65, 66; V. Mantovani, Vita di Girolamo Cardano, Milan, 1821;
Gherardi, Facolta Mat. di Bologna, 47 seq.

296

ITALY

son of a jurist, Facio Cardano (1444-1524), who was profes-
sor of jurisprudence and medicine in Milan and who edited
Peckham's Perspective, communis. Girolamo was a man of

remarkable con-
trasts. He was
an astrologer and
yet a serious stu-
dent of philoso-
phy, a gambler
and yet a first-
class algebraist,
a physicist of ac-
curate habits of
observation and
yet a man whose
statements were
extremely unre-
liable, a physi-
cian and yet the
father and de-
fender of a mur-
derer, at one
time a professor
in the University
of Bologna and
at another time
an inmate of an
almshouse, a vic-
tim of blind su-
perstition and yet

the rector of the College of Physicians at Milan, a heretic
who ventured to publish the horoscope of Christ and yet a
recipient of a pension from the Pope, always a man of
extremes, always a man of genius, always a man devoid of
principle. A certain bitter rival said of Voltaire, "Ce coquin-la
has one vice worse than all the rest; he sometimes has vir-
tues." So it was with Cardan.

PORTRAIT OF CARDAN

From the title-page of the first edition (1539)
of his arithmetic

CARDAN AND TARTAGLIA 297

His Ars Magna, 1 the first great Latin treatise devoted solely
to algebra, appeared at Nurnberg in 1545 and set forth the
theory of algebraic equations so far as it was then known,
including the solution of the cubic, which he seems to have
secured from Tartaglia under pledge of secrecy and then
dishonorably to have published, and the solution of the biquad-
ratic which had been discovered by his pupil Ferrari. He
also wrote on arithmetic, 2 astronomy, 3 physics, 4 and various
other branches of knowledge, 5 proving himself a man of re-
markable versatility and learning.

Tartaglia. Nicolo Tartaglia/' one of the greatest mathemati^
cians of Italy in the i6th century, was born at Brescia. Al-
though known as Tartaglia, we learn from his will 7 that his
brother's name was Fontana. It is said that he was present as a
child at the taking of Brescia by Gaston de Foix (1512 ) and at
that time received a saber cut in the face which caused an
imperfection in his speech. This gave him the nickname of
Tartaglia ("the stammerer"), which name he formally used
in his published works. He was self-educated but acquired
such proficiency in mathematics that he earned a livelihood
by teaching the science in Verona, Vicenza, Brescia, and
Venice (1535).

Tartaglia seems to have substantially completed the solu-
tion of the cubic equation and, as already stated, to have
imparted the secret to Cardan, who, in violation of his oath,
published it in I54S. 8

*Artis Magnae, she de regvlis algebraicis, liber vnvs, Nurnberg, 1545; Basel,
1570. 2 Practica arithmetice, Milan, 1539.

&De revolutione annorum, mensium et dierum . . . liber, Nurnberg, 1547;
De temporum et motuum erraticarufn restitutione, Nurnberg, 1547; Aphoris-
morum astronomicorum segmenta septem, Nurnberg, 1547.

*De subtilitate, Nurnberg, 1550, with a Paris reprint in 1551.

5 His Opera appeared in ten volumes, Lyons, 1663.

6 Born c. 1506; died at Venice, December 13/14, 1557- The name is also
spelled Tartalea. It is spelled in the text as it appears on the title-page of his
work of 1556.

7 Published by Boncompagni in 1881.

8 On his possible indebtedness to Ferro see G. Enestrb'm, in Bibl. Matk. t VII
(3), 3&

298

ITALY

Tartaglia was the first to apply mathematics to artillery
science, 1 a subject just being perfected by the great French
masters Galiot de Genouillac and Jean d'Estrees. He also
wrote the best treatise on arithmetic" that appeared in Italy

in his century, con-
taining a very full
discussion of the nu-
merical operations
and the commercial
rules of the Italian
arithmeticians. The
life of the people,
the customs of mer-
chants, and the ef-
forts at improving
arithmetic in the i6th
century are all set
forth in this remark-
able work. Tartaglia
also published (1543)
editions of Euclid
and Archimedes.

On the question of
the publication by
Cardan of the Tar-
taglia solution of the
cubic after a pledge
of secrecy, a biographer of the former, but one who could not
feel the influences of the i6th century, has this to say:

The attempt to assert exclusive right to the secret possession of a
piece of information, which was the next step in the advancement of
a liberal science, the refusal to add it, inscribed with his own name,

1 Nitova scienza, doe Invenzione nuovamente trovata, utile per ciascuno
speculative matematico bombardiero . . . , Venice, 1537; Qvesiti ed invenzioni
diverse, Venice, 1546.

2 General Trattato di nvmeri, et misvre, 2 parts (volumes), Venice, 1556-
1560; Tutte V opere d' arithmetica del famosissimo Nicolo Tartaglia, Venice,
1592, being substantially Volume I of the General Trattato.

From the title-page of La Prima Parte del General
Trattato, Venice, 1556

TARTAGLIA 299

to the common heap, until he had hoarded it, in hope of some day,
when he was at leisure, of turning it more largely to his own advan-
tage, could be excused in him only by the fact that he was rudely
bred and self-taught, and that he was not likely to know better.

TARTAGLIA APPLIES MATHEMATICS TO ARTILLERY SCIENCE
From // Primo Libro delli Qvesiti, et Inventioni diverse, Venice, 1546; 1562 ed.

Any member of a liberal profession who is miserly of knowledge,
forfeits the respect of his fraternity. The promise of secrecy which
Cardan had no right to make, Tartalea had no right to demand. 1

As already remarked, however, it is difficult to imagine con-
ditions in the year 1545, and it is hardly just to apply modern
ethics to a situation so different from our own.

1 Morley, Cardan, I, 270; Rixner and Siber, Leben und Lehrmeinungen
bertihmter Physiker am Ende des XVI. und am Anfange des XVII. Jahr-
hunderts, Sulbach, 1820; Firmiani, Girolamo Cardano, Naples, 1904. There is
a brief but good biography of Tartaglia in D. Marlines, Origine aritmet., p. 61 n.
See also A. Favaro, "Per la biografia di Niccolo Tartaglia," Archivio storicv
Italiano, 1913, and "Di Niccol6 Tartaglia," in Isis, I, 329.

300 ITALY

Ferrari. Lodovico Ferrari, 1 born in humble circumstances,
was taken into Cardan's household in Milan at the age of fif-
teen. Cardan soon recognized his remarkable ability and made
him his secretary. In spite of his ungovernable temper and
his blasphemous habits he was later accepted by Cardan as his
pupil and friend. Mathematical Italy would have given much
to be in his place in Cardan's household ; but, such were their
quarrels, it would have given more to be out again. At the
age of eighteen even Ferrari was glad to sever all relations
with his patron and to begin teaching by himself in Milan. He
was so successful there and in the mathematical contests of
the day as to attract the attention of the court and of the
Cardinal of Mantua. Through the favor of the latter he se-
cured a position that brought him abundant means. He then
became professor of mathematics at Bologna, but died there
in the first year of his service, at the age of thirty-eight, prob-
ably poisoned by his only sister.

Zuanne de Tonini da Coi had proposed a problem which
involved the equation

x 4 4- 6.tr + 36 = 6ox.

This problem Cardan attempted to solve, and having failed
he gave it to Ferrari. The latter succeeded in finding a method,
and thus the solution of the equation of the fourth degree was
discovered. Ferrari left no written works on mathematics, but
Cardan published in the Ars Magna (1545) this noteworthy
contribution to the theory of equations.

Bombelli. Rafael Bombelli (born c. 1530), a native of
Bologna, was the last of those Italian mathematicians of the
1 6th century who contributed in any noteworthy way to the
solution of the cubic and biquadratic equations. He wrote
U Algebra parte maggiorc dell' arimetica divisa in tre libri and

1 Born at Bologna, 1522; died r. 1560. The date of his death is also given as
1562 and 1565. Morley gives 1560. The Christian name is also written Luigi.
Cardan left an unpublished Vita Ludovici Ferrarii Bononiensis. See also G. de'
Sallusti, Storia dell' Origine e de' Progressi delle Matematiche, I, ^8 (Rome, 1846).

FERRARI AND BOMBELLI

301

published the work in Bologna in I572. 1 In this work Bom*
belli set forth the reality of the three roots of a cubic equation
in the case in which the cube root of an imaginary expression
is involved in the result
secured by the Tartaglia-
Cardan rule. The book
contained the most teach-
able and the most system-
atic treatment of algebra
that had appeared in Italy
up to that time.

Of Bombelli's life al-
most nothing is known.
He is thought, from the
introduction to his alge-
bra, to have been an en-
gineer in the service of
the patron to whom he
dedicates his book, Ales-
sandro Rufini, bishop of
Melfi.

Francesco Maurolico.
Of the mathematicians of
this period who were in-
terested in the Greek
writers the most prom-
inent was Frances'co
Mauroli'co,* a native of
Messina, Sicily, but of

Greek parentage. He was a priest, at one time an abbot, and
for some years professor of mathematics at Messina. He

J A second edition appeared at Bologna in 1570 with another title, L' Algebra
Opera, with the dedicatory letter reset, but otherwise using the same sheets as
the 1572 edition. The name is spelled as above in both editions of this work.

2 Born at Messina, Sicily, September 16, 1494; died at Messina, July 21, 1575.
Latin, Franciscus Maurolycus or Maurolykus; also known as Marullo. D. Scina,
Elogio di Francesco Maurolico (Palermo, 1808) ; Martines, Origine aritmet., 65.
A life of Maurolico, written by his nephew, was published at Messina in 1613.

FRANCESCO MAUROLICO
Engraved after a portrait from life

302 ITALY

translated into Latin the works of Theodosius and Menelaus,
the treatise of Autolycus on the sphere, 1 and the Phaenontena
of Euclid, and published works on Apollonius 2 and Archi-
medes/ 1 He also wrote various general works on mathematics
and arithmetic, 4 wrote on mathematical induction, 5 and was a
man of some creative power. A few of his more prominent
contemporaries are listed below, with a brief statement of
their contributions.

Italian Geometers. Federigo Commandino of Urbino (1509-
1575) is known as one of the leading translators and editors
of the Greek classics in mathematics. His editions of Euclid,
Archimedes, Apollonius, Aristarchus, Heron, Ptolemy, and
Pappus are highly esteemed.

Frances'co Baroz'zi (c. istf-post 1587), a Venetian noble-
man, edited the commentary of Proclus on the first book of
Euclid. 7 He also wrote on cosmography and geometry 8 and
translated Heron's works.

Giambattista Benedetti 1 ' (1530-1590), a Venetian by birth,
wrote on the geometry of a single opening of the compasses, 10
on the gnomon, on optics, and on the theory of numbers, and
gave excellent graphic treatments of various problems. 11

1 Published at Messina, 1558. 2 Messina, 1654.

3 Palermo, 1670 (the edition being lost in a shipwreck) and 1685.

*Opuscula Mathematics 2 vols., Venice, 1575, although written in 1553.
The Arithmetic or urn libri duo (Venice, 1575, but written in 1557) was the
second volume of the Opuscula and was republished in 1580. See also the
Elogia above mentioned, p. 114.

5 G. Vacca, "Maurolycus, the first discoverer of the principle of mathemati-
cal induction," Bulletin of the Am. Math. Soc., XVI, 70; W. H. Bussey, "Origin
of Mathematical Induction," Amer. Math. Month., XXIV, 109.

Franciscus Barocius, Francesco Barocci.

7 Prodi Diadochi . . . in primum Euclidis . . . librum comment., Padua, 1560.

8 Geometricum problema tredecim modis demonstratum, Venice, 1586.

9 Giovanni Battista Benedetti, Joannes Baptista Benedictus.

10 De resolutione omnium Euclidis problematum aliorumque . . ., Venice,
I 553- O n the general history of this important subject see W. M. Kutta, "Zur
Geschichte der Geometric mit constanter Zirkeloffnung," Nova Acta . . . Abh.
der K. Leop.-Carol. Deutschen Akad. der Naturforscher, Halle, LXXI, 71, and
Bibl. Math., X (2), 16.

11 Diversarvm Specvlationvm Mathematicarum t & Physicarum Liber, Turin,
1580. See Kara Arithmetica, p. 364.

MINOR WRITERS 303

Cosimo Bar'toli (1503-1572), a Florentine geometer, trans-
lated into Italian the works of the French mathematician
Oronce Fine 1 and wrote a popular work on mensuration/
Among the features of the book is a table of squares to 662 *.

Pietro Antonio Catal'di 3 (1548-1626) was a native of Bo-
logna and spent the closing years of his life there. He was
professor of mathematics and astronomy at Florence (1563)3
Perugia ( 1572 ), and Bologna (1584). He wrote several mathe-
matical works and to him are due the first steps in the theory of
continued fractions, although not the first idea of these forms.
His Prima Parte delta Pratica Aritmetica (Bologna, 1602)
and Trattato del numcri perfctti (Bologna, 1603) were printed
under the pseudonym of Perito Annotio, formed by transpos-
ing the letters in his given names. The second part of the
Pratica appeared under his own name in 1606, and similarly
for the third and fourth parts ( 1617, 1616). He also edited the
first six books of Euclid's Elements * wrote a brief treatise
on algebra, 5 and contributed to the theory of roots (1613),
the quadrature of the circle (1612), and various other subjects,

Matteo Ricci. Among the contemporaries of Cataldi the one
who did most for the spread of mathematics in remote lands
was Matteo Ricci, a man of remarkable energy and of great
influence through his work in China. He entered the Jesuit
order in 1571, left Rome for China in 1577, and reached

di Orontio Fineo del Delfinato; Diuise in cinque Parti: Arimetica
Geometria, Cosmografia, & Oriuoli, Tradotte Da Cosimo Bartoli, Gentilhuome
& Academico Fiorentino, Venice, 1587 (posthumous). See infra, page 308.

2 Del Modo di Misvrare le distantie, le superficie, i corpi, le piante, . .
Venice, 1564, with possibly an earlier edition.

3 Although the spelling Cattaldi is given on the title-page of his Dve Lettion.
di Pietr' Antonio Cattaldi (Bologna, 1577), the name is generally given in hL
other works as Cataldi. The above-named book is curious because the printer
not having fraction forms, was obliged to insert all fractions by hand. The namt
also appears as Cataldo.

4 Bologna, 1620.

& Regola delta Quantita, o Cosa di Cosa, Bologna, 1618.

6 Born at Macerata, Ancona, October 6, 1552 ; died at Peking, May 8 (or n)
1610. His Chinese name was Li-ma-to, derived from Ri (Chinese Li, for Ricci)
and Matteo. H. Bosnians, Revue des Quest, sclent., January, 1921.

304 ITALY

Canton in I578. 1 Here he did more than any of his prede-
cessors to make known in that country the mathematics and
astronomy of the West. With the help of native scholars, the
most prominent of whom were two learned mandarins, Hsii
Kuang-ching (1562-1634) and Li Chi Ts'ao (died 1631),
he translated (1603-1607) into Chinese the first six books
of Euclid's Elements? He also wrote an arithmetic, 8 which
he dedicated to his assistant, Li Chi Ts'ao, and compiled various
astronomical works/

Minor Writers. In the second half of the century Silvio
Belli (died 1575) wrote on practical geometry 5 and on the
theory of proportion." His geometry was very popular, six
editions appearing in the i6th century.

Another writer whose work attracted considerable attention
in the i6th and i?th centuries was Petrus Bongus, 7 to take the
Latin form of his name as it appears in the first edition of his
work. He was a native of Bergamo and became canon of the
cathedral in that city. His work of nearly 500 pages on the
mystery of numbers 8 went through several editions. It con-
tains a mass of information upon such subjects as the religious
significance of three, seven, and other numbers.

The Italian Arithmeticians. In all the leading countries many
arithmetics were printed in the i6th century, 9 but for our
present purposes it suffices to mention only a few of the more
prominent Italian writers.

Girolamo and Giannantonio Tagliente, 10 Venetian arithmeti-
cians of c. 1 500, wrote a work on commercial arithmetic 11 which

3 Pietro Tacchi Venturi, S. J., L'Apostolato del P. Atatteo Ricd, 2d ed.,
Rome, 1010 2 Ki-ho ^-yuan-pen. 3 Tung-wen-siian-ki

4 P. F. S. Vella, "Del P. Matteo Ricci," Memorie, Pontificia Accad. del
Nuovi Lincei, Rome, XXVIII, 51.

^Libro del misurar con la vista, Venice, 1565, with later editions in 1566,
1560, 1570, 1573, and 1595.

6 Delhi proporthne, et prop or tion alita, Venice, 1573.

7 Died at Bergamo, September 24, 1601.

*Mysticae Nvmerorvm significationis liber in dvas divisvs paries, Bergamo,
1583-1584.

9 For the list, consult Rara Arithmetica. 10 Pronounced ta lyen'ta.

13 Opera che insegna A fare ogni Ragione de Mercatia.

FROM RICCI^S TRANSLATION OF EUCLID, 1603-1607

From a manuscript copy of this translation, made in the I7th century. This
page shows the first proposition of Book I

306 FRANCE

appeared in Venice in 1515 and was so popular that it went
through more than thirty editions in the i6th century.

Francesco Ghaligai, 1 a Florentine arithmetician, published in
1521 a mercantile work entitled Summa De Arithmetical of
which two other editions appeared later.

Francesco Feliciano da Lazesio ! published an elementary
work on arithmetic, algebra, and practical geometry at Venice
in 1517/18.' This work went through at least fourteen editions
(including the revision of 1526) in the i6th century and sev-
eral in the century following.

Giovanni Sfortunati 5 published at Venice in 1534 a work
on commercial arithmetic which was well received and went
through several editions.

The first mercantile tables that had great popularity were
published in 1535 by a Venetian arithmetician, Giovanni
Mariani, 7 under the title Tariffa pcrpctva. The work was
often reprinted.

3. FRANCE

Centers of Activity. The centers of mathematical activity in
France in the i6th century were Paris and Lyons, the former
because of its ancient importance in all matters intellectual and
the latter because of its commercial supremacy and the desire
to cultivate some of the idealism of the northern capital. The
theoretical books appeared more frequently from the Paris
presses, while the output of what is known as practical mathe-
matics was fairly large among those whom a recent writer has
called "the morose and inhospitable Lyonese, ... in whose

1 Pronounced galega'e. He died February 10, 1536.

2 The edition of 1548 had the title Practica d' Arithmetica.

* 5 Born at Lazisa, near Verona, c. 1490; was living in 1536. The family name
is pronounced fa le'che a no. Lazesio also appears as Lazisio.

4 Libro de Abaco. There was a second work (Venice, 1526), called the
Libra di Arithmetica & Geometria . . . Intitulato Scala grimaldelli, but this was
only a revision of the work above mentioned.

5 Pronounced sf or too na'tg. He is also known by the Latin form Johannes
Infortunatus. Born at Siena, c. 1500.

6 Nvovo Lvme Libro di Arithmetica.

7 Born at Venice, c. 1500. The name also appears as Zuane Mariani.

THEORETICAL WORKS 307

esteem the pick of humanity is the prosperous silk merchant."
Paris was not yet a metropolitan city in the ecclesiastical sense,
the ancient Roman political divisions having been retained by
the Church, and the capital city being then and until 1622 sub-
ordinate to the metropolitan city of Sens. All this had no
effect, however, upon her intellectual and political supremacy
over all France.

Theoretical Works. The first of the French writers who
sought to maintain the standing of the Greek mathematics in
the intellectual atmosphere of France was Jacques le Fevre
d'Estaples, 3 known in his Latin works as Jacobus Faber Stapu-
lensis. He was a "Doctor Sorbonnicus," a priest, vicar of the
bishop of Meaux, lecturer on philosophy at the College Lemoine
in Paris, and tutor to the son of Frangois I. He wrote an intro-
duction to the arithmetic of Boethius 2 and a work on geometry,
edited (1499) Sacrobosco's Sphere and a description of the
number game of Rithmomachia, and published various other
works. His own writings were heavy and theoretical, and ex-
pressed the dying body of medieval mathematics. 3

Charles de Bouelles, 4 canon and professor of theology at
Noyon, wrote on geometry"' and the theory of numbers." The
latter work includes a book on perfect numbers. He is par-
ticularly worthy of attention, however, because of his work

1 Born at Estaples, near Amiens, c. 1455; died at Ne"rac, c. 1536. The French
forms also appear as Febvre and Etaples.

2 Introdnctio Jacobi fabri Stapulesis, in Arithmecam Diui Seuerini Boelij,
published in a volume with other works, Paris, 1503, the above title being from
the edition of c. 1507. In these two editions his geometry and perspective also
appear. The first edition of his compendium appeared in 1488. For detail?,
see Rara Arithmetic^ pp. 27, 30, 80, 82.

3 Baldi's estimate of his merits is exaggerated: "D'ingegno felicissimo attese
con gran frutto ad ogni sorte di dottrina, e giunse all' eccellenza di maniera che
fu giudicato meraviglia del suo secolo." Cronica, 107.

4 Born at Saucourt, Picardy, c. 1470; died at Noyon, c. 1553. The name ap-
pears also as Charles Bouvelles, Bouelles, Bouilles, and in the Latin form of
Carolus Bovillus.

*Geometricae introductionis libri VI, Paris, 1503; Livre singulier *t utile,
touchant I'art et pratique de Geometric, Paris, 1511, with several later editions.

6 Liber de duodecim numeris, part of his general work published at Paris
in 1509/10.

308 FRANCE

on the cycloid, he being one of the first to consider this figure
from the scientific standpoint. He also wrote on regular con-
vex and stellar polygons.

Minor Writers. One of the most pretentious of the mathema-
ticians of his time, and one of the least worthy, was Oronce
Fine, 1 more commonly known by the Latin form of his name,
Orontius Fineus. 2 In his young manhood (1518-1524) he was
imprisoned on account of his opposition to the Concordat, an
agreement between France and the Pope. Upon being released
he devoted himself to teaching, and about 1532 became profes-
sor of mathematics in the newly founded institution which was
later known as the College de France. 3 He wrote extensively
on astronomy and produced several works on arithmetic and
geometry, 4 including one on the quadrature of the circle. 5 Some
of his works were translated into Italian by Cosimo Bartoli.
While he enjoyed some reputation, he died in poverty and his
works were soon forgotten.

Among several French physicians of the i6th century who
devoted much attention to mathematics the only one of great
distinction is Jean Feme!" (1497-1558). He received his
degree in medicine at Paris in 1530, and four years later he
had so risen in his profession as to be called to a chair in the
faculty. His admirers spoke of him as the modern Galen, and
his Universa Medicina went through more than thirty editions.
In the field of mathematics he published (1528) a work of the

1 Born at Briangon, 1404; died in Paris, October 6, 1555.

-As spelled in the first edition of his Protomathesis, Paris, 1530-1532. The
name is also spelled Finaeus. There is no warrant for the spelling Fine.

3 Ramus speaks of him as the one "qui primus regia professione in Galliam
mathematicas artes retulit." Introduction to his Libri dvo, p. 3 (Basel, 1560).

4 Both of these subjects and some parts of astronomy are considered in his
Protomathesis. Among his other works are In sex priores libros geometricorum
elementorum Euclldls demonstrationes, Paris, 1536; De re et praxi Geometrica
libri III, Paris, 1555; and De rebus mathematicis, hactenus desideratis, Libri
1IH, with a biography, Paris, 1556.

*De quadratura circuit, Paris, 1544. This was severely attacked by Buteo.
See Bibl. Math., XII (3), 250.

6 Joannes Fernelius, as the Latin form appears in his De Proportionibus Libri
duo, Paris, 1558.

RAMUS 309

Boethian type on proportion, and his computation of the length
of a degree of the meridian was so satisfactory 1 as to entitle
him to a worthy place in the history of geodesy.

Among those who may properly be called the dilettanti
mathematicians of the time was Claude de Boissiere, 2 who
wrote on poetry and music as well as on astronomy and arith-
metic. His arithmetic, 3 a combination of the medieval theory
and the contemporary practice in calculation, is one of the
many books of the time that, as a result of the Hundred Years'
War, related mathematics to the science of warfare.

At about the same time Frangois de Foix, Comte de Candale
(c. 1502-1594), another of the dilettanti and a bishop in south-
ern France, was interested in a better translation of Euclid's
Elements* but he contributed nothing to the general theory of
geometry.

Ramus. Pierre de la Ramee, 5 better known by the Latin
form of his name, Petrus or Peter Ramus, descended from a
noble but impoverished family. His grandfather had been
driven from his estates in Burgundy and had been forced to
become a charcoal burner, and his father was a humble peasant.
Pierre early showed unusual intellectual powers, and after
many struggles obtained employment as a servant to a rich
student in the College de Navarre at Paris. By working in
this capacity during the day and studying at night he made
his way to the master's degree. It was on his examination in
1536, when he was only twenty-one years old, that he attracted

1 He made it 56,746 French toises instead of 57,024.

2 Claudius Buxerius. Born in the province of Grenoble, c. 1500.

3 L'art d'Arythmetiqve contenant tovte dimention, tres-singvlier et commode,
tant pour Vart militaire que autres calculations, Paris, 1554.

4 Euclidis . . . Elementa geometrica, Lib. XV . . . restituta. His accessit
decimus se\tm liber . . ., Paris, 1566. There was another edition, "Novissime
collati sunt XVII us et XVIII US priori editione . . . ," Paris, 1578. His name
appears in the Latin phrase "Auctore Francisco Flussate Candalla." For
various other editors and translators of Euclid in this period see P. Riccardi,
Saggio di una Bibliografia Euclidea, Bologna, 1887; hereafter referred to as
Riccardi, Saggio Euclid.

5 Born at Cust (Cultia, Cusia, Cus, Cuz, Cuth, Cut), Picardy, 1515; died
August 26, 1572.

3io FRANCE

the attention of intellectual Europe by his audacious attack
upon one of its idols, his thesis being "All that Aristotle has
said is false." 1 Soon after this he began his career of teaching
and was not long in attaining a high position in his profession.
For many years (from 1546) he was principal of the College
de Presles and held a professorship (from 1551) in the Col-
lege de France. He was an orator of great power and a skill-
ful debater, but his brilliant career was closed at the massacre
of St. Bartholomew's Day, August 26, 1572. Although his work
was chiefly in philosophy and the humanities, he devoted much
attention to mathematics, 2 editing the Elements of Euclid 3 and
writing on theoretical arithmetic, 4 geometry, 5 and optics.

Vieta. The greatest of all the French mathematicians of the
1 6th century was Franqois Viete, Seigneur de la Bigotiere,
better known by the semi-Latin name of Vieta. As a young
man he practiced law in his native town, afterward taking up
a political career and becoming a member of the Bretagne
parliament. His first work on mathematics appeared in Paris
in 1579. In 1580 he became master of requests at Paris, and
later was a member of the king's privy council. Under these
circumstances he was able to devote much leisure time to the

*Qiiaequmque ab Aristotele dicta essent, commentia esse.

2 There is a good summary of his life in a work by F. P. Graves, Peter
Ramus, New York, 1912, with a bibliography (p. 219) of the publications of
Ramus and of secondary sources of information.

3 Paris, 1545 and 1549. See Boncompagni's Bullettino, II, 389.

4 P. Rami, eloquentiae et philosophiae projessoris regii, arithmeticae libri tres,
Paris, I555J Paris, 1557; Basel, 1567; Paris, 1584. Also a Libri Duo, Paris, 1569,
1577, and 1581; Basel, 1580; Frankfort, 1586, 1591, 1592, 1596, and 1599;
Lemgo, IS99; English translation, London, 1593. See Kara Arithmetica, pp. 263,
330, 335-

B Paris, 1577; English translation, London, 1636. See also his Scholarvm
Mathematicarvm, Libri vnvs et triginta, Basel, 1569, in which he criticizes
Euclid's arrangement of the Elements from the point of view of logic.

6 Born at Fontenay-le-Comte (Fontenay -Vendee), 1540; died in Paris,
December 13, 1603. The French form also appears as Viet, Viette, and de
Viette, and the Latin as Vietaeus. There is a good sketch of his life in the
Penny Cyclopaedia, London, 1843, by De Morgan, with a summary of all his
works. See also J. L. F. Bertrand, "La vie d'un savant au XVI. siecle, Frangois
Viete," in his Sloges academiques r v> *43 (Paris, 1902) ; F. Ritter, Francois Viete,
Paris, 1905; G. Gambier, Le mathematicien Francois Viete, La Rochelle, 1911.

VIETA

study of mathematics, and the results were such that he ranks
as one of several notable instances of a man attaining high
standing in this science
although not devoting
himself chiefly to it until
rather late in life. Vieta,
indeed, remarked in a
letter to Adriaen van
Roomen that he did not
profess to be a mathe-
matician, but was merely
one to whom mathemati-
cal studies were delight-
ful in his hours of leisure.
Vieta wrote chiefly on
algebra, 1 but he was also
interested in geometry, 2
the calendar, and mathe-
matics in general. In
connection with the Gre-
gorian reform of the cal-
endar he acquired much
unfortunate notoriety
through his bitter antag-
onism to Clavius and
through his wholly un-
scientific attitude. He was an expert in deciphering, for the
government, the cryptic writing of diplomatic correspondence.

l lsagoge in artem analyticam y De aequationum recognition et emendatione
libri duo, De mtmerosa poteslatum purarum atque adjectarum ad exegesiu
resolutione tractatus, all published privately by Vieta, and republished by
Frans van Schooten, Leyden, 1646, in the Ofiera mathewrt'ra of Vieta. See
also Boncorapagni's Bullettino, I, 223, 245. There was a French translation of his
treatment of conations made by one j. I., de Vau'ezard, Les C*nq Livres des
Zetetiques, published at Paris in 1630. There was an Algebre de Viete d'vne
methode novvelle, daire, et Facile, by one James Hume (lac Hvmivs), a Scotch-
man, Paris, 1636, but it is only after the style of Vieta.

2 Effectionum geometricarum canonica recensio, and Supplementum Geo-
metriae, Paris, 1593.

FRANCOIS VlfeTE (VIETA)

From an old lithograph of a portrait
from life

312 FRANCE

Vieta's Work in Algebra. It will be shown in Volume II that
Vieta contributed extensively to the development of algebra
and trigonometry, but a brief reference to his work is appro-
priate in this connection. He was among the first to employ
letters to represent numbers in algebra, often using vowels for
the unknowns and consonants for the knowns. He found the
formula for sin n$ in terms of sin <; made an advance towards
proving that an equation of the nth degree is made up of n
linear factors ; showed how to increase, decrease, multiply, or
divide the roots of the equation j(x} = o by k ; gave one of the
earliest methods of evaluating TT by infinite products ; applied
algebra to geometry in such way as to lay a foundation for
analytic trigonometry ; indicated powers more simply than his
predecessors had done, using Aq for the square of the unknown,
Ac for its cube, Aqq for its fourth power, and so on; and
showed clearly the relation between the problems of the trisec-
tion of an angle and the solution of a cubic equation. His in-
teresting combination of infinite products and series is seen in
his statement that

Minor Writers. Of the writers on the theory of mathematics
at this period the only one of any note who published his works
at Lyons was Joannes Buteo, 1 a brother and afterward general
of the order of St. Anthony. He wrote chiefly on geometry 2
and arithmetic. 3 His geometry refuted various pretensions of
Oronce Fine as to the quadrature of the circle.

1 Born at Charpcy, r. 1485-1402; died at Caam, c. 1560-1572. These two
places are in Dauphine, France. The dates given by various writers differ
greatly. See also Boncompagni's BullettinOj XIII, 258, 265 n. This is the form
in which the family name appears in his Logistica, Lyons, 1559, that is, loan.
Bvteonis Logistica. It is also given as Boteo, Jean Buteon, Batcon, Borrel, and
Borell.

-Opera Geometric^ Lyons, 1554; De Quadrature, Circuit, Libri II, Lyons,
I5SO.

8 Logistica qvae & Arithmetica vulgd dicitur in libros quinque digesta, Lyons,
1559 ; the most original of his works. See Kara Arithmetica, p. 292 ; G. Wert-
heim, "Die Logistik des Johannes Buteo," Bibl. Math., II (3), 213.

PRACTICAL MATHEMATICS 313

Another instance of a French mathematician of some ability
publishing outside of Paris, and in this case outside of France,
is that of Francesco dal Sole (born c. 1490). He wrote in
Ferrara, Italy, and published an arithmetic in Venice. 1 The
only feature of the book worth mentioning is the combination
which it makes of the concepts of number and space."

Perhaps the most elaborate arithmetics published in France in
the 1 6th century, and among the least practical, were those of
Pierre Forcadel. 3 Little is known of this writer except that
he lived for a long time in Italy and finally, through the efforts
of Ramus, was called to Paris as professor of mathematics in
the College Royal. He translated luiclid I-VI and parts of
the works of Proclus, Archimedes, and other writers.

Practical Mathematics. The earliest of the Lyons school of
arithmeticians, and one of the most brilliant as well as most
unscrupulous, was Estienne de la Roche, 4 known as Ville-
franche, although a native of Lyons. He was a pupil of Chu-
quet, and in his arithmetic 5 he appropriated a large amount of
material from a manuscript of the latter which has since been
published. Perhaps no other French arithmetic of the i6th
century gives a better view of the methods of computation and
of the commercial applications of the subject.

The second noteworthy writer of the southern school was
Jacques Peletier, 7 a native of Le Mans, who is also known by
the Latin name of Peletarius. He was head of a college at
Bayeux (1547), secretary to the bishop of Le Mans, a physi-
cian at Bordeaux (1550), Poitiers, Lyons, and Paris, and finally
head of a college at Le Mans. He contributed to general

1 Libretti nvovi con le regole Di Francesco Dal Sole Gallo, Ferrara, 1546.
The first edition was Venice, 1526. There was a third edition in 1564.

2 Thus he has "Regola delle additione in generalita, tanto geometrica, quanto
arithmetica."

3 Born at Beziers; died in Paris, 1574. For description of the various works
see Kara Arithmetica, p. 284. 4 Born at Lyons, c. 1480.

5 Larismethique nouellement composee par maistre Estienne de la roche diet
Villefrdche natif de Lyo, Lyons, 1520. Kara Arithmetica, 128; Cantor, Ge-
schichte, II, chap. 59.

6 A. Marre, Boncompagni's Bullettino, XIII, 573.

7 Born at Le Mans, 1517; died in Paris, July, 1582.

314 ENGLAND

literature and to elementary mathematics. His arithmetic 1 was
published both at Poitiers (1549) and at Lyons (1554). He
also wrote on algebra, 2 Euclid's Elements, the geometry of lines
and angles, and the circle. He equated the terms of an equa-
tion to zero and stated that, when all roots are integral, any
root is a factor of the last term.

Another Lyons arithmetician of considerable note appeared
in the person of Ian Trenchant (born c. 1525). His arithmetic 3
includes the usual commercial applications and the operations
with counters as well as with common numerals.

Among the practical tables published at Lyons are a set pre-
pared by Monte Regal Piedmontois, professor of mathematics
in the University of Paris. These tables 4 are beautifully printed
on vellum and copies are very rare. The work contains the
products of numbers to 100 x 1000, and the author speaks of
having published part of the tables in Venice in 1575.

4. ENGLAND

English Writers. England was later than Italy or France in
her appreciation of mathematics, or at least in her publication
of works on this subject. 5 Although there is some mention of
arithmetic in the early works, it was not until 1522 that a
book devoted wholly to mathematics was printed in Great
Britain, the erudite but dull arithmetic 7 of Tonstall.

1 The 1607 edition has the title L'Arithmetiqve de lacqves Peletier dv Mans y
Departie en quatre liures. There were other editions. Kara Arithmetica, 245.

2 L'algebre departie en deux livres, Lyons, 1554.

3 L'Arithmetiqve de Ian Trenchant, Departie en trots liures, Lyons, 1566.
This title is from the 1578 edition. There were several editions. On an edition
of 1558 see Bibl. Math., II (3), 356.

^Invention novvelle et admirable, pour faire toute sorte de copte, Lyons, 1585.

s See also R. C. Jebb, in the Cambridge Mod. Hist., Vol. I, chap, xvi, and
J. Gairdner, ibid. t chap. xiv.

6 For example, in Caxton's Mirrour of the World or Thy mage of the same,
London, 1480, translated from the French, there is a chapter (10) beginning,
"And after of Arsmetrike and whereof it proceedeth," but this cannot be called
a treatise on the subject. There is a recent edition by O. H. Prior, London,

1913 (for IQI2).

T DC Arte Svppvtandi libri qvattvor Cvtheberti Tonstalli, London, 1522.

ARTE SVPPVTANDI
LIBRI QVATTVOR
CVTHEBERTI
TONSTALLI.

TITLE-PAGE OF THE TONSTALL WORK OF 1522

ENGLAND

Tonstall. Cuthbert Tonstall 1 was recognized as a man of
great learning and influence. He was born in Hackforth, York-
shire, went to Oxford in 1491, left on account of the plague
and entered Cambridge, and afterward went to Padua, where
he took the degree of doctor of laws. In 1511 he became vicar-
general to the Archbishop of Canterbury, was presented at
court, and soon thereafter received various ecclesiastical and
diplomatic appointments. Erasmus speaks in the highest terms
of his remarkable attainments at this period of his career. In
1522 he was promoted to the bishopric of London, but con-
tinued for some time in his diplomatic career. In 1530 he was
made bishop of Durham. In the troublous times of Edward
VI, however, no man of prominence was safe, and Tonstall was
accused of conspiracy in 1552, was deprived of his bishopric,
and was imprisoned in the Tower. On the accession of Mary
he was restored to his bishopric of Durham, but under Eliza-
beth he was again deprived of the honor (1559) and died a
few months later.

Tonstall relates in his work that in his dealing with certain
goldsmiths he suspected that their accounts were incorrect and
therefore renewed his study of arithmetic so as to check their
figures. Having been educated in part at Padua, in a country
that was still the leader in commercial arithmetic, he was fa-
miliar with business methods of computing and was quite pre-
pared to write a book on the subject. On his appointment
to he See of London he bade farewell to the sciences by pub-
lishing this work. He dedicated the book to one of the great-
est scholars and one of the noblest men of his generation, Sir
Thomas More, who, in the Utopia (1516), "the only work of
genius that she [England] can boast in this age," as Hallam
characterized it, had spoken of him as his " colleague and com-
panion . . . that incomparable man Cuthbert Tonstal," whose
"learning and virtues are too great for me to do them justice."

1 Born 1474; died at Lambeth Palace, November 18, 1559. The name is also
spelled Tonstal and Tunstall. The first name is commonly spelled Cuthbert,
but the spelling in the first edition of his work is Cuthebert. The border of the
title-page of this edition, shown on page 315, was engraved by Holbein, whose
initials appear at the left. The border had been used in an earlier book.

TONSTALL AND RECORDE

3*7

Some further idea of the intellectual group in which Tonstal]
moved may be obtained from the fact that it was his friend
Margaret Roper, More's daughter, whom Erasmus addressed
as the u ornament of
thine England." 1

The arithmetic of Ton-
stall was not original,
the material being con-
fessedly drawn from such
Italian writers asPacioli,
but the arrangement was
good and the presenta-
tion was clear even if
unnecessarily extended.
As for an arithmetic
written in Latin, how-
ever, the time had passed
for such a book to be
popular in England. 2

Recorde. The most
influential English math-
ematician whose works
were published in the
1 6th century was Robert
Recorde. 3 He entered
Oxford c. 1525 and be-
came a fellow of All
Souls College in 1531.
In 1545 he received the
degree of M. D. at Cambridge. He taught mathematics in pri-
vate classes both at Oxford and at Cambridge, but after receiv-
ing his degree in medicine he went to London and became
physician to Edward VI and Queen Mary, perhaps absorbing

1<r Margareta Ropera Britanniae tuae decus."

2 John Gill, Systems of Education, p. i. London, 1876.

3 Born at Tenby, Pembrokeshire, c. 1510; died in London, 1558. In some
editions of his works the name is spelled Record.

ROBERT RECORDE

From a recently discovered oil portrait on
wood, apparently made from life, and now
in the possession of W. F. Bushell, Fleet-
wood, Lancashire. The painting bears the
inscription "Rob* Record. M.D. 1556," but
this is now so darkened by age as not to
show in the photograph

3i8 ENGLAND

some of his educational ideas from Roger Ascham (1515-1568);
who was then the Latin secretary to each of these rulers. He
was also versed in the law, but this did not keep him from dying
in prison. In his will he describes himself as " Robert Recorde,
doctor of physicke, though sicke in body yet whole of mynde."
The cause of his imprisonment is not known. Although a
doubtful tradition says that it was for debt, it is more prob-
able that it was for some misdemeanor in connection with the
mines in Ireland, where he was for a time " Comptroller of Mints
and Monies." Recorde may be said to have been the founder
of the English school of mathematics, inasmuch as he wrote
in the English language and showed originality in the treat-
ment of his subjects and in his method of presentation.

at*

FROM THE PROBATE OF RECORDE ? S WILL

It is often stated that the will bears the date June 28, 1558. The will is not in

existence, so far as known, but the official copy is preserved, and the record

shows that this date of probate should be read as xviii die mensis Junii

Recorders Mathematical Works. Recorde published four
books on mathematics and one on medicine, all of which are
now extant, and he seems to have written others which have
been lost and which were probably never printed. The four
mathematical works were written in dialogue, a custom not
uncommon at that time, 1 and bore various fanciful names, as
shown in the following list :

i. The Ground of Artes, printed in London between 1540
and 1542, of which no copy of the first edition is known to be
extant. This was one of the most popular arithmetics printed
in the i6th century. It went through at least eighteen editions

custom appears in the Middle Ages as well, for example, in a loth
century MS. at Munich (S. Gunther, Geschichte des mathematischen Unterrichts
im deutschen Mittelalter, p. 26 n. (Berlin, 1887), hejftafter referred to as
Gunther, Math. Unterrichts) , but of course goes back to Plato. It appears in the
various editions of the anonymous Rithmimachia (Rara Arithmetica, p. 63), in
the arithmetic of Thierfelder (Niirnberg, 1587), and in numerous other works.

ROBERT RECORDE

319

oftottte,

totycbe is tlK feconfte par tc of

Arithmctiketcontotnpng tljmtac*

ttoiiof ttocte*: &be /&{* p;acrife,
toi ft the rate of Sputim :atiD

before 1601 and at least eleven more in the next century. Well
did a writer of 1662 remark that this book was "entaiPd upon
the People, ratified and
sign'd by the approba-
tion of Time." The
work includes compu-
tation by counters as
well as by written fig-
ures, and contains the
usual commercial top-
ics which European
countries north of the
Alps had derived from
Italy. It is commonly
but incorrectly referred
to as the first arithme-
tic printed in the Eng-
lish language, and it
is in fact the earliest
one of much lasting
influence ; but Recorde
mentions the existence
of other works of this
kind, saying:

I doubt not but some
will like this my booke
aboue any other English
Arithmetike hitherto writ-
ten, & namely such as
shal lacke instructers, for
whose sake I haueplain-ly
set forth the exaples, as
no book (that I haue
scene) hath hitherto.

Homlers.

Tbwgb ntany /tones doe learegreate price,
Ibe fthetftone is for ever/See
jfs neadefulljndin T*oorl(e atftrtunge:
Quite tbinga and karde it Wllfo cbattogc,
jfndmah tbem/barpejto rigttgoodvfc:
jfllartefmen tywejbei cannot cbufe,
SHt+febttbetpewtamtnfce,
K9efl>arpenejfeftnH& in it to he.

Xfogroundleofartes dtdtrtdetbitjlone
His \>fi isgrfatfjondmoare then one.
Hen if* ou lift your *//to to ^b^tU,
MockeJharfeneJJe tkertyjballiotigctt*.
Quite toittetberebytlM great elymtnde,
Sbarpe wtes are fined to tbeiffulle evde.
No to prone jm4j>ralfe f ai)ou doefinde,
jfndtojwfel/te not

Cffiftefe UBoofte^ are te bee (bloe,at
tbe 3&eft*00o;e of ldouU5,

TITLE-PAGE OF RECORDERS ALGEBRA, 1557

The first work devoted chiefly to algebra and
the theory of numbers to be printed in England

2. The Castle of Knowledge, printed in London in 1551, a
work on astronomy, and one of the first to bring the Copernican
system to the attention of English readers.

320 ENGLAND

3. The pathewaie to knowledge, printed in London in 1551,
containing an abridgment of Euclid's Elements.

4. The whetstone of witte, printed in London in 1557, "con-
tainyng the extraction of Rootes: The Cossike practise, with
the rule of Equation: and the woorkes of Surde Nombers."
The "cossic art" was another name for algebra, or, as Recorde
says, the subject that begins with "The rule of equation, com-
monly called Algebers Rule." In this work the modern sign of
equality first appears in print.

Such fanciful names as he uses for his titles were then the
fashion, as is seen in numerous other works of the time. 3

As to his arithmetic, England was just beginning to feel the
need for such a book. In Elizabeth's reign, extending from
1558 to 1603, mercantile England came to the front, the native
powers of the country were developed, new manufactures were
introduced, and artisans from all over Europe were encour-
aged to enter the employ of her nobles and her merchant
princes. 2 Never was there a better opportunity for a com-
mercial arithmetic, and never was the opportunity more
successfully met.

As already stated, the Grovnd of Aries was not the first
popular arithmetic in the English language. In 1537 there
appeared at St. Albans an anonymous work entitled An Intro-
duction for to lerne to reken with the Pen and with the
Counters, after the true cast of arismetyke or awgrym in
hole numbers, and also in broken, and this was reprinted in
1539, 1546, 1574, 1581, and 1595, although it never ranked
with Recorders work either in scholarship or in popularity.

Recorde's other works were, naturally enough, not so success-
ful, but they filled the needs for which they were written. It
was no slight honor to have it said that "he was the first that
ever writ of astronomy in the English tongue," 3 even though
the statement is exaggerated.

1 E.g. t Mirror for Magistrates, A Gorgeous Gallery of Gallant Inventions,
Groat's Worth of Wit, Pap with a Hatchet. See G. Saintsbury, History of
Elizabethan Literature, p. n (London, 1887).

2 W. Cunningham, in the Cambridge Mod. Hist., Vol. I, chap. xv. London,
IQ 02. a Aubrey, Brief Lives, II, 200.

MINOR WRITERS

The first rival to Recorders Grovnd of Artes, and the only
serious one that appeared in Great Britain for a hundred years,
was The Well spring of Sciences, 1 written by one Humphrey
Baker 2 in 1562 and published in 1568. It went through five
editions before 1601 and several after that date. The work is
a commercial arithmetic, is evidently under many obligations
to Recorde, and was written to meet the criticism of continental
scholars on the backward state of the subject in England."

Leonard and Thomas Digges. For about a hundred years,
beginning in the second half of the i6th century, the effect of

MILITARY MATHEMATICS IN 1572

From the Arithmeticall Militare Treatise, named Stratioticos, by Leonard and
Thomas Digges, London, 1572. This cut is from the 1579 edition

the continental wars showed itself in the textbooks on arith-
metic and practical geometry, particularly in England and
France. Problems relating to military affairs became more
numerous, and even the titles of mathematical works bore
evidence of this tendency. The most notable example in Eng-
land is seen in a work by Leonard Digges (died c. 1571) and

x The form in which the title appears in the 1580 edition.
2 He was a native of London and died after 1587. In some editions the name
is spelled Humfrey.

3 See the quotations in Kara Arithmetica, p. 327.

322

ENGLAND

his son Thomas Digges (died 1595), entitled An Arithmeticall
Militare Treatise, named Stratioticos (London, 1572). Leon-
ard Digges came of an ancient Kentish family. He studied
at University College, Oxford, but took no degree. He was
a mathematician of ability, his chief interest lying in the

application of the science
to surveying, military
engineering, and archi-
tecture. He published in
1556 a work on men-
suration, the Tectonicon.
He also wrote another
work on geometry which
his son Thomas pub-
lished in 1571 and again
in 1591 under the title
A Geometricall Practise,
named Pantometria.^
The Stratioticos was be-
gun by him and com-
pleted by his son.
Thomas matriculated at
Cambridge as pensioner
of Queens' College in
1546, taking his B.A.
in 1551 and his M. A.
in 1557, and attained

first rank among the English mathematicians of his time.
His own works related chiefly to astronomy and navigation,
and, like Recorde, he was probably a believer in the Copernican
theory, although he did not openly advocate the doctrine.

Of the 1 6th century writers on applied mathematics one of
the earliest was Richard de Benese, whose Boke of Measuring
of Lande was published between 1562 and 1575, probably the
earliest book on surveying printed in England.

JOHN DEE

After a portrait from life

ltl At the end he discourses of regular solids, and I have heard the learned
Dr. John Pell say it is donne admirably well." Aubrey, Brief Lives, I, 233.

JOHN DEE

323

line f P namely, t
for. CD ft o

The 8- Tltoreme. The

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r,ght Unr wh,J> Wil reared ob

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eqtult&onetothe other.

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therm thtpeint E Thtnlfay, thjl the angle A EC, u <</wtf(ki At
't) '& Forfira/muib u tht rijht lini At .jlnJith+fgink
f Imt <D C, making thtfeatfUs CE A^ndA E > thtitjtnfo
\tio jthtantflts C t A,&nJi AE. D^re f^iuiito tTtu n^jtui^Lt

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loth

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r,chtl,ntscuttheone theother,th ^Jjn^tsJ/'ull>f^uaU .'i

John Dee. Toward the close of the century there lived one
of the most curious characters to be found among the scientific
men of his time. This man was John Dee (1527-1608), a
native of London and a
student of St. John's Col-
lege, Cambridge. He took
his B.A. in 1546 and was
selected as one of the
original fellows of Trin-
ity College. His studious
nature may be inferred
from his own record of his
rules, "only to sleep four
houres every night ; to al-
low to meate and drink
(and some refreshing
after) two hours every
day; and the other eight-
een hours (except the
tyme of going to and being
at divine service) was
spent in my studies and
learning." He afterwards
traveled on the continent,
forming friendships with
Gerard Mercator, Gemma
Frisius, Jean Fernel, and
other scholars, taking
courses for two years at
Louvain, and publicly lecturing on Euclid before large audiences
in Paris. He became interested in alchemy and astrology and his
relations to the occult made his life a romantic one. 1 He wrote
the preface to the first English translation of Euclid's Elements?

1 Charlotte Fell Smith, John Dee, London, 1000.

2 The Elements of Geometrie of the most auncient Philosopher Evclide of
Megara. Faithfully (now first) translated into the Englishe toung, by H.
Billingsley, Citizen of London. . . . With a very fruitfutt Praeface made by
M. I. Dee, London, 1570.

TA^A/.''/.i"thePhilofopherwisthefirft inuenter o An JjJjJJJ
To conamQionatall Forthecxpo(itionolthcthinggeue,i$lufficiertnisfc

//^^/.<:appofitcnglet,oufcd ofthe taterfedionol vo ngjitlm*
iJarefoalIcd,beufethehfdde$ of the two ngl< ioyMdt<nw
onepo.nte.

T*r /< oftt/aprofofibnafter'Pelttanut.

FROM

r7ght'uoe$ riuti&thetwo nghtK

THE FIRST ENGLISH EDITION
OF EUCLID

The Billingsley or Dee translation,
London, 1570

324 GERMANY

and may, indeed, have made the translation in whole or in part
himself, although it is attributed to Sir Henry Billingsley, who
was later (1596) Sheriff and Lord Mayor of London. 1 Dee
was a man of erudition and of remarkable powers of exposition,
and his influence on mathematics in his day must have been
considerable.

Minor Writers. Among the books containing algebra and
printed in England in the i6th century there was an arithmetic
written by Thomas Masterson and published at London in
three books in 1592-1595. Concerning Masterson himself
nothing is known. He planned a treatise in six books but com-
pleted only three, the third consisting of that part of algebra
that has to do with the powers and roots of numbers.

The century closes with the name of Thomas Blundeville,
who published (1594) a work entitled Exercises, containing
sixe Treatises? one of these parts being on arithmetic and the
rest being on cosmography. The arithmetic contains "a brief e
description of the tables of the three speciall right lines belong-
ing to a circle, called sines, lines tangent, and lines secant," the
first fairly complete treatment of trigonometry in England. 3
He also published several other books, partly on cosmography.

5. GERMANY

Nature of the German Mathematics. When we compare the
mathematics of Germany with that of France in the i6th cen-
tury, we are struck by the same difference that existed in the
art of the two countries. The mathematics of Germany was
Gothic, unpolished, but virile ; the mathematics of France was
Renaissance, polished, but generally weak. Germany produced
a notable group of arithmeticians; France produced hardly
more than one. Germany produced two strong algebraists;
France produced one, a dilettante but a brilliant one. Ger-
many made a definite advance in geometry, in the study of

this question see A. De Morgan, British Almanac and Companion for
, 1837, p. 38 of the Companion. 2 The seventh edition appeared in 1636.

8 A. De Morgan, British Almanac and Companion for 1837, p. 42 of the
Companion; Arithmetical Books, p. 30.

NATURE OF THE MATHEMATICS

325

higher plane curves; France was content to contemplate the
past and possibly to dream of the century just ahead.

Of course the greatest influence for advance in the i6th cen-
tury was printing; but there was also Erasmus (1467-1536),

FROM THE MARGARITA PHYLOSOPHICA (1503)

Showing Arithmetica between the ancient counter reckoning and the modern
algorism. This is from the edition of 1504

who, although born in Rotterdam, lived in Germany, England,
France, and Italy, and was the world's scholar of the first third
of the 1 6th century; and there was also Martin Luther (1483-
1546), who set Germany thinking, not always for the best. It
was a century of intellectual awakening and of breaking away
from traditions, and all this showed itself in the mathematical

326 GERMANY

activity of the time. As in other countries, the number of
names connected with this activity now becomes so great as
to allow for only a limited selection.

The Margarita Phylosophica. The first modern encyclopedia
of any note, based upon the late Latin models, was the Marga-
rita Phylosophica? first published at Freiburg in 1503. The
author was Gregorius Reisch (died 1523), who studied at Frei-
burg in 1487 and took his bachelor's and master's degrees
there. He became a Carthusian and was made prior at Frei-
burg and confessor to Maximilian I. The work consists of
twelve books, and includes considerable material upon arithme-
tic, geometry, and astronomy. Its popularity is shown by the
fact that there were sixteen editions in the course of a century.

Albrecht Diirer. It is not often that the artist of today is
confessedly a mathematician ; the mathematician is more fre-
quently an artist. But the i6th century has upon its roster
the names of several great artists who did something in mathe-
matics, generally in architecture or in perspective. One man,
however, stands out with special prominence as a great artist
and at the same time as a mathematician with distinctly new
interests, and this is Albrecht Diirer. 2 His work as a painter
and engraver is well known, but it was in his treatises on geom-
etry^ fortification, 4 and human proportion 5 that he showed his

ir l'he title of the second (1504) edition, printed at Strasburg, reads, Aepltoma
omnis phylosophiae. alias Margarita phylosophica tractans de omni genere
scibili: Cum additionibus: Que in alijs non habentur.

2 Born at Nurnberg, May 21, 1471; died at Nurnberg, April 6, 1528. Cantor,
Geschichte, II, chap. 63 ; S. Giinther, Die geometrischen Ndherungskonstruk-
tionen Albrecht Durers, Ansbach, 1886; S. Giinther, "Albrecht Diirer, einer der
Begriinder der neueren Kurvenlehre," Bibl. Math. (1886), p. 137; H. Staig-
miiller, Diirer als Mathematiker, Prog., Stuttgart, iSqi; F. Amodeo, "Albrecht
Diirer precursore di Monge," Atti della R. Accademia delle Soc. Fis. e Mat.,
XIII (2), No. 16; W. B. Scott, Albert Durer, his Life and Works, London, 1869.

3 Underweysung der messung mit dem zirckel und richtscheyt in Linien,
ebnen, vnnd gantzen corporen, Nurnberg, 1525, with Latin editions, Paris, 1532
and 1555, and Arnheim, 1605; Institutiones Geometricae, Paris, 1532.

*Etliche vnderricht zu befestigung der Stett, Schloss und Flecken, Nurnberg,
J S 2 7> 1530, and 1538, with a Latin edition, Paris, 1535.

G Hierin sind begriffen vier Bticher von menschlicher Proportion, Nurnberg,
1528, with a Latin edition, Numbers, 1532,

STIFEL 327

mathematical powers. The geometry was the first printed work
to consider the subject of higher plane curves, and the first
to discuss scientifically the question of such approximate con-
structions as that of the regular heptagon.

Johann Stoffler, 1 although professor of mathematics at the
University of Tubingen, can be ranked as a mathematician
only because of his computation of astronomical tables. He
was one of the first to show how the Julian calendar could be
brought into harmony with astronomical events. His calcula-
tions led him to the absurd prediction, however, that the
Deluge would be repeated in the year 1524. The announce-
ment stirred all Europe, and the number of schemes to protect
the race was legion. The people of Toulouse even went so far
as to build an ark. Stoffler, however, seems to have survived
the storm of protest that ensued upon the failure of his predic-
tion, for he published in the year of his death a new ephemeris,
and left a commentary on the Sphere of Proclus, which was
published posthumously in 1534."

Stifel. The first German writer of the century to devote
his life to mathematics and to acquire an enviable reputation
in this field was Michael Stifel. 3 A lover of mathematics from
his childhood ; brought up in Esslingen, a veritable bulwark of
the ancient faith ; trained in the local Augustine convent and
taking holy orders ; giving promise of success in the Church ;
he was finally captured by the eloquence of Luther, and thought
himself a reformer when he was really a fanatic. Starting for
heaven with a group of peasants on the day which he had
prophesied would see the blotting out of this world, he ended
ignominiously behind the bars of a jail. Since Luther had
launched him on a career that landed him in prison, it was
proper that he should get him out, which he did. The state of

1 Born at Justingen, Swabia, December 10, 1452; died at Blaubeuern,
February 16, 1531.

2 The curious case is discussed in Bayle's Dictionaire, under Stofler.

3 Stiefel, Styfel, Stiffelius, Born at Esslingen, April 19, 1487 (some say 1486) ;
died at Jena, April 19, 1567. See Th. Miiller, Der Esslinger Mathematiker
Michael Stifel, Prog., Esslingen, 1897; Cantor, Geschichte, II, chap. 62.

328 GERMANY

StifePs mind may further be seen by the following line of
reasoning in which he indulged :

1. The Latin for Leo Tenth is Leo Decimus.

2. This may be written Leo DeCIMVs.

3. These capitals may be arranged thus: MDCLVI.

4. We may take away M for Mystery, and add X because
it is Leo X, and we then have DCLXVI.

5. But this is 666, the "Number of the Beast" in the Book
of Revelations ; and hence Leo X is the Beast.

Yet this is the man who, in the next few years, produced
some of the most original and vigorous mathematical works to
be found in the i6th century.

Stifel wrote five works on mathematics, these works treating
chiefly of the mysticism of numbers, 1 arithmetic, 2 and algebra. 3
His arithmetic, which is largely on algebra, is a more scholarly
work of the kind than any that had yet appeared in Germany,
doing for that country what Cardan's and Tartaglia's treatises
were doing for Italy.

Christoff Rudolff. StifePs chief work on algebra was his
edition (iS53- I 554) of a book known as the Coss, which ap-
peared in 1525 and was the first algebra of any moment to be
published in Germany. It was written by Christoff Rudolff, 4
concerning whose life very little is known. Rudolff published
three books, the Coss (152 5), the Kunstliche rechnung* (1526),

* Ein Rechen Biichlein, Wittenberg, 1532.

2 Arithmetica Integra, Niirnberg, 1544, to which Mclanchthon wrote the
preface; Deutsche Arithmetica, Nurnberg, 1545; and Rechenbuch von der
Welschen vnd Deutschen Practick, Nurnberg, 1546. Before the World War of
1914-1918 there was preserved in the University of Louvain a copy of the
Arithmetica Integra with marginal notes by Gemma Frisius. Fortunately these
notes were copied by H. Bosnians, S. J., and several of the most important
ones were published before the destruction of the library.

3 Die Coss Christoffs Rudolfis, Konigsberg, 1553-1554, 1571, and 1615. While,
as stated above, this is Rudolff 's work, it contains Stifel's commentary. The
Arithmetica Integra also contains the treatment of radicals that is now con-
sidered part of algebra. The word Coss is from cosa (causa), which, as explained
in Volume II, refers to the first power of the unknown quantity.

4 Born at Jauer, c. 1500.

6 The title of the third edition (Nurnberg, 1534) is Kunstliche rechnung mit der
zifier vnnd mit den zal pjenninge. There were eleven editions in the i6th century.

RUDOLFF AND GRAMMATEUS 329

and a collection of problems 1 ( 1 530) . Of these the Coss was the
most important, doing for algebra in Germany what Pacioli's
Suma had done for the subject in Italy. All such works, how-
ever, were for the scientific elite of the universities, not for the
elementary Latin schools. Of forty-six Schulordnungen of the
1 6th century only twenty-four gave mathematics any place, and
most of these were issued only in the second half of the century.

Johann Scheubel. In marked contrast to Stifel was his con-
temporary, but slightly his junior, Johann Scheubel. 2 Stifel was
brilliant, Scheubel was scholarly ; Stifel was popular, Scheubel
was heavy ; Stifel was eccentric, Scheubel was balanced ; and
Stifel was effusive, while Scheubel was a man of dignity and
poise. The University of Tubingen called Scheubel to a pro-
fessorship of mathematics at about the same time that Stifel
was sent to prison, and at Tubingen he wrote his arithmetics/ 5
Latin works that were too heavy for commercial purposes and
too light for his own students. Here, too, he wrote his algebras,
one of which he published, 4 leaving the other in manuscript, 5
and here he edited the seventh, eighth, and ninth books of
Euclid's Elements (1558). He also left in manuscript a copy
of Robert of Chester's translation of al-Khowarizmi's algebra/'
He gave the so-called Pascal Triangle a century before Pascal
wrote upon it, and extracted roots as high as the 24th by a
process similar to the one which employs the Binomial Theorem.

Grammateus. Of about the age of Scheubel and with similar
interests, but not so learned, Heinrich Schreyber 7 made for

1 Exempel-Buchlin, Augsburg, 1530, a commercial arithmetic.

2 Scheybl, Scheubelius, Scheybel. Born at Kirchheim, Wiirttemberg, Au-
gust 18, 1494; died February 20, 1570.

z De Nvmeris et Diversis Rationibvs sen re&uiis computationum opusculum,
a loanne Scheubelio compositum, Leipzig, 1545; Compendium Arithmeticae
Artis, Basel, 1549 (this title from the 1560 edition).

4 Algebrae compendiosa jadlisque descriptio, Paris, 1551.

5 This is now in the 1 library of Columbia University, New York City. His
other manuscripts were left to the University of Tubingen.

6 This is bound with the Columbia MS. above mentioned, and the translation
by Professor Karpinski was published in 1915.

7 Born at Erfurt as early as 1496. The name appears as Schreyber in one of
his works of 1523. but is given by modern writers as Schreiber.

330 GERMANY

himself a worthy place in German history. He was better
known in maturity by his Latinized Greek name of Henricus
Grammateus, and in his young manhood by the Latin name
of Henricus Scriptor. He studied at Cracow and was later
(iSo?) 1 enrolled in the University of Vienna, where he was
afterwards an instructor. He took his bachelor's degree in
1511 and his master's degree in 1518, thereafter teaching for
a time in the University and privately, and having Rudolff
for one of his pupils.- Driven from Vienna by the plague
( 1521 ), he went to Niirnberg and Erfurt, but returned a little
later ( 1525 ) and devoted himself to writing. His best-known
work was an arithmetic in the German language/ It includes
arithmetic computations with counters and figures, a little
work in the theory of numbers, a chapter on bookkeeping,
a few of the simplest rules of algebra, and a brief treatment
of the gaging of casks. He is the first German writer to make
free use of the signs 4- and in the treatment of algebraic
expressions, although the symbols had long been used for other
purposes. He also published other works on arithmetic, 4 and
wrote on the theory of proportion 5 and on mensuration , (J

Ludolf van Ceulen. One German writer, Ludolf van Ceulen, 7
may quite as well be classified among the Dutch mathemati-
cians, since he spent most of his life in Holland. He seems to

*For a record of the time re^ds: "Anno domini millesimo quingentesimo
septimo . . . Henricus Scriptoris de Erfordia."

2 Rudolff writes: "Ich hab von meister Heinrichen so Grammateus genennt
der Coss anfengklichen bericht emphangen. Sag im darumb danck."

*EYn new kunstlich behend vnd gewiss Rechenbuchlin vff alle Kauffman-
schafft . . . QBuchhalten durch das Zornal . . . <\Vister ruten. . . . M. Henri-
cus Grammateus, Vienna, 1518, with later editions. This title is from the
1535 edition. C. F. Miiller, Henricus Grammateus, pamphlet, Zwickau, i8q6.

4 Behend unnd khunstlich Rechnung nach der Regel und welkisch practic,
Niirnberg, 1521; Algorismus de integris Regula de tri cum exemplis, Erfurt,
1523; Eynn kurtz newe Rechenn unnd Visyrbuechleynn, Erfurt, 1523, a Visier-
buch being a work on the gaging of casks.

5 Algorithmus proportionum, Cracow, 1514.

6 Libellus de compositions regitlarum pro vasorum mensuratione, Vienna, 1518.

7 Born at Hildesheim, January 18, 1540; died at Leyden, December 31, 1610.
The name also appears as Ludolph van Collen, Cuelen, and Keulen. See
D. Bierens de Haan in Boncompagni's Bullettino, XIV, 571.

THE CLASSICAL GROUP 331

have left Germany in his childhood and to have been educated
under Dutch influences. He taught mathematics in Breda,
Amsterdam, and Leyden, and became professor of military
engineering in the University of Leyden in 1600. He is known
chiefly for his value of TT, at first given 1 to 20 and then to 35
decimal places. 2 He also published (1615) a work on arith-
metic and geometry.

Mention should also be made of Johann Werner, 3 a priest,
who was interested chiefly in astronomy but who wrote the
first original work on conies to appear in the i6th century. 4

Pitiscus. The last of the German writers on mathematics
whose work falls chiefly in the i6th century and whose con-
tributions entitle him to special mention is Bartholomaus Pitis-
cus. 5 He was a clergyman by profession but a mathematician
by preference. His trigonometry was the first satisfactory
textbook published on the subject and the first book to bear
this title. He also edited and perfected the table of sines of
Rhaeticus. 7

The Classical Group. The name of Philip Melanchthon,* the
friend and colleague of Luther and professor of Greek at Wit-
tenberg, does not ordinarily suggest the science of mathematics.
His wide range of human interest, however, led him not only

1 Van den Circkel, Delft, 1506, with editions in 1615 and (Latin) ibiq.

-De arithmetische en geometrische jondamenlen, Leyden, 1015, with a Latin
edition by Snell, Fundamemta (sic) Arithmetica et Geomctrica, Leyden, the
same year.

3 Born at Nurnberg, February 14, 1468; died at Nurnberg, 1528.

4 Libellus super viginti duobvs dementis conicis, Nurnberg, 1522. He also
wrote De Triangulis Libri IV \ see Abhandlungen, XXIV.

: ' Horn at Schlaun, near Grunberg, Silesia, August 24,1561; died at Heidelberg,
July 3, 1613. The name also appears as Petiscus.

" Trigonometriae sive de dimension e trtangttlorum libri quinque, Frankfort,
iSQS, as an appendix to the astronomy of Abraham Scultetus, or Abraham
Schultz (1566-1625). Complete editions by Pitiscus were published at Frank-
fort in 1599, 1608, and 1612, and at Augsburg in 1600. An English translation
appeared in London in 1630. In the edition of 1612 Pitiscus makes use of a
decimal point, sin 10" being given as 4.85 for r = 100,000.

7 Thesaurus mathematicus sive canon sinuum ad radium 1,0000 oooo oooo
. . . Frankfort, 1 1 593. On Rhaeticus, see page 333.

8 Greek form for Schwartzerd, his family name. He was born at Bretten,
Baden, February 16, 1497; died at Wittenberg, April 19, 1560.

332 GERMANY

into the fields of philosophy and religion, but also to the
study of astronomy and mathematics. He edited (1521-1560)
several works on astronomy by Aratus, Peurbach, Schoner, al-
Fargani, Sacrobosco, and Ptolemy. His activity in pure mathe-
matics is shown in a preface to StifePs Arithmetica Integra
(1544) and in a work on the mathematical disciplines. 1 He
is to be valued, however, for his influence rather than for his
contributions in the field of the exact science.

With the name of Melanchthon is naturally connected that
of his friend and associate, Joachim Camerarius, 2 a distin-
guished classicist and a professor at Tubingen and Leipzig.
His edition of Nicomachus'' appeared at Augsburg in 1554,
and a work on computation, De logistica, was published there
at the same time. He also wrote some astronomical verses,
which were published with Melanchthon's work of 1540.

Another one of the classical group who did a certain amount
of work in mathematics was Jacobus Micyllus. 4 His arith-
metic 5 was written for the Latin schools, and it contains a
considerable amount of ancient material. His treatment of sexa-
gesimal fractions is unusually extensive.

A little later than Micyllus there lived the well-known scholar
Michael Neander," who became professor of mathematics and
Greek (1551) in the University of Jena, and later (1560)
professor of medicine in the same institution. 7 He wrote an

^Mathematicarvm disciplinarvm, tvm etiam astrologiae encomia, Leyden, 1540.

2 Born at Bamberg, April 12, 1500; died at Leipzig, April 17, 1574. The
family name was Liebhard, but the office of chamberlain to the Prince-
Bishop of Bamberg being hereditary in this family, he took the Latin name
of Camerarius (chamberlain).

3 The title of the Daventcr (1667) edition is Explicatio loachimi Camerarii
Papebergensis in dvos libros Nicomachi Geraseni Pythagorei Deductionis Ad
Scientiam Numerornm.

4 Born at Strasburg, April 6, 1503; died at Heidelberg (?), January 28, 1558.
The name also appears as Moltzer, Molshem, Molsehm, and Molshehm.

^Aritkmcticae logisticae libri duo, Basel, 1555.

r >Born in the Joachimsthal, April 3, 1529; died at Jena, October 23, 1581.
The family name was Neumann, whence the classical form Neander.

7 On the relation of medicine to mathematics at this time, see D. E. Smith,
M Medicine and Mathematics in the Sixteenth Century," Annals of Medical
History, New York, 1917, p. 125.

MATHEMATICAL ASTRONOMERS 333

excellent work on metrology (Basel, 1555), one of the first to
treat of the subject historically and in a scholarly manner.
He also wrote on spherics. 1

Still another of the classical group was Guilielmus Xylander, 2
professor of Greek at Heidelberg. He translated the first six
books of Euclid's Elements into German (Basel, 1562), and
various works from Greek into Latin, including the Arithme-
tica of Diophantus (Basel, 1575) and the work of Psellus
(Basel, 1556). His Opuscula Mathematica appeared at Heidel-
berg in 1577 and contains a certain amount of work on astron-
omy, arithmetic, algebra, 3 and geometry.

Mathematical Astronomers. Although mathematics and as-
tronomy were no longer synonymous terms, the i6th century
produced one or two scholars whose interests in the two sciences
were apparently about equal. The first of these was Petrus
Apianus. 4 He wrote chiefly on astronomy, but his arithmetic 5
is interesting because it contains the first triangular arrange-
ment of the binomial coefficients (the Pascal Triangle) to
appear in print. This appeared some years before Stifel men-
tioned the subject. Apianus was professor of astronomy at
Ingolstadt, was interested in the teaching of trigonometry, and
was one of the few university professors of his time to give
instruction in the German language.

The leading mathematical astronomer in the Teutonic coun-
tries in the middle of the i6th century was Georg Joachim
Rhseticus/' He studied at Zurich and Wittenberg and was

1 There was another Michael Neander of the same period (1525-1595), who
wrote on physics, theology, and philology.

2 Wilhelm Holzmann. Born at Augsburg, December 26, 1532 ; died at Heidel-
berg, February 10, 1576.

3 "Dc svrdis, qvos vocant, nvmeris iis, qvi a qvadratis primo nascuntur,
Institutio docendo explicanda."

4 Born at Leisnig, 1495; died at Ingolstadt, April 21, 1552. Known also by
his German name, Peter Bienewitz or Bennewitz.

5 Eyn Newe Vnnd wolgegriindte vnderweysung alter Kauffmanss Rechnung,
Ingolstadt, 1527. His Cosmographia appeared in 1524.

6 Georg Joachim von Lauchen, his last name being derived from his home
region, the ancient Rhaetia. Born at Feldkirch, in Vorarlberg, February 16, 1514;
died at Kaschau, Hungary, December 4, 1576.

334

GERMANY

professor of mathematics in the latter university from 1537
to 1542. He published an arithmetic at Strasburg in 1541,
but most of his work was on astronomy and trigonometry. 1
He visited Copernicus in 1539, studied with him, and did

much to make his
theories known.

Clavius. Proba-
bly the man who
did the most of all
the German schol-
ars of the 1 6th
century to extend
the knowledge of
mathematics, al-
though doing little
to ex tend its bound-
aries, was Chris-
topher Clavius, 2 a
Jesuit, who passed
the later years of
his life in Rome.
He was an excel-
lent teacher of
mathematics, and
his textbooks were
highly esteemed
because of their
arrangement, par-
ticularly in the Latin schools. Sixtus V testified to his standing,
saying: "Had the Jesuit order produced nothing else than
this Clavius, on this account alone should it be praised." His
arithmetic 11 was published in Rome in 1583, was translated
into Italian in 1586, and went through several editions. His

*Opus Palatinum de Triangulis, Neustadt a. Hardt, 1596; Thesaurus Mathe-
maticus, Frankfort a. M., 1613, both posthumous.

2 Christoph Klau. Born at Bamberg, 1537; died at Rome, February 6, 1612.
3 Epitome Arithmeticae Practicae* Rome, 1583.

CHRISTOPHER CLAVIUS
Engraved after a portrait from life

CLAVIUS 335

algebra 1 appeared in 1608 and was one of the best textbooks
on the subject that had been written up to that time. He pub-
lished an edition of Euclid in 1574.^ While not precisely a
translation, the book proved very valuable because of the great
erudition shown in the extensive scholia which it contains.

Clavius was one of the mathematicians engaged in the reform
of the calendar (1582) under the direction of Pope Gregory
XIII. His collected works 5 contain, in addition to his arith-
metic and algebra, his commentaries on Euclid, Theodosius,
and Sacrobosco, his contributions to trigonometry and astron-
omy, and his work on the calendar.

Johann Schoner. Among the minor German writers on as-
tronomy and mathematics in this period Johann Schoner 4 and
his son Andreas Schoner (1528-1590) are perhaps the best
known. Johann was for a time a preacher at Bamberg, and
was later a teacher in the Gymnasium at Niirnberg. He edited
a well-known medieval arithmetic 5 and wrote on geometry,
astrology, and astronomy. Andreas wrote on astronomy and
dialing and edited his father's works.

German Arithmetic. The German arithmetics in the i6th cen-
tury were very practical. 7 Commerce was active, and Frank-
fort was one of the great trading centers of Europe, her agents
pushing out through the Hansa towns to England, France, and
the northern countries, as well as through Austria and Italy to
the Orient. The great house of Fugger multiplied its capital
tenfold be ween 1511 and 1527, possessing five times the
wealth which made the Medici so powerful in the preceding

1 Algebra Christophon Clavii Bambergensis, Rome, 1608.

2 Euclidis Elementorum Libri XV, as the title appears in the Frankfort edition
of 1654. The earlier editions were 1574, 1589, 1591, 1603, 1607, 1612, which
show the popularity of the work.

B 0pera Mathematica, 5 vols., Mainz, 1611 and 1612.

4 Born at Karlstadt, near Wiirzburg, January 16, 1477; died at Niirnberg,
January 16, 1547. The name also appears as Johannes Schonerus or Schoner.

6 Algorithmvs demonstrates, Niirnberg, 1534.

(l } Opera Mathematica, Niirnberg, 1561.

7 Hugo Grosse, Historische Rechenbticher des 16. und 17. Jahrhunderts,
Leipzig, i oo i.

336 GERMANY

century. But whereas the Medici had been the patrons of art
and of letters, the Fuggers were little beyond accumulators of
wealth. As a result of this national spirit, the German mathe-
matics of this period was more largely commercial than it
might have been under more favorable circumstances.

While the classical group was carrying mathematics to the
intellectual aristocracy a group containing a few skillful writers
was carrying it to the intellectual democracy. One of them,
Johann Boschensteyn, 1 like Stifel a native of Esslingen, taught
both Luther and Melanchthon, a sufficient honor for any
man. He was a professor of Hebrew at Ingolstadt, Heidelberg,
Niirnberg, and Antwerp, but his mathematical contribution
was of a humble nature, consisting merely of an arithmetic of
a commercial kind, printed at Augsburg in 1514.

Contemporary with Boschensteyn, and indeed but two years
his senior, was one whose influence on the people was far
greater, Jakob Kobel, 2 a native of Heidelberg. He and Coper-
nicus were fellow students at Cracow, and each had varied
lines of interest. Kobel began as a teacher of arithmetic, a
printer, a woodcarver, a poet, and a student of law, and ended
as a petty officeholder; while Copernicus began as a priest,
physician, and astronomer, and ended as a giant among the
thinkers of the world. Kobel wrote three arithmetics, although
the editions varied so much as to give an impression of a larger
number. These arithmetics were the Rechenbiechlin* (1514),
Mit der Krydcn 4 (1520), and the Vysierbuch 5 (s.a. but 1515).
Of these the first was the most important, passing through no
less than twenty-two editions in the i6th century. It is purely
commercial but shows more vigor and originality than any
arithmetic that had yet appeared in Germany. Among other

1 Among the variants of the name are Beschenstein, Boeschenstain, Bos-
senstein, Boechsenstein, Buchsenstein, Poschenstein, and Besentinus. Born at
Esslingen, Swabia, 1472; died 1540.

2 Kobel, Kobelius, Kobelinus. Born at Heidelberg, 1470; died at Oppenheim,
January 31, 1533.

3 As spelled on the title-page; Rechenbiichlein as spelled in the colophon.
4 Mit der Kryde od' Schreibfedern dutch die zeijerzal zu reche Bin neiiw
Rechepuchlein, Oppenheim, 1520.

5 0n gaging. In the 1537 edition, Eynnew Visir Buchlin.

ADAM RIESE

337

features are the crude illustrations, although similar ones
had already been used by Widman as early as 1489.

Adam Riese. The greatest of all the Rechenmeisters of
this century, however, was Adam Riese. 1 He was the most
influential of the
German writers
in the movement
to replace the
old computation
by means of
counters ("auff
derLinien") by
the more mod-
ernwrittencom-
putation ("auff

Federn"). He
wrote four arith-
metics, 2 all of
a commercial
nature and the
second ranking
as one of the
most popular

schoolbooks of Germany's best-known Rechenmeister. From the title
fh *~ntnr Tf P a & e ^ hk Rechenung nach der lenge/auff den Linihen

me century, it vnd Feder ^ Leipzigj ISSO

was to Germany

what Borghi's book was to Italy, Recorders to England, and

Gemma's to the Latin schools of the Continent. So famous

ADAM RIESE

possibly at Staffelstein, near Bamberg, c. 1489; died at Annaberg,
March 30, 1550. The name also appears as Ryse, Ris, and Reis. See B. Berlet,
Adam Riese, sein Leben, sein Rechenbucher und seine Art zu rechnen, new ed.,
Leipzig, 1892.

2 Rechnung aufi der linihen, printed in 1518 and again in 1525 and 1527. Title
from the 1525 edition.

Rechnung auff der Lynihen vn Federn in zal/mass/vnd gewicht auff
allerley handierung gemacht, Erfurt, 1522, with at least thirty-seven editions
before 1600.

Bin Gerechent Buchlein, Leipzig, 1536.

Rechenung nach der lenge/auff den Linihen vnd Feder, Leipzig, 1550.

338 THE NETHERLANDS

was the author that the phrase "nach Adam Riese" is still
used in Germany to signify arithmetical accuracy or skill.

It was due in no small degree to the encouragement given
by Luther that books like Riese's met with their great success.
At the very time that the great Rechenmeister was preparing
the second edition of his most successful work 1 Luther was
laying down his famous doctrine that all children should study
mathematics, 2 a thing unheard of before.

The only other Rechenmeister of the century to deserve
special mention is Simon Jacob/' who wrote two commercial
arithmetics. Blirgi mentions Jacob's treatment of series, and
apparently the former's table of antilogarithms, the Progress
Tabulen, 4 was suggested by the nature of exponents as laid
down in these and similar books of the i6th century.

6. THE NETHERLANDS

Geographical Limits. When we speak of the Low Countries,
the Netherlands, it should be borne in mind that boundaries
and governments were constantly shifting at this time, and
that when Charles V (died 1558) inherited this territory from
his grandmother, Mary of Burgundy, it included seventeen
provinces, each with its own government. When Philip II (died
1598) inherited it from Charles, the new ruler was accepted
with a pronounced expression of discontent, and finally (1568)
the provinces rebelled under William of Orange. The period
was therefore one of uncertain geographical limits, and in
speaking of the Netherlands it must be understood that \ve
speak of territory of which a part was under Spanish rule
during the i6th century. In general, therefore, we shall include
for our present purposes what is now Holland and Belgium.

n That is, in 1524, the edition appearing in 1525.

2 This is in his Schrijt an die Ratsherren alter Stddte Deutschlands, wherein
he says: "Wenn ich Kinder hatte, und vermochts, sie miissten mir nicht allein
die Sprachen und Historien horen, sondern auch singen und die Musika mit der
ganzen Mathematika lernen."

3 Born at Coburg; died at Frankfort a. M., June 24, 1564. His arithmetics
appeared at Frankfort a. M. in 1557 and 1565.

* Discussed in Volume II, Chapter VI.

GENERAL CONDITIONS 339

General Conditions. The general conditions in the country,
aside from those imposed by dynastic ambitions, were favor-
able to commercial and intellectual advance. The Netherlands
had shown considerable commercial activity in the isth cen-
tury. The economic decline in the latter part of that period,
owing to British rivalry, had been somewhat overcome by their
success in navigation and in the fishing industry. 1 Their arith-
metics reflect this commercial activity," showing an extensive
trade with Niirnberg, Frankfort, Augsburg, Danzig, and other
German towns ; with Cracow, Venice, and Lyons ; and with
Spain.

In matters of scholarship the work of that great humanist of
the North, Erasmus of Rotterdam (1467-1536), had told in
Holland as it had told in every other intellectual center of
Europe. Leyden was becoming one of the great forces of the
world. The Netherlands were ready to do their part.

Mathematics in the Netherlands. The first of the writers of
consequence in this territory in the i6th century was Joachim
Fortius Ringelbergius, 3 a student at Louvain and a teacher of
philosophy and mathematics in various places in France and
Germany. He wrote on astronomy, optics, and arithmetic,
and his collected works, 4 encyclopedic in character, were pub-
lished at Leyden in 1531. The book was filled with the usual
erudition of the time but contained nothing that advanced the
bounds of science.

Late in the i6th century Adriaen van Roomen/' a student of
both medicine and mathematics in Italy and Germany, became

J A. W. Ward, in the Cambridge Mod. Hist., I, chap, xiii; W. Cunningham,
ibid., chap. xv.

2 For a list of these books, not complete, see D. Bieren* de Haan, "BibMo-
Eraphie neerlandaise historico-scientifique," in Boncompagni's Bidlettino, XIV,
519; XV, 355; hereafter referred to as Bierens de Haan, Bibliog. See also the
Kara Aritkmetica for a list of books on arithmetic. There is also a list of Dutch
arithmetics in the preface to the 1600 edition of Coutereers Cyfier-Boeck.

3 Born at Antwerp, c. 1490; died c. 1536. In the vernacular the name
appears as Joachim Sterck van Ringelbergh.

4 loachimi Fortii Ringelbergij Andouerpiani opera, Leyden, 1531.

6 Born at Louvain, September 29, 1561; died at Mainz, May 4, 1615. The
Latin form Adrianus Romanus and the French form Adrien Romain are often

340 THE NETHERLANDS

professor of these two sciences in Louvain. He then became
professor of mathematics at Wiirzburg, and finally was ap-
pointed royal mathematician (astrologer) in Poland. While at
Louvain he published the first part of a general work on mathe-
matics, 1 and in this he gave the value of TT to seventeen decimal
places, an unusual achievement at the time. His other works
include one on the treatment of the circle by Archimedes
(1597) and one on spherical triangles (1609).

There lived in Holland in the middle of the century an en-
gineer who also had the name of Adriaen. His father's name
was Anthonis, and so he was called Adriaen Anthoniszoon.
From some connection with Metz, 2 he was known as Metius, a
name also used by his sons. He suggested -|ff as a con-
venient value of TT, probably oblivious of the fact that it had
already been given (1573) by a minor writer in Germany, 3
and certainly ignorant of its use in China several centuries
earlier. This Adriaen had a son, also named Adriaen, who was
called, after the custom of the time, Adriaen Adriaenszoon, but
was also called by his father's geographical name of Metius.
This younger Adriaen Metius 4 studied both law and medicine
and became (1598) professor of mathematics and medicine in
the University of Franeker. Although he wrote on mathemat-
ics, 5 his chief contributions were to astronomy. He published
his father's value of TT, and hence it is commonly attributed to
the son, although not due in the final analysis to either.

used. The first name is also spelled Adriaan. See H. Bosnians, Annales of the
Sociitt Scientifique, XXVIII and XXIX (Brussels), and Bibl Math., V (3), 342.

*Ideae mathematicae pars prima, seu Methodus polygonorum, Antwerp,
IS93-

2 He himself was probably born at Alkmaar, c. 1543. He died at Alkmaar,
November 20, 1620. This reason, commonly given for the name Metius, is doubt-
ful. The family name was Van Schelvan (haycock; Latin, meta).

3 Valentinus Otto, or Valentin Otho, also called Parthenopolitanus, a native
of Magdeburg. See Bibl. Math., XIII (3), 264.

4 Born at Alkmaar, December 9, 1571; died at Franeker, September 6 (16, or
18), 1635. The Latin form, Adrianus, is also used.

6 Praxis Nova Geometrica per usum circini, Franeker, 1623; Arithmetica et
Geometrica nova, Leyden, 1625; Maet-Constigh Liniael, Franeker, 1626, de-
scribing a kind of slide rule; Doctrmae Sphericae Libri V, Franeker, 1501; and
other works.

GEMMA FRISIUS

341

A more humble writer but a more progressive teacher ap-
peared early in the century in the person of Giel Vander
Hoecke. Although of no marked scholarship, his arithmetic 1
is worthy of study because of its early use of the plus and minus
signs as symbols of operation. Widman had already used
them as signs of ex-
cess and deficiency, and
Grammateus had used
them with their modern
significance, but they
now appear for the
first time in the Low
Countries.

Gemma Frisius. The
most influential of the
various Dutch mathe-
maticians of this century
was Gemma Regnier. 2
Having been born in
Friesland ( Frisia ) , he
was called the Frisian,
and was known as
Gemma Frisius. He was
thirty-two years old when
his arithmetic 3 was pub-
lished, and so favorably
did this work strike the popular taste, combining as it did the
commercial with the theoretical, that it went through at least
fifty-nine editions in the i6th century, besides several there-
after. He also wrote on geography and astronomy, suggesting
the present method of obtaining longitude by means of the

GEMMA FRISIUS
From a contemporary engraving

sonderlinghe boeck in dye edel conste Arithmetica, Antwerp, 1514;
2d ed., 1537.

2 Born at Dockum, East Friesland, December 8, 1508; died at Louvain,
May 25, 1555. The name is variously spelled, as Rainer, Renier, Reinerus.

3 Arithmeticae Practicae Methodvs Facilis, Antwerp, 1540. From certain
internal evidence (1575 ed., fol. B, 5, v.) it is probable that the book was
written c. 1536.

342 THE NETHERLANDS

difference in time, and taking one of the first steps toward the
modern methods of triangulation. 1 He became professor of
medicine at Louvain in 1541, and his son, Cornelius Gemma
Frisius (1535-1577), carried on his work, becoming professor
of medicine and astronomy in the same university. Cornelius
edited one of his father's works and wrote on astronomy and
medicine.

While Gemma Frisius wrote for the Latin schools, a man of
the people was needed to write for the common schools, and
this man was found in the person of Valentin Menher, a native
of Kempten. He wrote in the French language three or four
arithmetics 2 that occupied the same position in the Netherlands
that Borghi's did in Italy and Recorders in England. His
arithmetic of 1573 includes a certain amount of work in geom-
etry and trigonometry.

Among the Belgians who brought a high degree of scholar-
ship into their work in the editing of the early classics of mathe-
matics the one who stood highest at the beginning of the i6th
century was Jodocus Clichtoveus, a native of Nieuport, in
Flanders. He spent most of his time in France, assisted in
editing Boethius on arithmetic, and wrote a Praxis Numerandi
which was merely an edition of the algorism of Sacrobosco.
He died at Chartres in 1543.

The last of the Dutch writers of the century was Jacob Van
der Schuere of Meenen (c. 1550-1620), a teacher in a French
school in Haarlem. His arithmetic," commercial in character,
was one of the first of a long series of popular textbooks of this
type that appeared in Holland from 1600 to about 1750.

Stevin. The most influential of all the mathematicians pro-
duced by the Low Countries in the i6th century was Simon
Stevin. 4 In his younger days he was connected with the gov-
ernment service in Bruges. He traveled in Prussia, Poland,

1 Libellus de locorum describendorum ratione, Antwerp, 1533.

2 See Rara Arithmetica, pp. 250, 346.

3 Arithmetica, Oft Reken-const, . . . Door lacqves Van Der Schvere van
Meenen, Nu ter tijdt Francoysche School-meester tot Haerlem, Haarlem, 1600.

4 The name also appears in such forms as Stevinus, Steven, Stephan, Stevens.
Born at Bruges, c. 1548; died at Leyden or The Hague, c. 1620.

SPAIN 343

and Norway and later became a quartermaster general in the
Dutch army and director of certain of the public works. 1 His
most influential but not his most popular work was an arith-
metic," first published in Flemish at Leyden in 1585, and
republished the same year in a French translation. The im-
portance of this work lies in the fact that it was the first one
to set forth definitely the theory of decimal fractions, a sub-
ject that had been slowly developing for a century. Stevin also
made the first translation into a modern language of the work of
Diophantus, apparently from the Latin text of Xylander. 3

What made Stevin best known among his contemporaries,
however, was his contribution to the science of statics and
hydrostatics, 4 a subject naturally occupying much attention in
a country like Holland.

7. SPAIN

Spanish Writers. Spain furnished several native mathemati-
cians of considerable merit in the i6th century.' The intellectual
atmosphere was not favorable to the development of mathe-
matics, however, and many Spanish scholars settled in France
and Italy or at least published their works abroad. It is no
reflection upon the honesty of purpose of the Church to say

1 F. V. Gocthals, Notice historique sur la vie de S. Stevin, Brussels, 1842 ;
M. Steichcn, Memoire sur la Vie et let travaux de Simon Stevin, Brussels, 1841,

2 The title of the 1585 French edition, published at Leyden, is L'Arith-
metiqve de Simon Sterin de Brvges: Contenant les computations des nombres
Arithmetiques ou vulgaires : Aussi I'Algebre, auec les equations de cine quantitez.
Ensemble les quatre premiers Hitres d'Algebre de Diophante d'Alexandrie,
main tenant premierement traduicts en Francois.

3 Xylander's edition, Basel, 1575; Stevin's translation, Leyden, 1585.

*De Beghinselen der Weeghconst, Leyden, 1586; De Beghinselen des Water-
wichts, Leyden, 1586; Weeghdaet, Leyden, 1586. See also his Wisconstighe
Gedachtenissen Inhoudende f t ghene dacr hem in gheoeffent heeft . . . , Leyden,
1605-1608, with a Latin edition by W. Snell (Leyden, 1608) and a French edi-
tion by A. Girard (Leyden, 1634). He a'so wrote Problematum Geometricorum
Libri V (Antwerp, 1583) and other works.

J. Rey Pastor, Los matemdticos espanoles del siglo XVI, Oviedo, 1913;

F. Picatoste y Rodriguez, Apuntes para una Biblioteca Cientifica Espanola
del Siglo XVI, Madrid, i8Qi (hereafter referred to as Picatoste, Apuntes) ;
Acisclo Fernandez Vallin, Cultura Cientifica de Espana en el Siglo XVI, Madrid,
1893. The first and second of these works are particularly valuable. See also

G. Loria, "Le Matematiche in Ispagna," Scientia, XXV (May, June, 1919).

344 SPAIN

that the religious fervor of Spain from the isth to the i8th
century turned the thoughts of the intellectual class from
mathematics, although such work as was done was due to the
clergy. This was especially the case after the compact made
at Bologna in 1530 between Charles V and Clement VII. 1 To
this influence there should be added that which came from the
expulsion of the Jews, a race which had done so much in the
Middle Ages to foster the science of mathematics, at least with
respect to astrology and the theory of the calendar.

Ciruelo. The earliest Spanish mathematician of the century
was Pedro Sanchez Ciruelo, 2 who was professor of theology
and philosophy at Alcala, and was later canon of the cathedral
of Salamanca. He published an arithmetic 3 at Paris in 1495,
and a general work on mathematics 4 in 1516. He also edited
the theoretical work on arithmetic of Bradwardine in 1495 an ^
the Sphaera of Sacrobosco in 1498, and in general was a learned
exponent of the old school of mathematicians that was then
in favor in Paris.

Ortega. The second of the early Spanish writers was Juan
de Ortega, a Dominican from Aragon, concerning whose career
we know little except that he was living in 1512 and in 1567.
He wrote an arithmetic 5 which was published in 1512, both
in Barcelona and in Lyons, being the first book on commercial
computation known to have been printed in France. It was
a popular work, being reprinted in Rome, Messina, Seville,
Paris, and Granada. It is purely commercial and includes the
usual treatment of computation and the common applications
of the time.

!R. C. Jebb, in the Cambridge Mod. Hist., Vol. I, chap. xvi.

2 Born at Daroca, Aragon, c. 1470; died 1560.

8 The 1505 edition has the title Tractatus Arithmetice Pratice qui dicitur
Algorismus.

*Cursw quattuor mathematicarvm artiu liberaliu, Paris, 1516. First Spanish
edition, Alcala, 1516, but wanting the geometry, at least in the copy examined
by the author.

5 The title of the Rome edition of 1515 is Svma de Arithmetica: Geometria
Pratka vtilissima. The geometry consists simply of a little mensuration. In
the privilege of Leo X (1515) in this edition, the author is addressed as
"Dllecto filio lohani de Ortega Hispano Clerico Paletino."

LOW STATE OF MATHEMATICS 345

Joannes Martinus Blasius, 1 a Spanish astrologer and arith-
metician, published in Paris in 1513 a work on computation. It
was popular enough to warrant four editions. The author was
one of the earliest writers whose works appeared in print with
the spelling substractio for " sub traction/' a custom followed
quite generally by the Dutch and English arithmeticians for
several generations. He showed a good knowledge of the
classical writers in the domain of mathematics.

Another Spanish scholar who found the scientific work more
stimulating in Paris than in his native country was Caspar
Lax. 2 He took a course in theology at Saragossa, taught in the
University of Paris, and finally returned to Saragossa as a
teacher. His principal work was a prolix treatment of theoreti-
cal arithmetic 3 based on Boethius. He also wrote on the
Greek and medieval theory of proportion. 4

One of the most noteworthy treatises on mathematics pro-
duced in Spain in the i6th century is the work 5 of Juan Perez
de Moya (1562). This writer was born in San Stefano, in
the Sierra Morena, studied at Alcala and Salamanca, and be-
came a canon at Granada. His Arithmetica includes calcula-
tion, applied arithmetic, algebra, and practical geometry, and
contains a considerable amount of interesting historical material.

The last of the Spanish writers of any note, in the i6th
century, Jeronimo Mufioz, received his bachelor's degree at
Valencia in 1537. He traveled in Italy, taught at Ancona,
returned to Spain, and taught for ten years in the University of
Valencia. He wrote on arithmetic (1566) and Euclid, but his
chief work was on astrology.

is the name as it appears in the 1513 edition of his arithmetic. In
the 1519 edition the name appears as "Joannes Martinus, Scilicevs." The usual
Spanish form is Juan Martinez Siliceo. The original name was Juan Martinez
Guijeno, the word guijeno meaning silex, whence Sileceus, Sileceo, or Siliceo.
See V. Reyes y Prosper, "Juan Martinez Siliceo," Revista of the Soc. matem.
espanola, I, 153.

2 Born at Sarinena, c. 1487; died at Saragossa, 1560.

3 Arithmetica speculatiua magistri Gasparis Lax Aragonensis de sarinyena
duodecim libris demonstrate,, Paris, 1515.

*Proportiones magistri Gasparis lax, Paris, 1515.

B Arithmetica practica, y specvlatiua del Bachiller luan Perez de Moya,
Salamanca, 1562. 8 Hieronymus Munyos.

346 OTHER EUROPEAN COUNTRIES

Loss of the Jews. As for the Jews who had once added to the
brilliancy of Spain, hardly one of their descendants remained
in that country at the opening of the i6th century. The edict
of banishment of 1492 had driven out hundreds of thousands
of this race, some to slavery, some to death at the hands of
pirates, some to the plague-stricken towns of Italy, and some
to starvation. Persecution after persecution had accomplished
the purpose of those in power, but the result had sapped the
strength of Spain, and some of the best thought that would
have made for the advance of mathematics was turned to the
solving of the problem of self-preservation. 1

8. OTHER EUROPEAN COUNTRIES

Poland. In the i6th century Poland was one of the most
progressive countries of Europe in the field of arithmetic, pro-
ducing several works by native writers and reprinting a num-
ber by foreign scholars. 2 The first of her own arithmetics was
the Algoritmus of Tomas Klos, which appeared at Cracow in
1538.' Later in the century (1561) Benedictus Herbestus
( 1531-1593), a Jesuit priest, published an arithmetic in Latin. 4
One in the Polish language, chiefly commercial in nature, was
written by Girjka Gorla z Gorlssteyna and published at Czerny
in 1577. It was not through works like these, however, that
Poland contributed to human knowledge, but through those
of her greatest astronomer, or perhaps we should say the world's
greatest astronomer.

Copernicus. Not all of those who have aided in the progress
of mathematics have been primarily mathematicians. As we

a For a description of one phase of this movement see J. A. Symonds,
Renaissance in Italy: The Age of Despots, p. 399 (New York, 1883). On the
unpublished material relating to the Jewish contributions to mathematics and
astronomy, beginning in the nth century and closing in the i6th century, see
B. Cohen, "Ueber unveroffentlichte Schriften jiidischer Astronomen des Mit-
telalters," Jahrbuch der Judisch-LUerarischen Gesellschaft, XII (1918), i.

-Kara Aritkmetica, pp. 32, 97, 123, 190, 260, 303, and 353.

3 A reprint, edited by M. A. Baraniecki, was published at Cracow in 1889.

4 Arithmetica Linearis, eiq^ adiuncta Figvrata, cum quibusdam ex compvto
necessarijsy Cracow, 1561. This title is from the edition of 1577.

COPERNICUS

347

have already seen, the science of astronomy has always con-
tributed to her sister science, not merely in those centuries in
which the name " mathematician " meant an astronomer or an
astrologer, but in more recent times, when mathematics and
astronomy each outgrew the possibility of mastery by the
disciples of the other.
Among those whose in-
terest was primarily in
astronomy but who stim-
ulated the mathematician
to seek for new applica-
tions of his science none
stands higher than Nich-
olas Copernicus. 1 He was
educated at the Univer-
sity of Cracow (1491-
1495), spent some time
in the study of law, med-
icine, and astronomy in
the universities of Padua
and Bologna, and went
to Rome for the purpose
of continuing his work
in astronomy under the
patronage of the pope,
Alexander VI. He re-
turned to Poland in 1505, took holy orders, and obtained a
canonry at Frauenburg. By 1530 he had completed his theory
of the universe, the most significant step ever taken in the
science of astronomy, but it was not until 1543 that he pub-
lished his doctrines. 2 Gutenberg made the free spread of

1 Born at Thorn, on the Vistula, February IQ, 1473 ; died at Frauenburg,
May 24, 1543. He was named after his father, Niklas Koppernigk (died 1483)1
a native of Cracow. The English form of the name, Copernicus, is so familiar
that it is used throughout this work, although the spelling Coppernicus is
nearer the original. On his life consult L. Prowe, Nicolaus Coppernicus, 3 vols.,
Berlin, 1883-1884. The literature relating to him is extensive.

2 De revolutionibus orbium coelestium, Nurnberg, 1543.

COPERNICUS
From an early engraving

348 OTHER EUROPEAN COUNTRIES

thought possible, but Copernicus gave the thought ; Columbus
opened a new world, but Copernicus opened millions of worlds.

The work of Copernicus necessitated the improvement of trig-
onometry, and for this reason he wrote a treatise on the subject, 1
his single contribution to the literature of pure mathematics.

A generation after Copernicus a Danish mathematician,
Thomas Fincke (1561-1656), whose name also appears as
Finck, Fink, and Finchius, published a work called Gcometria
Rotundi (Basel, 1583), in which he made a number of con-
tributions to trigonometry.

Switzerland. The best known of the Swiss mathematicians
of this century was Henricus Loritus Glareanus," whose last
name probably comes from the name of his native canton
of Glarus. He was a professor in Basel (1515-1521), at
the College de France (1521-1524), and later in both Frei-
burg and Basel. He wrote on arithmetic, metrology, and music.

The only other Swiss writer of the period who need be men-
tioned is Cunradus Dasypodius/ 5 He was professor of mathe-
matics at Strasburg and canon of St. Thomas's Church in that
city. He had in mind the editing of all the Greek mathematical
works, and made a beginning in that direction. His edition of
Euclid's Elements appeared at Strasburg in 1564, and he wrote
a mathematical dictionary. 4

Portugal. Only a single Portuguese mathematician acquired
any considerable reputation in the i6th century. This man was
Pedro Nunes (better known by his Latin name of Nonius but

1 Z> lateribus et angulis triangular um libellus, Wittenberg, 1542.

2 Heinrich Loriti Glarcan. Born at Mollis, in Glarus, June, 1488; died at
Freiburg, Breisgau, May 28, 1563. H. Schreiber, Hemrich Loriti Glareanus,
Freiburg, 1837. The date of his death may have been March 27, 1563.

3 Name as given in his Lexicon. This is the Latin-Greek form for his family
name of Rauchfuss (rough-foot), Greek 8a<rvir6deios (of a hare), a Sao-i/Troi's
(rough foot) being a hare. Born at Frauenfeld, c. 1530; died at Strasburg,
April 26, 1600.

*\etKov sen Dictionarium Mathematicum, M. Cunrado Dasypodio, Strasburg,
1573. He also wrote Institutionum Mathematicarum voluminis primi Erote-
mata Logistic ae Geometriae Spherae Geographiae, 2 vols., Strasburg, ISQ3,
iSQ^; Volvmen primum : mathematkum disciplinarum principia, 2 vols., Stras-
burg, 1567, 1570; Brevis Doctrina de Cometis (on astrology), Strasburg, 1578.

SWITZERLAND 349

often called by the Spanish name of Nunez, 1 a scholar of Jew-
ish origin. He studied at the University of Lisbon and later
(1530) became professor of moral philosophy in that institu-
tion. He was (1544-1562) also professor of mathematics in
the University of Coimbra and held the posts of cosmographer
to the king, Don Joao III, and tutor to the royal princes. 2
His only mathematical work was devoted to algebra, arith-
metic, and geometry/ 5 but he is best known for his works on
navigation and astronomy, 4 and for the instrument for the
reading of small angles, often called the " nonius," which was
the forerunner of the vernier that is seen on transits and cali-
pers. 5 He left several manuscripts on geometry and navigation.
In the early part of the i6th century the interest in the
great voyages of the Portuguese led to the publication of vari-
ous treatises on the sphere and the use of the astrolabe for
nautical purposes. Among the earliest of these works was an
anonymous one on the astrolabe cind the quadrant, the instru-
ments then chiefly used in navigation.

at Alcacer do Sal, 1502 ; died at Coimbra, August n, 1578. The year
of his birth is often given as 1402 and that of his death as 1577. Nuncs himself
records, however, "... sit anno Domini 1502 quo ego natus . . . ," and a i6th
century MS. note in a book in the National Library at Lisbon reads: "Natus est
hie Doctor ano Dni 1502. Obiit vero tertio iclus Augusti ano Dni 1578."
Nothing more authoritative is known. See Guimaraes, Les Math. Portug., p. 16,
with authorities given. The name also appears as Pedro Nunez Salacicnse, and
in French as Pierre Nugne. See also Bensaude, Astron. Portug., p. 59; Picatoste,
Apuntes, p. 218.

2 His dedicatory epistle of 1564 speaks of himself as "Cosmographo Mayor
del Rev de Portugal, y Cathedratico Jubilado en la Cathedra de Mathematicas
en la Vnivcrsidad de Coymbra."

3 Livro de algebra em arithmctica y geometria, written c. 1532, published at
Antwerp in 1564, and reprinted there in 1567 with variations in the spelling of
words in the title. The above title is substantially as in Guimaraes, Les Math.
Portug., pp. 21, 105, 306; but sec H. Bosnians, "Sur le 'Libro de algebra' de
Pedro Nunez," Bibl. Math., VIII (3), 154; Cantor, Geschichte, II, chap. 59;
Picatoste, Apuntes, p. 221.

*Tratado da Esphera com a Theorica do Sol e da Lua, Lisbon, I537J Tratado
sobre certas duvidas da Navega$ao, an appendix to the preceding work.

G This is described in his De Crepusculis Liber unus, Lisbon, 1542.

QRegimento do estrolabio & do quadrante, published c. 1509. This was re-
produced in facsimile in Volume I of J. Bensaude, Histoire de la Science N antique
Portugaise, Munich, 1914. See this publication for other works of similar
nature in that period.

350 THE ORIENT

Jewish Writers in the East. With the expulsion of the Greek
Christians at the fall of Constantinople (1453) there returned
to the city many Jews who had been subject to persecution
under their regime. This movement continued as occasion of-
fered, and notably when the Jews were driven from Spain in 1492
and from Portugal in 1496. The sultan Mohammed II made
Moses Kapsali chief rabbi over all the Turkish Jews, and on the
latter's death he was succeeded in office by Elia Misrachi. 1
This learned rabbi wrote an arithmetic 2 based to a considerable
degree on a work by Rabbi ben Ezra. Knowing both Greek
and Arabic, he also drew from each of these sources, particu-
larly from the latter. He was interested in making arithmetic
practical and in having the processes thoroughly understood. 3
He seems to have been the first Hebrew writer to treat of
finding the sum of the cubes of the first n natural numbers.
He also wrote commentaries on the works of Euclid and
Ptolemy.

Russia. The only mathematical works known to have been
written in Russia in the i6th century were a geometry and an
arithmetic, both of which date from about 1587 to 1594.*
Translations of monographs on western mathematics also ap-
peared in this century, but they only show the low state of
science at that time in Russia. This is seen, for example, in
the fact that the Origines of Isidorus of Seville (570-636) was
thought worthy of translation.

9. THE ORIENT

Close of the Dark Ages. We have seen that the East had its
dark periods in history just as the West had them, and with
each the darkness was greatest just before the dawn. In Asia
the gloom was particularly oppressive in the i6th century.

1 Born probably at Constantinople, c. 1455; died at Constantinople, 1526.

2 Se}er ha-Mispar (Book of Numbers}.

:J G. Wertheim, Die Arithmetik des Elia Misrachi, p. 6. Frankfort, 1893.

4 N. M. Karamzin, History of the Russian Empire (in Russian), X, 25^, 436
(Petrograd, 1824); Lavrovsky, Ancient Russian Schools (in Russian), p. 180
(Kharkov, 1854); Russian Encyclopedia (in Russian).

CLOSE OF THE DARK AGES 351

India was intellectually dead. China was just becoming aware
of the extent of Western learning, and seemed discouraged in
the effort to advance independently. Japan was not yet awake.
For these reasons the i6th century was a dark one for Oriental
mathematics.

Islam. Of the heirs to the glory of the scholars of Islam in the
field of mathematics the name of only one of any note is found
in the records of the century. Beha Eddin, 1 as he is generally
called, was probably a Persian. He wrote on a variety of sub-
jects and among his works was an elementary textbook on arith-
metic 2 and the first part of an exhaustive treatise on the subject. 3

A single other writer may properly be mentioned with Beha
Eddin, namely, Mohammed ibn Ma'ruf ibn Ahmed, Taqi ed-
din (1525/1526-1585), who seems to have lived in Constanti-
nople. He wrote on algebra, arithmetic, and astronomy.

India. After the death of Bhaskara (c. 1175) there was no
great interest shown in the advance of mathematics in India
except so far as it related to astronomy. From the i6th cen-
tury only two names of any note have come down to us. Surya-
dasa, who flourished c. I535, 4 and Ganesa, 5 who lived about the
same time, were both commentators on the works of Bhaskara.
Suryadasa refers to Srldhara's method of finding the area of a
cyclic quadrilateral, and Ganesa quotes one of Srldhara's rules
for the area of a segment of a circle.

China. China at this time was experiencing a kind of calm
before that influx of European mathematics which was heralded
by the great Jesuit leader, Matteo Ricci. K'u Ying-hsiang
(c- I 5S)? governor of Yunnan, wrote on algebra and geometry ;
T'ang Shun-ki (1507-1560) wrote on the mensuration of the

J Beha ed-dm al-'Amili, Mohammed ibn Hosein. Born probably at Amul
(Amol), near the Caspian Sea, 7547 ; died at Ispahan, 1622. Nazam ed-dm
Ahmed, in his biography, says that he was born at Baalbek. See A. Maare,
Beha Eddin, 2d ed., Rome, 1864.

2 The Kholdsat al-Hisdb (Essence of Arithmetic). The work has no par-
ticular merit. There was an Arabic-Persian edition, Calcutta, 1812; an Arabic-
German edition, Berlin, 1843; and a French translation by Marre, Rome, 1864.

3 The Bdhr al-Hisdb (Ocean of Arithmetic), left unfinished.

*Bibl. Math., XIII (3), 205. *Ibid., 205. Also written Ganecji.

352 THE ORIENT

circle; Hsin Yun-lu (c. 1590) contributed to the subject of the
calendar, more in quantity than in quality ; and Ch'eng Tai-wei
(1593) wrote a work on arithmetic 1 in which we find the earliest
description of the suan-pan computation. These writers lacked
the genius of some of their immediate predecessors ; and they
contributed nothing that was commensurate in importance with
the productions of the century following.

Japan. The i6th century did not see the awakening of the
intellectual Japan as it saw the awakening of Europe. The
East was at this time about a century behind the West in this
great world movement. Perhaps it is more nearly accurate to
say that the i6th century in Japan corresponds more closely to
the I3th century in the West ; it was a century of preparation.

Probably the chief cause which contributed to this prepara-
tion in the field of mathematics was the journey to China made
by one Mori Kambei Shigeyoshi, a scholar in the service of
two of the powerful lords of Japan. The story goes that the
great hero Toyotomi Hideyoshi, better known as Taiko, having
subdued all of the country, decided that he would make his
court a great intellectual center. In pursuance of this purpose
he sent Mori to China to acquire and bring back that mathe-
matical knowledge which was so lacking in Japan. Mori,
being a man of humble birth, was not well received in China,
and for this reason Taiko made him Lord of Dewa, 2 hoping thus
to give him high standing among Chinese scholars. Owing to
political and military difficulties, chiefly Taiko's invasion of
Korea (1592), Mori's mission was not successful, but he
brought back with him a considerable amount of material and
is said by some to have made the Chinese abacus 3 known in
Japan. His last years were spent in Kyoto in teaching the
use of this instrument.

Although this is the story as often told, there is a question
as to whether Mori really visited China or went only to Korea.

1 Suan-fa Tong-tsung (A systematized treatise on arithmetic). His name
also appears as Tch'eng Ta-wei. 2 Dewa no Kami.

3 The suan-pan. It developed later into the Japanese soroban. Both of these
instruments are discussed in Volume IL

JAPAN 353

It seems certain, however, that he knew something of Chinese
mathematics, that he was an expert with the abacus, that he
advertised himself as " the leading instructor in division in the
world/' and that he was a very successful teacher. Among
his pupils were three men, 1 known to their contemporaries as
"The Three Arithmeticians," who will be mentioned later.
That Mori was the first who took the abacus from China to
Japan is very doubtful, but he seems to have made it popular.

10. THE NEW WORLD

General Conditions. One would not expect to find a treatise
on mathematics printed in the New World within sixty-four
years of its discovery by Columbus, and still less would he
think that only forty-five years after this great discovery there
was set up a press for the disseminating of knowledge among
the inhabitants of the western hemisphere. Each of these
events stands out, however, as a historic fact, a testimony to
the zeal and foresight of the early conquerors.

Among the adventurous band organized by Cortes for his
first expedition to Yucatan in 1518 was a young chaplain whose
name appears in his work of 1556 as Juan Diez. He was of
a literary turn of mind, as is shown by three or four books
which he published. One of these works was on mathematics,
and this appeared in Mexico under the following title : 3

teplata.? 020 Denies rermo8sel]$ira Ton neceflariaaa
Io8inercadere8:todo geherosetratantes.
reglas tocanteaal SHrt>metica.

It should be stated, however, that there were several writers
at this time by the name of Juan Diez (Diaz), two of them

iYoshida Shichibei Koyu, Imamura Chishd, and Takahara Kisshu.

2 D. E. Smith, The Sumario Compendioso of Brother Juan Diez, Boston, 1921.

354 THE NEW WORLD

apparently being in Mexico, and there is much uncertainty
among the best Spanish biographers as to which was the author
of the Sumario.

Printing Established in Mexico. The first viceroy of New
Spain, which included the present Mexico, was a man of re-
markable genius and of prophetic vision, Don Antonio de
Mendoza. He assumed his office in 1535, and for fifteen years
administered the affairs of the colony with such success as to
win for himself the name of "the good viceroy." He founded
schools, established a mint, ameliorated the condition of the
natives, and encouraged the development of the arts. In his
efforts at improving the condition of the people he was ably
assisted by Juan de Zumarraga, the first bishop of Mexico.
Among the various activities of these leaders was the arrange-
ment made with the printing establishment of Juan Crom-
berger of Seville whereby a branch should be set up in the
capital of New Spain.

The idea of setting up a press in Mexico seems to have been
considered as early as 1534, even before Mendoza became
viceroy, doubtless at the suggestion of Juan de Zumarraga ;
but it was not until 1536 that the plan was carried out. Juan
Cromberger then sent over as his representative one Juan
Pablos, a Lombard printer, and so the "casa de Juan Crom-
berger" was established, prepared to spread the doctrines of
the Church to the salvation of the souls of the unbelievers.
Cromberger himself never went to Mexico, but his name
appears either on the portadas or in the colophons of all
the early books. From and after 1545, however, the name
is no longer seen, Cromberger having died shortly before
this time.

Nature of the Sumario Compendioso. The Sumario Compen-
dioso consists of one hundred and three folios, generally num-
bered. After the dedication there is an elaborate set of tables,
including those relating to the purchase price of various grades
of silver, to per cents, to the purchase price of gold, to assays,
and to monetary affairs of various kinds.

JUAN DIEZ

3SS

The mathematical text consists of twenty-four pages be-
sides the colophon. Of these pages, eighteen relate chiefly to
arithmetic and six to ^ ^ ^

tTna^rta 00 trtvf.mfd f

media on pa j.t.rrrvif. fj

ion $9 i9.tftri0.<

lion

algebra. The arithme-
tic includes problems
in the reduction of
maravedis to pesos, of
ducats to crowns, and
the like; but it also
considers questions re-
lating to the preceding
tables and to simple
commercial transac-
tions in general. There
are also problems relat-
ing to the theory of
numbers, some of them
involving rules similar
to those found in the
works of Fibonacci and
Diophantus. Of the
latter type the follow-
ing is an example:

ihf.on

^ r '*r m

vij.cn

I'lnfod

ij*mfo0

itimfo0

iiif.mfos

v.mfoa

,
tttjrvrj.

vilfmfoa

it).

iii pa if.trwrif Ii
V) p* \\t-fVi\l ii)

r. p0 t

ti.s i|.tncifvli/ ij
s v.Mnnij. lij.
c.

wiii,p0 ij.trfrvi)- ii
e v.tj:wf|. II).
t

nnPosdii,p0
ir mfoo It. vj.p

cir.mFo0c. p0

ii. mfo0 cr

Fl j-mroB cr^vjga

rtii.mFo0crl. p0

cliijmroa

rliitjmroa

Give me a number
which, increased by 15,
is a square number ; and
decreased by 4 is also
a square number. Rule:
Add 15 and 4, making
19 ; then add i to this
result, making 20. Now
take the half of this
number 20, which is 10 ;
square this result thus:
10 times 10 is 100. From
this subtract 15, and we have 85, and this is the number required, that
is, the one from which if you subtract 4 you have 81, the root of

PART OF THE TABLE OF THE SUM A RIO

The entire table covers nearly one hundred and

eighty pages, and the above facsimile fills half

of one page

356

THE NEW WORLD

.25.He.toa,

S4i.1Ke.Koa.
us6.iae.jroa.

3025.1Rc.voa.
}3(>4,7Rc.?o4,
?6oo.11\e.i?o4.

24*

UO*

240.

384.

4&0,.

804,

960,

1080*
. 72o.

2400..
2i6o*
1920*

.2905.

^6o

345$.

CONGRUENT AND CONGRUOUS NUMBERS
FROM THE SU MARIO OF JUAN DIEZ

He defines the terms thus: "A congruous
number is such a square number that, sub-
tracting from or adding to it another number,
called a congruent number, it will still be a
square." There are five errors in the table
here given

which is 9. The same thing
happens if you add the 15,
the result being a hundred,
the root of which is 10 ; for
10 times 10 is 100, which
checks.

Some idea of the work
in the theory of numbers
may be obtained from
the table of congruous
and congruent numbers
here shown in facsimile,
and of the nature of the
algebraic problems from
the following example:

A man has mares and cows
in quintuple proportion, in
such a way that if you
square the number of mares
and square the number of
cows, the products added
will be 1664. Required the
number of mares and the
number of cows.

Apparently the author
had some taste for mathe-
matics and was fairly
well versed in algebra as
it was then known in the
best schools of Spain and
Italy. When we reflect
that only two important

treatises on algebra had
at that time been issued from the European presses, and that
the Sumario was the first mathematical work to be published on
another continent, the credit due to Diez is the more apparent.

DISCUSSION 357

TOPICS FOR DISCUSSION

1. Reasons for the great advance made in mathematics in the
1 6th century.

2. Causes of the prominence of Italy in mathematicil research
in the i6th century.

3. Influences leading to the predominance of algebra in the field
of mathematics in the i6th century.

4. The leading features in the progress of algebra in the i6th
century, with a consideration of the countries and individuals most
closely connected with this progress.

5. A comparison of the mathematics of England, France, Italy,
and the Teutonic countries in the i6th century.

6. Influences leading to the advance in mathematics in the
Netherlands at this time.

7. The leading treatises on algebra in this century, with a con-
sideration of the important features of each.

8. The life and works of Robert Recorde. His influence upon
British mathematics.

9. The way in which the arithmetics of the various countries met
the commercial needs of the time.

10. The life and works of Michael Stifel.

11. Influences leading to the development of trigonometry in
Europe at this time.

12. Effect of the Renaissance upon the nature of mathematics in
the 1 6th century.

13. The nature of Oriental mathematics in the i6th century.

14. A consideration of the causes of the backward state of mathe-
matics in Spain and Portugal in the i6th century.

15. The life and works of Copernicus and his influence upon
mathematics in general.

1 6. The field of mathematical activity developed by the Jewish
scholars in the i6th century.

17. The general nature of the literary activity in Mexico in the
1 6th century, with special reference to the need which produced
the work of Juan Diez.

1 8. The revival of the study of the works of the classical writers
in the i6th century, with special reference to its influence upon
mathematics.

CHAPTER IX
THE SEVENTEENTH CENTURY

i. GENERAL CONDITIONS

Political Situation. In order to comprehend the causes of
the remarkable advance of mathematics in the iyth century
it is necessary to consider briefly the political and social in-
fluences that made this century conspicuous in science, in
letters, and in the development of human rights. This was
the century that saw broken forever in the Anglo-Saxon civi-
lization the doctrine of the divine right of kings; that saw
the beginning of the end of this same doctrine in France
in the brilliant reign of Louis XIV; and that saw Russia
amalgamated into a powerful nation by a powerful leader,
Peter the Great. In this century, too, we may possibly find
one of the early steps toward the World War of 1914-1918,
namely, the founding of the military machine of Prussia at
the hands of the Great Elector of Brandenburg. This was also
the period in which Europe saw the turning back of the Turks
by the Hapsburgs in Austria ; in which the New World was
definitely opened to colonization and trade ; and in which the
Thirty Years' War (1618-1648) disturbed the political and
religious life of a considerable part of Europe. Great world
activities like these could not but affect natural science as well
as political, and abstract science as well as natural.

The Trend to the North. There was also the general reason
why mathematics, like all intellectual pursuits, should trend
to the north, the reason that had slowly led it away from
the warmer countries from time immemorial, the ability to
conquer the cold and the darkness of the long winter nights.
Heat and light have always joined with the soil and the

MATHEMATICAL SITUATION

359

distribution of moisture to make conditions favorable for intel-
lectual work. While coal was known in England as early as
the Qth century, the i6th and lyth centuries saw a great ad-
vance in the comforts of living north of the Alps, and hence in
the ability to utilize the long winter nights in intellectual
pursuits. Other influences were evidently more powerful, but
this one must be recognized as being somewhat significant.

MATHEMATICS IN MILITARY AFFAIRS

Measuring the distance to a castle by the aid of a form of quadrant. From
Mario Bcttino's Apiaria, Vol. I, Bologna, 1645

Mathematical Situation. Mathematics does not develop by
centuries any more than by political units or by religious
faiths. Vieta and Harriot were quite as much of the i6th as
of the i yth century, just as Erasmus was quite as much of
England and Germany and France as of the Low Countries,
and as Stifel the Lutheran was quite as good an algebraist as
Stifel the monk. Nevertheless it is convenient to mark off
blocks of time, and centuries serve the purpose fairly well
in mathematics as in art and in politics.

It is impossible to say with truth that this century or that
is the greatest in the development of any human interest, but
it is entirely within the range of truth to assert that few if
any centuries did so much for mathematics as that one which

360 GENERAL CONDITIONS

saw Fermat begin the modern theory of numbers and, with
Descartes and Harriot, invent the analytic geometry ; Cavalieri
pave the way for Newton and Leibniz, who, in their turn,
established the calculus ; Pascal and Desargues open new fields
for pure geometry; Napier reveal to the world a new method
of computation ; and a large number of brilliant scholars apply
the theories thus developed to the study of curves, to difficult
problems of mensuration, and to the science of celestial
mechanics.

Moreover, the i?th century was characterized by a new
spirit, that of intellectual internationalism, of a free exchange
of ideas among members of the learned class, and of the call-
ing of scholars from one country to another, sometimes by
universities, sometimes by academies, sometimes by royal com-
mand. This spirit was even more manifest in the i8th century,
when men like Euler, the Bernoullis, and Lagrange were looked
upon no longer as national assets, but as Europeans in the
largest sense of the term.

It seems unreasonable to separate these intellectual activi-
ties from the general Spirit of the Times. Printing had begun
to show its power ; not merely the intellectual aristocracy but
the people in general were beginning to think; the scholar
could now make his discoveries known, and his audience was
no longer composed literally of those who heard his voice;
and with the breaking of religious and political canons came
also the breaking of the canons of science.

As an incident in the military activity of this period the
mathematics of warfare became even more prominent than it
was in the i6th century, a symptom of the terrible part that
it was to play some three hundred years later in the great
World War.

Effect of Skepticism in General. Skepticism (not religious
skepticism in particular, but skepticism with respect to tradition
in general, skepticism which Buckle has called "hardness
of belief") has been the sine qua non of progress. This is as
true in mathematics as it was in the combat with the physics
of Aristotle, with the music of Boethius, with the canons of

IMPORTANCE OF THE CENTURY

Giotto, or with the divine right of kings. Just so long as Euclid
was sacrosanct in elementary geometry, or Apollonius in
conies, or Ptolemy in astronomy, or Boethius in arithmetic, the
world could not progress in mathematics. It was only when
men began to doubt the infallibility of these ancient leaders
that they developed a new conception of geometry, a new way
of handling conies, a new system of the universe, and a new

MATHEMATICS IN MILITARY AFFAIRS
From Leonhard Zubler's work on geometric instruments, Zurich, 1607

view of arithmetic. The world began to get skeptical of author-
ity in the i6th century, and the leaven of new ideas worked so
rapidly that the opening of the iyth century saw the time ripe
for an entire recasting of mathematical theories.

2. ITALY

Shifting the Center. The iyth century saw a shifting of the
center of mathematical activity from Italy northward. One
cause has already been mentioned, and other causes are not
difficult to see, being largely political. In general, mathe-
matics flourishes where the environment is favorable, and in

362 ITALY

the i ?th century the political environment was more favorable
in France and England than in Italy. The glory of Venice
was rapidly fading out; Pisa was no longer a seaport; Flor-
ence had ceased to be the source of business customs ; Rome
had lost her hold upon the most vigorous parts of Europe ; and
the rivalry between the Italian states was not of that healthy
nature which makes for intellectual progress. Italy was still
a mathematical power, but it was no longer the world's intel-
lectual center.

Cavalieri. From the standpoint of mathematics alone the
Italian writer who influenced the science most in the i?th
century was probably Bonaventura Cavalieri/ a Jesuit, a pupil
of Galileo's, and professor of mathematics at the University
of Bologna from 1629 to the time of his death. He wrote on
conies,- trigonometry/ 5 optics, astronomy, and astrology, and
was one of the first to recognize the great value of logarithms.
His greatest contribution, however, was his principle of indi-
visibles, a principle announced by him in 1629 but not set
forth in printed form until six years later. 4 The theory is based
upon the assertion that a line is made up of an infinite number
of points, a plane of an infinite number of lines, and a solid of
an infinite number of planes. The theory thus forms the basis

a Born at Milan, 1598; died at Bologna, November 30, 1647. The year of
his birth may, however, be 1591. He may have died December i, 1647, but it
was the night of November 3o-December i. See P. Frisi, Elogio del Cavalieri,
Milan, 1778; G. Piola, Elogio di Bonavenlura Cavalieri, Milan, 1844; F.
Predari, Delia Vita e delle Opere di Bonaventura Cavalieri, Milan, 1843 ;
A. Favaro, Bonaventura Cavalieri nello studio di Bologna, Bologna, 1888.
The name appears also as Cavallieri, Cavaglieri, Cavalerius, and de Cavalleriis.
His place of birth is given by Piola as Milan, although others give it as
Bologna. The tomb records the date of his death and the fact that he was
a native of Milan.

2 Lo specchio ustorio, overo Trattato delle settioni coniche, Bologna, 1632.

3 Directorium generale uranometricum in quo Trigonometriae logarithmicae
fundament a ac regulae demonstrantur , astronomicaeqne supputationes ad solam
fere vidgarem eruditionem reducuntur, Bologna, 1632 ; Compendia delle regole
dei triangoli colle loro dimostrazioni, Bologna, 1638; Trigonometria plana et
spherica, Bologna, 1643.

4 Geometria Indivisibilibus continuorum nova quadam ratione promota t
Bologna, 1635 ; 2d ed., Bologna, 1653 ; Exercitationes Geometricae sex, Bologna,
1647.

CAVALIERI

363

of a crude kind of calculus, and by its aid Cavalieri found it
to solve manv nroblems in mensuration that would now

be solved by the more
scientific methods of in-
tegration. The term " in-
divisible" is ancient, and
Cavalieri made no claim
to originality in its use.

Galileo. Much more
widely known than Ca-
valieri, more widely
known, indeed, than
most men of his time,
Galileo Galilei 1 was des-
tined to bring great glory
to Italy in general and to
Florence in particular.
Born on the day of
Michelangelo's death,
and dying in the year
of Newton's birth, he
seemed to fill the gap be-
tween the lives of these
two great leaders, a stir-
I ring period in the history

of art, letters, politics,
science, and religious
thought ; and in the nu-
merous controversies re-
lating to all these lines
he played a major part.

He was the son of a certain Florentine nobleman, a dilettante
in music and mathematics, whose estate had become so greatly

iBorn at Pisa, February 18, 1564; died at Florence, January 8, 1642. The
literature relating lo Galileo is extensive. A good bibliography, particularly
as to his life and the controversy over the inquisition question, is given by
Karl von Gebler, Galileo Galilei and the Roman Curia, English ed., London,

BONAVENTURA CAVALIERI

From an engraving by G. A. Labus after a
drawing by A. Alfieri

364 ITALY

reduced as to indicate for Galileo a life devoted to the restora-
tion of the family fortune, the life of a cloth merchant. By

GALILEO GALILEI, 1564-1642

From the painting in the Pitti Gallery in Florence, school of Sustermans (Sutter-
mans, 1597-1681). Robinson's Medieval and Modern Times

some good chance, however, he was sent to the convent of
Vallombrosa, and here he displayed such unusual powers that

1879, p. xxix. See also P. Frisi, Elogio del Galileo, Milan, 1774; G. P. C. de'
Nelli, Vita e commerdo letterario di Galileo Galilei, 2 vols., Lausanne, 1793;
A. Favaro, Galileo Galilei e Suor Maria Celeste, Florence, 1891, with valuable
material relating to his contemporaries. Viviani's sketch of the life of his great
teacher may be found in the Milan edition of the Opere di Galileo Galilei of
1808-1811, Vol. I. The best edition is the one published under the editorship of
A. Favaro, Florence, 20 vols., 1890-1909, with a biographical index in Volume XX.

GALILEO 365

his father changed his mind and decided that he should study
medicine. He entered the University of Pisa in 1581, in a
period of great intellectual upheaval, but his medical studies
were soon put aside by an incident that was to help change the
scientific thought of the world. The beautiful lamp that still
hangs in the cathedral at Pisa had been moved from its
vertical position in order the more readily 'to light it, and
Galileo noticed that the oscillations were at first considerable
but gradually became less and less. They seemed, however,
to be made in equal periods of time, and this inference he con-
firmed by comparing them with his pulse. Thus he was able to
establish the approximate isochronism of the vibrations ( a fact
which had been asserted by the Arabs) and to make a be-
ginning in medical diagnosis by accurately timing the arterial
beats. Galileo was also led at this time, it seemed by chance,
and certainly against his father's wishes, to the study of geom-
etry. His success was such that at last he secured parental
consent to give up medicine and devote himself to science.
He soon became well known throughout Italy, and in 1589 was
made professor of mathematics at Pisa. It is indicative of the
low esteem in which mathematics was then held that, whereas
the professor of medicine received the equivalent of about
$2150 a year, Galileo was rewarded by a salary equivalent to
only $65. It was here that he began his experimental work in
physics, but owing to local controversies he resigned his chair
in 1591. The next year he was offered the professorship of
mathematics at Padua, and here he was enabled to carry on
some of his most important scientific work.

Of his controversies in astronomy, of his construction of
the first satisfactory telescope, 1 of his invention of the mod-
ern type of microscope, and of his work in physics this is
not the place to speak. It should be said, however, that
his interest in mathematics was maintained throughout his
stormy life. While at Padua he invented the proportional

1 On the invention of the telescope the first and one of the best of the
standard works is P. Borel, De Vero Telescopii Inventory The Hague, 1655-
1656.

366

ITALY

compasses, 1 an instrument which was in great favor for a cen-
tury or more but which of late has been generally discarded.

Torricelli. Of those physicists who sat at the feet of Galileo
and who also showed an interest in pure mathematics no
one was more celebrated than Evangelista Torricelli. 2 Although

more than forty years the
junior of Galileo, he survived
trim by only five years, dying
it the age of thirty-nine.
He had studied under a pupil
)f this great master, but was
ilso privileged to receive
nstruction from the latter
limself, then blind and en-
"eebled by age, in the last
yrear of his life. Not only
lid Torricelli write on phys-
ical questions and give to
the world the barometer, but
be contributed to geometry
is well, 3 anticipating the
srork of Roberval on the
method of tangents. Pere
After a contemporary engraving Mersenne had announced

to Galileo in 1638 that

Roberval had squared the cycloid, a curve to which Galileo
had first called attention. Galileo thereupon sent the letter to
various friends, and Torricelli responded by squaring the
cycloid, and Viviani by determining the tangent.

*Le Operazioni del Compasso Geometrico et Militate, Padua, 1606. See
also A. Favaro, "Per la storia del compasso di proporzione," in the Atti of the
Istituto Veneto, LXVII, 2, 723.

2 Born in or near Faenza, possibly in the village of Modigliana, October 15,
1608; died at Florence, October 25, 1647. See the Opere of Torricelli, ed. Loria
and Vassura, Introduziore, Faenira, 1919; G. Loria, Atti d. R. Accad. dei
Lincei, XXVIII (5), 409.

8 Opera geontetrica, Florence, 1644; F. Jacoli, "Evangelista Torricelli ed il
metodo delle tangent! detto Metodo del Roberval," Boncompagni's Bullettino,
VIII, 265.

EVANGELISTA TORRICELLI

TORRICELLI AND VIVIANI 367

Vincenzo Viviani. This Vincenzo Viviani 1 was also a disciple
of Galileo's. He was interested in physics and in the applications
of mathematics, but his tastes led him even more strongly to
geometry. 2 His first work (1659) established his reputation,
and he was honored by the Medici, made mathematician to
Ferdinand II, the grand duke of Tuscany, elected to member-
ship in various learned societies, and invited to France by
Louis XIV and to Poland by King Casimir, invitations which
he felt compelled to decline. In 1692 he proposed to scholars
a problem which attracted wide attention, and which may be
stated briefly as follows : There is among the ancient monu-
ments of Greece a temple dedicated to Geometry. The plan is
circular and the temple is surmounted by a hemispherical dome
which has four equal windows of such size that the rest of the
surface can be exactly squared. Required to find how this
is possible.

The problem appeared in the A eta Eruditorum under a
designation which is an anagram of the words "A postremo
Galilei Discipulo," a title which Viviani was always proud to
bear. 3 Of this problem there were submitted correct solutions
by Leibniz, Jacques Bernoulli, 1'Hospital, Wallis, and David
Gregory, but Viviani himself gave the simplest one of all.
He also solved the trisection problem by the aid of the equilat-
eral hyperbola.

Minor Writers. Among the minor Italian writers of the i7th
century Giovanni Antonio Magi'ni, 4 a friend of Kepler, was
widely known as the maker of the most perfect maps of Italy
up to that time, and as professor of astrology, astronomy, and
mathematics in the University of Bologna for nearly thirty

1 Born at Florence, April 5, 1622; died at Florence, September 22, 1703.

2 De Maximis, ct Minimis Geometrica Divinatio in Qvintvtn Conicorum
Abollonii Pergaei adhvc desideratvm, Florence, 1650; Quinto libra de?li Elementi
d' Evclide, Florence, 1674; De Locis Solidis Secunda Druinatio Geometrica,
Florence, 1673 and 1701; and other works.

3 Viviani T s solution appeared in his Formazione e misure di tittti i deli, con
la struttura e quadratura esatta dell' inter 0, e . . . uno degli antichi delle volte
regolari degli architetti, Florence, i6q2.

4 Born at Padua, June 13, 1555; died at Bologna, February n, 1617 or
1615.

368 ITALY

years (from 1588). While his interest was chiefly in astron-
omy, on which he wrote extensively, he also contributed to the
theory of numbers' and to trigonometry. 2

Marino Ghetaldi,* another of the minor writers of the period,
was descended from a patrician family and divided his time
between scientific and diplomatic pursuits, working in the
field of mathematics and acting as ambassador for Venice to
Rome and to Constantinople. In his chosen field of science he
still further divided his interests, this time between pure mathe-
matics and its physical applications. He wrote upon geometry 4
and algebra 5 but contributed little that was original.

Giovanni Alfonso Borelli," a third in the list of lesser mathe-
maticians, studied at Pisa and then taught philosophy and
mathematics at Messina (1649). In 1656 he was recalled to
Pisa to take the chair of mathematics. He was also a physician,
and his posthumous work De motu animalium (1680-1685) was
highly esteemed. He edited some of the Greek classics on
mathematics. 7

The Cassini Family. Although classified among the minor
writers from the standpoint of their contributions to pure
mathematics, several of the members of the remarkable Cas-
sini family would stand among the leaders if we considered

^Tabula tetragonica sen quadratorum nnmerorum, cum suis radicibus,
Venice, 1502.

2 De Plants Triangvlis Liber Vnicus, and De Dimetiendi ratione per Quad-
rantem, 6- Geometricum Quadratum, Libri Qvinqve, Venice, 1502 ; Tabulae et
canones primi mobilis; item calculus triangulorum sphaericorum, Venice, 1604.

3 Born at Ragusa, 1566; died probably at Constantinople, but possibly at
Ragusa, 1626 or 1627.

4 Nonnullae propositiones de parabola, Rome, 1603; Apollonius redivivus seu
restituta Apollonii pergaei de inclinationibus Geometria, Venice, 1607; and two
other works. See also H. Wieleitner, Bibl. Math., XIII (3), 242; E. Gelcich,
Abhandlungen, IV, 191.

5 De resolutione et compositione mathematica, libri quinque, Opus post-
humum, Rome, 1630, a work on algebra applied to geometry.

6 Born at Castelnuovo, near Naples, January 28, 1608; died at Rome,
December 31, 1679.

7 Euclides restitutus, Pisa, 1658; Apollonii Pergaei conicorum libri V, VI, et
VII, Florence, 1661; Elementa conica Apollonii et Archimedis opera, Rome,
I67Q.

THE CASSINI FAMILY

369

their work in the field of astronomy. At any rate they deserve
to be mentioned in this connection because of their skillful ap-
plication of mathematics in their chosen field of science. The
founder of the astronomical line was Giovanni Domenico
Cassini, 1 professor of astronomy at Bologna (1650). Louis
XIV asked that he be sent to Paris (1669), and soon after

Ci/nf /Ifa&t tfcsjvnti fimJM***.

AUTOGRAPH OF GIOVANNI DOMENICO CASSINI

From a receipt written on parchment, February 27, 1706. The French form of
the name was adopted after he went to France in 1669

he became (1671) the first astronomer royal of France. Since
he became naturalized in France and his son Jacques Cassini 2
was born there, the line now ceases to be Italian. Jacques, the
second son, succeeded his father (1712) as astronomer royal,
and, like him, wrote numerous monographs on astronomy.
Cesar-Francois Cassini de Thury, 3 the son of Jacques, suc-
ceeded his father (1756) as astronomer royal, and was in turn

a Born at Perinaldo, June 8, 1625; died in Paris, September 14, 1712.
2 Born in Paris, February 18, 1677; died at Thury, April 16, 1756.
3 Born in Paris. June 17, 1714; died in Paris, September 4, 1784.

370 FRANCE

succeeded by a son, Jacques Dominique Cassini de Thury. 1
Each of these members kept up the traditions of the remark-
able family with respect to its contributions to science.

3. FRANCE

France and England. It is of little moment whether we say
that the center of mathematical activity in the iyth century
rested in France or in England ; perhaps it would be more just
to speak of each as one of the foci of an ellipse, as was the
case with Bagdad and Cordova, so strong are the claims that
each may fairly adduce. When we try to balance such names
as Harriot, Napier, Oughtred, Wallis, Barrow, and Newton
against so remarkable a group as Fermat, Desargues, Des-
cartes, Pascal, Mersenne, and 1'Hospital, it is like comparing
two infinities. Perhaps England's mathematics was more
usable in the natural sciences, while that of France was more
of the nature of I' art pour Vart ; but any such distinction is
easily attacked. If we speak of France before England, it
means only that we may consider at random either focus of
the ellipse with equal justice.

As to France, Paris had risen in political, intellectual, and
artistic splendor, while Lyons had fallen. The Lyonese still had
their four great fairs annually, and were content with their lot.
The two cities still represented, however, the mathematics of
the country.

Early Writers. Among the early French writers of this cen-
tury, Denis Henrion 2 published the first logarithmic table to
appear in France. 3 Contemporary with him was the learned
scholar Claude Richard, 4 who entered the Jesuit order in 1606,
taught mathematics in Lyons for a number of years, was called
to Spain in 1624, and became professor of mathematics at

1 Born in Paris, June 30, 1748; died at Thury, October 18, 1845.

2 Born c. 1590; died c. 1640.

*Traicte des logarithmes, Paris, 1626. He also published a Collection . . . de
divers Traictes Mathematiques, Paris, 1621; the Logocanon, ou Regie Propor-
tionelle, Paris, 1626; and other works.

4 Born at Ornans, Burgundy, 1589; died at Madrid, October 20, 1664.

DESCARTES 371

Madrid. He wrote commentaries on Euclid's Elements ( 1 645 ) ,
the Conies of Apollonius (1655), and other Greek works.

Another writer of this period, Pierre Herigone, whose life
was one of comparative obscurity, published a work on general
mathematics 1 which stands out as a good summary for the
time, and which displayed considerable originality in the field
of algebra.

Descartes. If one were asked to name the man who was most
influential in the revolutionizing of mathematics in the iyth
century, he would naturally find it difficult to answer. Prob-
ably the name of Newton would lead in any ballot among
scholars. Newton's modest assertion that he had seen farther
only by standing on the shoulders of giants 2 is capable of easy
proof, as is the case with most men of eminence. It is the
genius, however, who can pick out his giants. Certainly
among those selected by Newton or by Fate was Rene Des-
cartes,"' a man whose varied genius led philosophers to rank
him primarily as one of themselves, physicists to claim him for
their guild, and mathematicians to look upon him as one of
the greatest geniuses in their domain, each group being fully
justified in its own opinion.

Descartes was fortunate in his birth, his father, Joachim
(died 1640), being a counselor in the parliament 4 of Bretagne
(from 1586) and possessed of sufficient means to give the son

^Cursus mathematicus, 5 vols., Paris, 1634-1637, with a Supplementum,
1642-1644.

2 D. Brewstcr, Life of Newton, 1, 142. Edinburgh, 1855.

3 Born at La Haye, Touraine, March 31, 1596; died at Stockholm,
February n, 1650. Latin, Renatus Cartesius, whence we speak of the Cartesian
geometry and the Cartesian philosophy. The earlier English and French
writers frequently used the form Des Cartes. Other branches of the family
used the older spelling Des Quartes and Des Quartis. Ren6 was also called,
against his wish, M. du Perron, from a small seigneurie belonging to the family
and situated near Poitou. This name was used by the family in his younger
days to distinguish him from his brother.

The most recent biographies are those of G. Miihaud, Descartes, Savant,
Paris, 192 1, containing various essays theretofore published; Elizabeth S. Haldane,
Descartes, His Life and Times, London, 1905; and C. Adam, Descartes, Paris,
1910. See also J. Millet, Descartes, sa vie, ses Travaux, ses decouvertes, avant
1637, Paris, 1867. *Parlement, little more than a local court.

372 FRANCE

an opportunity for early advance. When only a child Rene
was placed in a recently founded Jesuit school at La Fleche,

RENE DESCARTES
After the painting by Franz Hals, now in the Louvre

in Maine, where he remained for eight years (1604-1612) and
where he came to know Mersenne (p. 380), who was seven or
eight years his senior. The two established there a friendship

AUTOGRAPH LETTER OF DESCARTES

fas completing tl
r as the father of
of three years

374 FRANCE

that endured through life. He was sixteen years old when he
finished his course at La Fleche and returned to his home
"overwhelmed by the blessings and praises of his teachers."
There he remained for a year, his father then deciding that he
should complete his education in Paris. Here he renewed his
acquaintance with Mersenne, now become a Minimite, but the
acquaintance was soon interrupted by his friend's departure for
Nevers. He now set about to cultivate the study of geometry,
made the acquaintance of Mydorge (p. 378), and came in
contact with the best exponents of the mathematical thought
of the day. It was not long, however, before he felt the need
for a broader knowledge of the world and so decided to enlist
in army service. Not sympathizing with the aims of the French
nobility, he joined the army of Maurice, Prince of Orange,
a man who had already begun to attract to himself a group of
scholars, and in this way he became acquainted with Stevin
(p. 342). He soon became weary of army life, however, and
in 1621 withdrew from the service. He now devoted four years
to travel, visiting the German states, Denmark, Holland, Swit-
zerland, and Italy, returning to Paris in 1625. His old friend
Mersenne had also returned, Mydorge was still there, De-
sargues (p. 383) had joined the group, and besides these men
of the mathematical coterie he met Balzac and the literary
world generally, and was presented to Richelieu. Paris was
not the place for his meditations, however, and in 1628 he de-
cided to take up his residence in Holland, staying there until
1649, when, at the invitation of Christina of Sweden, he went
to Stockholm, where he remained until his death (1650) a few
months later.

His first years in Holland (1629-1633) were given to the
preparation of a work on philosophy. 1 On finishing this trea-
tise he learned of the condemnation of Galileo for daring to tell
the truth, and so he, quite unlike his Italian contemporary,
decided that discretion was the better part of valor. As a re-
sult, the work was not published during his lifetime. He then
devoted himself to the preparation of his great treatise on

*Le Monde* Paris, posthumously published in 1664.

DESCARTES'S GEOMETRY 375

method in science, 1 a work which included three appendixes,
one of which bore the modest title La Geometric.

Descartes's Geometry. It was in this appendix, a small hand-
book of only about a hundred pages, that analytic geometry
first appeared in print. 2 The fundamental idea in Descartes's
mind was not the revolutionizing of geometry so much as it
was the elucidating of algebra by means of geometric intuition
and concepts ; in a word, the graphic treatment of the equation.
His imagination extended far beyond this, however, to the
establishing of a universal mathematics in which algebra, geom-
etry, and arithmetic should be closely related members. He
began by extending the ancient idea of latitude and longitude,
showing that any point in a plane is uniquely determined by
two coordinates, x and y, the equation F(x, y) = o expressing
a property which is true for every point of the curve. By
studying the equation, therefore, he could, through the princi-
ple of a one-to-one correspondence, transfer his results at any
time to the curve itself. It constituted what John Stuart Mill 3
says was "the greatest single step ever made in the progress
of the exact sciences." The Geometric is divided into three
books. In the first book he relates the fundamental operations
of arithmetic to geometry, his use of a certain unit of length
being the only novelty in this feature ; in the second book 4 he
classifies curves and considers the methods of finding tangents
and normals ; and in the third book 6 he deals with the nature

*Discours de la methode pour bien condui/e sa rahon et chercher la veritt
dans les sciences, Leyden, 1637. The appendixes were La Dioptrique, Les
Meteores, and La Geometric. A Latin translation of the geometry, by Frans
van Schooten, appeared at Amsterdam in 1640.

2 The Opera Omnia of Descartes appeared at Amsterdam in 9 volumes, i6go-
1701, with a second impression in 1713. It also appeared at Paris in 13 volumes,
1724-1726. The V. Cousin edition of Les CEnvres appeared at Paris in n vol-
umes, 1824-1826, with a new edition by J. Simon in 1844. The latest edition,
edited by Charles Adam and Paul Tannery, was printed at Paris, Volume I
appearing in 1807. For a bibliography of the most important works relating to
Descartes see Haldane, loc. cit., p. 387.

s An Examination of Sir W. Hamilton's Philosophy, p. 617. London, 1878.

4 In the Latin edition, ;< De natura linearum curvarum."

6 In the Latin edition, "De constructione Problematum Solidorum, & Solida
excedentium."

376 FRANCE

of the roots of equations, considering among other things the
rule of signs that has since been known by his name. The idea
of analytic geometry had already been worked out by Fermat,
or at least was conceived by him and Descartes at about the
same time ; but Fermat only thought, while Descartes not only
thought but wrote. Some conception of the plan was probably
also in the mind of Harriot, and Descartes was familiar with
Harriot's work ; 1 but, after all, the real idea of functionality
as shown by the use of coordinates was first clearly and publicly
expressed by Descartes.

Some idea of his range of knowledge and interest may also
be obtained from his work on anatomy, begun in 1634, and of
which a Latin edition appeared at Leyden in i664. 2

Descartes the Man. Descartes inherited, apparently from his
mother, a feeble constitution. He speaks of his "dry cough
and pale complexion," which remained with him until he en-
tered the army, and which led his physician to predict that
his life would be a short one. He overcame his early ailments,
however, and although he died at fifty-four he was able to
say in his later life that for thirt}' years he had been free
from any illness that deserved the name. John Stuart Mill re-
marked of him : " Descartes is the completest type which his-
tory presents of the purely mathematical type of mind that
in which the tendencies produced by mathematical cultivation
reign unbalanced and supreme.' 7 Although the statement may
well be questioned, it is interesting as a striking assertion if
for no other reason.

Early Commentators on Descartes. Of the two leading com-
mentators on Descartes in the iyth century the first (1649)
was his warm personal friend, Florimond de Beaune, 3 an office-
holder at Blois. He also wrote on algebra, 4 being one of the

1 Artis Analyticae Praxis, London, published posthumously in 1631.

2 De Homine figuris, et Latinitate Donatus a Florentine Schuyl, Leyden,
1664. This edition has some engravings of great merit.

3 Born at Blois, 1601; died at Blois, 1652.

4 De aequationum constructione et limitibus, published posthumously at
Amsterdam in 1659. There was also an edition, Amsterdam, 1683.

FERMAT

377

first to treat scientifically of the superior and inferior limits of
the roots of a numerical equation. He endeavored to deduce
the nature of curves from the properties of their tangents and
was also interested in the improvement of astronomical instru-
ments. The second and more prominent of the commentators
was Frans van Schooten 1 (c. 1615-
1661), who will be considered later.

Fermat. The life and works of
Pierre de Fermat 2 illustrate the fact
that no one can account for a genius.
Why should the greatest writer on
the theory of numbers, at least after
the time of Diophantus, suddenly
appear in the person of a modest,
retiring, punctilious counselor of the
parliament of Toulouse, and in the
1 7th century? Why should this ob-
scure officeholder succeed in making
such a name for himself while ap-
parently giving no serious attention
to mathematics until he was over thirty ? And why, aware of
his powers as he must have been, was he content to make his
results known chiefly through letters to men like Mersenne,
Roberval, Pascal, and Descartes instead of publishing them for
the benefit of scholars in general ? One answer to each of these
questions is that genius is eccentric.

It may have been Bachet's translation (1621) of Diophantus
that directed Fermat's attention to the theory of numbers, for

1 Known also as Franciscus van Schooten.

2 Born at Beaumont de Lomagne, near Toulouse, c. 1608; died at Castres or
Toulouse, January 12, 1665. The date of his birth as given by different writers
varies from 1590 to 1608. His tombstone in the church of the Augustines in
Toulouse (later in the museum) gives the date of death as above and the
age as fifty-seven years.

E. Brassinne, Precis des wuvres mathematiques de Pierre Fermat, Paris,
1853 ; (Euvres de Fermat, edited by P. Tannery and C. Henry, Paris, 1891-
1912. See also C. Henry in Boncompagni's Bullettino, XII, 477 and XIII, 437 ;
P. Tannery in Darboux's Bulletin, VII (2) ; G. Libri, "Fermat," in Revue des
Deux Mondes, May 15, 1845.

FERMAT
After an old lithograph

378 FRANCE

he left a series of notes and letters on this work which were
published in the form of a commentary (Toulouse, 1670) after
his death. At any rate he showed remarkable ability in this
field of mathematics. He asserted that no integral values of
x, y, and z can be found to satisfy the equation x"+y" = s n if n
is an integer greater than 2, and this is commonly known as
Fermat's Theorem. No satisfactory demonstration has ever
been published, and it is not known whether Fermat himself
demonstrated it, few of his proofs having been preserved.
What knowledge we have of these proofs is due in part to
marginal notes made by him in his reading, and in part to a
certain manuscript of Huygens found at Leyden in 1879. He
seems to have claimed that he had proved this particular theo-
rem. Fermat 's letters show th?t he had developed the idea of
analytic geometry before Descartes published (1637) his work
upon the subject. Descartes proposed to represent a curve
by an equation, to study this equation, and in this way to dis-
cover the properties of the curve itself ; while Fermat did
substantially the same thing, designating the equation as the
" specific property" of the curve and deriving all other prop-
erties from it.

In connection with his study of curves Fermat proceeded to
apply the idea of infinitesimals to the questions of quadrature
and of maxima and minima as well as to the drawing of tan-
gents. In this he seems to have anticipated the work of Cava-
lieri, but the date of his discovery is unknown. 1

Mydorge. Of the influential members of the brilliant group
of mathematicians that brought the science prominently to the
attention of Paris in the first half of the i7th century Claude
Mydorge, 2 a friend of Descartes, must be named among the
first. He was a man of means, an official of the government, a

ir The esteem in which he was held by Descartes is shown in one of the
letters which the latter wrote to him : " Je n'ai jamais connu personne, qui
m'ait fait paraitre qu'il sut tant que vous en geometric." Cantor, Geschichte,
II, chap. 79. Format's Varia Opera Mathematica, 2 vols., edited by his son,
Samuel Fermat (Toulouse, 1670), was reprinted in facsimile in Berlin in 1861.

2 Born in Paris, 1585; died in Paris, July, 1647.

THEORY OF NUMBERS

379

physicist of recognized standing, and a mathematician of fair
attainments, writing upon optics, conies/ and the recreations
of mathematics. 2

Other Writers on the Theory of Numbers. Fermat was not
alone in his interest in the theory of numbers at this time.
Indeed, he was not the
earliest French scholar of
the century to consider
the subject, for Diophan-
tus had been made
known in France before
Fermat showed any in-
terest in the subject.
The man responsible for
this initial step in the
theory of numbers was
Claude-Caspar Bachet,
Sieur de Meziriac, 3
mathematician, philoso-
pher, theologian, poet,
and one of the ablest
writers of his day. He
came from an ancient

MERSENNE

Engraving by Duflos

and noble family 4 and
passed some part of his
youth in Italy. He took
the first steps toward entering the Jesuit order, but aban-
doned the idea and went to Paris, where he became a member
of the Academic des Sciences. His work on mathematical

*De Sectionibus Conicis Libri IV, Paris, 1631. See also the "Problemes de
Geometric Pratique" in Boncompagni's Bullettino, XVI, 514.

2 He edited the popular Recreations Mathematiqves (title as in the 1628 ed.)
of Leurechon (ist ed., 1624). Mydorge's editions were Paris, 1630, 1634, 1639,
and later. See Boncompagni's Bullettino, XIV, 271.

8 Born at Bourg-en-Bresse, October 9, 1581 ; died at Bourg-en-Bresse,
February 25, 1638.

4 For particulars and for biographical information in general see P. Bayle,
Dictionaire historique et critique, Paris, 1734, with numerous bibliographical
references; hereafter referred to as Bayle, Dictionaire.

380 FRANCE

recreations 1 was the best of all that appeared in the iyth cen-
tury and is still looked upon as a classic in that field, both in
style and in content. His well-known translation of Diophan-
tus from the Greek into Latin was published in i62i. 2

*> 7-

"0J+& rw^ ^It/^-uX c>x^'V >

~^fr^ .

AUTOGRAPH LETTER OF MERSENNE
Written about 1640

Another contributor to the theory of numbers at this time
appeared in the person of Marin Mersenne," a Minimite friar,
who taught philosophy and theology at Nevers and Paris and was
in constant correspondence with the greatest mathematicians

^Problemes plaisans ft delect ables, qui se font par les nombres, Lyons, 1612.
There were later editions, Lyons, 1624; Paris, 1874; Paris, 1879; Paris, 1884.

2 Diophanti Alexandrini Arithmeticorum libri sex, Paris, 1621. The text
appears in Greek and Latin. Fermat's edition appeared at Toulouse in 1670.
Xylander's translation had already appeared at Basel in 1575.

3 Born at Oiz6, Maine, December 8, 1588 ; died in Paris, September i, 1648.
F. H. D. C., La Vie dv R. P. Mersenne, pp. 2, 6 (Paris, 1649).

PASCAL

of his day. He was a voluminous writer, editing some of the
works of Euclid, Apollonius, Archimedes, Theodosius, Mene-
laus, and various other Greek mathematicians. He also wrote
on a variety of other subjects, including physics, mechanics,
navigation, geometry, and mathematical and philosophical
recreations. It is in the
theory of numbers, how-
ever, particularly with
respect to prime num-
bers and perfect num-
bers, 1 that he made con-
tributions of real value.

Pascal. Blaise Pascal, 2
whom Bayle 3 appre-
ciatively calls "one of
the most sublime spirits
in the world," was
blessed in having a
father who could and
did start him in the
right direction. Etienne
Pascal was an able
mathematician and was
so desirous of giving the
best advantages to his
only son that he relinquished his post of President a la Cour des
Aides of his province and went to Paris in 1 63 1 . Educated solely
under his care, Blaise showed phenomenal ability in mathematics
at an early age, and although his father wished him first to have

BLAISE PASCAL
After a contemporary drawing

Cogitata Physico-Mathematica, Paris, 1644, appeared four years before
his death. On the nature of Mersenne's Numbers see W. W. R. Ball, Messenger
of Mathematics, XXI, 34, 121 ; but consult also L. E. Dickson, History of the
Theory of Numbers, I, 12 and 31 (Washington, 1919), hereafter referred to as
Dickson, Hist. Th. Numb.

2 Born at Clermont-Ferrand in Auvergne, June 19, 1623 ; died in Paris,
August 19, 1662.

3 Bayle, Dictionaire, IV, 500, with an unusually good biography for Bayle,
who is commonly more erudite than helpful. See also A. Maire, L'ceuvre scien-
tifique de Blaise Pascal Paris, 1912 ; A. Desboves, Etude sur Pascal, Paris, 1878.

382 FRANCE

a thorough grounding in the ancient languages, and therefore
took from him all books on mathematics, he succeeded in be-
ginning geometry by himself and in making considerable prog-
ress before his efforts were discovered. Various anecdotes of
his youthful activities in mathematics are told by his sister,
Madame Perier, who wrote his biography. She relates that he
discovered independently most of the first book of Euclid, that
his intuition in mathematics seemed miraculous, and that geom-
etry was simply his recreation. He played with conies as other
children play with toys, but with the divine enjoyment of dis-
covering eternal truths. When Descartes was shown a manu-
script which Pascal wrote on conies at the age of sixteen, he
could hardly be convinced that it was not the work of the
father instead of the son. At the age of nineteen he invented
a computing machine that served as a starting point in the
development of the mechanical calculation that has become so
important in our time. That he should have been permitted to
present one of these machines to the king and one to the royal
chancellor shows the esteem in which he must have been held.
At the age of twenty-three he became interested in the work
of Torricelli in atmospheric pressure, and soon established for
himself a reputation as a physicist. Among his discoveries was
the well-known theorem which bears his name, that the three
points determined by producing the opposite sides of a hexa-
gon inscribed in a conic are collinear, a theorem from which
he deduced over four hundred corollaries. He also wrote
( X 6S3) so extensively on the triangular arrangement of the co-
efficients of the powers of a binomial, which had already at-
tracted the attention of various writers, that this arrangement
has since been known as Pascal's Triangle. In connection with
Fermat he laid the foundation for the theory of probability. 1
He also perfected the theory of the cycloid and solved the
problem of its general quadrature.

1 I. Todhuntcr, A History of the Mathematical Theory of Probability,
Chap. II (Cambridge, 1865), hereafter referred to as Todhunter, Hist. Proba-
bility. On Pascal's use of induction in this theory see W. H. Bussey, Amer.
Math. Month., XXIV, 203. Pascal's work on the triangular arrangement of
coefficients was published posthumously in 1665.

PASCAL AND DESARGUES 383

His contributions to science and letters often appeared under
the nom de plume of Louis (Lovis) de Montalte (as in his
Lettres provinciates} and the anagram on the same name, Amos
Dettonville (as in various problems which he proposed). It
is on this account that Leibniz occasionally speaks of him as
Dettonville.

Having already, at the age of twenty-five, made for himself
an imperishable reputation in mathematics and physics, he
suddenly determined to abandon these fields entirely and to
devote his life to a study of philosophy and religion. The life
of penance which he lived thereafter seems strange, since all
must feel that he had little of which to repent, but at any rate
it speaks well for the faith of men that he was sincere in his
belief and irreproachable in his conduct.

Desargues. In the line of pure geometry the most original
contributor of the i7th century was Gerard Desargues, 1 of
whose life not much is known except that he was for a time an
officer in the army, that he then lived in Paris (1626), where
he gave some public lectures, that he was an engineer, and
that his later years were spent upon his estate near Condrieux.
He published several works, but is known chiefly for his treatise
on conies. 2

Perhaps because it appeared at about the same time as the
great work of Descartes, perhaps because the chief interest in
mathematics shown by Desargues had been in its applications
to the study of perspective, this masterpiece seems to have
attracted little general attention, although appreciated by both
Pascal and Descartes. At any rate the work was soon for-
gotten and remained almost unknown until Chasles happened
to find a copy in 1845, since which time it has been looked
upon as one of the classics in the early development of
modern pure geometry. In this work he introduced the no-
tions of the point at infinity, the line at infinity, the straight

iBorn at Lyons, 1593 J died at Lyons. 1662. N. G. Poudra, Desargues . . .
CEuvres, 2 vols. Paris, 1864.

2 Brouillon pro jet d'une atteinte aux euinemens des rencontres d'un cone
avec un plan, Paris, 1639; M. Chasles, Aperfu, 74; Poudra, loc. cit. t pp. 97, 303.

FRANCE

line as a circle of infinite radius, geometric involution, the tan-
gent as a limiting case of a secant, the asymptote as a tangent
at infinity, poles and polars, homology, and perspective, thus

laying a substantial
basis for the modern
theory of projective
geometry.

L'Hospital. Guil-
laume Frangois An-
toine de PHospital, 1
Marquis de St.-
Mesme, a man of an-
cient and honorable
family, was one of
the world's infant
prodigies in mathe-
matics. When only
fifteen he was one
day at the Due de
Roanne's and heard
some mathemati-
cians speaking of a
difficult problem of
Pascal's. To their
surprise he said that
he thought he could
solve it, and in a few
days succeeded. A

career which he sought in the army proved impossible owing
to his defective sight, and the latter part of his life was given
to his favorite study. He was a pupil of Jean Bernoulli's and
introduced the ideas of the new analysis into France. 2 He also
wrote on geometry, algebra, and mechanics, most of his works
being published after his death.

iBorn in Paris, 1661 ; died in Paris, February 2, 1704. He is also known
as the Marquis de 1'Hospital. The family also spelled the name Lhospital and,
somewhat later, FHopital. 2 Analyse des infiniment petits, Paris, 1696.

MARQUIS DE L'HOSPITAL

He had much to do with the introduction of
Newton's mathematical ideas into France

L'HOSPITAL AND ROBERVAL 385

Frenicle de Bessy and De la Loubere. Among the corre-
spondents of Fermat, Bernard Frenicle de Bessy, 1 an office-
holder and a member of the Academic des Sciences at Paris,
was known for his work on Pythagorean numbers, that is, num-
bers which form the sides of a right-angled triangle, 2 and for
his interest in magic squares. At about the same time Antoine
de la Loubere, 3 a Jesuit and a lecturer on mathematics, rhetoric,
theology, and the humanistic subjects, was showing much in-
terest in the study of curves. This interest is seen in his quad-
rature problem 4 and in his study of the cycloid. r> His method
of tangents, in which the tangent is taken as the direction of a
moving point, was quite forgotten until its value was recognized
in its applications in kinematics. 6

Roberval. Among the contemporaries of De la Loubere, Gilles
Persone de Roberval 7 became well known for his discoveries
in the field of higher plane curves and for his method of draw-
ing a tangent to a curve (already suggested in substance by
Torricelli), which was a definite step in the invention of the
calculus. He was professor of philosophy in the College Ger-
vais at Paris, and later professor of mathematics in the College
Royal. His chief interest was in physics, but he also wrote on
the cycloid (his "trochoid") and other curves, on algebra and
indivisibles, 8 and (1644) on the astronomy of Aristarchus.

iBorn in Paris, c. 1602 ; died 1675.

2 Traite des triangles rectangles en nombres, Paris, 1676 (posthumous).

3 Born at Ricux, Languedoc, 1600; died at Toulouse, 1664. The name also
appears as Laloubere, Laloucre, Lovera, Lalovera, Lalouverc.

*Elementa telragonismica sen demonitratio quadraturae circuit et hyperbolae
ex datis ipsorum centris gravitatis, Toulouse, 1651. See also Montucla, Histoire,
II (2), 77-

zpropositio 36 a excerpt a ex quarto libro de cycloide nondum edito, Toulouse,
1659 J Veterum geometria promota in septem de cycloide libris, Toulouse, 1660.

6 See Chasles, Aper$u, 58, 96; F. Jacoli, Boncompagni's Bullettino, VIII, 265.

7 Born at Roberval, near Beauvais, August 8, 1602; died in Paris, October 27,
1675. The name Roberval, by which he is commonly known, was merely that
of his birthplace. The family name, Persone, appears also in the Latin form
of Personerius, whence a derived French form is Personier.

8 De geometrica planarum et cubicarum aequationum resolutione, De recog-
nitione aequationum, and Traite des indivisibles. These and other of his
memoirs were collected, published in 1693, and republished in the Memoir es de
Vancienne academie, Vol. VI.

386 FRANCE

Other Writers of the Period. At about the same time Claude
Frangois Milliet Dechales, 1 for some time a Jesuit missionary
in Turkey, taught in the schools o f his order in Marseilles,
Lyons, and Chambery. He is chiefly known for his editions
of Euclid's Elements* and for a general work on mathematics/''
but his original contributions to the subject were slight.

Noted as a voluminous writer on theology, Antoine Arnauld
(1612-1694) "the great Arnauld" as the Jansenists called
him, hated by the Jesuits and the Calvinists alike because of
his bitter attacks upon their beliefs deserves at least some
slight mention for his encouragement of mathematics. He
was interested in the works of his great contemporaries, such
as Descartes and Pascal, wrote on geometry ( 1667) an d magic
squares, and showed interest in the theory of numbers. 4

Among those who in the latter part of the century did much
to make geometry popular was Philippe de Lahire/' a pupil of
Desargues's and a man of scattered genius. He was at first a
painter and architect, then a pensionnaire astronome of the
Academic des Sciences at Paris, then professor of mathematics
in the College Royal and the Academic de V Architecture, and
in his later years (from 1679) was connected with the geodetic
survey of France. He wrote several works on conies, algebra,
and astronomy, besides contributing a large number of memoirs
on mathematics, astronomy, and physics to the Academic des
Sciences. He also wrote on epicycloids (1694) and roulettes
(1694, 1706), and summarized what was then known on magic
squares (1705).

ifiorn at Chambery, Savoy, 1621; died at Turin, March 28, 16*78. The
name also appears as Deschales and De Challes.

2 Latin ed., Lyons, 1660; French ed., Paris, 1677.

3 Cnrsus sen Mundus Mathematicus, Lyons, 1674, with a later edition in i6qo.

4 K. Bopp," Antoine Arnauld . . . als Mathematiker," Abhandlungen, XIV, 187.

6 Born in Paris, March 18, 1640; died in Paris, April 21, 1718. The Biogra-
phic Universclle gives the date of his death as 1710, but most authorities give
it as 1718. The name is often written La Hire. See Chasles, Aper$u, 118, 550,
553; Curtze, in Bibl. Math., II (2), 65.

ti Theorie des coniques, Paris, 1672; Nouvelle Methode de Geometrie, Paris,
1673; Nouveaux element des sections coniques ; Les Lieux Geometriques\ La
Construction ou ejection des equations, Paris, 1679, with an English translation,
London, 1704; Sectiones conicae in novem libros distributae t Paris, 1685.

MINOR FRENCH WRITERS 387

Among the contemporaries of de Lahire, but a few years
his junior and like him a pensionnaire of the Academic des
Sciences, Michel Rolle 1 made for himself a worthy name. He
was connected with the war department and apparently was not
concerned with teaching. He wrote on both geometry 2 and
algebra, his publications on geometry appearing in the form
of numerous memoirs, and those on algebra in memoirs and
in two books. 3 To him is due the theorem that /'(#) = o has at
least one real root lying between two successive roots of
/(*) = o

As a representative of the other French mathematicians of
this period there may be named Pierre Nicolas, 5 a pupil of
De la Loubere's and rector of the Jesuit college at Beziers, who
wrote on the logarithmic spiral and on conchoids. 6

In the field of textbook making the most popular French
writer of this time was Frangois Barreme, a native of Lyons,
who died in Paris in 1703. His Arithmetique (Paris, 1677)
went through many editions, and his name is still a synonym
for a ready reckoner (bar&mc).

4. GREAT BRITAIN

Great Britain in the Seventeenth Century. As already re-
marked, the two foci of the ellipse that bounded mathematical
Europe in the i7th century were located in France and Great
Britain. The British center was Cambridge, although the con-
tributions of Edinburgh in the notable discoveries of Napier

a Born at Ambert, Auvergne, April 21, 1652; died in Paris, November 8,
1719. The date of his death is also given as July 5.

2 A certain curve, xy 2 = a (y mx) 2 , has recently come to bear his name,
but apparently with no justification.

3 TraiU d'algebre, Paris, 1690; Methode pour resoudre les questions
indeterminees de Valgebre, Paris, 1699. F. Cajori, "What is the origin of the
name 'Rolle's Curve' ?"Amer. Math. Month., XXV, 291, and Bibl. Math., XI
(3) 300- See also Bibl. Math., IV (3), 399, and U Intermediate des Mathema-
ticiens, V, 76, and XVI, 244 (Paris), hereafter referred to as V Inter medicare.

4 Demonstration d'une Methode pour resoudre les Egalitez de tons les degrez,
Paris, 1691. 5 Born at Toulouse, c. 1663; died r. 1720.

6 De novis spiralibus exercitationes, Toulouse, 1693; De Lineis logarithmicis
spiralibus hyperbolicis t Toulouse, 1696; De conchoidibus et cissoidibus, Tou-
louse, 1697.

388 GREAT BRITAIN

and Gregory, and of Oxford in the works of such men as Har-
riot, Briggs, Halley, Wren, and Wallis were such as to chal-
lenge the supremacy of the Cambridge school.

Harriot. It is rather surprising to think that the man who
surveyed and mapped Virginia 1 was one of the founders of
algebra as we know the science today. Such, however, is the
case, for Thomas Harriot 2 was sent by Sir Walter Raleigh to
accompany Sir Richard Grenville (1585) to the New World,
where he made the survey of that portion of American terri-
tory. He returned to England (1587) and published (1588)
a report upon the colony, and some years later wrote a work
that helped to establish the English school of algebraists. Har-
riot took his B.A. at Oxford in 1579. After his return from
America he was introduced to the Earl of Northumberland and
was received with other scholars into his household. The earl
allowed him a pension of 300 a year for the rest of his life.
He was prominent as an astronomer and corresponded with
Kepler. He discovered the solar spots, and his observations
of the satellites of Jupiter were independent of those made by
Galileo at the same time. His great work on algebra 3 was pub-
lished ten years after his death. In this work he assisted in
setting the standard for a textbook in algebra which has been
generally recognized since that time. The work includes the
formation of equations with given roots, the law as to the num-
ber of roots, the relation of the roots to coefficients, the trans-
forming of equations into equations having roots differing from
the original roots according to certain laws, and the solution
of numerical equations. He used small consonants for the
known quantities and small vowels for the unknown quantities.

a Or rather North Carolina, since the present boundaries had not yet been
fixed. On his map of Virginia see P. L. Phillips, Virginia Cartography, Wash-
ington, 1896. His report was published in London in 1588. A second edition
appeared in Hakluyt's The Prindpall Navigations, London, 1589. F. V. Morley,
The Scientific Monthly, XIV, 60.

2 Born at Oxford, 1560; died near Isleworth, July 2, 1621. The name appears
as Hariot in several early works.

3 Artis Analyticae Praxis, ad Mquationes Algebraicas noua . . . Methodo re-
soluendas, London, 1631. It was probably written c. 1610.

HARRIOT AND NAPIER

389

He also took some steps in the direction of analytic geometry.
In the further matter of symbols, he used V3 for the cube
root, 1 and the characters > and < for "is greater than" and
"is less than" respectively.

Napier. When Hume, the historian, wrote his appreciation
of Napier as "the person to whom the title of ' great man' is
more justly due than to
any other whom his
country has ever pro-
duced," he spoke without
exaggeration. Burns can
be appreciated only in
the vernacular, Scott ap-
peals chiefly to a single
period in the develop-
ment of the individual,
the fame of those whose
effigies justly have place
along Princes Street in
Edinburgh is generally
only national; but Na-
pier's remarkable inven-
tion affects the whole
world with constantly
increasing power. The
artisan who carries in
his pocket the slide rule
is relatively as much indebted to the genius of the Laird of
Merchiston as the astronomer, the engineer, the physicist, and
the mathematician.

John Napier 2 was descended from a strong ancestry which
included men who had held positions of prominence because
they deserved to do so. Merchiston Castle, built in the

JOHN NAPIER

Engraved by Stewart after the original
painting in Edinburgh

i As in

for

2 Born at Merchiston Castle, now in the city of Edinburgh, 1550; died there,
April 4, 1617. The name also appears as Naper, Naperus, Neper, and Neperius.

3QO GREAT BRITAIN

century, was one of the two strongholds on the outskirts of
Edinburgh, and was enlarged from time to time until it became
an imposing structure, symbolic of the Napier house.

Napier was born in the period of greatest strife between
Protestantism and Catholicism in Scotland. John Knox began
his mission only three years before Napier's birth, and the
seeds of the bitter antagonism which the latter felt towards
Rome were early planted in his soul. In 1563, when only
thirteen years old, Napier was sent to the University of
St. Andrews, but left without taking a degree. He probably
studied abroad, but in 1571 was back again in Scotland, this
time at Gartness, in Stirlingshire, where his father had some
property. It was probably here that he wrote his popular
theological work, A Plaine Discouery of the whole Reuelation
of Saint lohn (Edinburgh, IS93), 1 a bitter attack upon the
Church of Rome. The common people accused him of deal-
ing in the black art, while the intellectuals recognized him as a
man of remarkable ingenuity ; but the end was the same. He
planned to use burning mirrors, like those of Archimedes but
so potent that they should destroy an enemy's ships "at what-
ever appointed distance" ; a piece of artillery that should, as an
early writer described it, " clear a field of four miles circumfer-
ence of all the living creatures exceeding a foot of height"; a
chariot which should be like "a living mouth of mettle and
scatter destruction on all sides"; and "devises of sayling
under water," all of which is of interest in view of the engines
of destruction used first in the World War of 1914-1918. It is
of interest to compare the mind of Napier, as seen in his vision
of future achievements, with that of Roger Bacon. Each seemed
to many of his contemporaries, and perhaps to most of those
who knew him, as mentally unbalanced and as a mere visionary
in his contemplation of future warfare, and yet each prophesied
with remarkable success respecting many inventions of the
present time.

1 Its great popularity is shown by the fact that it was reprinted in London
in 1504, 1611, and 1641; in French translation at La Rochelle in 1602, with a
fourth edition in 1607 ; in Dutch translation at Middelburgh in 1600 ; and in
German translation at Frankfort a. M. in 1611.

NAPIER AND BRIGGS 391

Napier on Logarithms. Napier wrote two works on loga-
rithms/ besides one on computing rods 2 and one on algebra.*
Of all his works he probably thought the Plaine Discouery of
the whole Reuclation of Saint lohn the most important, but
the world has long since forgotten it. The popular verdict of
his day was that the Rabdologia was his greatest work, but it
is now looked upon only as one of the curiosities of history.
The scientific world looked upon his Dcscriptio as epoch-
making, and the scientific world was right. 4

Briggs. The man who did most to start the invention of
Napier on its road to success was Henry Briggs, 5 whom Ought-
red rather absurdly called the English Archimedes.** He en-
tered St. John's College, Cambridge, in 1577, took the degrees
of B.A. in 1581 and M.A. in 1585, and was made a fellow in
1588. He was the first professor of geometry at Gresham
College, London (1596-1619), after which he became pro-
fessor of astronomy at Oxford. He saw at once the great im-
portance of Napier's invention, and in a letter to James Ussher,
archbishop of Armagh, dated March 10, 1615, he speaks of
himself as being " wholly employed about the noble invention

l Mirifici Logarithmorum Canonis descriptio . . . Authore ac Invent ore
loanne Nepero, Barone Merchistonii, &c. Scoto, Edinburgh, 1614, with other
editions in 1616, 1619, and 1889; Leyden, 1620; London, 1616 and 1618; Mirifict
ipsius canonis constructio, which appeared posthumously and was added to the
edition of 1610 mentioned above. On all these editions and on the subject in
general see C. G. Knott, Napier Tercentenary Memorial Volume, London, 1015.

2 Rabdologiae, sev nvmerationis per virgulas libri dvo> Edinburgh, 1617,
published the year of his death. For description, see Volume II. There was an
edition at Leyden, 1626; an Italian translation, Verona, 1623, in which Napier
is quoted as ascribing all glory and honor "alia Beatissima Vergine Maria,"
which is the last thing he would have dreamed of doing; and a German transla-
tion, Berlin, 1623. There was a free English translation of part of the work
by John Dansie (.1 Mathematicall Manuel, London, 1627), the first English
version to appear.

3 Preserved in the Napier family and published in Edinburgh in 1839 under
the title De Arle Logistka Joannis Naperi Merchistonii Baronis Libri Qid
Super sunt.

4 The subject is considered at length in Volume II.

5 Born at Warley Wood, Yorkshire, Februaiy, 1560/61 (1561 N. S.) ; died
at Oxford, January 26, 1630/31. The date of birth is often given as 1556,
but the parish register shows that it was 1560/61.

e Aubrey, Brief Lives, I, 124.

392 GREAT BRITAIN

of logarithms." In the following year (1616) he made a visit to
Edinburgh for the purpose of meeting with Napier, and re-
peated his visit in 1617. It was on the first of these visits that
Briggs suggested the base 10, of which Napier had already
thought, this being the base of the common system of log-
arithms that has been in use ever since that time, and on his
return to Oxford he prepared a table accordingly.

Briggs published ten works and left six others unpublished.
The published works include treatises on navigation, Euclid's
Elements? logarithms, 2 and trigonometry. 3

Gellibrand. Briggs's friend Henry Gellibrand, 4 who edited
his trigonometry (1633), entered Trinity College, Oxford, in
1615, was granted the degrees of B. A. in 1619 and M. A.
in 1623, took holy orders, and entered upon church work-
Having heard one of Sir Henry Savile's lectures, he was so
impressed that he gave up his curacy and devoted himself
entirely to mathematics. Besides editing the trigonometry left
unpublished by Briggs he also wrote on navigation and the
variation of the magnetic needle, and composed a trigonometry
of his own/ 1

Oughtred. One of the greatest of the writers of the early
part of the i?th century in his influence upon English mathe-
matics was William Oughtred." As in the case of Harriot,

1 Elementorum Euclidis libri VI priores, London, 1620.

2 Arithmetica logarithmica, London, 1624. The final (French) edition ap-
peared at Gouda in 1628.

^Trigonometria Britannica, sive de doctrina triangulorum libri duo, Gouda,
1633, published posthumously by his friend Henry Gellibrand.

4 Born in the parish of St. Botolph, Aldersgate, London, November 17,
1597; died in London, February 16, 1636/37.

An Institution Trigonometricall wherein . . . is exhibited the doctrine of the
dimension of plain and spherical triangles . . . by tables . . . of sines, tangents,
secants, and logarithms, London, 1638. This title is from the second edition
(1652).

6 Born at Eton, March 5, 1574; died at Albury, June 30, 1660. He also
wrote his name Owtred, and John Locke preferred the form Outred, from which
spellings we infer the pronunciation. On his life and works see F. Cajori,
William Oughtred, Chicago, 1916, hereafter referred to as Cajori, Oughtred.
The date of his birth is from Aubrey, Brief Lives, "Gulielmus Oughtred
natus 5 Martii 1574, 5 h. P. M."

OUGHTRED 393

he was not a professor of mathematics ; but, like Harriot also,
he knew more mathematics than most professors of his day.
He entered King's College, Cambridge, in 1592, became a
fellow in 1595, and received the degrees of B. A. in 1596 and
M. A. in 1600. Speaking of his college work he says :

The time which over and above those usuall studies I employed
upon the Mathematicall sciences, I redeemed night by night from
my naturall sleep, defrauding my body, and inuring it to watching,
cold, and labour, while most others tooke their rest.

He vacated his fellowship in 1603 and in the following year
began his ministry. He gave much of his time, however, to
mathematics and to correspondence with mathematicians.
Aubrey, 1 whose gossiping biographies are always entertaining,
gives this description of him :

He was a little man, had black haire, and blacke eies (with a
great deal of spirit). His head was always working. He would
drawe lines and diagrams on the dust . . . did use to lye a bed till
eleaven or twelve a clock . . . Studyed late at night ; went not to
bed till ii a clock; had his tinder box by him; and on the top of
his bed-staffe, he had his inke-horne fix't. He slept but little. Some-
times he went not to bed in two or three nights.

He thus seems to have violated many of the usual canons of
health, and probably continued to do so until his death at the
ripe old age of eighty-six.

The Clavis Mathematics. Oughtred's best-known work is his
Clavis mathematics? a brief treatise on arithmetic and algebra,
composed (c. 1628) for the purpose of instructing the son of
the Earl of Arundel. It was published in London in 1631.
He had already written (c. 1597) a treatise on dialing, but this
was not published until 1647, when it appeared in an edition

1 Aubrey, Brief Lives, II, 106.

2 For the full title and for a study of the various editions see Cajori,
Oughtred, p. 17; H. Bosnians, "La premiere Edition de la 'Clavis Mathematica '
d'Oughtred," Annales de la Societe scientifique de Bruxelles, XXXV, 2 me
partie, p. 24.

394 GREAT BRITAIN

of the Clavis? The influence of the latter work was very great.
In it appear contracted multiplication and division, the dis-
tinction between the two uses of the signs + and , the
symbol ( : : ) for proportion, and the symbols x for multipli-
cation (already known) and ^ for the absolute value of a
difference.

The Slide Rule. The invention of the slide rule seems unques-
tionably due to Oughtred," and there also seems to be good
reason for believing that he is the author of the Appendix to
the Logarithmcs printed with the English translation of Na-
pier's work (London, 1618) and containing the first natural
logarithms. He also wrote a trigonometry 4 to which reference
will be made later, and a work on gaging, and he translated
and edited Leurechon's French work on mathematical recrea-
tions. Among his many pupils were John Wallis and Sir
Christopher Wren, the former of whom wrote of the Clavis
that it "doth in as little room deliver as much of the funda-
mental and useful part of geometry (as well as of arithmetic
and algebra) as any book I know."

Gunter. Connected with the general movement to simplify
calculation, initiated by Napier and Briggs, there stands out
prominently the name of Edmund Gunter, 4 who left the minis-
try to become professor of astronomy (1619) at Gresham Col-
lege, London. He published the first table of logarithmic sines
and tangents to the common base, 5 suggested to Briggs the use
of the arithmetic complement, invented the surveyors 7 table
and the chain which until recently was commonly known as

1 It was translated into Latin by Sir Christopher Wren. Another of his
works on dialing (1600) was published in 1632.

2 The Circles of Proportion and The Horizontal} Instrument, London, 1632.
See F. Cajori, History of the Logarithmic Slide Rule, New York, IQOQ, and
" On the History of Gunter's Scale and the Slide Rule," University of California
Publications in Mathematics, I, p. 187.

8 Trig on o me trie* or, The manner of calculating the Sides and Angles of
Triangles, by the Mathematical Canon, demonstrated, London, 1657.

4 Born in Hertfordshire, 1581; died in London, December 10, 1626.

*Canon triangulorum or Table of Artificial Sines and Tangents, London,

GUNTER 395

Gunter's chain, and devised a kind of slide rule known as
Gunter's scale. Aubrey relates the following incident:

When he was a student at Christ Church, it fell to his lott to
preach the Passion sermon, which some old divines that I knew did
heare, but they sayd that 'twas sayd of him then in the University
that our Saviour never suffered so much since his passion as in that
sermon, it was such a lamentable one Non omnia possumus omnes. 1

The Savilian and Lucasian Professorships. Mention is so often
made of the Savilian professorships of geometry and astronomy
at Oxford and of the Lucasian professorship at Cambridge that
a brief statement concerning them is desirable.

Sir Henry Savile, 2 the founder of two chairs at Oxford,
was warden (1585) of Merton College, Oxford, and in 1596
was appointed provost at Eton. He lectured on Euclid," and
although he contributed little to mathematics, he contributed
much to the extending of the knowledge of the science through
the professorships which he founded in 1619, and which have
been held by a distinguished line of scholars for three centuries.

Henry Lucas 4 was for a time a student at St. John's College,
Cambridge, but seems not to have matriculated. He was, how-
ever, admitted M.A. in 1635-1636 and was elected to represent
the university in parliament in 1639-1640. By his will he di-
rected that lands yielding ioo 5 a year be purchased to found
the professorship that bears his name. It was in the year after
the founding of the professorship that Barrow was elected
(1664) as the first occupant of the chair, and six years later
he was succeeded by Newton (1670). In his inaugural address
Barrow speaks of Lucas as "a new and benignant star, shining
with a ray both true and propitious, such as has not for many
years risen above the academical horizon." 6

1 Brief Lives, I, 276.

2 Born at Over-Bradley, near Halifax, Yorkshire, November 30, 1540; died
at Eton, February IQ, 1622.

3 Praelectiones XIII in principium element orum Euclidis, Oxoniae habitae
1620, Oxford, 1621.

4 Died in London, June 22, 1663. 6 In 1860 it had risen only to 155.
6 WhewelPs translation. For Kirby's translation see the English edition of the

Mathematical Lectures, p. vii (London, 1734).

396 GREAT BRITAIN

Barrow. To Isaac Barrow 1 it was given to rank as one of
the best Greek scholars of his day, to attain the highest honors
as professor of mathematics, to be looked upon as one of the
leading theologians of England , to be recognized as a pro-
found student of physics and astronomy, to acquire fame as
a preacher and controversialist, and to be perhaps the first to
recognize the genius of Newton and to develop his great talents,
but his reputation might have been higher had he worked
exclusively in only one of his fields of interest, particularly in
optics or geometry. He was prepared for the university at
Charterhouse, London, and at Felstead School, in Essex. He
entered Trinity College, Cambridge, in 1644, took his degree
of B. A. in 1648 (when only eighteen), was elected fellow the
year following, and left in 1655. His ready wit is illustrated by
an incident that occurred in his examination for holy orders,
the dialogue being said to have run as follows:

Chaplain. Quid est fides?
Barrow. Quod non vides.
Chaplain. Quid est spes ?
Barrow. Magna res.
Chaplain. Quid est car it as ?
Barrow. Magna raritas.

The chaplain is then said to have given up in despair and to
have reported the candidate's lack of reverence to the bishop.
Fortunately the latter had a sense of humor and Barrow was
duly admitted. 2

Barrow traveled for some time after taking orders and in
1662 was elected professor of geometry at Gresham College.
Soon after, as already stated, he was elected (1664) the first
Lucasian professor. Six years later (i67o) 3 he resigned in
favor of Newton and devoted his attention to theology. He

1 Born in London, October, 1630; died in London, May 4, 1677.

2 W. W. R. Ball, Cambridge Papers, p. 109. London, 1918.

8 These two dates are often given as 1663 and 1669, on account of the style
of calendar. His inaugural lecture was given March 14, 1664. See W. Whewell,
The Mathematical Works of Isaac Barrow, p. 5 (Cambridge, 1860).

BARROW

Statue in Trinity College Chapel, Cambridge. It stands just north of the statue

of Newton

398 GREAT BRITAIN

became chaplain to Charles II in 1670, master of Trinity Col-
lege in 1673,* and vice-chancellor of the university in 1675.
He edited Euclid's Elements 2 and the Data, 3 the works of
Archimedes, 4 the conies of Apollonius, and the spherics of
Theodosius. His own contributions to science appear chiefly
in his general lectures 5 and in his works on optics and geome-
try. 7 He also gave a new method for determining tangents,
one that approached the methods of the calculus. 8 In this work
he made use of a " differential triangle/' which is still essentially
the basis of the initial work in differentiation and which we shall
consider at some length in Volume II.

Newton. Isaac Newton 9 was born in the stirring times of the
Cromwell rebellion and was the posthumous son of a farmer in
Lincolnshire. He was a small and feeble child, and as a boy
he gave little promise of success in the battle of life. At the
age of twelve he was taken from the local day school and
placed in a school at Grantham. According to his own state-
ment he was at first extremely inattentive to his studies and
ranked among the lowest in the school. His chief interests
seemed to be in carpentering, mechanics, the writing of verses,
and drawing. Later, however, he began to show considerable

*W. W. R. Ball, Cambridge Papers, p. 171.

2 Eudidis Elementorum Libri XV, Cambridge, 1655, with many later editions.
3 Euclidis Data, Cambridge, 1657.

4 The Archimedes, Apollonius, and Theodosius were published together at
London in 167$.

s Lectiones Mathematicae XXIII, London, 1683 (posthumous).

6 Lectiones XVIII, . . . in qvibvs Opticorum Phaenomenun . . investigantur
. . ., London, 1669.

7 Lectiones Geometricae, London, 1670. There is a translation with com-
mentary by J. M. Child, Chicago, 1916.

8 For a good summary, see Ball, Hist. Math., 6th ed., p. 311.

9 Born at Woolsthorpe, near Grantham, Lincolnshire, December 25, 1642;
died at Kensington, March 20, 1727. One of the best books with which to
begin a study of Newton's life is A. De Morgan, Essays on the Life and Work
of Newton, 2d ed., Chicago, 1914, not so much because of the essays them-
selves as for the bibliographical and critical notes of the editor, P. E. B. Jour-
dam. The best biography of Newton is that of Sir David Brewster, The
Memoirs of Newton, 26. ed., Edinburgh, 1860, hereafter referred to as Brewster,
Memoirs of Newton. A new biography of Newton and a definitive edition of
his complete works are both sadly needed.

NEWTON

401

Newton's work was so
highly appreciated that
about this time (1668) he
was invited to revise the
lectures of Barrow, his
former teacher. In 1669
the latter resigned the
Lucasian professorship of
mathematics at Cambridge,
and Newton was appointed
his successor. Thus at the
age of twenty-seven he had
begun his great work on
the calculus and mathe-
matical physics, and held
already one of the high-
est academic honors in the
world.

For nearly sixty years
after receiving his profes-
sorship he was looked upon
as one of the greatest lead-
ers in the fields of physics
and mathematics, receiving
all the honors that could
be hoped for by a man in
academic life. He was
elected fellow of the Royal
Society in 1 6 7 2 , was chosen
in 1689 to represent the
university in parliament,
and was appointed warden
of the mint in 1696 and
master in 1699. He was
again elected to parliament
in 1701, but he took no
interest in politics. He was

ROETTIERS S MEDAL OF NEWTON

Of the numerous portrait medals of New-
ton, those by Croker and by Roettiers
(first struck in 1730) are the best. This
particular medal was struck in 1774. The
reverse bears the motto from Vergil's
^Eneid (Lib. V, 378), "Another is sought
for him." Its significance appears from the
context: "Another is sought for him, nor
does any one from so great a band dare
approach the man, and draw the gauntlets
on his hands"

402 GREAT BRITAIN

knighted by Queen Anne in 1705. The latter part of his life
was spent in London and was mathematically unproductive.

Newton always hesitated to publish his discoveries. His
greatest work, the Principia* was begun in 1685 but was not
published until 1687, and then only under pressure from his
friend Halley, the astronomer. In this work he sets forth
his theory of gravitation, "indisputably and incomparably the
greatest scientific discovery ever made/' 2 following the methods
of the Greek geometry as being more easily understood by stu-
dents of his time. ;j It was the third of the three great discov-
eries in the field of mathematical astronomy , the heliocentric
theory, finally established by Copernicus ; the elliptic orbits of
the planets and the laws relating thereto, finally established
by Kepler ; and the law of universal gravitation, with which
Newton's name will always be connected.

Referring to this great work, there is humor as well as jus-
tice in the remark of Lagrange to the effect that Newton was
the greatest genius that ever lived, and the most fortunate,
since we can find only once a system of the universe to be
established. Whether Einstein's Theory of Relativity invali-
dates this statement remains to be seen.

Newton's Arithmettca Univer sails, a work on algebra and the
theory of equations, was written in lecture form in the period
1673-1683 but was not published until 1707.* He wrote a
work on analysis by series 5 in 1669, but it was not published
until 1711. His work on the quadrature of curves was writ-
ten in 1671 but was not published until 1704, and his other

1 Philosophiae Naturalis Principia Mathematica, London, 1687, reissued the
same year with a new title-page ; 2d ed. by Roger Cotes, Cambridge, 1713 ;
3d ed. by Pemberton, London, 1726. For a list of editions of all of Newton's
works see G. J. Gray, A bibliography of the Works of Sir Isaac Newton,
Cambridge, 1888; 2d ed., Cambridge, 1907.

2 W. Whewell, History of the Inductive Sciences, Bk. VII, ii, 5.

3 For a resume of the work see W. W. R. Ball, Essay on the Genesis, Con-
tents, and History of Newton's Principia, London, 1893.

4 > Arithmetic -a universalis sive de compositione et resolutione arithmetica liber,
Cambridge, 1707; English translation, London, 1720.

5 Analysis per Mquationes numero terminorum infinitas, London, 17x1.

*Tractatus de quadratura curvarum f printed in the first edition of his
Opticks, London, 1704.

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AUTOGRAPH OF NEWTON

The document was written after he had retired from his work at Cambridge
and is interesting politically

404 GREAT BRITAIN

works were similarly held from the reading public until after
Newton had given up his lectures at Cambridge. His Fluxions*
treating of the subject by which he is probably the best known,
did not appear until nine years after his death, although the
theory had by that time become well known through the pub-
lication of such works as that of Charles Hayes (1704).

It was a noble and a generous tribute that Leibniz paid
when he said that, taking mathematics from the beginning of
the world to the time when Newton lived, what he did was
much the better half.

Newton had the eccentricities of genius, always being so
absorbed in his work as to be oblivious of life about him. Many
stories are related of his absent-mindedness, some of them also
told of other mathematicians and very likely most of them
apocryphal.

Of these stories one relates that, when giving a dinner to
some friends, he left the table to fetch a bottle of wine ; but on
his way to the cellar he forgot all about the errand, went to his
room, put on his surplice, and ended up in chapel. It is also
related that once in riding he dismounted and started in his
absent-minded way to lead his horse up the hill, but found
when he came to remount that the horse had wandered off,
leaving only the bridle in Newton's hand.

Newton died in 1727 and was buried in Westminster Abbey,
where his tomb is still seen.

Voltaire attended the funeral, and we shall see later that
when he was very old he did much to make Newton's philoso-
phy known in France. It is said that "his eye would grow
bright and his cheek flush" when he said that he had once lived
in a land where "a professor of mathematics, only because he
was great in his vocation," had been buried "like a king who
had done good to his subjects." 2

Method of Fluxions and Infinite Series . . . from the Author's Latin
Original not yet made publick . . ., London, 1736, translated and edited by
John Colson. There was another translation published shortly after this,
London, 1737.

2 S. G. Tallentyre, Life of Voltaire, p. 57. New York, n. d.

HALLEY

405

Halley. In his youth Edmund Halley 1 attended St. Paul's
School, London, distinguishing himself in both mathematics
and the classics. He entered Queen's College, Oxford, in 1673,
and before he was twenty he communicated a paper to the
Royal Society. So noteworthy had been his progress that
in the very month
in which he reached
his twentieth birthday
(November, 1 676) he
set out for St. Helena
for the purpose of
making astronomical
observations. On the
day before he was
twenty-one he made
the first complete ob-
servation of a transit
of Mercury. So re-
markable was his
work at St. Helena
that Flamsteed 2 called
him the " southern
Tycho " and the Royal
Society elected him
to a fellowship when
he was only twenty-
two (1678). Although

his supposed materialistic views prevented his election as Sa-
vilian professor of astronomy at Oxford in 1691, he followed
Wallis as Savilian professor of geometry in 1 703. He succeeded
Flamsteed as astronomer royal in 1721. He was the first to
predict the time of the return of a comet, and the comet of
which he announced the period has since then been known by
his name.

EDMUND HALLEY
Friend of Newton, illustrious as an astronomer

in London, November 8 (October 29, o.s.), 1656; died at Greenwich,
January 14, 1742.

2 John Flamsteed (1646-1719), first astronomer royal of England.

406 GREAT BRITAIN

While Halley's tastes were chiefly astronomical, he was
deeply interested in geometry/ algebra, 2 and the construction
of logarithmic tables. 3 He solved the problem of the con-
struction of a conic, given a focus and three points.

It was due to Halley's insistence as well as his financial
help that Newton published his Principia in 1687. Besides
editing the conies of Apollonius (1710) he edited the works
of Serenus 4 and Menelaus. 5 He also compiled a set of mor-
tality tables, thus giving a practical basis to the subject of
life insurance.

Wallis. Of the contemporaries of Newton one of the most
prominent was John Wallis/ 1 He studied theology at Emmanuel
College, Cambridge, and took the degree of B.A. in 1637 and
that of M.A. in 1640, the year in which he was ordained. He
became a fellow of Queens' College in 1644. His tastes, how-
ever, were in the line of physics and mathematics, and in 1649
he was elected to the Savilian professorship of geometry at
Oxford, a position which he held until his death. He was
awarded the degree of doctor of divinity in i653, 7 became chap-
lain to Charles II in 1660, and was one of the founders of the
Royal Society (1663).

Wallis was a voluminous writer, and not only are his writ-
ings erudite, but they show a genius in mathematics that
would have appeared the more conspicuous had his work
not been so overshadowed by that of his great Cambridge

1 Apollonii Pergaei Conicorum Libri octo, Oxford, 1710. His version from
the Arabic of the treatise of Apollonius De Sectione Rationis, with a restoration
of the two lost books De Sectione Spatii, appeared at Oxford in 1706.

2 A new and exact Method of finding the Roots of any Equation, London,
1720; "De numero radicum in aequationibus solidis ac biquadraticis," Phil.
Trans., 1687; "Methodus nova, accurata et facilis inveniendi radices aequa-
tionum," Phil. Trans., 1694. See also the abridgment of the Phil. Trans., 4th ed.,
I, 63, 68, 81, 137 (London, 1731).

3 "A most compendious and facile method for constructing the logarithm?
. . .," Phil. Trans., 1695.

4 De Sectione Cylindri et Coni, Oxford, 1710.

^Menelai Sphaericorum Libri III, published by Dr. Costard, Oxford, 1758.

6 Born at Ashford, Kent, November 23, 1616; died at Oxford, October 28.
1703.

7 According to the Diet. Nat. Biog. Older authorities give 1654.

JOHN WALLIS, 1616-1703

From D. Loggan's drawing from life, Published in the Opera Mathematica,

Oxford, 1695

408 GREAT BRITAIN

contemporary. He was one of the first to recognize the signifi-
cance of the generalization of exponents to include negative
and fractional as well as positive and integral numbers. He
recognized also the importance of Cavalieri's method of indi-
visibles, and employed it 1 in the quadrature of such curves as

y x t y = x" y and y == x* + x^ + * 2 + He failed in his
efforts at the approximate quadrature of the circle by means
of series because he was not in possession of the general
form of the binomial theorem. He reached the result, how-
ever, by another method. He also obtained the equivalent of

l-^-] for the length of an element of a curve,

\CIX /

thus connecting the problem of rectification with that of quad-
rature. In 1673 he wrote his great work De Algebra Tractatus ;
Historicus & Fractious, of which an English edition appeared
in 1 68s. 2 In this there is seen the first serious attempt in Eng-
land to write on the history of mathematics, and the result
shows a wide range of reading of the classical literature of
the science. This work is also noteworthy because it contains
the first record of an effort to represent the imaginary number
graphically by the method now used. The effort stopped short
of success but was an ingenious beginning. Wallis was in
sympathy with the Greek mathematics and astronomy, edit-
ing parts of the works of Archimedes, Eutocius, Ptolemy,
and Aristarchus; but at the same time he recognized the
fact that the analytic method was to replace the synthetic,
as when he defined a conic as a curve of the second degree
instead of a section of a cone, and treated it by the aid of
coordinates. His writings include works on mechanics, sound,
astronomy, the tides, the laws of motion, the Torricellian tube,
botany, physiology, music, the calendar (in opposition to
the Gregorian reform), geology, and the compass, a range
too wide to allow of the greatest success in any of the

1 Arithmetica Infinitornm, sive Nova Methodus Inquirendi in CurviUneomm
Quadraturam, aliaque difficiliora Matheseos Problemata, Oxford, 1655.

2 The Latin edition is in Volume II of his Opera Mathematka, Oxford,
1693-1695-

WALLIS AND GREGORY 409

lines of his activity. He was also an ingenious cryptologist and
assisted the government in deciphering diplomatic messages. 1
Among his interesting discoveries was the relation

4 3- 3* 5 5 7- 7-

one of the early values of TT involving infinite products. 2

Gregory. Scotland seems frequently to attempt to conceal
her great men by a kind of protective coloring. Probably it
is because she does not wish to seem to boast of her success.
She had in the iyth century, and still has, a distinctive school
of mathematical teaching, but she seems reluctant to proclaim
the fact. Occasionally, however, she fails to keep her scholars
and writers under cover, and so in mathematics we have such
names as Napier, Gregory, and Kelvin.

James Gregory 3 was one of the first Scotchmen to make for
himself a great name both in mathematics and in physics. He
lived for some years in Italy, but in 1668 returned to Scotland
to assume the professorship of mathematics at St. Andrews.
In 1674 he became professor of mathematics at Edinburgh,
where he died a year later. Like most of the leading British
mathematicians of this period, he was equally interested in
mathematics and physics. In 1661 he invented but did not
practically construct the reflecting telescope which bears his
name. 4 He also originated the photometric method of esti-
mating the distances of stars. In the field of pure mathematics
he expanded tan- 1 ^, tan0, and sec- 1 ^ in series (1667), dis-
tinguished between convergent and divergent series, calcu-
lated the areas of successive polygons as the number of sides

*D. E. Smith, in the Bulletin of the Amer. Math. Soc., XXIV (2), p. 82.

2 See his Opera Mathematica, I, 441, where he explains that he uses the
symbol Q for the ratio of the square on the diameter to the area of the circle.
In our symbolism this is q^iirr 2 , or 4:77. On page 469 he gives the value
stated above. See also his Volume II, cap. Ixxxiv.

3 Born at Drumoak, near Aberdeen, November, 1638; died at Edinburgh,
October, 1675.

4 Described in his Optica promota, sen abdita radiorum refiexorum et refrac-
torum mysteria geometrice enucleata, London, 1663.

4io

GREAT BRITAIN

was doubled, and gave an ingenious but unsatisfactory demon-
stration of the incommensurability of n. 1 The series

= tan - I taii*0 -f J tan r '0

is commonly known by his name. He died in the thirty-seventh
year of his age, shortly after being stricken with blindness as a
result of the strain upon his eyes in carrying on his astronomical
observations.

AUTOGRAPH OF LORD BROUNCKER

Part of a document dated December 31, 1661, just before Brouncker's appoint-
ment as Chancellor of Queen Catherine

Other British Writers. First among the other British writers
of the period made memorable by the great work of Barrow,
Wallis, and Newton, was William, Viscount Brouncker. 2 He
was one of the founders and was the first president of the
Royal Society, a linguist with the genius of a mathematician,

1 Vera circuli et hyperbolae quadratura, Padua, 1667, and again in 1668.
Geometriae pars universalis, Venice, 1667; "De circuli quadratura," in the
Phil. Trans., London, 1668. See also G. Heinrich, Bibl. Math., II (3), 77.

2 Born at Castle Lyons, Ireland, c. 1620 ; died at Westminster, April 5, 1684.
The name is also given as Brounker. He was the second viscount.

MINOR WRITERS 411

and a man who was justly esteemed by the mathematical
world of his day. He was interested in the rectification of the
parabola 1 and the cycloid" and in the quadrature of the hyper-
bola l (1668) and the circle. The last of these investigations
led him to state that

4 = _J__ i 2 _r__ 5 2

7T 1+2+2+2+'"' '

a form which he derived from the continued product which
Wallis had discovered. This was the first use of continued
fractions by an English writer. In 1662 he was appointed
Chancellor of Queen Catherine, and in 1681 he became Mas-
ter of St. Catherine's Hospital in London. Aubrey, who knew
him, relates that "he was of no university, he told me. He
addicted himselfe only to the study of the mathematicks, and
was a very great artist in that learning." 4

John Pell, 5 who was about ten years older than Brouncker,
was admitted to Trinity College, Cambridge, at the age of thir-
teen, and also studied at Oxford. At the age of twenty he had
mastered Latin, Greek, Hebrew, Arabic, Italian, French,
Dutch, and Spanish. He became a professor of mathematics
at Amsterdam (1643-1646) and at Breda (1646-1652), was
Cromwell's representative in Switzerland (1654-1658),
entered the ministry (1661), was given the degree of D.D. at
Lambeth (1663), and in the same year was made a Fellow of
the Royal Society. He spent the latter part of his life in
parish work and in writing on mathematics.

He wrote on the quadrant (1630), on other matters relating
to astronomy, on the magnetic needle, and on the quadrature
problem, 6 and computed a table of 10,000 square numbers

ia On the Proportion of a curved line of a paraboloid to a straight line,"
Phil. Trans., Ill, 645.

2 "Of the Finding of a straight Line equal to that of a cycloid," Phil.
Trans., VIII, 640.

3 "An algebraical paper on the squaring of the hyperbola," Phil. Trans.
abridgment, 4th ed., I, 10. London, 1731. 4 Oxford ed., 1898, I, i2&.

G Born at Southwick, Sussex, March i, 1611; died in London, December 12,
1685.

6 Refutation of Longomontanus's Pretended Quadrature of the Circle, 1646;
in Latin, 1647.

412 GREAT BRITAIN

(1672). He edited (1668) the algebra of Rhonius 1 (1622-
1676), which had been translated by Thomas Branker 2 and
which contains the first mention of the Anglo-American sign
( -s- ) of division. Through an error on the part of Euler,
Pell's name is commonly connected with a certain equation of

the form ,

,r- ./r=i,

although he had but little to do with it. 3

One of the most promising pupils of Wallis was William
Neile/ who gave (1660) the rectification of the semicubical
parabola y :} ax 2 . He died too soon, however, to fulfill his
early promise.

If it had not been for the great fire of London (1666), Sir
Christopher Wren 5 would have been known as a mathemati-
cian rather than as the architect of St. Paul's Cathedral. He
was educated at Westminster School and in 1649 or if>5 was
entered at Wadham College, Oxford. He was graduated B.A.
in 1650-1651 and M.A. in 1653. * n J 653 he was elected fellow
of All Souls College, Oxford, where he resided until 1657.
From 1657 to 1660 he was professor of astronomy at Gresham
College, London, and from 1661 to 1673 was Savilian professor
of astronomy at Oxford. He was a Fellow of the Royal Society
and was its president from 1680 to 1682. He wrote on "The
law of nature in the collision of bodies," on the grinding of
hyperbolic mirrors, on perspective, and on the rectification

1 J. H. Rahn, An Introduction to Algebra, London, 1668. The book was
first published in Zurich in 1659, in German.

2 Or Brancker. He was a clergyman, born in Devonshire, 1636; died at
Macclesfield, 1676. He also taught at Macclesfield. He wrote a work of no
merit on astronomy.

3E. E. Whitford, The Pell Equation, New York, ipi2, with a good bibliog-
raphy ; H. Konen, Gesckichte der Gletchung t' 2 Du~= i, Leipzig, IQOI ;
G. Enestrom, "Ueber den Ursprung dcr Benennung Pcll'sche Gleichung," Bibl.
Math, III (3), 204.

4 Born at Bishopsthorpe, December 7, 1637; died in Berkshire, August 24,
1670.

5 Born at East Knoyle, Wiltshire, October 20, 1632; died in London,
February 25, 1723.

9 Phil. Trans., 1669, but originally separately printed in Latin as Lex naturae
de cottisione corporum.

TEXTBOOK WRITERS 413

problem, and he discovered (1669) the two systems of gener-
ating lines on a hyperboloid of one sheet. After the great fire
he took a prominent part in the rebuilding of St. Paul's Cathe-
dral and more than fifty other churches and public buildings
in London. His noble epitaph in the cathedral is well known. 1

Among the other pupils of Wallis who acquired some reputa-
tion was John Caswell. 2 He matriculated at Wadham College,
Oxford, at the age of sixteen. Six years later he began teach-
ing mathematics in Oxford, and so marked was his success that
in 1709 he was elected Savilian professor of astronomy. He
published a trigonometry in 1685 and at one time thought of
publishing the work of Menelaus on spherics, but gave up the
plan. 3

James Gregory's nephew, David Gregory, 4 was professor of
mathematics at Edinburgh from 1684 to 1691, after which he
became Savilian professor of astronomy at Oxford. He pub-
lished a work on geometry 5 when he was only twenty-three
years of age, and in a work on optics 1 ' set forth the possibility
of achromatic lenses. He also wrote on the Newtonian theory.'"
In 1703 he brought out at Oxford an elaborate edition of
Euclid's works.

Textbook Writers. In the i7th century we reach a time in
the development of elementary mathematics when textbooks
became so standardized and numerous as to require in this
connection the mention of only the most important. Their
mission henceforth was to improve the method of presenting
theories already largely developed and to adapt the applica-
tions of these theories to the needs of the world. From that
time on they ceased to be a great factor in the presentation
of mathematical discoveries.

*" Lector, si monumentum requiris, circumspice"
2 Born at Crewkherne, Somerset, 1655; died April, 1712.
3 L. C. Karpinski, Bibl. Math. XIII (3), 248.

4 Born at Kinnairdie, Banffshire, June 24, 1661 ; died at Maidenhead, Berk-
shire, October 10, 1708.

5 Exercitatio Geometrica de dimensione figurarum, Edinburgh, 1684.
*Catoptricae et Dioptricae Sphaericae Elementa, Oxford, 1695.
7 Astronomiae Physicae et Geometriae Elementa, Oxford, 1702.

414 GREAT BRITAIN

The most prominent British textbook writer on elementary
arithmetic in the i7th century was Edmund Wingate. 1 This
is somewhat strange, because he entered the profession of the
law after leaving Oxford, 2 went to Paris in 1624, where he
taught English to the princess Henriette-Marie, future wife of
Charles I, and returned to England in 1650 and became member
of parliament for Bedford, none of which activities related to
mathematics or even to textbooks. His tastes, however, seem
to have led him to look upon mathematics as an avocation, for
in the same year that he went to Paris he published a work on
Gunter's scale, 3 two years later a work on logarithmic arith-
metic, 4 and in 1630 his popular work Of natural and artificial
Arithmetick (London), a work that went through many editions
and was popular for more than a century. He also prepared
a set of logarithmic tables 5 and a work 6 on a mathematical
instrument which he had invented.

Of the textbook makers of this period in the domain of ele-
mentary algebra the best known was John Kersey, 7 a self-made
teacher. He was highly esteemed in London as an instructor
in mathematics and was a friend of Wingate's. His algebra s
presents the subject in a logical and teachable manner. One
of the most interesting features is the preliminary explanation
of the analogies between proportion, which then held a high
place, and the modern treatment of equations. The work was,
however, altogether too elaborate to meet with great success.

William Leybourn (1626-^. 1700) was well known in Lon-
don in his day, not only as a writer of textbooks but as a

3 Born at Flamborough, Yorkshire, 1596 ; died in London, December 13, 1656.

2 He edited the second edition of Britton's famous Collections on English
law, London, 1640.

3 Construction, description et usage de la regie de proportion, Paris, 1624.
4 Arithmetique logarithmique, Paris, 1626; English edition, London, 1635.
5 Tables of the Logarithms of the sines and tangents, London, 1633.
G Ludns Mathematicus, London, 1654.

7 Born probably at Bodicote, near Banbury, Oxfordshire, 1616 (at any rate
he was baptized there November 23, 1616) ; died May, 1677. See the Bibl.
Math. XII (3), 263, with evidence for this date.

% The Elements of that Mathematical Art commonly called Algebra, London,
1673.

TEXTBOOK WRITERS 415

teacher of mathematics and a surveyor. He wrote on astron-
omy, 1 surveying, 2 arithmetic, ' the logarithmic rule, 4 the Napier
computing rods, 5 and mathematical recreations. In 1690 he
published a Cursus Mathematicus containing the substance of
his other works. Three years later he published the best known
of his works, a ready reckoner, 7 the first elaborate book of the
kind to appear in English.

Leybourn was not, however, as popular as Edward Cocker, 8
who is described in 1657 as living "on the south side of St.
PauPs Churchyard, over against St. PauPs Chain . . . where
he taught the art of writing and arithmetick in an extraordinary
manner." He was also a bibliophile, and in his " public school
for writing and arithmetic" where he " takes in boarders" he
had a large library of manuscripts and printed books in various
languages. He wrote three or four books on arithmetic and
penmanship, but it was his Arithmetick, being a Plain and
Easy Method, edited by John Hawkins in 1678, that fixed the
expression "according to Cocker" in the common speech of
England. This remarkable book went through upwards of a
hundred editions and had great influence upon British text-
books for more than a century. He also wrote on algebra, 9 but
there has always been a question as to how much of this work

1 Urania Practica, London, 1648, written with Vincent Wing. One of the
first books on astronomy written in English, but not the first, Recorde's Castle
of Knowledge having appeared in 1551.

2 Planometria, or the Whole Art of Surveying of Land . . . by Oliver
Wallinby, London, 1650, the pseudonym being formed by transposing the letters
of his name, with minor changes.

3 Arithmetick, Vulgar, Decimal, and Instrumental, London, 1657.

*The Line of Proportion or Numbers, commonly called Gunter's Line,
made easie, London, 1667.

5 The Art of Numbering by Speaking-Rods: Vulgarly termed Nepeirs Bones,
London, 1667 and 1685. Leybourn, as will be shown in Volume II, mistakes the
etymology of Rabdologia, and translates the word as "Speaking- Rods."

6 Pleasure with Profit; consisting of Recreations of divers kinds, Numerical,
Geometrical, Mechanical, Statical, Astronomical, London, 1694.

7 Panarithmolog ia, being a Mirror Breviate, Treasure Mate for Merchants,
Bankers, Tradesmen, Mechaniks, . . . London, 1693.

8 1631-1675, probably a descendant of the Northamptonshire family of
Cokers.

9 Algebraical Arithmetic, or Equations, London, 1684.

41 6 GERMANY

as well as the arithmetic was due to Cocker and how much to
Hawkins. The world is as well off either way, the important
thing being that in them we have two books that represent the
popular view of the elementary science at the period in which
they were written.

5. GERMANY

Germany in the Seventeenth Century. In the i6th century
Germany moved toward the front in the mathematical prog-
ress of Europe, but with Italy always in the lead. In the i yth
century she fell behind, producing only a single mathematician
of the first class. The first part of the century was the period
of the Thirty Years' War (1618-1648), and indeed the whole
period was one of unrest in the Teutonic countries, a period
quite unsuited to intellectual progress.

Kepler. Although known chiefly for his work in the domain
of astronomy, Johann Keppler, 1 or Kepler, as the name more
often appears in English, ranked high as a mathematician. He
studied in the cloister school at Maulbronn and at the Univer-
sity of Tubingen. At the age of twenty-two he began teaching
mathematics and moral philosophy in the Gymnasium at Gratz
in Steyermark. 2 For two years (1599-1601) he was an assist-
ant of Tycho Brahe, and in 1601 became court astronomer to
Kaiser Rudolph II and later to his successors Matthias ( 1612 )
and Ferdinand III (1615). His life was, however, made almost
unendurable through domestic infelicity, troubles at court, and
financial difficulties. In his work in mathematics he shows
himself an excellent geometer and was also much interested in
algebra. He set forth (1604) the idea of continuity in elemen-
tary geometry, 3 made (1615) some advance in the use of the
infinitesimal, 4 and did much to further the cause of logarithms/'

ifiorn at Weil der Stadt, Wiirttemberg, December 27, 1571; died at Regens-
burg, November 15, 1630. From the family of von Kappel.
-Or Graz, in the former Austrian crownland of Styria.

3 In his Ad Vitellionem Paralipomena. See the Frisch edition of his Opera
Omnia, II, 119, 187 (Frankfort, 1858-1871).

4 In his Stereometric, ; e. g., see the Frisch edition of his Opera Omnia, IV, 583.

5 His Chilias Logarithmorum appeared at Marburg in 1624, with a Supple-
mentum in

LEIBNIZ 417

Tschirnhausen. Ehrenfried Walther, Graf von Tschirnhau-
sen, 1 often known as Tschirnhaus, is one of the few minor
German mathematicians of the time to require special mention.
He served for a year or two (1672-1673) with the Dutch army,
traveled extensively, and then settled down on his estates, de-
voting his time largely to mathematics and physics. He is
known chiefly for his study of curves, 2 including caustics and
catacaustics, 3 and for his work in maxima and minima 4 and the
theory of equations. 5

Leibniz. Gottfried Wilhelm, Freiherr von Leibniz, was the
only pure mathematician of the first class produced by Ger-
many during the i7th century. He early showed great pro-
ficiency in mathematics, having read the most important
treatises on the subject before he was twenty. He studied
law at Nurnberg, entered upon a diplomatic career, traveled ex-
tensively, and made the acquaintance of leading mathemati-
cians in Holland, France, and England. He finally went to
Hannover and became librarian to the duke. He lived in good
style, and the visitor to Hannover today may see his palatial
house, now used as a museum.

The leisure which his office allowed him gave Leibniz the
opportunity to develop the differential and integral calculus.
He seems to have begun to think about the subject in 1673^

ifiorn at Kiesslingswalde, near Gorlitz, April 10, 1651; died at Dresden,
October n, 1708. H. Weissenborn, Lebensbeschreibung dcs Ehrenfried Walther
von Tschirnhaus, Eisenach, 1866.

2 "Nova methodus tangentes curvarum expedite deterrninandi," Acta Eru-
ditorum, I (1682), and various other memoirs in the same publication.

3 All published in the Acta Eruditorum. The terms are due to Jacques and
Jean Bernoulli. See Jacques Bernoulli, Opera, I, 466, et passim.

4 "Nova methodus dctcrminandi maxima et minima," Acta Eruditorum, II
(1683).

5 "Methodus auferendi omnes terminos intermedios ex data aequatione,"
Acta Eruditorum, II, 204.

6 Born at Leipzig, June 21, 1646 (o.s.) ; died at Hannover, November 14, 1716
(N.S.). His father wrote his name Leibniitz. It is often written Leibnitz.
The Latin form used by our Leibniz was at first Leibniizius or Leibnuzius, but
later the name appears as Leibnitius. The preferred spelling in modern German
scientific works is Leibniz.

7 At least, this is his claim. See J. M. Child, The Early Mathematical
Manuscripts of Leibniz, p. 37 (Chicago, 1920).

41 8 GERMANY

some years after Newton had explained the fluxional calculus
to his pupils. Two years later he had his theory well developed,
but it was not until 1684 that he published, in the Ada Erudi-
torum, a description of the method and its possibilities.
There is no longer any doubt that Leibniz developed his cal-
culus quite independently, and that he and Newton are each
entitled to credit for their respective discoveries. The two lines
of approach were radically different, although the respective
theories accomplished results that were practically identical.
Leibniz knew or could easily have known what Newton was
doing, and this may have suggested the line of work ; and he
knew the contribution already made by Barrow 1 in the form
of the "differential triangle," but at any rate he was original
in much that he accomplished. In a word, it may be said that
he made Cavalieri scientific. He also laid the foundation for
the theory of envelopes and defined the osculating circle and
showed its importance in the study of curves.

Leibniz was a diplomat in the days when Machiavelli was a
model, and this is not a flattering way in which to characterize
him. Of no other man does the visitor to the portrait gallery
of the mathematical world carry away such varied impressions.
Some of the engravings show him as a man of great refine-
ment and dignity, while others show the mean, dishonest, dis-
appointed face of a man whose word would never be accepted.
In the old controversy as to the invention of the calculus, the
word of Leibniz would not have the weight usually given to
the statement of a scholar. Nevertheless it must be repeated
that he showed great originality in his theory and in the
symbolism of the calculus, and is entitled to a high degree of
credit for this work.

Minor Writers. Among the minor writers of the first half of
the i yth century Johann Faulhaber 2 is known as a successful
teacher of mathematics in Ulm. He published various works on
algebra and on the curious phases of elementary mathematics.

J J. M Child, loc. cit., p. ipn. et passim, and his Geometrical Lectures of
Isaac Barrow, Chicago, 1918.

2 Born at Ulm, May 5, 1580; died at Ulm, 1635.

>f*N*v*

* tf

, I/

LEIBNIZ
After an engraving by Ficquet

420 GERMANY

His contemporary, Benjamin Ursinus/ professor of mathe-
matics at the University of Frankfort a. d. O., was one of

r/? uW '

/723

CK|

AUTOGRAPH LETTER OF LEIBNIZ
First page of a letter to Christian Wolf, dated February 10, 1712

the first to introduce logarithms into Germany. 2 Still an-
other contemporary, Daniel Schwenter, 3 was professor of

1 Born at Sprottau, July 5, 1587 ; died at Frankfort a. d. O., September 27,
1633 01 1634. Ursinus is the Latin form for Behr, his family name.

2 Trigonometria logarithmica, . . . cum magno logarithmorum canone, Frank-
fort a. d. O., 1618 ; 2d ed. 1635 ; Magnus canon triangulorum logarithmicus,
Koln a.d. Spree, 1624.

3 Born at Niirnberg, January 31, 1585; died at Altdorf, January 19, 1636.

MINOR WRITERS

421

Hebrew (1608), oriental languages (1625), and mathematics
(1628) at the University of Altdorf. He wrote a well-known
work on mathematical recreations. 1 A fourth of the minor

AUTOGRAPH LETTER OF LEIBNIZ
Second page of the letter shown on page 420

writers of this period, Peter Roth, 2 was a Rechenmeister at
Nurnberg. He wrote an unimportant algebra 3 in which he
treats of equations of the third and fourth degrees.

*Deliciae physico-mathematicae oder Mathematische und philosophische
Erquickstunden, Nurnberg, 1636 (posthumous).

2 Born at Ingolstadt, c. 1580; died at Nurnberg, 1617.

zArithmetica philosophica, oder schone, neue, wohlgegrundete, iiberail-t
kunstliche Rechnung der Coss oder Algebra, Niirnberg, 1608.

422 THE NETHERLANDS

For his general learning and his great gift to the world through
the museum at Rome which bears his name, if not for any
important contribution to mathematics, Athanasius Kircher 1
should be mentioned, although he might quite as appropriately
be recorded in the list of Italian scholars, since he spent the
better years of his life in Rome. He was a Jesuit, professor of
mathematics and philosophy, and later of Hebrew and Syriac,
at the University of Wurzburg, after which he went (1635)
to Avignon and then to Rome, where he taught mathematics
and Hebrew. He was a voluminous writer on a great variety
of subjects and a zealous collector of curios. His study of
optics may have led to the invention of the stereopticon, or to
its improvement, but the claims of Kircher's friends are ques-
tionable. His mathematical works relate to instruments 2 and
to the occult in number, 3 and are not to be taken seriously.

6. THE NETHERLANDS

Geographical Limits. In speaking of the Netherlands we
must again mention the fact that reference is made to geograph-
ical rather than political boundaries. The xyth century was a
strenuous one for those countries which we now designate
as Holland and Belgium. Roughly speaking, the latter was
then under the sovereignty of Spain until Louis XIV began his
conquest in 1667, an d the former was nearly lost to the same
invader in 1672. In the midst of these strenuous times,
but generally before the wars, the Netherlands produced sev-
eral mathematicians who stood high in their respective lines.

Snell. There lived at Leyden at the opening of the i?th cen-
tury a professor of mathematics (1581) and Hebrew by the
name of Rudolph Snell. 4 He wrote a few unimportant works
on mathematics, all of which were published in the i6th

1 Born at Geisa, near Fulda, May 2, 1602 ; died at Rome, November 28, 1680.
2 Pantometrum Kirckerianum, h. e. Instrumentum geometricum novum, Wurz-
burg, 1660.

3 Arithmologia sive De occultis numerorum tnysteriis, Rome, 1665.

4 The Latin form, Snellius, is often used. Born at Oudewater, October 8,
1546; died at Leyden, April 2, 1613.

SNELL, GIRARD, AND HUYGENS 423

century. He had a son, Willebrord Snell van Roijen, 1 who
succeeded him as professor of mathematics at Leyden (1613),
devoting himself chiefly to astronomy, physics, and trigonom-
etry. He wrote on the mensuration of the circle, 2 and set
forth the properties of the polar triangle in spherical
trigonometry. 3 To him is due the name "loxodrome" for the
rhumb line in navigation, the latter name being due to the
Portuguese navigator and mathematician Nunes (Nonius).

Girard. Little is known of the life of the mathematician
Albert Girard. 1 He wrote on trigonometry, 5 fortifications,
practical geometry, and algebra, 7 and edited the works of
Simon Stevin. 8 He was one of the first to appreciate the sig-
nificance of the negative sign in geometry, and was successful
in his use of imaginary quantities in the theory of equations.
He inferred by induction, as others had done, that an equation
of the nth degree has n roots, expressed the sum of the first
four powers of the roots of an equation as functions of the
coefficients, and discussed general polygons, both cross and
simply convex. 9

Huygens. Although he was known chiefly as one of the
world's greatest physicists, particularly in relation to the study
of the pendulum, the invention of pendulum clocks, and the
laws of falling bodies, Christiaan Huygens 10 should be ranked

at Leyden, 1581; died at Leyden, October 31, 1626.

2 Cyclometricus, de Circuit Dimensions secundum Logistarum Abacos,
Leyden, 1621.

^Doctrinae Triangulorum Canonicae . . . Libri IV, Leyden, 1627 (post-
humously published). Vieta had already given them.

4 Born at St. Mihiel, Lorraine, 1505 ; died at The Hague, December 8/9, 1632.
He seems to have lived chiefly in Holland.

^Tables de sinus, tangentes et secantes, The Hague, 1626.

Q Geometrie contenant la theorie et la pratique d'icelle, escrite par Sam.
Marolois, revue, augmentee et corrigee, Amsterdam, 1627. Marolois was a writer
on fortifications.

7 Invention nouvelle en Valgebre, Amsterdam, 1629. Reprinted by Bierens de
Haan, Leyden, 1884. 8 This edition appeared posthumously, Leyden, 1634.

9 Thus he classified quadrilaterals as "la simple, la croisee et 1'autre ayant
Tangle renversee."

10 Born at The Hague, April 14, 1629; died at The Hague, June 8, 1695.
The name often appears as Huyghens or Hugenius, with several other variants.
There is a biography in his Opera Varia, Leyden, 1724.

424

THE NETHERLANDS

high among those who improved the new geometry and made
known the power of the calculus. He introduced the notion

of evolutes, 1 rectified
the cissoid, deter-
mined the envelope
of a moving line, in-
vestigated the form
and the properties of
the catenary, 2 wrote
on the logarithmic
curve, gave in mod-
ern form the rule for
finding maxima and
minima of integral
functions, wrote on
the curve of descent
(1687), published a
work on probability,
proved that the cy-
cloid is a tautoch-
ronous curve, and
contributed exten-
sively to the applica-
tion of mathematics
to physics.

Minor Writers. Among the minor writers the name of Gre-
goire de Saint-Vincent 3 is one of the best known. He was a
Jesuit, taught mathematics in Rome and Prag (1629-1631),
and was afterwards called to Spain by Philip IV as tutor
to his son Don Juan of Austria. He wrote two works on

iHorologium oscittatorium y Paris, 1673. It is in this treatise that his work
appears at its best.

2 "Solutio problematis de linea catenaria," Acta Emditorum, 1691.

s Born at Bruges, 1584; died at Ghent, January 27, 1667. Among the com-
mon variants of his name are Gregorius a Sancto Vincentio and Gregorius von
Sanct Vincentius. For a biographical sketch by Quetelet, see the Annales
Belgiques, VII (1821), 253, and one by Bosnians in the Annales of the Societe"
Scientifique of Brussels, XXXIV, i, 174.

CHRISTIAAN HUYGENS

From an engraving by Edelinck, after a drawing
by Drevet

MINOR WRITERS 425

geometry, 1 giving in one of them the quadrature of the hy-
perbola referred to its asymptotes, and showing that as the
area increased in arithmetic series the abscissas increased in
geometric series.

The Van Schooten family produced three generations of pro-
fessors of mathematics at Leyden, all sympathetic with the
science but no one of them of first rank. The first was Frans
van Schooten, 2 with a trigonometric table to his credit (1627).
The second, his son, was also called Frans van Schooten. 1 He
wrote on mathematics 4 and was a professor in the engineering
school at Leyden and the teacher of Huygens. He edited Vieta
(1646), wrote on perspective (1660), and is well known for his
Latin edition of the geometry of Descartes (1649). His half
brother, Petrus van Schooten, 5 occupied the chair of mathe-
matics at Leyden, being later (1669) transferred to the chair of
Latin, but he contributed nothing of permanent value in either
of his fields of interest.

Rene Frangois Walter, Baron de Sluze 6 was a man of stand-
ing in the Church, with a taste for mathematics. He contributed
to the geometry of spirals and the finding of geometric means,
and also invented a general method for determining points of
inflection of a curve. One of his contemporaries, but a few
years his junior, Johann Hudde 7 was an officeholder in Amster-
dam. He was interested in the theory of maxima and minima

I 0pus zeometricum quadraturae circuit et sectionum coni, 2 vols., Antwerp,
1647 ; Opus geometricum posthumum ad mesolabium per rationem, . . . Ghent,
1668. See also K. Bopp, Abhandlungen, XX, 87.

2 Born at Leyden, 1581; died at Leyden, December n, 1646. The name
often appears in the Latin form Franciscus. There are a number of interesting
MSS. of works by the various Van Schootens now in the library at Groningen.
For a list of these, see H. Brugmans, Catalogus codicum, No. 108 seq. (Gronin-
gen, 1898).

3 Born at Leyden, c. 1615; died at Leyden, May 29, 1660.

4 Principia Matheseos Universalis, Leyden, 1651; Exercitationum Mathe-
maticarum Libri quinque, Leyden, 1657. See Bibl. Math., XII (3), 156.

5 Born at Leyden, February 22, 1634; died at Leyden, November 30, 1679.

6 Born at Vise on the Maas, July 7, 1622; died at Lie"ge, March 19, 1685.
The name also appears as Sluse or Slusius.

7 Latin, Huddensis. Born at Amsterdam, probably in 1628 or 1629; died at
Amsterdam, April 16, 1704.

426 OTHER EUROPEAN COUNTRIES

(1658) and the theory of equations (1657), and in his work
on the latter subject he separated into factors the polynomial
which he equated to zero.

About this time Cornelis van Beugham wrote a Biblio-
gr aphia Mathematica which appeared at Amsterdam in 1688.
This seems to have been the first printed book devoted solely
to mathematical bibliography.

7. OTHER EUROPEAN COUNTRIES

Countries Considered. Of the other European countries that
exerted substantial influence on mathematics in the i7th cen-
tury Switzerland stands easily at the head. This seems due not
to any particular intellectual influences but to the efforts of
one of the most interesting families known in the history of
science, and to the labors of a man who came near being the
inventor of logarithms. 1

The other countries demanding attention are Spain and
Denmark.

The Bernoullis. Students of heredity have called attention
to the extraordinary number of distinguished scholars who de-
scended from the protestant population expelled from the cath-
olic countries in the i6th and i7th centuries." Presumably
the same result would be found among the descendants of
Catholics, Jews, political refugees, or others who maintained
their faith in any manner that tries the souls of men. 3 Of
those who descended from Belgian stock that was rooted up
during the reign of terror of the Duke of Alba were the mem-
bers of the Bernoulli family, a family that furnishes one of
the most remarkable evidences of the power of heredity or of
early home influence in all the history of mathematics. No

*L. Isely, Histoire des Sciences Mathematiques dans la Suisse Fran$aise t
Ncuchatel, IQOI.

2 A. dc Candolle, Histoire des sciences et des savants, 2d ed., p. 338. Paris,
1885.

3 Among similar cases of descendants of religious refugees are those of
Jean Trembley, Simon Lhuilier, Georges-Louis Le Sage, Louis Bertrand, and
Elie Bertrand,

THE BERNOULLIS

427

less than nine of its members attained eminence in mathematics
and physics, 1 and four of them were honored by election as
foreign associates of the Academic des Sciences of Paris.

Jacques Bernoulli.
The first of the fam-
ily to attain any
reputation in mathe-
matics was Jacques
Bernoulli. 2 He first
studied theology, but
his taste was in the
direction of astron-
omy, mathematics,
and physics, and he
traveled in France,
Holland, Belgium,
and England for the
purpose of devoting
his time to these
studies and to meet-
ing learned men. He
returned to Switzer-
land in 1682, took
up the study of the
new calculus as set
forth by Leibniz, and
in 1687 became pro-
fessor of mathemat-
ics in the University

JACQUES BERNOULLI

The elder of the two brothers who founded the
famous Bernoulli family of mathematicians

a For the relationships, see P. H. von Fuss, Correspondence mathematique
. . . de quelques celebres geometres du XVIII. siecle, Vol. I, p. xviii (Petrograd,
1843) (hereafter referred to as Fuss, Correspondence) ; also a note in Ter-
quem's Nouvelles Annales de Math., Suppl. Bulletin, Vol. 17, p. 85 (Paris,
1858) ; P. Merian, Die Mathematiker Bernoulli, Basel, 1860.

2 Born at Basel, December 27 (o.s.), 1654; died at Basel, August 16, 1705.
Jacobus Bernoulli, often called Jacques (I) Bernoulli to distinguish him from
another Jacques (1759-1789) in the i8th century. English writers often call
him James and the German writers Jakob. Since, however, he wrote in French
or Latin, the preferable form of bis given name is Jacques or Jacobus.

428 OTHER EUROPEAN COUNTRIES

of Basel. He wrote (1683-1701) a large number of memoirs
for the Acta Eruditorum. These memoirs include such lines
of research as series (1686), the quadrisection of a general tri-
angle by two normals (1687), conies (1689), lines of descent
(1690), mensuration (1691), cycloids (1692, 1698, 1699),
transcendent curves (1696), and isoperimetry (1700). He
wrote the second book devoted to the theory of probability, 1
although the subject had been studied in Italy and France
much earlier than this.

He solved by infinitesimal analysis the problem of the
isochronous curve which had already attracted the attention of
various writers and had been solved by Huygens, Leibniz,
L'Hospital, and Newton. He determined the length of the
catenary curve. Because of his study of the logarithmic spiral,
r =a e , he directed that this curve should be engraved upon his
tombstone, with the words Eadem mutata resurgo? and the
visitor to the cloisters at Basel may still see the rude attempt of
the stonecutter to carry out his wish. 3

The fact that his father was emphatically opposed to his
study of astronomy and mathematics, placing all possible ob-
stacles in his way, led him to choose for his device Phaethon
driving the chariot of the sun, with the legend, Invito patre
sidcra verso. 4

His brother Jean Bernoulli 5 was thirteen years his junior,"
which may account for an attitude of superiority on the part of
Jacques and one of resentment on the part of Jean which
caused the ill feeling that long existed between them. His
father learned no wisdom by his failure to have his brother

1 Ars conjectandi, Basel, 1713 (posthumous). With this was published his
Tractates de Scricbus Infinitis, Ear urn q lie sum ma Pint t a, et Usu in Quadrature
Spatiorum & Rectificationibus Curvarum, and a letter De Ludo pilae reti-
cularis. His Opera in two volumes appeared at Geneva in 1744.

2 "I shall arise the same, though changed."

3 There is an engraving of the design in J. J. Battierius, Vita . . . Jacobi
Bernoulli, p. 40 (Basel, 1705).

4 "I study the stars against my father's will "

r> Born at Basel, July 27 (o.s.), 1667; died at Basel, January i, 1748. The
first name also appears as Johann or John, and often as Jean (i).

6 Jacques was the fifth child in his father's family, and Jean was the tenth,
a fact which may interest those who have faith in the theory of primogeniture.

THE BERNOULLIS

429

Jacques become a theologian, and so he determined to make
Jean a merchant. The latter thought that he preferred medi-
cine or literature, but soon found that his real taste was for
mathematics, and so the world was saved the loss of a genius.

Jean first studied
medicine and wrote
his doctor's disserta-
tion De cffervcsccn-
tia et jermcntatione
(Basel, 1690). As
Jacques found theol-
ogy uncongenial, so
Jean found medicine
equally so, and each
sought in the study
of mathematics the
mental activity that
he required. Jean
became professor of
mathematics in the
University of Gro-
ningen in 1695, and
on his brother's death
(1705) was elected
to fill the place thus
vacated at Basel.
Each was made a
foreign member of
the Paris Academic
des Sciences in 1699.

JEAN BERNOULLI
The younger brother of Jacques (I) Bernoulli

Jean was even more prolific than his brother in his contribu-
tions to mathematics, writing on a wide range of topics, includ-
ing caustic curves (1692), differential equations (1694), the
rectification afid quadrature of curves by series (1694), the
cycloid (1695), catoptrics and dioptrics (1701), the multisec-
tion of angles and arcs (1701), isochronous curves and curves
of quickest descent (1718), and various related subjects. He

430 OTHER EUROPEAN COUNTRIES

wrote on the calculus and was one of the most influential
scholars on the Continent in making its power appreciated.
His collected works appeared six years before his death. 1

AUTOGRAPH OF JEAN (l) BERNOULLI

The "second fils a Petersbourg" mentioned in the letter was Nicolas (II)
Bernoulli, who died at Petrograd only a little over a month after this lettor

was written

To him is due the use of the term "integral" in its technical
sense in the calculus, the first attempt to construct an integral
calculus, and the invention of the exponential calculus. 2 He
was the first, except in such obvious cases as those known to
Cardan, to obtain real results by the use of V^T ; for example,
in finding tan n$ in terms of tan <b.

1 Opera omnia, 4 vols., Lausanne, 1742.

2 "Principia calculi exponentialium," Ada Eruditorum, 1697.

THE BERNOULLIS

431

The Later Bernoullis. The work of the later Bernoullis
falls in the i8th century, but it is appropriate to mention it
briefly at this time. The leading one of the descendants of
the two brothers was Dan-
iel Bernoulli. 1 He was the
son of Jean (I) Bernoulli,
who, curiously enough, made
the same mistake as his
own father in trying to force
his son into trade. Daniel
spent some years (1725-
1733) in Petrograd as pro-
fessor of mathematics in the
Academy of Petrograd, but
in 1733 returned to Basel,
where he became a professor
in the university. He was a
prolific writer, most of his
work appearing in the mem-
oirs of the Academy of Petro-
grad, but he published one
volume on mathematics be-
fore going to Russia. 2 Most

of the memoirs were upon physical questions, but a few related
to pure mathematics, including the computation of trigonometric
functions (1772, 1773), continued fractions (1775), and the
Riccati problem.

Dr. Hutton 3 relates these incidents concerning him:

He used to tell two little adventures, which he said had given him
more pleasure than all the other honours he had received. Travelling
with a learned stranger, who, being pleased with his conversation,
asked his name; "I am Daniel Bernoulli," answered he with great
modesty; "And I," said the stranger (who thought he meant to

1 Usually designated as Daniel (I) Bernoulli. Born at Groningen, February 9
(January 29, o.s ), 1700; died at Basel, March 17, 1782.

2 Exercitationes quaedam mathematicae, Venice, 1724.

* Philosophical and Mathematical Dictionary, I, 205. London, 1706.

DANIEL (l) BERNOULLI
After a portrait from life

432 OTHER EUROPEAN COUNTRIES

laugh at him), "am Isaac Newton." Another time having to dinner
with him the celebrated Koenig the mathematician, who boasted,
with some degree of self-complacency, of a difficult problem he had
resolved with much trouble, Bernoulli went on doing the honours
of his table, and when they went to drink coffee he presented Koenig
with a solution of the problem more elegant than his own.

Nicolas (I) Bernoulli 1 was a nephew of Jacques (I) and
Jean (I). In his younger days he was professor of mathe-
matics at Padua ( 1716-1719), but returned to Basel, where he
became a professor in the university. He was trained in the
law, and his first mathematical treatise was upon the use of
the theory of probability in legal matters." He wrote exten-
sively on differential equations and geometry.

Nicolas (II) Bernoulli* was the son of Jean (I). He
studied law, traveled extensively, became professor of law
at Bern (1723-1725), and was finally called to Petrograd as
professor of mathematics. He wrote on the geometry of curves,
but his death at the age of thirty-one closed a promising career.

Jean (II) Bernoulli 4 was the youngest son of Jean (I). He
studied law but spent his later years as professor of mathe-
matics in his native city. His work was chiefly on physics.

Jean (III) Bernoulli 5 was the son of Jean II. Like his
father, he studied law but soon turned to mathematics, be-
coming director of the mathematics class at the Academy of
Sciences at Berlin. He was much interested in the history of
astronomy but also wrote on the doctrine of chance (1768),
recurring decimals (1771), factoring (1771), and indeter-
minate equations (1772).

The other Bernoullis who were interested to a greater or
less degree in mathematics were Daniel (II) (1751-1834), son

1 Born at Basel, October 10, 1687; died at Basel, November 2Q, 1759. A note
in Terquem's Nouvelles Annales, Suppl. Bulletin, 1858, p. 86, says November 25,
1749, but this is an error.

2 De Usu Arils Conjectandi in Jure, Basel, 1709. He also edited the Ars
Conjectandi of his uncle Jacques (1713).

8 Born at Basel, January 27, 1695; died at Petrograd, July 26, 1726.

4 Born at Basel, May 18, 1710; died at Basel, July 17, 1790.

5 Born at Basel, November 4, 1744 ; died at Kopnick, near Berlin, July 13, 1807.

SWISS WRITERS 433

of Jean (II); Jacques (II) (1759-1789), son of Jean (II):
Christoph (1782-1863), son of Daniel (II); and Jean Gus-
tave (1811-1863), son of Christoph ; but none of these attained
great fame. The Bernoulli blood had lost its strength.

Other Swiss Writers. Two other Swiss mathematicians of
the 1 7th century deserve mention, one a genius, the other a
plagiarist. The genius was Jobst Biirgi, 1 from 1579 to 1603
court watchmaker to Landgraf Wilhelm IV of Hesse, and later
(until 1622) to Kaiser Rudolph II. He wrote on the propor-
tional compasses 2 and on astronomy, but is best known for his
invention of logarithms independently of Napier. He was led to
the idea by an entirely different route from that taken by the
latter, approaching it through the theory of exponents. He did
not publish anything upon the subject until after Napier had
made known his discovery, and when he finally concluded to
print his work it was in the form of a small table of anti-
logarithms, issued anonymously at Prag in i62O. :t The book
never attracted any attention and remained practically unknown
except to historians of mathematics.

The other Swiss writer was of a different character. He was
a professor while Burgi was a watchmaker ; his name has been
known for three centuries, while Burgi's has been almost for-
gotten ; but he was a plagiarist, while Biirgi was a genius. Paul
Guldin 4 began his work as a goldsmith. He later entered the
Jesuit order, lived for a long time in Rome, and became
professor of mathematics at the University of Vienna and

1 Born at Lichlensteig, February 28, 1552; died at Cassel, January 31, 1632.
The first name a'so appears as Joost and Justus; the last as Burgi, Byrgi,
Borgen, and Byrgius. See R. Wolf, "Zwei Kleine Notizen zur Geschichte der
Mathematik," Bibl. Math., Ill (2), 33.

-Invented by Galileo. They were also described (1607) by Levinu3 Hulsius
in his Dritter Tractat der mechanischen Instruments, Frankfort a. M., 20 pp.,
being the third of four tractates, 1603-1615. In the fourth of these tractates
Hulsius describes a pedometer ("Instrument Viatorii oder Wegzahlers " ) , the
earliest mention of this instrument.

3 Arithmetische und Geometrische Progress Tabulen, Prag, 1620.

4 Born at St. Gall, June 12, 1577; died at Gratz, November 3, 1643. His
name was originally Habakuk Guldin, but he changed it when he went from
Protestantism to Catholicism.

434 OTHER EUROPEAN COUNTRIES

later at Gratz. He wrote on physics and mathematics, 1 but is
chiefly known for the fact that his name attaches to a theorem
of Pappus on the volume of a solid generated by the revolution
of a plane about an axis, 2 a theorem which he included in his
works without credit, fully aware that it was in the works of
Pappus, to which he is known to have had access.

Denmark. In the i yth century Holstein, then a part of Den-
mark, produced one and only one mathematician of note,
Nicolaus Mercator. 3 He was one of the leading writers of the
time on cosmography/ and also wrote on trigonometry, 5 the
method of computing logarithms, and astronomy, besides
editing Euclid's Elements. 7 He lived for some time in London
and was one of the first members of the Royal Society. In his
Logarithmotcchnia he gives the series that bears his name,

Spain. Whatever may be said for mathematics in the i6th
century in Spain, less can be said for the century following.
Philip II was a bigot, but he was a great bigot. He commanded
the respect and won the loyalty of his people, and his reign was
one of great works in art and in letters. After his death ( 1 598 ) ,
however, there occupied the throne a sorry line of kings, and
the ruin of the country began. The population of Madrid fell
one half in a century, and Seville's sixteen thousand looms were
reduced to less than three hundred. From being a prosperous
world power the country became a wreck among nations, and
in the general destruction mathematics suffered with the other
sciences and with letters and art.

1 Problema arithmeticum de rerum combinationibus, Vienna, 1622.

-Centrobaryca sen de centra gravitatis . . . liber /, Vienna, 1635; Centra-
barycarum pars altera, Vienna, 1641.

3 Born near Cismar, in Holstein, c. 1620; died in Paris, February, 1687.

4 Cosmographia sive Descriptio coeli et terrae in circulos . . ., Danzig, 1651.

5 Trigonometria sphaericorum logarithmica . . ., Danzig, 1651.

6 Logarithmotechnia, sive Methodus construendi logarithmos nova accurata
et facttis . . ., London, 1668.

7 Euclidis Elementa Geometrica Libri VI nova ordine ac methodo fere demon-
strata. London.

DENMARK, SPAIN, RUSSIA 435

Russia. As for Russia, she had not yet awakened. The first
Russian arithmetic to appear with Hindu- Arabic numerals was
written by a teacher named Magnitzky and was printed in
1703. Under Peter the Great there was some development of
vocational mathematics, but it was only after the founding of
the Academy of Sciences at Petrograd in 1725 that pure mathe-
matics had any standing. 1

8. THE ORIENT

Effect of Western Civilization. The introduction of Western
civilization into India, China, and Japan is interesting because
of its diverse effects. As to India, mathematics was already
stagnant, and the European influence gave it no stimulus.
India has always been content to take her time. Not since
Bhaskara (i2th century) has she produced a single native
genius in this field. In the i7th century only one name, that
of Raganatha (c. 1621), attracted any general attention even
in India, and he contributed nothing that was original. China,
which had once done so much in algebra, was content in the
1 7th century to adopt the European astronomy while allowing
her own undoubted abilities to lie dormant. Japan alone of
all the Orient developed her native mathematics, although
with more or less suggestion from France, Belgium, Italy, and
Germany, through the Jesuit missionaries in China ; from Hol-
land, through the Dutch traders at Nagasaki; and possibly
through scholars who secretly visited the universities of the
Low Countries. As to Persia and Arabia, mathematics was
dead and forgotten.

China. The mathematical feature of importance at the open-
ing of the i7th century in China was the work of the Italian
Jesuit Matteo Ricci already mentioned in Chapter VIII. After
his death (1610) other missionaries carried on this work, not
only in a religious line but also in the introduction of Western

1 V. Bobynin, " De 1'etude sur PHistoire des mathematiques en Russie," Bibl.
Math., II (2), 103 ; A. N. Peepin, History of Literature (in Russian), I, 253
(Petrogradj 1911).

436 THE ORIENT

science. Among these were the Jesuit Nicolo Longobardi 1
(who went to China in 1596), Giacomo Rho, 2 Johann Adam
Schall von Bell/ Smogolenski 4 (1611-1656), and Ferdinand
Verbiest. 5 Of these Schall and Verbiest were particularly prom-
inent in astronomical work. Smogolenski made known the
use of logarithms/ 5 and his pupil, Sie Fong-tsu, published the
first Chinese work on the subject (c. 1650). It is interesting
to observe that Vlacq's tables (1628) were reprinted in Peking
in I7i3. 7

In this century there were several Chinese scholars who wrote
important works, but they were all inspired by the Jesuits,
and their works are based on European models. 8 Mention
should also be made of Mei Wen-ting (1633-1721), a pro-
found scholar, well versed in European as well as native science,
who wrote on a variety of subjects and to whom we are in-
debted for much information concerning the history of Chinese
mathematics.

The reason for the activity of the Jesuits in teaching the
Western astronomy to Chinese scholars is apparent. It is only
by establishing an intellectual superiority that a foreign religion

1 Born at Calatagirone, Sicily, 1565; died at Peking, December u, 1655.
The Chinese name was Lung Hua-ming.

-Born at Milan, 1503; died at Peking, April, 1638. He reached China in
1618. The name also appears as Jacomo Russ, whence the Chinese, Lo Ya-ku.

3 Born at Cologne, 1501; died at Peking, August 15, 1666. His Chinese name
was Tang Jo-wang. He arrived in China in 1622, and in 1630 was called upon
by the government to reform the calendar. He and his colleagues wrote a
large number of works. 4 Chinese name, Mu Ni-ko.

r 'Born at Pitthem, near Courtrai, Belgium, October o, 1623; died at Peking,
January 28, 1688. Chinese name, Nan Huai-jen. He arrived in China in 1659.
See H. Bosnians, "Ferdinand Verbiest," in the Revue des Quest, scientifiques,
PP- iQ5, 375 (Brussels, IQI2), and "Le probleme des relations de Verbiest avec
la Cour de Russie," in Annales de la Societe d Emulation pour V etude de Vhist.
. . . de la Flandre, p. 103 (Bruges, 1913). It was under his direction that most
of the large astronomical instruments were made (1674) for the emperor.

6 In his T'ien-pu Chen-yuan.

7 In the Lii-li Yuan-yuan. This work also contained a treatment of algebra
on European lines.

8 For example, Tu Chih-ching, who wrote a geometry, Chi-ho Lun-yiieh,
based on Euclid, and the Su-hsiao Tao, based on European mathematics;
Huang Tsung-i, who wrote on the calendar; and Ch'en Chin-mo (c. 1650) f
who gave 3.15025 as the value of TT.

JAPAN 437

ever makes permanent progress. The Jesuits were not long in
seeing that the two sciences in which Europe far surpassed the
East were geometry and astronomy, and on these they concen-
trated their attention. 1

The Intellectual Awakening of Japan. When Japan, in the
1 7th century, finally awoke to her intellectual possibilities, it
was in a blaze of glory not unlike that which characterized
her awakening to her national possibilities in the iQth century.
Her progress in mathematics was strangely comparable to the
remarkable progress that was going on at the same time in
Europe, for in this century she developed a native calculus at
almost the same time that Newton and Leibniz were working
out their epoch-making theories.

Of the pupils of Mori Kambei Shigeyoshi (p. 352), one
of those to achieve renown was Yoshida Shichibei Koyu, or
Mitsuyoshi (1598-1672), whose Jinko-kr was the first great
work on arithmetic to appear in Japan. In this the value of IT
is given as 3.16. So familiar was the name of this work that
it was often used subsequently as a synonym for arithmetic. J

The second of Mori's pupils to contribute to mathematics in
a noteworthy manner was Imamura Chish5, whose Jngai-rokn,
devoted to stereometry as well as to arithmetic, appeared in 1639.
In this work the value of TT is given as 3.162, the area of a
circle as { cd, and the volume of a sphere of radius \ as 0.51.

The third of Mori's celebrated pupils was Takahara Kisshu,
or Yoshitane, but he published nothing on mathematics.

The middle of the century saw also a number of minor
writers whose works show considerable ability in mathematics.
Among these was Isomura or Iwamura Kittoku, known also
as Yoshinori. In his Kctsugi-sho (1660) there are a number

1 " Ce fut alors que des Jesuites penctrerent dans la Chine pour y precher
1'evangilc. Us ne tarderent pas a s'apperccvoir qu'un des moyens les plus
efftcaces pour s'y maintenir . . . etoit d'etaler des connoissances astronomiques."
Montucla, Histoire, I (2), p. 468.

2 The full title means "Small number, large number, treatise," that is, a
treatise on numbers from the smallest to the largest.

3 Compare the word algorismux as synonymous with arithmetic, and the
name "Euclid" as synonymous with geometry.

THE ORIENT

of interesting problems proposed by Yoshida Koyu, of which
the following, referring to measurements, are typical :

There is a log of precious wood 18 feet 1 long, whose bases are 5
feet and 2\ feet in circumference. . . . Into what lengths should
it be cut to trisect the volume?

A circular piece of land 100 measures in diameter is to be
divided among three persons so that they shall receive 2900, 2500,

ONE OF ISOMURA KITTOKU'S PROBLEMS, 1660

His Ketsugi-sho appeared in 1660. This is from the 1684 edition. It represents

the early state o* the advanced native mathematics of Japan and shows the rise

of a crude integration

and 2500 measures respectively. 2 Required the lengths of the chords
and the altitudes of the segments.

There is also in this work a rough approach to an integral
calculus.

In his later years Isomura devoted much attention to magic
squares, magic circles, and magic wheels. 8 By a very ingenious
method he showed that the surface of a sphere, which he at
first thought was equal to irV, has the value Trd 2 .

I ln the original, "3 measures." For this and other problems see Smith-
Mikami, p. 66.

2 That is, square measures, by drawing parallel chords.

3 For particulars, see Smith-Mikami, p. 69.

SEKI KOWA 439

In 1663 Muramatsu Kudayu Mosei began the publication
of a work on arithmetic and mensuration which contributed to
the knowledge of the circle and the regular polygons, but only
in respect to measurement.

In 1664 Nozawa Teicho published a work called Dokai-sho,
in which some ingenious problems in mensuration appear and
a step is taken in advance of Isomura in the integral calculus.

In 1666 Sato Seiko wrote the Kongcnki, a work in which
the custom is continued of proposing and solving ingenious
problems. This is the first Japanese work in which the ancient
Chinese method of solving numerical higher equations appears.

In 1670 Sawaguchi Kazuyuki wrote a work entitled Old and
New Methods in Mathematics^ In this there again appears an
approach to an integral calculus, somewhat after the method
of Cavalieri, and also a treatment of numerical equations.

Seki Kowa. The most distinguished Japanese mathematician
of the i yth century, and in some respects the most distinguished
of all Japanese mathematicians, Seki Shinsuke Kowa, or Taka-
kazu (1642-1708), was born of a samurai 2 family and showed
his great mathematical ability at an early age. He acquired
his knowledge to a large extent without the aid of teachers,
showing great ingenuity in the affairs of life, in mechanics, in
mathematics in general, and in problem-solving in particular.
He improved upon the Chinese methods of solving higher
equations, systematized the early Chinese use of determinants,
possibly invented the circle principle (yenri method) which
was later developed into a kind of calculus, and proposed nu-
merous problems of an intricate nature. Two problems pro-
posed by Sawaguchi Kazuyuki and solved by Seki Kowa are
substantially as follows:

In a circle three circles are inscribed, each tangent to the other
two and to the original circle. They cover all but 120 square units
of the circumscribing circle. The diameters of the two smaller circles
are equal and each is 5 units less than the diameter of the next larger
one. Find the diameters of the three inscribed circles.

^Kokon Sampo-ki. 2 Feudal lords.

440 THE ORIENT

In a certain triangle AJ1C there is a point P such that PA = 4,
ppj = 6, and /'(; = 1.447 ", sucn that the sum of the cubes of the
longest and shortest sides is 637 ; and such that the sum of the
cubes of the other side and the longest side is 855. Find the lengths
of the sides. 1

The Chinese had some idea of the determinant, as we have
seen, but it is to Seki that the honor must be given of expanding
a determinant in solving simultaneous equations, a discovery
which anticipated the one made by Leibniz.

Seki's reputation was such as to attract to him a large
number of pupils, and his influence upon them was so great
as to make itself felt up to the time when the native mathe-
matics became absorbed in the Western science which was so
completely adopted in the igth century.

On the other hand, Seki made no great discovery in mathe-
matics, with the exception of his anticipation of determinants.
He was a great teacher, he did pioneer work in the awakening
of a scientific spirit in Japan, he showed ingenuity in improving
upon the work of his predecessors, but he was not the author of
any new method that is now recognized as valuable, and he
wrote no great treatise that stands forth today as anything more
than a historical document. Because of his efforts to give to his
people a knowledge of the mathematical sciences, however, His
Majesty the Emperor of Japan very justly paid honor to his
memory in 1907 by bestowing upon him the highest posthumous
honor ever awarded to such a scholar.

Space does not permit of the further mention of Japanese
scholars, with the exception of Nakane Genkei (1661-1733), a
contemporary and disciple of Seki, whose works on astronomy
(see the illustration on page 441) were influenced by European
treatises which had begun to find their way into Japan through
the Dutch traders at Nagasaki. He is also known to have been
familiar with certain works of the Jesuit missionaries in China
and to have recognized their superiority in the astronomical field.

^For the problems and suggestions as to solutions see Smith-Mikami,
pp. Q6, TOO.

I
I

JAPANESE ASTRONOMY INFLUENCED BY THE WEST

From a work by Nakane Genkci, printed in i6gb, showing the explanation of

the phases of the moon. Influenced by the Dutch astronomy which had begun

to be known in Japan

442 THE ORIENT

Contact with Europe. There were certain periods in the his-
tory of Japan when contact with the outer world was very
difficult. Even when the Dutch traders had a monopoly of
bartering with the country through the port of Nagasaki, it was
practically impossible for students to leave the island. We
find, however, mention of two Japanese students on the records
of Dutch universities in Seki's time. 1 Nothing further is known
of these men, but the important fact is that they represent
contact with intellectual Europe. The problem which this
suggests is to ascertain whether through these or similar chan-
nels any suggestion of the status of European mathematics
reached Japan in Seki's time. There is a tradition, too, that
Hatono Soha, a physician, went to "the Spanish lands 7 ' 2 at
this time, and that he later returned to Japan. If this is
true, European learning of some kind entered the country in
the second half of the i?th century. Whether or not it sug-
gested the calculus, which reached its highest native develop-
ment in Japan in the i8th century, is unknown.

There is a tradition that Seki made a pilgrimage to the
ancient shrines at Nara, having learned of certain treatises
which were carefully preserved in the Buddhist temples there,
and which no one was able to understand. These proved to be
Chinese works on mathematics, and Seki is said to have spent
three years in mastering their contents. But again, we do not
know whether they contained fragments of Western learning
that the Jesuits had brought from Europe, or the ancient alge-
braic science of the Chinese, or possibly traces of Hindu astrol-
ogy. No doubt, however, the Japanese scholars will in due time
search out the sources of the mathematics of Seki's school.

n ln the Album Studiosorum Academiae Lugduno Batavae (The Hague, 1875)
it appears that one "Petrus Hartsingius Japonensis," aged 31, was studying
philosophy at Leyden in 1654. He is also mentioned by van Schooten in his
Tractates de concinnandis demonstrationibus geometricis ex calculo alge-
braico, in Descartes's La Giometrie, 1661 and 1683 editions, p. 413. He also
appears on the roll in 1660, as a student of medicine, and again in 1669. In
the Album there is also the entry, under the date September 4, 1654, "Franciscus
Carron Japonensis." Of course the names are not Japanese, and Franciscus
Carron is that of a Christian missionary of a century earlier.

2 Which were then interpreted by the Japanese as including Holland.

DISCUSSION 443

TOPICS FOR DISCUSSION

1. Conditions particularly favorable to the development of mathe-
matics in the lyth century.

2. Causes of the decline of Italy's position in mathematics in the
jyth century.

3. Forerunners of Newton and Leibniz irt the development
of the calculus.

4. Cases of relatively late development of mathematical ability.

5. Cases of the influence of inheritance or of early environment
upon the development of mathematical power.

6. The relative standing of mathematics in France and in
England in the iyth century.

7. The rise of analytic geometry.

8. The greatest mathematical discovery in the i7th century,
with statements showing why it was the greatest.

9. The four greatest mathematical books of the century.

10. From the standpoint of the individual and his life, the most
interesting mathematical personage of the century.

11. The steps taken in the i7th century to improve geometry.

12. The five most interesting mathematicians of France in the
1 7th century.

13. The five most interesting British mathematicians in the i7th
century.

14. The influence of Newton's English predecessors upon him
and upon his work.

15. Certain early steps in the use of continued fractions, infinite
series, and infinite products in the i7th century.

1 6. The rise of books on mathematical recreations.

17. The influence of astronomy and mathematics upon each
other in the i7th century.

1 8. Men who were prominent in both mathematics and physics
in the i7th century.

19. The six leading mathematical countries in the i7th century, ar-
ranged according to their importance, with reasons justifying the
arrangement.

20. The introduction of Western mathematics into the East.

21. The work, the general standing, and the influence of the
Japanese scholars in the i7th century.

CHAPTER X
THE EIGHTEENTH CENTURY AND AFTER

i. GENERAL CONDITIONS

Status of Elementary Mathematics. Since this work is con-
cerned primarily with the history of elementary mathematics,
it would be quite justifiable to set its limit at the close of the
1 7th century. By that time arithmetic as we ordinarily speak
of it, referring to the operations with numbers for commercial
and industrial purposes, was practically what it is today. We
have changed the way of teaching it, and we have added new
applications from time to time as the requirements of business
dictated ; but the mathematical part of the subject has been
very nearly static. We even preserve certain traditional topics
and methods that might profitably have been discarded long
ago, rarely recognizing that logarithms are more easily handled
than roots, and that the algebraic equation is superior to the
method of proportion, which we still retain for certain purposes
where it might better be discarded.

The algebra that is taught in the secondary schools and in the
freshman course in college was practically all in use before
1700. The symbolism has changed but little, and although
the elementary textbook is more extensive, it contains no mathe-
matics that was not generally known before that date. The
changes that have been made relate chiefly to methods of
teaching and to the applications of the subject.

Elementary geometry as ordinarily taught to beginners has
made no advance, although, scientifically speaking, the founda-
tions have been explored with far-reaching results. The pupil
who studies geometry in a secondary school today is not getting
as good mathematics as the one who studied it in the 1 7th cen-
tury, simply because it was the selected boy who took the work

444

ELEMENTARY MATHEMATICS 445

at that time. Euclidean geometry is what it was then ; it has
been rearranged for educational purposes, but the modern text-
book of the popular type is not mathematically as scientific as its
predecessor. Geometry has made giant strides, but not in the
field that teachers generally cultivate in the secondary schools.

Elementary trigonometry and analytic geometry were well
known to the mathematical world at the close of the iyth cen-
tury, and even our modern geometry had made some progress
in the work of Desargues.

The calculus has been greatly improved with respect to its
foundation principles and the method of presentation, but the
elementary calculus that is taught in our colleges, both differen-
tial and integral, with its most important applications, was
familiar by the year lyoo. 1

The elementary theory of equations, the solution of numeri-
cal equations, the symmetric functions of the roots, such forms
as continued fractions, the actual handling of complex num-
bers, the use of infinite series, and even the elementary use of
determinants, all these and various similar topics were well
understood before the i8th century. From the standpoint of
elementary mathematics, therefore, a large part of the history
of the subject closes with the year 1700.

It would be a mistake, however, to suppose that the further
progress of the science has an interest only to the student of
higher mathematics. Great achievements in any line of work are
always stimulating, and some knowledge of these achievements
and of the men who made them is necessary to the well-
informed teacher or student of the science. 2

Limitation of the Study. It is the purpose, in this chapter,
to limit the study chiefly to a consideration of those mathema-
ticians whose achievements were so noteworthy that everyone
who is interested in mathematics should be informed concerning

the mathematics of the iSth century, particularly in Great Britain,
consult J. Leslie, "Dissertation Fourth . . . The Progress of Mathematical
and Physical Science, chiefly during the eighteenth century," in the seventh
edition of the Encyclopaedia Britannica.

2 For a summary of the history of mathematics in the iQth century see
J. Pierpont, Bulletin of the Amer. Math. Soc., XI, 136.

446 GREAT BRITAIN

them. In the mathematical world names like De Moivre, d'Alem-
bert, Euler, Laplace, Lagrange, Legendre, Gauss, Monge, Galois,
Poncelet, von Staudt, and Steiner are so often seen that all
teachers of mathematics should know something of the achieve-
ments of the men. The student of the higher branches will
know this in connection with his researches, but for the teacher
in the elementary field of the science a brief resume will be
helpful. It should be understood, however, that no mention will
ordinarily be made of certain names which, had they been met
in the formative period of elementary mathematics, would have
found place. Various men who might properly be mentioned in
this connection will be referred to in Volume II of this work.

Royal Patronage. With the i8th century the king of France
no longer stands out as the sole royal patron of science. 1 Queen
Anne bestows knighthood upon Newton; George I shows an
interest in scientific laboratories ; Peter the Great is at pains
to meet with learned men and founds (1724) an academy at
Petrograd to which there later come such mathematicians as
the Bernoullis and Euler; George III, in spite of his parsi-
mony, endows the observatory of Herschel ; and Frederick II
calls to the Berlin Academy Maupertuis, d'Alembert, a Ber-
noulli, and Lagrange, not to speak of Voltaire, who, as we shall
see, had some claim to the title of mathematician. In spite
of all this, France held high place in the fostering of all the
sciences, perhaps the highest; and no other nation could
boast, at the close of the i8th century, such a galaxy of stars
in the mathematical firmament.

2. GREAT BRITAIN

Nature of the Work. As would naturally be expected, the
influence of Newton determined to a large extent the nature
of the work in Cambridge, Oxford, London, and Edinburgh in
the 1 8th century. The improvement of the calculus and the
widening of the range of applications of the subject were the

1 A. Rambaud, Histoire de la Civilisation Franqaise^ i2th ed., II, 473. Paris,
1911.

COTES 447

characteristic features. A few of the leading names connected
with this movement will be mentioned, together with a brief
statement of the contributions of each.

It should be observed, also, that the i8th century saw mathe-
matics made popular for the first time in Great Britain. Schools
were established for the poor to attend on Sunday, the only
day that they could attend them at all, and this was done much
against the opposition of many in the Church ; circulating
libraries were established ; printing ceased to be largely a
London monopoly and spread throughout the country ; ele-
mentary handbooks appeared ; and such popularizers of mathe-
matics as the Ladies 9 Diary (1704-1840) and the Gentleman's
Diary (1741-1840) had a wide circulation. Even such classi-
cal works as Newton's Principia were printed in the vernacu-
lar. Mathematics had ceased to be aristocratic; democracy
had begun to assert its rights in intellectual as well as political
matters.

Cotes. Newton is credited with the statement, "If Cotes
had lived, we had known something," a remark that might, it
would seem, be made with greater force with respect to certain
others who, like Pascal, Galois, and Clifford, died at a rela-
tively early age. Nevertheless it is true that few scholars
showed such powers of analysis before the age of thirty-four,
the age at which Roger Cotes 1 died. He had shown a taste for
mathematics when only about twelve years of age. He was
educated at St. Paul's School, London, where he also developed
a taste for metaphysics, philosophy, and divinity, and thence
proceeded to the mathematical Mecca of England, Trinity
College, Cambridge. When only twenty-four years of age
he was appointed (1706) to the Plumian professorship of as-
tronomy, the first to fill the chair which had just been estab-
lished (1704) by Dr. Plume, archdeacon of Rochester. In
1713 he published at Cambridge the second edition of New-
ton's Principia. Only two of his memoirs appeared during his
lifetime, but most of his writings were collected and published.

1 Born at Burbage, Leicestershire, July 10, 1682 ; died at Cambridge, June 5,
1716.

448 GREAT BRITAIN

shortly after his death, by his cousin, Dr. Robert Smith (1689-
I768), 1 who succeeded him in the Plumian professorship.

He discovered an important theorem on the nth roots of
unity, partly anticipated the method of least squares, and dis-
covered a method of integrating rational fractions with binomial
denominators. His theorem on the harmonic mean between the
segments of a secant to a curve of the nth order, reckoned from
a fixed point, is well known.-

The Calculus in English. The first work on the Newtonian
calculus to appear in the English language was published in
London in I7O4. 3 It was written by Charles Hayes, 4 a member
of Gray's Inn, London. The purpose of the work is set forth
in the preface:

The Author has been well assured that there are in England as
many Lovers of the Mathematicks as in any part of the World ;
. . . that in other Nations the best pieces of Learning are written
in their own mother Tongues, for the good of their Country which
we seem purposely to slight, seeking a little empty applause by
writing in a Language not easily attained.

The work is clearly written, but was overshadowed later by
such treatises as those of James Hodgson r ' and John Rowe.
Hayes also wrote on the finding of longitude (1710) and
began but did not live to complete a CJtronographia Asiatica
et Acgyptica.

^Harmonia Mensurarum, sive Analysis et Synthesis . . ., Cambridge, 1722.
The second part of the volume comprised his Opuscula Mathematica.

2 W. W. R. Ball, A History oj the Study of Mathematics at Cambridge, p. 88
(Cambridge, 1880), hereafter referred to as Ball, Hist. Math. Cambridge;
Chasles, Aper$u, p. 147.

3 A Treatise of Fluxions: or, an Introduction to Mathematical Philosophy ,
London, 1704.

4 Born 1678; died in London, December 18, 1760.

5 Master of the mathematical school in Christ's Hospital, London. He was
born in 1672 and died in London, June 25, 1755. Among his several works
was The Doctrine of Fluxions, London, 1736, with an edition in 1758.

6 Introduction to the Doctrine of Fluxions, London, MXCCLT (sic for
MDCCLI, 1751), with editions in 1757, 1767, and 1809. This was the first
really popular presentation of the subject in English.

THE CALCULUS IN ENGLISH 449

In connection with these works Joseph Raphson's 1 history
of the calculus 2 should be mentioned. The purpose of the book
is thus stated :

To assert the Principal Inventions of this Method, to their First
and Genuine Authors; and especially those of Sir Isaac Newton,
who has vastly the Advantage of all others as well in respect of
Priority of Time, as the Great and Noble Nature of his Discovery.

Such a book, written in the heat of the controversy as to the
priority of the works of Newton and Leibniz, would naturally
be open to the charge of partisanship.

James Stirling. A brief list of some of the other important
1 8th century writers will tend to show the nature of the work
being clone in Great Britain.

James Stirling 3 was educated at Glasgow and at Balliol Col-
lege, Oxford. He left Oxford (1715), partly on account of his
relations to the Jacobites, and went to Venice to accept a pro-
fessorship. In Italy he formed the acquaintance of Nicolas
Bernoulli, who was then at Padua, and was probably encouraged
by him to write his well-known work on lines of the third
order. 4 Of these lines he added four to those already discussed
by Newton, and he also wrote a paper on the differential
method in the treatment of infinite series. 5 He returned to
London in 1725, devoting himself to mathematics, meeting
with Newton in the latter's closing years, writing several im-
portant memoirs, and corresponding with many noted mathe-
maticians of the day. His sojourn in Venice gave him the
nickname "the Venetian," and by this he was commonly

1 Or Ralphson. He wrote an Analysis aequationumuniversalis, London, 1690,
and died before 1715.

-The History of Fluxions, Shewing in a compendious manner The first Rise
of, and various Improvements made in that Incomparable Method, London,
1715 (posthumous).

3 Born at Garden, Stirlingshire, 1692; died at Edinburgh, December 5, 1770.

4 Lineae tertii ordinis Newtonianae sive illustratio tractatus Newtoni de
enumerations linearum tertii ordinis, Oxford, 1717.

^Methodus differential sive Tractatus de summatione et interpolations
serierum infinitarum, London, 1730. An English translation by Francis Holliday
appeared in 1749.

450 GREAT BRITAIN

known to his friends. He made an important survey of the
Clyde, and in later life ( 1 735 ) became the manager of a mining
company in Lanarkshire ; and, strange to say of such a scholar,
he made a great success of this venture.

Be Moivre. Although born in France, Abraham de Moivre 1
spent his life from the age of eighteen in London, and may
properly be ranked with the English school of mathema-
ticians. Compelled by narrow circumstances to forego the life
of a student, he supported himself by private teaching, by
lecturing, and by giving answers to mathematical puzzles. It
is said that he passed most of his time in a London coffee house,
where his genius in solving problems brought to him sufficient
return for his humble needs. Having come by chance upon
a copy of Newton's Principia, he discovered his weakness in
the higher range of mathematics and by assiduous application
soon became recognized as a man of genuine ability in research.
He was admitted to membership in the Royal Society and into
the academies of Paris and Berlin. His work was chiefly on
trigonometry, 2 probability, 3 and annuities. 4 In his discussion
of trigonometry he gave the theorem which bears his name,
(cos x 4- i sin x) n = cos nx + i sin nx, a relationship already stated
in substance by Cotes, 5 and one which leads to numerous inter-
esting identities in connection with complex numbers. Indeed, it
stands as one of the basic propositions in the theory of such
numbers. He is also known for having given the various quad-
ratic factors of ,r 2 " 2, kx" + i, for having stated the rule for
finding the probability of a compound event, for his work on
recurring series, and for his extension of the quadrature of the
lunes of Hippocrates.

1 Born at Vitry, Champagne, May 26, 1667; died in London, November 27,
1754-

^Miscellanea Analytica, de seriebus & quadraturis . . ., London, 1730.

3 Doctrine of Chances, London, 1718, with later editions in 1738 and 1756'.
II was dedicated to Newton. There was an Italian edition, Milan, 1776.

4 Annuities upon Lives, London, 1725, with later editions.

5 See Volume II, Chapter IV.

6 See the summaries of his papers in the Phil. Trans, abridgment, 4th ed.,
I, i, 29, 81, 90, et passim ; IV, 3, 25, 77 (London, 1731).

DE MOIVRE 451

There is often told a story of his death, to the effect that he
bad declared it to be necessary to sleep a quarter of an hour
longer each day than on the preceding one. If he was sleeping
six hours a day when he began this series, it is evident that
the first day thereafter he would sleep 6^ hours, and on the
73d day he would reach the limit.

Whiston. Born in the same year as De Moivre, but under
seemingly more favorable stars, and dying only two years
before him, William Whiston 1 lived a life as ideal as De
Moivre's was discouraging. He received many honors, wrote
numerous scientific and theological works, and held the Luca-
sian chair of mathematics in Cambridge (1703-1710); and
yet he left a name that is today by no means so well known
in the history of mathematics as that of his humbler contem-
porary. His chief interests were in astronomy.

Brook Taylor. The name of Brook Taylor 2 is familiar to
every student who knows the rudiments of the calculus, Tay-
lor's Theorem being one of the first instruments that he uses.
The discoverer of this theorem was educated at St. John's Col-
lege, Cambridge. He early gave great promise of success in
mathematics, wrote various papers for the Philosophical Tran-
sactions, was admitted to the Royal Society, and became its
secretary. When only thirty-four, however, he gave up his
secretaryship and devoted himself to writing. In 1715 he
published a work 3 in which is contained his well-known propo-
sition, /2

and some treatment of the calculus of finite differences, of
interpolation, and of the change of the independent variable.
He published two works on perspective, 4 giving the first

a Born at Norton, Leicestershire, December 9, 1667; died in London, Au-

gust 22, 1752.

2 Born at Edmonton, August 18, 1685; died in London, December 29, 1731.

*Methodm Incrementorum Directa et Inversely London, 1715. He had an-
nounced the discovery in 1712.

4 Linear Perspective, London, 171$; Principles of Linear Perspective, Lon-
don, 1719.

452

GREAT BRITAIN

general enunciation of the principle of vanishing points. He
was also the author of various memoirs on physics, logarithms,
and series. 1

From an engraving by J. Dudley

Maclaurin. Associated with Taylor's name, on account of
the theorem above mentioned, is that of Colin Maclaurin, 2 a
Scotch mathematician, who entered the University of Glasgow
(1709) at the age of eleven. He soon showed a taste for

1 For a biography see the preface to his posthumous work, Contemplatio
Philosophica, London, 1703.

2 Born at Kilmodan, Argyllshire, February, 1698 ; died at York, June u.
1746. See C. Tweedie, Math. Gazette, IX, 303.

MACLAURIN

453

mathematics, and at the age of twelve, having accidentally

run across a copy of the work, he mastered the first six books

of Euclid in only a few

days. At the age of fif-

teen he took the degree

of M. A., publicly defend-

ing with much success a

thesis on the power of

gravity. At the age of

nineteen he was elected

to the chair of mathe-

matics in the Marischal

College, Aberdeen, and

at the age of twenty-one

(1719) he took to his

London printer his first

important work. 1 After

traveling for some time

as tutor to the son of

Lord Polwarth he be-

came ( 1 72 5 ) an assistant

at the University of

Edinburgh, finally being

elected to aprof essorship.

To him is due a method of generating conies which bears his

name. His treatise on fluxions 2 contains the well-known identity,

COLIN MACLAURIN

Known chiefly for the formula which bears
his name

a relationship easily deduced from Taylor's Theorem, and one
which had been announced by James Stirling twelve years
earlier. Maclaurin greatly generalized the theory of the mystic
hexagram and was the first to publish a work on the subject,
for Pascal's essay, although written more than a century
earlier, did not appear in print until 1 779. He wrote an algebra

1 Geometric, Organica: sive Descripto Lmearum Curvarum Universalis,
London, 1720. Part of the propositions were worked out in his i6th year.
^Treatise of Fluxions, 2 vols., Edinburgh, 1742.

454 GREAT BRITAIN

that was published posthumously, 1 and various memoirs on
geometry and physics. Ball 2 has very well summed up his
influence in these words :

Maclaurin was one of the most able mathematicians of the i8th
century, but his influence on the progress of British mathematics
was on the whole unfortunate. By himself abandoning the use both
of analysis and of the infinitesimal calculus, he induced Newton's
countrymen to confine themselves to Newton's methods, and it was
not until about 1820, when the differential calculus was introduced
into the Cambridge curriculum, that English mathematicians made
any general use of the more powerful methods of modern analysis.

Saunderson. Nicholas Saunderson 3 deserves mention as one
of the mathematicians of this period, not so much because of
his great achievements in advancing the science as on account
of the inspiration that his history offers to those who labor
under difficulties such as discourage most men and lead them
early to abandon hope.

When only one year of age he became blind through an
attack of smallpox. He was a pupil of Whiston's, who was
then Lucasian professor of mathematics at Cambridge, and suc-
ceeded him in 1711. He was created doctor of laws in 1728
by command of George II, and became a Fellow of the Royal
Society in 1736. He was very successful as a teacher, and is
especially known for his Algebra, published posthumously in
1740-1741 and translated into French by E. de Joncourt
(Amsterdam, 1756). His Method of Fluxions also appeared
after his death (1751). Saunderson counted among his friends
such well-known scholars as Newton, Cotes, and De Moivre,
and did much to make the philosophy of Newton known to the
mathematicians of his time. He could carry on long and
complicated mathematical problems mentally, which partly
accounts for his success in spite of his misfortune. 4

1 Treatise of Algebra, London, 1748, with several later editions. On his at-
tempted proof of Taylor's Theorem see Bibl. Math., I (3), 438.

2 Hist. Math., 6th ed., 388.

3 Born at Thurlston, Yorkshire, January, 1682; died at Cambridge, April 19,
1739. * For his biography consult the Algebra, mentioned above.

,; ' .

v^ ' " : mJ'''-: :; *vf '$&;*#.: j
j^/r-r^ti^li,/ 4 ';

NICHOLAS SAUNDERSON
After Vanderbanck's painting

456 GREAT BRITAIN

Other British Writers of the Century. Among those who con-
tributed to the advance of the Newtonian philosophy at this
time Humphrey Ditton 1 deserves mention. He was a man
without university training but interested in Church work.
He left this work, however, and devoted the latter part of his
life to the study of mathematics and to teaching. He was much
esteemed by Newton, on whose recommendation he was elected
mathematical master at Christ's Hospital, London. He pub-
lished a number of memoirs on mathematics and physics, a
work on fluxions, 2 a revision of an algebra/' and a work on
perspective (1712).

All students of the history of geometry will recognize the
name of Robert Simson, 4 who, although educated as a physi-
cian, became professor of mathematics (1711) at the Uni-
versity of Glasgow. He was a thorough student of Greek
mathematics, 5 and most of the English editions of Euclid
are based upon his edition of the Elements. He was averse to
the use of algebraic analysis in geometry, and his methods are
those of the Greeks.

Another well-known Scotch writer of this period, Matthew
Stewart,' 1 entered the University of Glasgow in 1734, coming
under the instruction of Simson. He also attended Maclaurin's
lectures at Edinburgh, and succeeded him in his professorship
in 1747. He was particularly interested in geometry and in the
introduction of the simple form of the Greek synthetic demon-
stration into modern higher mathematics. 7 He also devoted

1 Born at Salisbury, May 20, 1675; died in London, October 15, 1715

-An Institution oj Fluxions, London, 1706. There was an enlarged edition
in 1726.

-Synopsis Algebraira of John Alexander, London, 1700.

*Born at Kirktonhall, Ayrshire, October 14, 1687; died at Glasgow, Oc-
tober i, 1768. See William Trail, Account of the Life and Writings of Robert
Simson, Hath, 1812.

ft Seclionnm Conicarum Libri F, Edinburgh, 1735, with an enlarged edition in
1750; Apollonii Pergaei Locornm Planorum libri 77, restitnti . . ., Glasgow,
1749; Euclid's Elements, Glasgow, 1756, with numerous editions. Some of his
works were published posthumously at Glasgow in 1776.

Born at Rothcsay, Isle of Bute, 1717; died at Edinburgh, January 23, 178$.

"General Theorems^ Edinburgh, 1746; Tracts, Physical and Mathematical,
Edinburgh, 1761.

MINOR WRITERS 457

much of his energy to astronomy, particularly to the problem
of the sun's distance from the earth. 1 Several propositions of
modern geometry bear his name. 2

Contemporary with these Scotch writers, but living in Eng-
land, there was that strange mathematical genius, Thomas
Simpson.' He was brought up by his father to be a weaver,
and hence his early education was confined to the reading and
writing of English. Since Thomas persisted in reading beyond
what his father thought necessary, resulting in a vigorous pater-
nal protest, the boy decided to run away from home. A peddler
having given him a copy of Cocker's arithmetic containing
an appendix on algebra, he began the study of mathematics.
His life was a turbulent one, and it suffices for our purposes to
say that he struggled against poverty in London with sufficient
success to allow him to publish works on the calculus, 4 prob-
ability/ 1 algebra, and various other subjects, and to prepare
numerous monographs of importance. He was a man of un-
doubted genius, and his abilities were recognized in his election
as professor of mathematics at the Woolwich Military Academy
in 1743 and as a Fellow of the Royal Society in 1745. As a
teacher he was a failure ; equally was he a failure in the home ;
and, as with many other human failures, drink finally asserted
the mastery.

Another self-made mathematician of this period appeared in
the person of John Landen, 7 known for his work on residual
analysis, 8 a theory by which he solved a number of problems
more simply than had been done by fluxions, and for his memoir

iScc Essays of the Phil. Soc. of Edinburgh, 1756.

-Chasles, Apcrc.u, 173.

3 Born at Market Bosworth, Leicestershire, August 20, 1710; died at Market
Bosworth, May 14, 1761.

4 A new Treatise of Fluxions, London, 1737 ; The Doctrine and Application
of Fluxions, London, 1750.

5 A Treatise on the Nature and Laws of Chance, London, 1740; The
Doctrine of Annuities and Reversions, London, 1742.

Q An Elementary Treatise of Algebra, London, 1745; 2d ed., 1755.

7 Born at Peakirk, near Peterborough, January 23, 1719; died at Milton,
near Peterborough, January 15, 1790.

8 Discourse concerning Residual Analysis, London, 1758 ; The Residual Analy-
sis, London, 1764.

458 GREAT BRITAIN

( I 7S5) on the rectification of the arc of a hyperbola. He also
wrote on astronomy, series, elliptic transcendents (1771), and
physics, and was one of the early contributors to the theory of
the top. He was admitted to the Royal Society in 1766.

Charles Hutton, 1 one of the best-known English writers on
mathematics at the close of the i8th century, owed his promi-
nence more to his perseverance than to his scientific ability.
He was professor of mathematics at the Military Academy at
Woolwich (1772-1807) and is chiefly known for his mathe-
matical tables" and his dictionary. 3

Among the contemporaries of Hutton one of the intellectual
leaders, but by no means one of the best-known, was William
Wallace. 4 As a young man he was a printer, later becoming
interested in bookselling. Meantime he developed a taste for
mathematics, gave private lessons, and at the age of twenty-
six became a teacher in the Perth Academy. He finally became
a professor in the University of Edinburgh. He wrote a num-
ber of memoirs on logarithms, trigonometry, the pantograph,
and geodesy, but his chief work was in relation to the quad-
rature of the hyperbola and to hyperbolic functions. 5

Contemporary with Landen and Hutton there was a writer
who did very much less but possibly gained somewhat more
in the estimation of many people, the accident of a single dis-
covery having given his name a place in the history of the
number theory. John Wilson was one of those men who did
one thing well in the field of mathematics and then failed to

ifiorn at Newcastle upon Tyne, August 14, 1737; died in London, January
27. 1823.

^Mathematical Tables, containing Common, Hyperbolic, and Logistic Loga-
rithms, with other Tables, and a large and original History of the Discoveries
and Writings relating to those Subjects, London, 1785.

8 Mathematical and Philosophical Dictionary, London, 2 vols., 1795, 1796,
with a second edition in 1815.

4 Born at Dysart, Fifeshire, September 23, 1768; died at Lauriston, near
Edinburgh, April 28, 1843. S. Giinther, William Wallace, tin Vorlaufer der
Lehre von den Hyperbelfunktionen, Prog., Ansbach, 1880.

B Giinther, ioc. cit., and the Trans, of the Royal Soc. of Edinburgh, VI, 269,
271, 302, etc.

6 Born at The How, Westmoreland, August 6, 1741 ; died at Kendal, October 18,
1703. See a note on his life by Cantor, Bibl. Math., Ill (3), 412.

EARLY NINETEENTH CENTURY 459

meet the expectations of his contemporaries. He entered Peter-
house, Cambridge, in 1759, and while still an undergraduate he
discovered that if p is a prime number, then i+(p i)! is a
multiple of p, a fact already known to Leibniz but not pub-
lished. The statement has generally been known as Wilson's
Theorem. 1 He was senior wrangler in 1761 and became a
Fellow of the Royal Society in 1782.

The Early Nineteenth Century. Of British writers born in
the 1 8th century but whose work was done in the century fol-
lowing, relatively few stand out as brilliant mathematicians.
Among those who contributed in some noteworthy way to the
progress of their science, one of the earliest was Robert Wood-
house, 2 who was educated at Caius College, Cambridge, became
Plumian professor in the university, and did much to replace
the calculus of fluxions by the differential calculus. He sought,
moreover, to put the latter subject on a firm scientific foundation,
and it is due in no small degree to his efforts that this was done.

Probably the name best known of all this group, in books
on elementary mathematics, is that of William George Horner. 3
Although not a man of great ability as a mathematician, he
succeeded in making for himself a name that is well known
to students of algebra. He was a teacher at Bath when he
came independently upon an ancient Chinese method of ap-
proximating the roots of a numerical equation. This method,
which had been practically forgotten in China, was made known
in a paper read before the Royal Society in 1819, and since that
time Horner's Method has become familiar in all parts of the
English-speaking world. 4

His contemporary, George Peacock, 5 was a very different
type of man. A student at Trinity College, Cambridge, he

the history of the theorem see Dickson, Hist. Th. Numb., I, 50.

2 Born at Norwich, April 28, 1773; died at Cambridge, December 23, 1827.

8 Born in 1786; died at Bath, September 22, 1837.

4 Republished in the Ladies' Diary for 1838 and again in revised form in
The Mathematician for 1843. His proof (1826) of Euler's Theorem (see
Volume II, Chapter I) should be mentioned as showing his ability in number
theory. As to the validity of the Chinese claim, see Volume II, Chapter VI.

5 Born at Denton, April 9, 1791 ; died at Ely, November 8, 1858.

460 GREAT BRITAIN

came to represent the solid, substantial mathematics of Eng-
land and to do much to improve its status. He was appointed
Lowndean professor of astronomy and geometry in 1836, and
three years later became dean of Ely cathedral, spending the
last twenty years of his life there. He was interested in the
movement to introduce the differential notation into the work
in the calculus, in the founding of the observatory at Cam-
bridge, and in the preparation of scholarly treatises on elemen-
tary mathematics. He was one of the prime movers in all
mathematical reforms in England during the first half of
the i Qth century, although contributing no original work of
particular value.

One of his contemporaries, and also a Trinity man, Charles
Babbage 1 was a worthy representative of the output of Cam-
bridge in this period. He became Lucasian professor (1828-
1839), assisted in founding the Astronomical Society (1820),
the British Association for the Advancement of Science
(1831), and the Statistical Society of London (1834), and did
much to introduce the differential notation into British mathe-
matics. He worked on an elaborate calculating machine, the
most noteworthy effort in this direction, from the point of view
of originality, since that of Pascal. 2

Perhaps the British mathematician of this period who
showed the greatest genius, or at any rate the greatest perse-
verance, was Peter Barlow. 3 Born of humble parents, he became
one of the leading writers of England on the theory of numbers, 4
professor of mathematics at the Woolwich Military Academy,
and a Fellow of the Royal Society (1823). His contributions
to the magnetic theory, the strength of materials, and optics
were also noteworthy, and his mathematical dictionary 5 is
still a valuable source of information. His tables of factors,

1 Born at Teignmouth, Devonshire, December 26, 1792; died in London.
October 18, 1871.

-His Calculating Engines, a work including much historical information,
was edited by his son, General H. P. Babbage, and published at London in 1889.

3 Born at Norwich, October 13, 1/76; died March i, 1862.

4 An Elementary Investigation of the Theory of Numbers, London, 1811.

*A New Mathematical and Philosophical Dictionary, London, 1814.

WILLIAM ROWAN HAMILTON 461

reciprocals, powers, roots, hyperbolic logarithms, and primes
should also be mentioned as the best of the earlier publications
of the kind, being still looked upon as standards. 1

Sir William Rowan Hamilton. Of those whose work added
to the prestige of Great Britain and Ireland in the mid-Victorian
period, only a few need be mentioned in a work devoted
chiefly to elementary mathematics. Of these, one of the best
known is Sir William Rowan Hamilton." He was one of the
great mathematical products of Ireland in the igth century.
Although descended from Scotch stock, he was proud to pro-
claim himself an Irishman. He was one of the infant prodigies
that occasionally arise in the history of mathematics, usually,
as we have seen, with disappointing results. At the age of
three he read English fluently and was somewhat advanced in
arithmetic; at five he could read Latin, Greek, and Hebrew;
at eight he could also write Latin and read French and Italian ;
at ten he was studying Arabic ard Sanskrit ; at twelve he had a
working knowledge of all these languages, together with Syriac,
Persian, Hindustani, and Malay, and was contesting with the
American prodigy, Zerah Colburn, in long mental calculations ;
and at thirteen he had written an algebra, which, fortunately
no doubt, was not offered for publication. At fourteen he was
able to write Persian, and at sixteen he had made known an
error in one of the demonstrations in the Mecaniqitc Celeste
of Laplace. At Trinity College, Dublin, he continued his bril-
liant record, receiving his appointment to the professorship of
astronomy while still an undergraduate. Hamilton was knighted
in 1835. l n T &43 h e made his great discovery of quaternions,
but his first work on the subject was not published until 1853.
His second work appeared posthumously. ' If the theory has
not led to the results anticipated by Hamilton and his friends,

Mathematical Tables, London, 1814. De Morgan publibhcd an edi-
tion in 1856.

2 Born at Dublin, August 3, 1805; died at Dublin, September 2, 1865. For
biography, see A. Macfarlane, Ten Krithh Mathematicians, p. 34 (New York,
iqi6) ; hereafter referred to as Macfarlane, Ten Brit. Math.

3 Lectures on Quaternions, Dublin, 1853; Elements of Quaternions, London,
1866 (posthumous).

462 GREAT BRITAIN

both it and the Ausdehnungslehre of Grassmann have greatly
extended the vision of both mathematicians and physicists.

Salmon, De Morgan, and Boole. Among the prominent math-
ematicians produced by Ireland in the igth century there
should also be mentioned George Salmon, 1 whose works on
conies, higher plane curves, analytic geometry of three dimen-
sions, and higher algebra are recognized as standard authorities.

Less well known than Hamilton for any single achievement,
less well known than Salmon for his thoroughness in science,
familiar to a much larger circle of readers because of his wide
range of interest and his skill in popularizing the science,
was that eccentric but brilliant teacher, Augustus De Morgan."
Educated at Trinity College, Cambridge, he became professor
of mathematics in London in 1828, displaying unusual gifts as
a teacher and scattering his energies recklessly. His Trigonom-
etry and Double Algebra (1849) contained certain features of
quaternions, but he did not follow this or any other theory to
the conclusion that seemed within his reach. He wrote various
textbooks, each a mine of information for the teacher and
entirely hopeless for the pupil. His contributions to the theory
of probability still rank as among the best in English, and the
same may be said for his contributions to logic. He devoted
considerable attention to the history of mathematics, but his
articles are not only eccentric but unreliable. His best work in
this line is to be found in Smith's Dictionary of Greek and
Roman Biography (London, 1862-1864), the Penny Cyclo-
pedia (London, 1833-1843), the Companion to the Almanac
for various years, and his Arithmetical Books (London, 1847).
His Budget of Paradoxes, edited by Mrs. De Morgan after
his death, 3 is an interesting satire on circle squarers and their
kind. Had he been able to confine himself to one line, he
might have been a much greater though a less interesting man.

If poverty, delayed education, and general lack of early
advantages were a bar to progress in abstract science, George

1 Born at Cork, September 25, 1819; died at Dublin, January 22, 1904.
2 Born at Madura, India, June 27, 1806; died in London, March 18, 1871.
3 London, 1872; 2d ed., by D. E. Smith, Chicago, 1915.

SYLVESTER 463

Boole 1 would never have been professor of mathematics at
Queen's College, Dublin, the theory of invariants and covari-
ants would not have been what it is today, and the mathemati-
cal theory of logic might not have reached its present position.
Boole's circumstances did not permit of his beginning any
serious study of mathematics until he was twenty, although
he picked up by himself some knowledge of Latin and Greek
and was able to do a little teaching to help him on with his
scholastic work. When he was twenty years old he decided
that he was able to open a school of his own, and this he did,
using the small income to assist his aged parents and to buy
books for himself. Thus, without any university training, he
advanced, and in 1849 was appointed to the chair of mathe-
matics in Queen's College. His Mathematical Analysis of
Thought (1847), Laws of Thought (1854), Differential Equa-
tions (1859), and Finite Differences (1860) are still looked
upon as standard authorities.

Sylvester. There were two men, companions for many years
but, like many companions, of very different character, who
stand out with special prominence at this period. The first of
these was James Joseph Sylvester. 2 Educated at St. John's
College, Cambridge, and one of the most gifted members of his
class, he was not allowed to take a degree because of his
Jewish faith, and for the same reason he was barred from a
fellowship. For the degree he went to Dublin, but after the
abolition of the theological tests in 1872 the University of
Cambridge awarded him both the bachelor's and master's
degrees. Soon after leaving the university he was appointed
(1837) professor of natural philosophy in University College,
London, and two years later (1839) was elected a Fellow of
the Royal Society. Opportunities for advancement not being
promising in England, he accepted an appointment (1841) as

iBorn at Lincoln, November 2, 1815; died at Cork, December 8, 1864. See
Macfarlane, Ten Brit. Math., p. 52.

2 Born in London, September 3, 1814; died in London, March 15, 1897.
See F. Franklin, in Bulletin of the Amer. Math. Soc., Ill, 299; M. Noether
Math. Annalen* L, 133 ; P. A. MacMahon, Nature, March 2 j, 1897.

464

GREAT BRITAIN

professor of mathematics in the University of Virginia. His
election took place on July 3, 1841, and he began his work in
the autumn. He made a failure of his teaching, had a serious
personal encounter with a student who is said to have attacked
him, and hurriedly left the university in the following March.

The official records of
March 22, 1842, contain
the following resolution :

Resolved, That the res-
ignation of Mr. Sylvester
be accepted, to take effect
from and after the 2Qth
day of the present month
or at any earlier period
that he may elect; that
a copy of this resolution
be forthwith communicated
to him by the Secretary,
and that he be informed
that in accepting his res-
ignation the Board has not
deemed it necessary to in-
vestigate the merits of the
matter in difference be-
tween himself and the
student Ballard, and does
not mean to impute to
Mr. Sylvester any blame
in the matter.

JAMES JOSEPH SYLVESTER

His influence was strongly exerted in estab-
lishing university research in mathematics in
the United States

Evidently, therefore, the university authorities were con-
vinced that Sylvester's further relations with the faculty were
not desirable.

He seems to have returned to London about three years
after leaving Virginia. Here he took up actuarial work, became
a student in the Inner Temple (1846), and was called to the
bar (1850). He became professor of mathematics at the Mili-
tary Academy in Woolwich in 1855 and remained there until

CAYLEY

465

his forced retirement on account of age (1869). In 1877 he
was called to Johns Hopkins University and did more than
any other man of his time to establish graduate work in mathe-
matics in America. Among his other contributions to the ad-
vance of the science in this country was his founding of the
American Journal of Mathematics. In 1883 he was elected to
succeed H. J. S. Smith in the Savilian professorship at Oxford,
but his lectures were not popu-
lar and in 1892 he gave place
to a deputy professor and spent
his last years in London.

Sylvester was often looked
upon as unsystematic, domi-
neering, impractical, conceited,
and unhappy, but those who
knew him well have testified to
his genial nature and his enthu-
siasm in his work with students.
His contributions show that he
was a genius in mathematical
investigation, his chief line of
interest being in higher alge-
bra, including the study of
invariants. ARTHUR CAYLEY

Cayley. The Companion of From a photograph made in 1870

Sylvester already referred to

was Arthur Cayley. 1 He was the son of an English merchant
who had settled in Petrograd and who looked forward to his
son's taking part in the business which he had established. Soon
after young Cayley, at the age of fourteen, was sent to King's
College School, London, it was found that he showed such
ability in mathematics that his father decided that he should
proceed to Cambridge. He accordingly entered Trinity Col-
lege at the age of seventeen, and his progress was such that he
graduated with the highest honors, secured a fellowship, and

1 Born at Richmond, Surrey, August 16, 1821; died at Cambridge, Janu-
ary 26, 1895. For biography, see Macfarlane, Ten Brit, Math., p, 64.

466 GREAT BRITAIN

devoted himself to the preparation of a number of important
memoirs. Forced to find some remunerative employment, he
then /took up the law, and for fourteen years made a specialty
of conveyancing, devoting his leisure to the preparation of
further scientific memoirs. Sylvester was at this time an actu-
ary in London, and the two were close friends and were in
frequent consultation. About 1860 the Sadlerian professorship
of pure mathematics was established at Cambridge, 1 and Cayley

u^.

UI,r'

FROM AN AUTOGRAPH LETTER FROM CAYLEY TO SYLVESTER

The Miller referred to is W. J. C. Miller, for many years the editor of the
mathematical columns in the Educational Times, London

was the first to occupy the chair (1863). Although he wrote
but one extensive work, the Treatise on Elliptic Functions
(Cambridge, 1876), he contributed a large number of impor-
tant memoirs to various scientific publications. In 1889 the
Cambridge University Press began the publication of his
papers, nearly a thousand in number, in collected form. Seven
volumes appeared under his own editorship, the remaining six
volumes being published under the supervision of Professor
Forsyth.

Cayley's papers cover a very wide range, but it may be
said that his chief interest was in the fields of elliptic functions,
the theory of invariants, and analytic geometry. Of these the
theory of invariants was the one which he did most to advance.

i Through funds bequeathed originally by Lady Sadler to found certain
lectureships (1710).

CLIFFORD 467

America is indebted to him for his course of lectures at
Johns Hopkins University in 1882 on the Abelian and theta
functions, whereby he again cooperated with Sylvester, who
was then helping to place mathematics in this country on a
university basis. Well might Sylvester say of him that what-
ever he touched he embellished. 1

H. J. S. Smith. Less well known because working in a nar-
rower field, but in the same class of genius as Sylvester and
Cayley, Henry John Stephen Smith 2 began his education under
the care of his mother, a woman. of unusual ability. In 1841
he went to Rugby, where he came under the influence of
Dr. Arnold. From Rugby he went (1844) to Balliol College,
Oxford, later spending some time at the Sorbonne and the Col-
lege de France. In 1860 he became Savilian professor of geom-
etry at Oxford and a year later was made a Fellow of the Royal
Society. His time was so taken up in public duties of various
kinds that he did not achieve the success in mathematics of
which he was capable. His interest in this science was chiefly
in the direction of the theory of numbers and the study of
binary and ternary quadratic forms. It was probably in rela-
tion to a problem in this theory that he is said to have re-
marked, "It is the peculiar beauty of this method, gentlemen,
and one which endears it to the really scientific mind, that
under no circumstances can it be of the smallest possible util-
ity. " His various writings were collected by Dr. Glaisher and
published in i8p4. 3

Clifford. Among the most promising mathematicians pro-
duced in England in the iQth century, but one whose early
death prevented the maturing of his genius, was William King-
don Clifford. 4 He was educated at King's College, London

^'Cayley, of whom it may so truly be said, whether the matter he takes
in hand be great or small, 'nihil tetigit quod non ornavit.'" Phil. Trans., XVII
(1864), 605.

2 Born at Dublin, November 2, 1826 ; died at Oxford, February 9, 1883. For
biography see Macfarlane, Ten Brit. Math., p. 92.

3 The Collected Mathematical Papers of H. J. S. Smith, 2 vols., Oxford, 1894.
4 Born at Exeter, May 4, 1845 ; died in Madeira, March 3, 1879.

468

GREAT BRITAIN

(1860-1863) and Trinity College, Cambridge. At the univer-
sity his mathematical genius was at once recognized, and in
1868 he was elected to a fellowship at Trinity. In 1871 he
was made professor of applied mathematics at University Col-
lege, London, and three years later was elected a Fellow of the
Royal Society. He was among the first to protest against the

analytic bias of the Cambridge
mathematicians, and he assisted
in introducing into England the
graphic methods of Mobius and
other German writers. His most
important works were in relation
to Riemann's surfaces, biqua-
ternions, and the classification of
loci. His Common Sense of the
Exact Sciences is a classic on
the foundations of mathematics,
and suggests, as other works
(including those of Copernicus
and Kepler) had already done,
the idea of relativity in all phys-
ical measurements. His Mathe-
matical Papers, edited by R.
Tucker, appeared in I882. 1

Todhunter. Of all the English
mathematicians of this period

the one most widely known to elementary students and to
teachers is Isaac Todhunter. 2 His name is familiar to everyone
who has studied the history of textbook making in the igth
century, although he was much more than a textbook writer.
As a young man he attended evening classes at the University
of London, where he came under De Morgan's influence, receiv-
ing the B.A. degree in 1842 and the M.A. degree two years
later. He then entered St. John's College, Cambridge, and

1 These contain (pp. xv, xxxiii) a brief biography.

2 Born at Rye, Sussex, November 23, 1820; died at Cambridge. March I,
1884. See Macfarlane, Ten Brit. Math., p. 134.

WILLIAM KINGDON CLIFFORD

From a photograph made shortly
before his death

OTHER BRITISH MATHEMATICIANS 469

in 1848 took his second B.A. degree. He remained at Cam-
bridge until 1864, beginning the textbook writing which made
him financially independent and resulted in a series of works
that exercised great influence on education in all the English-
speaking world. He also wrote (1865) a work on the history
of probability and one (1873) on the history of the mathe-
matical theories of attraction, each a classic in its line. He was
a good mathematician but not a great one, an excellent linguist,
and a man who stood for sound scholarship. As is commonly
the case with men in his line of work, numerous stories are told
of him, one being that he used to remark that he knew two
tunes, the first of which was "God save the Queen" and the
second wasn't, and that he recognized the former by the fact
that people stood up when it was sung.

Other British Mathematicians. Among the other British
mathematicians of prominence in this period it is possible
at this time to mention only a few of those whose names should
be familiar to the general student of mathematics. Others will
be found in the second volume of this work. Dr. George Berke-
ley (1684-1753) is known in the history of the calculus for his
work, The Analyst (London, 1734), in which he attacked the
foundations of the new science ; Edward Waring ( 1734-1798)
was interested in the theory of numbers, and a theorem relating
to powers is known by his name; Sir James Ivory (1765-
1842), with mathematics as an avocation, did much to advance
the progress of analytic methods in England and contributed
to the theory of attraction; James Booth (1806-1878) wrote
on modern geometry; Sir John Frederick William Her-
schel (1792-1871) contributed to the study of analysis but
finally followed in his father's steps and devoted himself to
astronomy; James MacCullagh (1809-1846) contributed to
the theory of quadric surfaces ; Thomas Penyngton Kirkman
(1806-1895) made the attempt to extend the theory of quater-
nions and was interested in the subject of " Analysis situs";
George Biddel Airy (1801-1892), Astronomer Royal of Eng-
land, contributed to the lunar and planetary theory; John
Couch Adams (1819-1892), independently of the French

470 FRANCE

astronomer Lcverrier, determined mathematically the position of
the planet Neptune ; Sir George Howard Darwin (1845-1912),
son of Charles Darwin the naturalist, contributed to the theory
of three bodies; Sir Robert Stawell Ball (1840-1913), Astron-
omer Royal of Ireland and later Lowndean Professor of Astron-
omy and Geometry at Cambridge, wrote on the theory of
screws (1876); Peter Guthrie Tait (1831-1901), sometime
professor at Belfast and later at Edinburgh, is well known for
his work in quaternions and physics; Lord Kelvin (William
Thompson, 1824-1907) contributed extensively to the applica-
tion of mathematics to physical problems ; James Clerk Max-
well (1831-1879) is especially known for his application of
mathematics to the study of electricity; and Lord Rayleigh
(John William Strutt, 1842-1919) is similarly known with
respect to mathematics and the study of vibrations. 1

3. FRANCE

Nature of the Work. France took the lead again in the i8th
century, as she did in the first half of the century preceding.
She may have had no more brilliant intellects than England,
but she found in the differential and integral calculus a set of
tools that she could use more deftly than the British mathema-
ticians could use the heavy machinery of the calculus of fluxions.
It was perhaps as well that the two nations should experi-
ment on different lines of approach, and it is evident that each
cultivated its own peculiar power of attack by so doing, but
the total results secured in France show for more than those
produced in Great Britain.

Early Eighteenth Century Writers. There will first be men-
tioned a few names of those who, although born in the 1 7th cen-
tury, completed their work after its close. Among the first of
these writers was Pierre Varignon, 2 professor of mathematics at
the College Mazarin (1688) and later at the College Royal, and

a For a list of biographies of mathematicians dying between 1881 and 1900,
see G. Enestrom, Bibl. Math., II (3), 326, covering all European countries.
2 Born at Caen, 1654; died in Paris, December 22, 1722.

EARLY WRITERS OF THE PERIOD 471

a member of the Academic des Sciences at Paris. Although
intended for the Church, he accidentally came across a copy of
Euclid, and thus was led, as so many others have been, to the
study of mathematics. He then read Descartes's Geometrie, and
thereafter devoted himself to the mathematical sciences, with
special emphasis upon physical problems. He was one of the
first of the French scholars to recognize the value of the new
calculus. His chief contributions were to the science of me-
chanics, although he wrote upon pure mathematics as well. 1

A little younger than Varignon, although dying before him,
Pierre-Remond de Montmort" rose to a position of some promi-
nence. He was born to fortune and thus had ample means to
enable him to follow his tastes in the study of law and phi-
losophy. He was made a canon of Notre-Dame at a time
when piety was not the chief qualification, but finally married
and devoted the rest of his life to travel, to Paris, and to
mathematics. He was chiefly interested in the doctrine of
chance, 3 a subject which brought him into cordial relations
with De Moivre and with Jean and Nicolas Bernoulli. He
also wrote on infinite series ' and summed to n terms the series

;/(;/ I) A ;/(;/ l)(;/ 2) ,

S = na 4- v -- ; - A # 4- - v ~- ~ kca H .

t \ i \

-* * 3 '

Antoine Parent, 5 a private teacher of mathematics in Paris
and (1699) a member of the Academic des Sciences, was an-
other of the minor writers who helped to advance his subject.
His interest was chiefly in mechanics and physics, although
he also wrote on arithmetic, the cycloid, geometry, and per-
spective. He is known in the history of mathematics for his

1 Eclair cissements sur I'analyse des infiniment petits, Paris, 1725 (post-
humous) ; Elements de matMmatiques, Paris, 1731 (posthumous) ; Maniere de
tronver nne infinite de portions de cercle toutes quarrables moyennant la seule
geometrie d'Euclide, Paris, 1703 ; and other works.

2 Or Monmort. Born in Paris, October 27, 1678; died in Paris, October 7,
1719. His family name was Remond, the "de Montmort" being assumed from
his estates.

*Essai d'anlyse sur les jeux de hazard* Paris, 1708; 2d ed., 1714.

4 "De seriebus infinitis tractatus," in the Philosophical Transactions for 1717.

Born in Paris, September 16, 1666; died in Paris, September 26, 1716.

472 FRANCE

work in analytic geometry of three dimensions. 1 His most im-
portant contributions were published in his collected works in

1 70S- 2

Among Parent's contemporaries, Joseph Saurin, 3 a priest,
wrote on the determination of tangents at multiple points of an
algebraic curve, on the curve of least descent, and on various
other geometric questions.

There was also Thomas-Fantel de Lagny, 4 who gave up the
law for the purpose of devoting himself to mathematics. He
wrote on new methods of extracting roots (1692), the cubature
of the sphere (1702), binary arithmetic (1703), and methods
of solving problems. 5 The story is told that Maupertuis, called
to his deathbed and finding him in a comatose state, asked him
suddenly for the square of 12 ; whereupon De Lagny started
up, gave the answer, and at once passed away.

Less of a mathematician but contributing worthily to the
science, Amedee Frangois Frezier, a French infantry officer
(1702-1707) and later an engineer in South America and San
Domingo, by means of his works on stereometry as applied
to stone cutting and architecture 7 laid part of the foundation
for the theory of descriptive geometry.

Among those who formed a brilliant group in Paris at this
time was one of the youthful prodigies that, as we have seen,
arise from time to time, Frangois Nicole/ He was a boy of
unusual promise, having shown his genius in geometry by rec-
tifying the cycloid at the age of nineteen. His interest in the

1 In a paper read before the Academic in 1700. See also Chasles, Aperqu
hhtoriquc, p. 138.

2 Reclierches de mathematiqucs et de physique, Paris, 1705 ; rev. ed., 3 vols.,
Paris, 1713.

3 Born at Courthezon, Vaucluse, September i, 1655; died in Paris, Decem-
ber 2Q, 1737.

4 Born at Lyons, November 7, 1660; died in Paris, April 12, 1734.

5 Analyse Generate, ou Methodes Nouvelies pour resoudre les Problemes de
tons les Genres et de tons les degrez a I'Infini, Paris, 1733.

6 Born at Chambery, 1682; died at Brest, October 16, 1773.

7 La theorie et la pratique de la coupe des pierres et des boh, ou Traite de
Stereotomie, Strasburg, 1738; 2d ed., 3 vols., Paris, 1754, 1768, 1769; Elements
de Stereotomie, a I'usage de V architect lire, Paris, 1750-1760.

8 Born in Paris, December 23, 1683; died in Paris, January 18, 1758.

MAUPERTUIS

473

study naturally led to a consideration of roulettes in general, 1
a subject in which he showed great insight. He also wrote on
the calculus of finite differences, 2 lines of the third order
(1729), probability (1730), conies (I73 1 ); cubics (1738,
1741, 1743), and the trisectio^
cases of unusually early
development, however,
his work was not of the
highest order.

Maupertuis. The work
of Pierre Louis Moreau
de Maupertuis 3 was of
a more stable kind. In
his younger days (1718)
he was a captain of dra-
goons in the French
army, but he later retired
to private life and de-
voted himself to the study
of mathematics. He was
made a member of the
Academic des Sciences
in 1731, directed the
measurement of a me-
ridian degree in Lapland
in 1736, became presi-
dent of the physical class
in the Berlin Academy ^745-1753), basked in the sunshine of
the favor of Frederick tie Great for the usual brief period,
learned that he could "climb, but heights are cold," and after
falling from favor spent the last six years of his life in his
native country. His chief work was in astronomy and geodesy,

iMethode generate pour determiner la nature des courbes jormees par le
roulement de toutes sortes de courbes sur une autre courbe quelconque, Mem.,
Paris, 1707; Maniere de determiner la nature des roulettes jormees sur la super-
fide convexe d'une sphere , Paris, 1708, 17^2.

*Traite du calcul des differences finies, Paris, 1717, 1723, 172^-
s Born at St. Malo, July 17, 1698; died at Basel, July 27,

PIERRE LOUIS MOREAU DE MAUPERTUIS

For some years a favorite at the court of

Frederick the Great, interested chiefly in

geodesy

474 FRANCE

but he also wrote on maxima and minima (1724), quadrature
problems (1727), curves in general (1727-1730), and various
physical questions. 1 He taught mathematics to his friend the
Marquise du Chatelet and, considering her friendship for Vol-
taire, deserved something better than the harsh treatment which
the latter gave him in his Diatribe du Dr. Akakia, written with

cs/f**4f

AUTOGRAPH LETTER OF MAUPERTUIS

This loiter was written to Frederick the Great in 1750, while Maupertuis was
still president of the physical class in the Berlin Academy

the desire to defend a learned but indiscreet Swiss mathemati-
cian, Samuel Koenig (died 1757), who had accused Maupertuis
of plagiarism. One of the biographers of Voltaire speaks of
Maupertuis as "the pompous and touchy mathematician," and
the phrase is probably appropriate.

Minor Writers. Alexis Fontaine des Bertins 2 was more prom-
ising in his youth but less successful in his later years than his
contemporary, Maupertuis. He was a man of means, was

*CEuvres de Mr. de Maupertuis, 4 vols., Paris, 1752 ; Lyons, 1768.

-Born at Bourg-Argental, Loire, c. 1705; died at Cuiseaux, August 21, 1771.

THE CLAIRAUT FAMILY 475

therefore able to devote his life uninterruptedly to study, and
became a member of the Academic des Sciences in 1733, when
only about twenty-eight years of age. His early promise was
not fulfilled, however, his efforts not being directed in lines of
probable success. He wrote on tautochronous curves (1734
and 1768) and differential equations, and proposed various
problems in geometry and astronomy. He suggested the com-
mon notation of partial derivatives of a function of several
variables.

Among his contemporaries, and a man of considerable
influence in the scientific circle of Paris, there should be
mentioned Jean Paul de Gua de Malves. 1 He belonged to
a family that had been impoverished by John Law's Mis-
sissippi scheme, and, seeing no career open to him, he
entered the Church and secured a benefice which enabled
him to live comfortably and to devote his life to study. He
wrote a work on the Cartesian analysis 2 which gave him admis-
sion to the Academic des Sciences (1740) and a professorship
(1743) of philosophy in the College de France. He seems to
have suggested the idea of the Encyclopedic which was finally
carried out by Diderot, d'Alembert, and Voltaire. He per-
fected the proof of Descartes's Law of Signs (1741) and wrote
on geometry and trigonometry.

Clairaut Family. One of several noteworthy instances in the
history of mathematics, showing the influence of heredity or
early environment, is seen in the case of the Clairaut family.
Jean Baptiste Clairaut 3 was a teacher of mathematics in Paris
about the middle of the i8th century. He was a correspondent
of the Berlin Academy and published three memoirs on geom-
etry in the Miscellanea (1734, 1737, *743)- One of his sons
was Alexis Claude Clairaut, 4 the most prominent member of
the family, an infant prodigy who read PHospitaPs Analyse des

iBorn at Carcassonne, c. 1712 ; died in Paris, June 2, 1786.

2 Usage de I'analyse de Descartes pour decouvrir, sans le secours du calcul
different iel, les proprietes . . . des lignes giomttriques de tous les ordres t Paris,
1740.

3 Died soon after 1765. The name is also spelled Clairault.
4 Born in Paris, May 7, 1713; died in Paris, May 17, 1765.

476

FRANCE

infiniment petits and his Traite des sections coniques at the age
of ten, presented a paper on geometry before the Academic des
Sciences when he was thirteen, and was admitted to member-
ship in the Academic and published a work on curves of double
curvature 1 when he was only eighteen. His solutions of the

&Jl^pt***S*^^

^-spLem^ fcve<i^*<Jv &**

AUTOGRAPH OF ALEXIS CLAUDE CLAIRAUT

Written about twenty years before his death and while he was working on

his algebra

problem of tangents drawn to such curves and of the quadra-
ture of the curves themselves arc still found in current treatises.
Clairaut was only twenty-three when he was made a mem-
ber of the commission which went to Lapland to measure the
length of a degree. He now began to devote most of his
attention to problems of celestial mechanics, but still found
time to write on geometry (1741), algebra (1746), algebraic

iRecherches swr les courbes a double courbure, Paris, 1731.

VOLTAIRE AND DU CHATELET 477

curves on a cone (1732), maxima and minima (1733), the
calculus (1739), and similar topics, most of his contributions
being in the form of memoirs presented to the Academic. He
demonstrated Newton's theorem that all curves of the third
order are projections of one of five parabolas. Possibly it was
his interest in Newton that first brought him under the spell
of the Marquise du Chatelet in the months just preceding her
death, when she was hastening to complete her translation of
the Principia.

Arago, in his eulogy of Laplace, remarked :

Five geometers Clairaut, Euler, d'Alembert, Lagrange, and La-
place shared among them the universe of which Newton had dis-
closed the existence. They explored it in all directions, penetrated
into regions which had been thought inaccessible, pointed out there
a multitude of phenomena which observation had not yet detected,
and finally, and herein lies their imperishable glory, they brought
within the domain of a single principle, a single law, all that is
most refined and mysterious in the movements of the celestial
bodies.

Alexis had a brother 1 who died when he was sixteen, but
who at the age of fourteen had read a memoir on geometry
before the Academic des Sciences, and who published a work
on geometry 2 when he was only fifteen.

Voltaire and the Marquise du Chatelet. The world does not
often connect the name of Voltaire with mathematics, and when
it connects that of the Marquise du Chatelet with the science, it
is largely by courtesy. Each, however, did something to make
the Newtonian theory known, and each absorbed enough mathe-
matics to make the labor fairly serious.

Frangois Marie Arouet, n known tc the world as Voltaire and
as the foremost leader of the i8th century in the contest for
human liberty, 4 was interested in mathematics chiefly because
he was interested in all things English, was interested in Newton,

iBorn in Paris, 1716; died in Paris, 1732.

z Traite des quadratures circulates et hyperboliques, Paris, 1731.

3 Born in Paris, November 21, 1694; died in Paris, May 30, 1778.

4 Among his monographs is the Essai sur Us ProbabiliUs en fait de Justice.

478

FRANCE

was interested in getting out a work on Newton's philoso-
phy, 1 and was interested in Emilie, Marquise du Chatelet.-
Daughter of the Baron de Breteuil, the marquise married at

nineteen, and turned her
brilliant mind to Euclid,
to Newton, to the liter-
ary classics of Greece
and Rome, to Locke, and
to Voltaire. She had
studied mathematics
under Maupertuis and
Koenig, read Newton
and understood him,
at least in part, and in
due time translated the
Principia, 3 completing it
a few days before her
death. Frederick the
Great, who loved an epi-
gram far more than he

EMILIE, MARQUISE DU CHATELET loved the COUrtesieS of

After a lithograph from a contemporary draw- Iife ? suggested this epi-

ing by N. H. Jacob taph : "Here lies one

who lost her life in giv-
ing birth to an unfortunate child and to a treatise on philos-
ophy." Madame du Chatelet also wrote on physics, but at
best she was only an amatrice in science. 4

la philosophic de Neuton, Amsterdam, 1738.

2 Gabrielle Emilie Le Tonnelier de Breteuil; born in Paris, December 17, 1706;
died at Commercy, September 10, 1740.

8 It was published posthumously at Paris in 1759. There is a bibliography of
her works in A. Rebiere, Les femmes dans la science, 2d ed., p. 6$ (Paris, i8Q7).

4 Voltaire, in one of his many epigrams about her, wrote:

"Son esprit est tres philosophic,

Mais son coeur aime les pompons."

In his work on Newton he addresses a poem to her, beginning:
"Tu m'appelles k toi, vaste & puissant G6nie,
Minerve de la France, immortelle Emilie,
Disciple de Neuton, & de la V6rit6."

D'ALEMBERT

479

B'Alembert There are certain names in the history of
mathematics to which there attaches a special human inter-
est apart from the mere recital of a list of discoveries. One
of these is d'Alembert. 1

On the night of No-
vember 16, 1717, a gen-
darme, while making his
rounds in Paris, found
near the church of Saint-
Jean le Rond a newly
born infant who had
been abandoned to the
fate of winter, and had
him hurriedly christened
with the name of his first
resting place, Jean Bap-
tiste le Rond. Foster
parents were found and
Jean grew up, known
but unrecognized by his
mother, pitied and some-
what helped by his
father, and soon showed "
remarkable intellectual JEAN BAPTISTE LE ROND D'ALEMBERT

powers that spoke for From an engraving made by P. Maleuvre
intellectual parentage. in I 77S after a drawing made by A. Pujos

His mother, Madame de in I774

Tencin, sister of a cardinal, has been described by one of
d'Alembert's biographers as "'small, keen, alert, with a little
sharp face like a bird's, brilliantly eloquent, bold, subtle, tire-
less, a great minister of intrigue, and insatiably ambitious."
His father, General Destouches, was a man of large heart, and
at his death in 1726 left enough to provide for the boy's educa-
tion. When Jean was eighteen (1735) he took his bachelor's
degree and soon, for reasons unknown, adopted the name of
d' Alembert. He prepared for the bar, then took up medicine, and

iBorn in Paris, November 16, 1717; died in Paris, October 29, 1783.

/U*^ *>^*" *": ^

M ";.%

.- uw '.' "'. "V,,

480 FRANCE

finally devoted his life to mathematics. Friend of Voltaire,
collaborator on the Encyclopedic, admirer of Madame du Def-
fand, and lover of her companion Mademoiselle Julie de
Lespinasse, he knew those in France who were best worth
knowing and experienced all the joys and sorrows that Paris
affords. One of his biographers says :

In himself d'Alembert was always rather a great intelligence than a
great character. To the magnificence of the one he owed all that has
made him immortal, and to the weakness of the other the sorrows
and the failures of his life. For it is by character and not by intellect
the world is won. 1

D'Alembert wrote upon mathematics in general, 2 the
calculus and its applications, 3 the theory of differential equa-
tions/ and dynamics/'

Minor Writers. Among the minor writers of the middle of
the 1 8th century one of the best known is Johann Heinrich
Lambert. 6 Born in humble surroundings, leading a roving life,
acting as bookkeeper, secretary, private tutor, and architect,
and living in Germany, Holland, France, Italy, and Switzer-
land, he was, in spite of such an unsettled life, a voluminous
writer, his fields of interest being as varied as his occupations
and his places of abode. He wrote on perspective, light, astron-
omy, logarithms, pyrometry, transcendent quantities, theory of
equations, the slide rule, psychology, ballistics, photometry, and
a variety of other subjects, most of his efforts displaying re-
spectable mediocrity, all save one, hyperbolic trigonometry,
and this gave him an enduring place in history. 7

1 S. G. Tallentyre, The Friends of Voltaire, chap, i, London, 1907.

-Opuscules mathematiques, 8 vols., Paris, 17611708.

3 In the memoirs of the Berlin Academy, 1746 and other dates.

4 Cantor, Geschichte, II, chap. 118. r 'J. Bertrand, D'Alembert, Paris, iSSqT

Born at Mulhousc, Alsace, August 26, 1728; died at Berlin, September 2=5,
1777. Mulhou?e was then Swiss territory, so that he may also properly be
ranked among Swiss scholars.

7 For bibliography see Engel and Stackel, Die The one der Parallellinien*
p. 151 (Leipzig, 1805). Consult also F. Rudio, Archimedes, Huygens, Lambert,
Legendre, Leipzig, 1892; D. Huber, Johann Heinrich Lambert, Basel, 1829;
J. Lepsius, Johann Heinrich Lambert, Munich, 1881 ; F. Schur, Johann Hem-
rich Lambert als Geometer, Karlsruhe, 1905.

MINOR WRITERS

481

Alexandre Theophile Vandermonde, 1 member (1771) of the
Academic des Sciences at Paris and director (1782) of the
Conservatoire des Arts et Metiers, was another of the rela-
tively minor writers of this period. He contributed to the
theory of equations through two memoirs (1771, 1772), and
to the general theory of
determinants.

Etienne Bezout 2 was also
one of the writers of this
period on the theory of
equations. He was an ex-
aminer for the navy and is
known for several memoirs
and textbooks. He was
among the first to recognize
the value of determinants.
His method of elimination
by the aid of symmetric
functions (1764 and 1779)
is well known to students in
the theory of equations.

During the Reign of Ter-
ror the revolutionists spared
most of those whose math-
ematical genius is now
recognized, but they did so
reluctantly in the case of
Marie-Jean-Antoine-Nicolas Caritat, Marquis de Condorcet. 3
Brought up under the Jesuits, he admired their learning
and hated their doctrines. He was admitted to the Aca-
demie des Sciences when only twenty-six (1769) and at
thirty became its secretary. At an age when all his family

1 Born in Paris, February 28, 1735; died in Paris, January i, 1796. On his
given name see Zeitschrift (HI. Abt.), XLI, 83.

2 Born at Nemours, March 31, 1730; died in Paris, September 27, 1783.

3 Born at Ribemont, near Saint-Quentin, September 17, 1743; died near
Bourg-la-Reine, in the vicinity of Paris, March 29, 1794. See Arago's Bio-
graphic, read before the Academic des Sciences, December 28, 1841.

ANT01NE-NICOLAS CARITAT
MARQUIS DE CONDORCET

After an engraving from a drawing
from life

482 FRANCE

was demanding that he should be a captain of cavalry
he was making for himself a name by his essay on the inte-
gral calculus and by his work on the problem of the three
bodies. 1 He then took up the theory of differential equations,
wrote extensively on the calculus, applied himself to the study
of probability, wrote various eulogies on deceased academicians
which are still read as classics in French literature, and lived the
life of a scholar and, in Voltaire's words, of u the man of the
old chivalry and the old virtue." D'Alembert spoke of him as
a volcano covered with snow ; in other words, he was an intel-
lectual aristocrat, and that was enough to condemn him in the
days of 1 794. " If I have one night before me," he had said, " I
fear no man ; but I will not be taken to Paris." When the jailer
to whom the gendarme had taken him for the night opened the
door of his cell, Condorcet was dead, with an empty poison ring
by his bed. He had kept the faith with himself.

Lagrange. Joseph Louis, Comte Lagrange, 2 was Tourangean
by descent, Italian by birth, German by adoption, and Parisian
by choice. He began his teaching as professor of mathematics
in the artillery school at Turin (1755) when only nineteen
years of age, succeeded Euler (1766) as mathematical director
in the Berlin Academy, and was called to Paris (1787), where
he became a member of the Academic des Sciences and, some-
what later (1795), was professor of mathematics at the newly
founded Ecole Normale and (1797) at the Ecole Poly technique.
Under Napoleon he was made a senator and a count, and was
awarded other honors appropriate to his genius.

Lagrange was not one of the infant prodigies in mathe-
matics. Indeed, it is said that he showed no interest in the

i-Essai sur le calcul integral, Paris, 1765; Analyse de la solution du probleme
des trois corps, Paris, 1768, the memoir which first called attention to his powers,

2 Born at Turin, January 25, 1736; died in Paris, April 10, 1813. As an
Italian by birth, the name might be given as Giuseppe Luigi; but the family
was originally French, affiliated to that of Descartes, and Lagrange spent the
best years of his life in France. His complete works were published in Paris,
14 vols. (1866-1892), with a biography by Delambre in Volume I. See also
G. Loria, " G. L. Lagrange nella vita e nelle opere," in the Annali di Matematica
pura cd appHcata, XX (3), p. ix.

LAGRANGE 483

subject until he was seventeen ; but from that time on he made
such marvelous progress that in a few years he became recog-
nized as the greatest living scholar in his science. When he
was twenty-three years old he published two memoirs 1 which
at once attracted attention. Euler wrote (October 2, 1759)
an enthusiastic letter to him about the problem of isoperimetry
which is here solved and on which the great Swiss mathemati-
cian had long been working, and d'Alembert was equally appre-
ciative of its importance. It is here that we find the beginning
of the calculus of variations, and it is here that Lagrange
took the first step toward Berlin and Paris, although it was
not until 1766 that Frederick the Great wrote that "the
greatest king in Europe" wanted "the greatest mathema-
tician of Europe" at his court. As a result of this letter,
Lagrange went to Berlin and remained there more than twenty
years. At about the time that Frederick was urging him to
go to Berlin he solved Fermat's problem relating to the equa-
tion nx 2 4- i = y 2 , n being integral and not a square," an intel-
lectual feat that added greatly to his reputation. He now
began a series of investigations on partial differential equa
tions, numerical equations, the theory of numbers, the calculus
of variations, and the application of mathematics to physical
problems, and made some progress in the theory of elliptic
functions. To one of his memoirs (1773) may be traced the
first important step in the theory of invariants, and in another
there is evidence that the notion of a group was in his mind.
At this time, too, he composed his monumental work on ana-
lytic mechanics, 3 although this was not published until a year
after he left Berlin.

The death of Frederick (1787) brought many changes to
Prussia. Lagrange, whose frail constitution had never found
the climate of Berlin salutary, and whose sensitive nature now

l Recherches sur la mfohode de maximis et minimis, Turin, 1759; Sur Vinte-
gration d'une Equation differentielle a differences finies, qui contient la theorie
dcs suites recurrentes, Turin, 1759. These appeared m the Miscellanea Tauri-
nensia.

2<t Sur la solution des problemes indtterminfe du second degr," published
in the Miscellanea Tatirmensia in 1767. s Mecanique analytique, Paris, 1788.

484 FRANCE

found the intellectual atmosphere far from agreeable, decided
to accept the invitation of Louis XVI to take up his residence
in Paris. It was about the time of the agitation for the

**fJ c"> )m<J Jic*** &J et*/t4 Cff' lldt-4 JC. /)'^') /J*"/J0ie

O4M-

DOCUMENT SIGNED BY LAGRANGE

This official document was written by Laplace and was signed by Lagrange

and himself

metric system, and Lagrange was made president of the com-
mission to carry out the work. The value of such an under-
taking could appeal even to a Sans-culotte, and so, although
all foreigners were banished from France, the Committee of
Public Safety expressly excepted Lagrange from the decree.
Nevertheless the fate of Lavoisier and Baillv. both of whom

LAGRANGE 485

met their death by the guillotine, led Lagrange to decide on
leaving France. He spoke bitterly to Delambre of the death
of Lavoisier, saying that the mob had removed in an instant
a head that it would take a century to reproduce. Prussia
knew his genius and seriously wanted him back ; Paris knew
his name and vaguely wished him to remain. The Prussia of
that day wished to be scientific ; the Paris of that day merely
wished to be thought so. But just as Lagrange was reaching
a decision, new forces were created in France, and these forces
were more potent than any fear of the guillotine, than any dis-
couragement at the acts of the revolutionary leaders, or than
any call of the successor of Frederick. France had decided to
establish a school with the humble name of Ecole Normale,
and a little later she established a second one, the Ecole
Polytechnique, and to each school Lagrange was called and to
each he gave a mathematical impetus that it has never lost.
In the first he saw a chance to found the training of teachers
on the most thorough scholarship, and no similar institution
either before or since that time has so thoroughly recognized
the value of this principle, and none has ever stood so high in
the esteem of the world. Similarly, in the Ecole Polytechnique
he saw the opportunity for basing the technical work on a
foundation of the highest type of mathematical skill, and this
institution, like the other, has ever since been a constant
inspiration to the world of science. It was at the Ecole Nor-
male that he gave those lectures on algebra and arithmetic 1
that he had to temper to the revolutionary demand before he
could bring the work up to the standard that permitted him
to present the calculus of functions. 2 It was at the Ecole
Polytechnique that he lectured on analytic functions, 3 setting
forth in new fashion the differential and integral calculus and
the calculus of fluxions. Here, too, he expounded his note-
worthy work on numerical higher equations. 4

d'arithmetique et d'algebre, 1794-1795.
2 Lemons sur le calcul des fonctions, Paris, 1801.
^Theorie des fonctions analytiques contenant Us principes du calcul dif-
fer entiel, Paris, 1797.

*Traite de la resolution des Equations numeriques de tous degris, Paris, 1798.

486

FRANCE

It is probable that his work more profoundly influenced later
mathematical research than did that of any of his contem-
poraries, although it was an era of giants in this field.

Laplace. Pierre-Simon, Marquis de Laplace, 1 was born in
poverty and owed his early education to the interest which his

promise excited in men
of intellectual power.
Of these days of strug-
gle he never spoke. Al-
most the first reliable
records that we have
of his life show him
studying and afterwards/
teaching mathematics
in the military school
at Beaumont and mak-
ing such a reputation as
to lead to his call to
succeed (1784) Bezout
as examiner of the ar-
tillery corps. He later
took part in the or-
ganization of the Ecole

Polytechnique and the
Ecole Normale. Napo-
leon made him a count
and appointed him min-
ister of the interior (1799). After standing his eccentricities
for six months, the consul dismissed him with the remark that
he carried into his work the spirit of the infinitesimal. 2 Laplace
then entered the senate but made no worthy record. After
the restoration, Louis XVIII raised him to the peerage and
(1817) made him a marquis.

PIERRE-SIMON LAPLACE

From Goutiere's engraving after a painting by
Naigeon

at Beaumont-en- Auge, Calvados, March 23, 1749; died in Paris,
March 5, 1827. For Arago's eulogy on Laplace, see the English translation in
the Smithsonian Institution Report for 1874, p. 129 (Washington, 1875).
2 "L'esprit des infiniment petits."

LAPLACE 487

Laplace was a political opportunist. At heart he was a
royalist, but for his personal interests he became a follower of
Napoleon. He was a friend of the people but not a believer
in the people's judgment.

His name is chiefly connected with astronomy and celestial
mechanics, 1 but he also wrote on probability, 2 the calculus,
differential equations, and geodesy. ' As a master of the theory
of celestial mechanics he stands unrivaled. 4 As to his style of
exposition, Nathaniel Bowditch (1773-1838), the sslf-made
American astronomer, remarked : " I never come across one of
Laplace's 'Thus it plainly appears' without feeling sure that I
have hours of hard work before me to fill up the chasm and
find out and show how it plainly appears." 5

Legendre. The third of the great trio, of which the first
two were Lagrange and Laplace, appeared in the person of
Adrien-Marie Legendre." He was educated at the College
Mazarin in Paris, where he early showed his taste for mathe-
matics, and with the help of his teacher, the Abbe Marie, 7 and
of d'Alembert, he became (1775) professor of mathematics
in the Ecole Militaire at Paris, resigning in 1780. Two years
later (1782) he won the prize of the Berlin Academy for his
essay on the path of a projectile. 8 In elementary mathematics

i Exposition du systeme du monde, 2 vols., P^ris, 1796; Traiti de mecanique
celeste, 5 vols. and suppl., Paris, 1700-1825.

2 Theorie analytique des probaHlite^ Parib, 1812 ; Eswi philowphique sur
les probability, Paris, 1814.

{ His collected works were published in seven volumes in Paris in 1843-1847.
A later and better edition was published in fourteen volumes by the Academic
dcs Sciences, 1878-1912.

4 With that felicity of speech which characterizes the French, Fourier enu-
merated his great discoveries, and added: "Voila des titres d'une gloire veri-
table, que rien ne pcut aneantir. Le spectacle du ciel sera change; mais a ces
epoques reculees, la gloire de Finventeur subsistera toujours; les traces de son
genie portent le sceau de I'imniortalite."

5 For an appreciation of Laplace and for his influence upon the century
tollowing, see R. S. Woodward, Bulletin of the Amer. Math. Soc., V, 133.

r >Born at Toulouse, September 18, 1752 ; died in Paris, January 10, 1833.

7 Joseph Francois Marie (1738-1801), who edited Lacaille's Tables de
Logarithmes (1768) and his Lecons elementaires de Mathematiques (1798),
which had appeared in 1760 and 1741 respectively.

*Recherches sur la trajectoire des projectiles dans les milieux resistants

488

FRANCE

he is known chiefly for his geometry, 1 a work which had a
generous reception in various countries and which justly ranks
as one of the best textbooks ever written upon the subject.
In it he sought to rearrange the propositions of Euclid, sepa-
rating the theorems from the problems and simplifying the
proofs, without lessening the rigor of the ancient methods of

treatment. To Legendre
is largely due the aban-
doning of Euclid as a text-
book in American schools.
In higher mathematics
Legendre is known for his
works on the theory of
numbers 2 and on elliptic
functions.' He is also
known for his treatises on
the calculus, higher geom-
etry, mechanics, astron-
omy, and physics. To him
is due the first satisfac-
tory treatment of the
method of least squares, 4
although Gauss had al-
ready discovered the
method. In his theory of numbers appears the law of quad-
ratic reciprocity which Gauss called the "gem of arithmetic."
The treatise on elliptic functions appeared almost simultaneously
with the works by Abel and Jacobi on the same subjects ; and
although Legendre had spent thirty years on the theory, he

1 Elements de giometrie, Paris, 1794.

2 Essai sur la theorie des nombres, Paris, 1798; 2d ed., Paris, 1808, with
supplements in 1816 and 1825 ; 3d ed. under the title Theorie des nombres, Paris,
2 vols., 1830.

*Memoire sur les transcendantes elliptiques f Paris, 1794 ; Traite des jonctions
elliptiques et des integrates euUriennes, Paris, 1827-1832,

4 On the history of this subject see M. Merriman, Method of Least Squares,
p. 182 (New York, 1884); Transactions of Connecticut Academy, IV, 151
(1877), with complete bibliography to that date; Todhunter, Hist. Probability,
Cambridge, 1865.

ADRIEN-MARIE LEGENDRE
After a lithograph by Delpech

AUTOGRAPH LETTER OF LEGENDRE

In some of his letters the form "Le Gendre" appears, as in this case. In general
the name is spelled Legendre

490

FRANCE

recognized at once the superiority of the treatment given to it by
these younger men, and posterity has agreed with his judgment.
Failing to yield to the government in its desire to dictate to
the Academic, he was deprived of his pension, and his last days
were spent in poverty. His letters of this period are depressing,

showing how one of the
greatest scientists of France
had lost heart at the failure
of a nation to recognize his
honesty of purpose and his
powers of intellect.

Beginning of the Nine-
teenth Century. Of those who
made France a great mathe-
matical center in Napoleon's
day, Gaspard Monge 1 was,
after Lagrange, Laplace, and
Legendre, one of the leaders.
He was the son of an itiner-
ant tradesman, and was one
of many in the history of sci-
ence who early showed prom-
ise of success. At the age
of fourteen he constructed
a fire engine which was
put into service, and at six-
teen he was teaching in a

secondary school (college) in Lyons. At twenty-two he was
professor of mathematics in the military school at Mezieres,
and from this time on he continued to progress, with the
excitement of just escaping the guillotine, until he reached
a professorship (1794) in the Ecole Poly technique. He also
became a member of Napoleon's staff in Egypt, a senator

1 Born at Beaune, May 10, 1746; died in Paris, July 28, 1818. See M. Bris-
son, Notice historique sur Gaspard Monge, Paris, 1818 ; F. Arago, "Gaspard
Monge," in Arago's CEuvres completes, II, 427 (Paris, 1854) ; Ch. Dupin, Essai
historique s;/r . . . Monge, Paris, 1819.

GASPARD MONGE, COMTE DE PELUSE
After a lithograph of a drawing by Hesse

MONGE 491

(1799), and, as Comte de Peluse, a member of the nobility.
With the restoration, however, all his honors were taken from
him, and his last years were a period of disappointment.

He is known chiefly for his elaboration of descriptive geom-
etry, a theory which, as we have seen, was suggested by Frezier
in 1738, but which Monge worked out independently while
at Mezieres. It was some years before anything was published
on the subject, the idea being held as a military secret of great
value in the designing of fortifications. The opening of the
Ecole Polytechnique gave him an opportunity to lecture upon
the theory, and finally, in 1799, he published his treatise upon
it. 1 He also wrote numerous memoirs on differential equa-
tions, curves on various surfaces, and physical problems.

Among the most unfortunate scholars on the roll of the
world's mathematicians there will always rank the name of
Pierre-Frangois-Andre Mechain/ a man who rose from the posi-
tion of private tutor to become one of the leading astronomers
of France, charged with duties of greatest importance in con-
nection with the metric system, a collaborator with the great
Delambre in the field work on which the units were based,
and one who was recognized as a scientist of genuine ability.
It was his duty to measure that part of the meridian lying be-
tween Rodez and Barcelona. After his report was sent to
Paris he discovered that he had made an error of 3" in the
latitude of Barcelona. In his endeavor to conceal this error,
which he knew would ruin his scientific reputation, he sought
to extend the meridian, cutting out Barcelona altogether, but
died from yellow fever while carrying out the plan. Instead of
being known as a scientist of repute he is thought of as the
man who made the chief mistake in the determining of the
standard meter. It should be said, however, that the fault was
not really his, for the obstacles placed in his way were such as
to make accurate observations almost impossible.

3 Geometrie descriptive. Lemons donnies aux Ecoles normales, Van 3 de la
Repuhlique (17^4-1705), Paris, Tan VII (1708-1709).

2 Born at Laon, August 16, 1744; died at Castellon de la Plana, near

492

FRANCE

Sylvestre Frangois Lacroix, 1 whose work falls in this period,
was one of those men who succeed by persevering rather than
by distinguished scientific ability. In his early years he occu-
pied various positions in the naval and military schools, but
finally became connected with the Ecole Normale, the Ecole

Polytechnique, and the Col-
lege de France, positions of
highest prominence. He was
a voluminous writer on higher
algebra, geometry, probabil-
ity, and the calculus, but he
is not known for the original
development of any great
theory. The translation
(1816) of his calculus into
English by Charles Babbage,
Sir John Frederick William
Herschel, and George Pea-
cock, however, did much to
introduce the Continental
methods and notation into
the work of the Cambridge
school of mathematicians.

JEAN-BAPTISTE- JOSEPH DELAMBRE

After a drawing made two years before
Delambre's death

Delambre. Jean-Baptiste-
Joseph Delambre 2 furnishes
one of the interesting cases
of a man who turned late to
the study of mathematics and yet rose to be a leader in the
science. As a young student in Amiens he was steeped in the
classics, and it was not until he was thirty-six years of age that
he seems to have even begun the serious study of astronomy, a
subject which required him at the same time to begin his mathe-
matical work. He was forty before he published any thing on the

iBorn in Paris, 1765; died in Paris, May 25, 1843. See the eulogies pro-
nounced at the Institut in 1843 by Libri and Despretz, and various biographical
articles of the time.

2 Born at Amiens, September 19, 1749; died in Paris, August 19, 1822.

Mrd***

AUTOGRAPH LETTER OF DELAMBRE

With computations relating to the survey made for establishing the
standard meter

494 FRANCE

subject, 1 and it was some years later that he was awarded a prize
by the Academic for his tables of Uranus. 2 From this time
on he was known as one of the leading astronomers of France,
and his various works on the history of astronomy are still
looked upon as authorities. In the history of elementary math-
ematics he is chiefly known for his work in measuring the arc
of the meridian between Dunkirk and Barcelona for the pur-
pose of establishing the basis for the metric system. He was a
scholar, a persistent worker, and a man of highest character/ 1

Carnot. Another interesting illustration of the development
of mathematical talent rather late in life is seen in the case of
Lazare-Nicolas-Marguerite Carnot, 4 a member of an old and
respected family of France. After the manner of so many
sons of the well-to-do landowners, he studied for the army,
-ind was thus led to the military school at Mezieres. Here he
came under the influence of Monge, and thus his tastes were
turned toward geometry. He developed into one of the great
military leaders of France, held various important offices, voted
for the execution of Louis XVI, suffered in the general up-
heavals of the Revolution, was exiled by Napoleon, and spent
his later years in Magdeburg.

His scientific work showed itself in various lines, but espe-
cially in his contributions to geometry. 5 It was in these con-
tributions that he assisted in laying the foundations for modern
synthetic geometry.

1 Tables de Jupiter et Saturne, Paris, 1789.

2 Published with other tables in 1792.

3 The great scientist Cuvier, in his address at the burial of Delambre in the
cemetery of Pere-Lachaise, said of him: "Qu'il me soit permis, au moment oil je
vous dis ce triste et dernier adieu, de rendre temoignage a cet admirable carac-
tere que, pendant vingt ans de liaison intime et de rapports journaliers, je n'ai
pas vu se d&nentir un instant. Jamais, pendant ce long intervalle, un seul
mouvement n'a trouble votre inalterable douceur, . . . il ne vous est echappe
une parole qui ne fut dieted par la justice et la raison."

4 Born at Nolay, Cote d'Or, May 13, 1753; died at Magdeburg, August 2,
1823. F. Arago, Biographic de . . . Carnot, eulogy delivered before the Aca-
d6mie des Sciences, August 21, 1837.

*Geomttrie de Position, Paris, 1803 ; Sur la relation qui existe entre les dis-
tances respectives de cinq points quelconques pris dans I'espace, suivi d'un Essai
sur la thlorie des transfer sales, Paris, 1806.

CARNOT AND GERGONNF 495

Gergonne and his Time. With this work is also connected
the name of Joseph-Diez Gergonne. 1 In his younger days he
was lieutenant in the artillery, then becoming a teacher of
mathematics at Nimes and later a professor at Montpellier.
His great work, however, was as editor (1810-1831) of the
mathematical journal" which commonly bears his name. In
his later years he gave himself up to the life of a retired stu-
dent. He was a prolific writer, chiefly on questions of geom-
etry, the terms ' polar' (1810) and ' class' of a curve (1827)
originating with him.

While not, like Gergonne and many other men of Napoleon's
time, himself an army man, Simeon-Denis Baron Poisson 3 was
the son of a soldier. He showed unusual abilities in mathe-
matics when very young, and on this account was sent (1798)
to the Ecole Polytechnique, where his powers came to the
attention of men like Lagrange and Laplace. Soon after finish-
ing his prescribed work he was given a place on the faculty and
devoted the rest of his life to teaching there and in the univer-
sity, and to contributing to the literature of mechanics, mathe-
matical physics in general, and pure mathematics. In the field
of mathematics his chief contributions were to the theory of
probability, algebraic equations, differential equations, definite
integrals, surfaces, and the calculus of variations.

There have been several instances in the history of mathe-
matics where a man's name has become known for a single
discovery, not in itself remarkable, but striking in its peculiar
interest. Such an instance is seen in the case of Charles-Julien
Brianchon, 4 a student in the Ecole Polytechnique (1804) and
later (1808) an artillery officer. Brianchon had the ingenuity,
when only twenty-three (1806), to take the dual of Pascal's
proposition concerning a hexagon inscribed in a conic. The

1 Born at Nancy, June IQ, 1771; died at Montpellier, May 4, 1859. M. A.
Lafon, Gergonne, Sa vie et ses travaux, reprint (n. d.) from the Extraits des
Mem. de VAcad. de Stanislas.

2 Annales de Mathematiques pures et appliquees.

3 Born at Pithiviers, June 21, 1781 ; died at Sceaux, April 25, 1840.

4 Born at Sevres, December ig, 1783 ; died at Versailles, April 29, 1864.
J. Boyer, "Charles-Julien Brianchon d'apres des documents in&iits," Revue
scientifique, I (4), 592.

496 FRANCE

result is Brianchon's Theorem with respect to the concurrence
of lines joining opposite points of a circumscribed hexagon. 1
He became a professor in the artillery school and wrote several
memoirs on geometry, particularly on curves of the second
degree (1806) and lines of the second order (1817).

Poncelet. The life of Jean-Victor Poncelet 2 illustrates the
military activity of many mathematicians of the disturbed
Napoleonic period. A pupil of Monge's in the Ecole Poly-
technique (1807-1810), he entered the army (1812) as lieu-
tenant of engineers. On the French retreat from Moscow he
was captured by the Russians and was taken to Saratoff, on
the Volga River. Here he devoted his time to the contempla-
tion of certain possibilities in the domain of mathematics, and
on his return to Metz (1814) he began to put the results of his
thoughts into form for publication. The result was his great
contribution to the theory of projective geometry. 3 He devoted
the latter part of his life to military duties, his leisure being
given to writing on mechanics, hydraulics, series, and geometry.
He was one of the founders of modern geometry, probably the
most important one. The Germans were more strongly in-
fluenced by his works, however, than were his own country-
men. It was Chasles who awakened France to the importance
of his contributions and to a recognition of his genius.

Cauchy. The great technical and military schools founded
or encouraged by Napoleon began at this time to enroll the
most brilliant scientists of France. Among these was Augustin-
Louis Cauchy, 4 who entered the Ecole Polytechnique in Paris
at the age of sixteen, proceeding thence to the Ecole des Fonts
et Chaussees. After a certain amount of engineering experience
he was elected to the chair of mechanics in the Ecole Polytech-
nique and to membership in the Academic des Sciences. On

1 Chasles, Apergti, p. 370.

2 Born at Metz, July i, 1788; died in Paris, December 23, 1867.

*Traite des proprietes projectives des figures, Paris, 1822 ; Applications
d'analyse et de geomttrie, 2 vols., Paris, 1862, 1864.

, 4 Born in Paris, August 21, 1789; died at Sceaux, May 23, 1857. J. Bertrand,
"Eloge" in the Memolres of the Academic d. Sci., Paris, Vol. 47, pp. clxxxiii-ccv

FONCELET AND CAUCHY

497

account of the political situation he went to Turin in 1830,
where he became professor of mathematics in the university.
Two years later he went to Prag and in 1838 returned to Paris
and taught in certain Church schools. In 1848 he was made
professor of mathematical astronomy in the university. His
life was one of unrest on account of his own marked eccen-
tricities as well as because of the changing political situation in
France; but in spite of
this fact he published
upwards of seven hun-
dred memoirs on mathe-
matics and showed him-
self a man of uncommon
scientific ability. Al-
though usually display-
ing an affable manner,
he was not a man of
good breeding, being pos-
sessed of an unfortunate
conceit, narrow in his
views, and disposed to ar-
gue endlessly over trifles.
He was an indefatigable
worker, and his contri-
butions to mathematics
include researches into
the theory of residues,
the question of convergence, differential equations, the theory
of functions, the elucidation of the imaginary, operations with
determinants, the theory of equations, the theory of probability,
the foundations of the calculus, and the applications of mathe-
matics to physics. He was one of the first to use the imaginary
as a .fundamental instead of a subsidiary quantity, was the
first to use Gauss's word "determinant" in its present sense,
did much to establish the modern theory of convergence, and
perfected the theory of linear differential equations and the
calculus of variations.

AUGUSTTN-LOUIS CAUCHY

One of the foremost mathematicians of

France in the igth century

498 FRANCE

Chasles. Michel Chasles, 1 one of the leading French geom-
eters of the igth century, was, like his leading contemporaries
in the field of mathematics, a student at the Ecole Polytech-
nique (1812-1814). He went into business for a time but
again returned to scientific work. He began publishing im-
portant memoirs on geometry as soon as he left school (1814),
but it was his semihistorical work on the development of geom-
etry 2 and his treatise on higher geometry 3 that gave him a
world-wide reputation. These works were followed by various
important memoirs on the different branches of geometry. 4
Chasles became professor of geometry and mathematics at the
Ecole Poly technique in 1841 and professor of geometry in the
faculty of sciences in 1846. In 1867 he prepared a noteworthy
report on the progress of geometry in France. 5 He also re-
ceived (1865) the Copley medal of the Royal Society for his
work in conies.

Galois. The mathematician has not always been as conserva-
tive or as engrossed in his studies as the world seems to think.
As an illustration of this fact one of the most interesting is that
of Evariste Galois. 6 Educated at the Lycee Louis-le-Grand and
the Ecole Normale, at Paris, a rabid republican, twice impris-
oned for his political views, a hot-blooded lover who fought a
duel at twenty which cost him his life, he was able in the space
of three or four years, even in his boyhood, to make for him-
self a lasting reputation as a genius. His life was mentally bril-
liant, but physically, politically, and morally it was a failure.

1 Born at Epernon, November 15, 1793; died in Paris, December 18, 1880.
For an obituary notice see Boncompagni's Bullettino, XIII, 815.

2 Aper$u historique sur Vorigine et developpement des methodes en geometric,
Paris, 1837; 2d ed., Paris, 1875; 3d ed., Paris, 1889; German ed., Halle,
1839. Some of the editions of his works bear the imprint of both Brussels
and Paris.

*Traite de geometrie superieure, Brussels, 1852; Traite des sections coniques,
Paris. 1865,

* For example, "Construction de la courbe du troisieme ordre determinee
par neuf points" (1853), Journal de math, pures el aPpliguees* XIX (1854).

5 Rapport sur les progres de la geom6trie y Pans, 1871.

*>Born in Paris, October 26 1811; died in Paris, May 30, 1832. P. Dupuv,
"La vie d'fivariste Galois," 6cole normale t Annales, XIII (3), 197; G. Sarton.
The Scientific Monthly, XIII, 363.

LATER WRITERS 499

To him is due, however, one of the first important modern
advances in the theory of groups, and hence to him we owe
much of our modern theory of algebraic equations of higher
degree. His most important memoir 1 was written the year
before his death but was not published until 1846."

Poincare. Of the French mathematicians of 'the close of the
1 9th century no one ranked so high in the estimation of his
contemporaries as Henri Poincare. 3 There was hardly a branch
of mathematics, pure or applied, to which he did not contribute
in one way or another. His reputation was first made in his
treatment of Abelian functions and in the more general type to
which he gave the name of Fuchsian functions. His memoirs on
these subjects began to appear in the Comptes rendus in 1880
and in the first volume of the Ada Mathematica. His contri-
butions to elliptic functions, modular functions, double inte-
grals, and the general theory of analysis are well known. He is
equally well known for his important contributions to astron-
omy and physics and for his profound researches in the field of
philosophy.

Other Contributors. Among the many others who added to
the reputation of France in the field of mathematics during this
period there may be mentioned Jean-Baptiste-Marie-Charles
Meusnier de la Place (1754-1793), usually known as Meus-
nier, who wrote on the theory of surfaces; Jean-Baptiste
Biot (1774-1862), who successfully applied mathematics to
problems in physics and astronomy; Jean-Nicolas-Pierre
Hachette (1769-1834), who wrote on algebra and geometry;
Sophie Germain (1776-1831), known for her work on the
theory of elastic surfaces; Louis Poinsot (1777-1859), who

^'Memoire sur les conditions de resolubilite des equations par radicaux,"
Liouville's Journal, XI (1846). His Manuscrits, edited by J. Tannery, appeared
at Paris in 1908.

2 On the history of the group theory see the bibliography by C. Alasia in
the Rivista di fisica, matematica e scienze naturali, XVIII-XXII. See also
Miller, Introduction^ p. 97.

8 Born at Nancy, April 29, 1854 > died in Paris, July 17, 1912. E. Lebon,
Henri Poincari, Paris, 1909; V. Volterra, "Henri Poincare," Rice Institute
Pamphlets, I, 133 (Houston, Texas).

SCO FRANCE

contributed to the theory of numbers, to geometry, and to
mechanics; Gabriel Lame (1795-1870), primarily a physicist
but writing on probability and surfaces; Theodore Olivier
(1793-1853), especially concerned with descriptive geometry;
Louis Arbogast (1759-1803), whose Cakul des Derivations
appeared in 1800; Jean Robert Argand (1768-1822), who
wrote on the graphic representation of V^T ; Joseph Fourier
(1768-1830), known for his work in series, particularly with
respect to Fourier's series, which is used in studying the flow
of heat; Charles Dupin (1784-1873), prominent because of
his works on mechanics and differential geometry; Georges-
Henri Halphen (1844-1389), who contributed to the theory
of invariants; Jean Gaston Darboux (1842-1917), contributor
to differential geometry, one of the editors of the Bulletin des
sciences mathematiques et astronomiques, and permanent sec-
retary of the Academic des Sciences; Edmond Laguerre
(1834-1886), a contributor to the theory of equations ; Charles
Hermite (1822-1901), who proved the transcendence of e and
who wrote on the theory of functions ; Joseph Liouville (1809-
1882), long the editor of Liouville's Journal ; Charles Meray
(1835-1911), original in his ideas of the foundations upon
which elementary geometry should be built; Joseph Alfred
Serret (1819-1885), best known for his Cours d'algebre supe-
rieure (1849) but also a prolific writer on the function theory,
groups, and differential equations ; Joseph-Louis-Frangois Ber-
trand (1822-1900), professor of mathematical physics in the
College de France and secretary of the Academic des Sciences,
a writer on the theory of probability, the calculus of variations,
and differential equations as applied to dynamics ; Pierre Du-
hem (1861-1916), contributor to mathematical physics and
especially, by his study of original sources, to the history of sci-
ence; and Louis Couturat (1868-1914), writer upon the inter-
relation of mathematics and logic. Any such list is necessarily
fragmentary, and the student who wishes to carry his investiga-
tions farther should consult such works as the French or
German editions of the encyclopedia of mathematics.

CHRISTIAN VON WOLF

SOI

4. GERMANY

General Survey. Germany began to show her real strength
in mathematics at the close of the i8th century. Theretofore
she had depended largely on imported men, such as Euler,
the Bernoullis, and Lagrange. Now she produced Gauss, and
his influence on German
mathematics made Got-
tingen a focus for schol-
ars; it placed Germany
among the leading na-
tions in the cultivation
of this science, and gave
her a position of suprem-
acy during part of the
i gth century.

Of the work accom-
plished in the i8th cen-
tury a fair example is
that of Freiherr Chris-
tian von Wolf, 1 a phi-
losopher of merit and a
mathematician of erudi-
tion if not of brilliancy.
He took his master's de-
gree at Leipzig in 1703
and at once became

Dozent in the university. CHRISTIAN VON WOLF

Soon after this (1706) After a mezzotint by Jacob Haid

he went to Halle as

professor of mathematics and, somewhat later, of physics.
Because of his religious views he was banished from the uni-
versity in 1723, but was immediately invited to accept the
professorship of philosophy at Marburg. He was recalled to
Halle by Frederick the Great, to whom a religious question
was not a matter of much moment, became chancellor of the

1 Born at Breslau, January 24, 1679; died at Halle, April 9, 1754.

502 GERMANY

university in 1743, and was raised to the rank of baron (Frei-
herr) in 1745. He was a member of various learned societies,
did much to popularize the theories of his friend Leibniz, and
was a voluminous writer but not an original thinker. His most
extensive works are his Elementa and Anjangsgrunde? but he
also prepared an unimportant set of logarithmic tables (1711)
and wrote a mathematical dictionary (1716)."

In the field of elementary education, Germany produced a
number of important writers, but few whose names can be
rated as international. Among the most industrious of the
group was Christian Pescheck, 3 who wrote a large number of
textbooks and was one of the first of the German writers to
consider seriously the methods of teaching the subject.

Gauss. The real founder of modern German mathematics,
however, is Carl Friedrich Gauss, 4 one of the many mathema-
ticians who rose to highest eminence from very humble birth.

AUTOGRAPH OF GAUSS
Signed as Director of the Royal Society of Sciences at Gb'ttingen

The son of a day laborer, his abilities showed themselves so
early as to attract attention, and he was sent to the Carolineum
at Braunschweig (1792-1795) and thence to the University of
Gottingen (1795-1798). During his university career he con-
ceived the idea of the theory of least squares, 5 discovered the

1 Elementa matheseos universae, 4 vols., Halle, 1713, with later editions;
Anfangsgriinde aller mathematischen Wissenschaften, 4 vols., Halle, 1710, with
several editions.

2 J. C. Gottsched, Historische Lobschrijt des . . . Christians . . . Freyhesm
von Wolf, Halle, 1755; W. Arnsperger, Christian Wolffs Verhaltnis zu Leibniz,
Weimar, 1897.

3 Born at Zittau, July 31, 1676; died at Zittau, October 28, 1747.

4 Born at Braunschweig (Brunswick), April 30, 1777; died at Gottingen,
February 23, 185 5- In his autographs the first name begins with C. The name
was originally Johann Carl Friedrich Gauss.

6 Legendre (1805) was the first to write upon the subject, introducing it in
his work on the orbit of comets. The deduction of the law was effected by an
Irish- American writer, Adrain, in 1808.

LATER \VR1TERS 507

at the University of Berlin, became Privatdozent in 1825, and
two years later was made professor of mathematics at Konigs-
berg. He was a prolific contributor in various lines of mathe-
matics, but his chief work was in the fields of elliptic functions, 1
determinants, the theory of numbers, 2 differential equations,
the calculus of variations, and infinite series.

Belonging to about the same period as Jacobi, Peter Gustav
Lejeune-Dirichlet 11 was educated at Gottingen, studied under
Gauss, and became professor of mathematics at Breslau,
Berlin, and Gottingen. He was chiefly interested in algebra,
the number theory, and quadratic forms.

That a university professorship is not a sine qua non to suc-
cess in mathematics is a fact again illustrated in the case of
Hermann Giinther Grassmann, 4 who was the son of a teacher
of mathematics in the Gymnasium at Stettin and himself
occupied a similar position in the same school. The father
wrote some textbooks of no particular moment, and both he and
the son gave much attention to physical questions. There was
also another son, Robert, with whom Hermann collaborated in
writing an arithmetic (1860). The entire output of the family,
however, was as nothing compared with Hermann's Ausdeli-
nMngslchre? In this he set forth a theory that covered much
the same ground as the theory of quaternions, then being inde-
pendently developed by Sir William Rowan Hamilton.

Ernst Eduard Kummer is another instance of a man of
genius who spent some years as a Gymnasium teacher before
being called to a university chair. Educated for theology as
well as mathematics, he began his teaching at Sorau, after-
wards going to Liegnitz, where he taught for ten years

l Fundamenta nova theoriae functionum ellipticarum^ Konigsberg, 1820.
2 Canon arithmeticus, Berlin, 1839.

3 Born at Diiren, February 13, 1805; died at Gottingen, May 5, 1859.
4 Born at Stettin, April 15, 1809; died at Stettin, September 26, 1877. See
F. Engel, Jahresbericht of the Deutsche Afath.-Verein.> XIX, i.

5 Die Wissenschaft der extensiven Grosse oder die Ausdehnungslehre, Leipzig,
1844; completed in 1862. A list of his works may be found in the Maths-
matische Annalen, XIV, 43.

6 Born at Sorau, Nieder-Lausitz, January 29, 1810; died at Berlin, May 14,
1893.

508 GERMAN?

(1832-1842) in the Gymnasium, having Kronecker for one of
his pupils. He then became (1842) professor of mathematics
in the University of Breslau, later (1855) being transferred to
Berlin, where he remained until 1884. In Crelle's Journal may
be found his valuable contributions to the theory of hyper-
geometric (Gaussian) series (1836), the Riccati equation
(1834), the question of the convergency of series (1835), the
theory of complex numbers (1844, 1850), and cubic and bi-
quadratic remainders (1842, 1848). He created the theory
of ideal prime factors of complex numbers (1856) and laid
down the principles applicable to Kummer surfaces. 1

His contemporary, Georg Friedrich Bernhard Riemann, 2
also proved himself a genius in the study of surfaces. He
studied at Berlin and Gottingen, receiving his doctorate at the
latter university in 1851. His dissertation 3 has since been
recognized as a genuine contribution to the theory of functions.
Three years later (1854) he became a Privatdozent in Got-
tingen and in 1857 became a professor 4 of mathematics in the
university. His introduction of the notion of geometric order
into the theory of Abelian functions, and his invention of the
surfaces which bear his name, led to a great advance in the
function theory. He also set forth (1854) a new system of
non-Euclidean geometry, 5 and wrote on partial differential
equations, 6 elliptic functions, 7 and physics. 8

1 Surfaces of the fourth degree, 16 knot points, 16 singular tangent planes.
See "Allgemeine Theorie der gradlinigen Strahlensysteme," Crelle's Journal,
LVII (1860), 189. For a biographical sketch and a list of his works see the
Jahresbericht of the Deutsche Math.-Verein., Ill, 13.

2 Born at Breselenz, Hannover, September 17, 1826; died at Selasca, Lago
Maggiore, July 20, 1866.

^Grundlagen jur eine allgemeine Theorie der Functionen einer verdnderlichen
complexen Grosse, Gottingen, 1851 ; 2d ed., Gottingen, 1867.

4 He succeeded Dirichlet as ordentlicher Professor in 1859.

B Ueber die Hypothesen welche der Geometric zu Grunde liegen, Leipzig,
1867. Like several of his works, this appeared posthumously.

6 Partielle Differentialgleichungen, Braunschweig, 1869; 2d ed., Hannover,
1876; 3d ed., Braunschweig, 1882; 4th ed., Braunschweig, 1900-1901.

7 Elliptische Functionen, Vorlesungen mil Zusatzen, Leipzig, 1899.

8 See his Gesammelte mathematiscke Werke und wissenschaftlicher Nachlass^
Leipzig, 1876; 2d ed., Leipzig, 1892; French translation, Paris, 1898; contains a

LATER WRITERS 509

One of the most brilliant and promising mathematicians of
Germany in the middle of the igth century appeared in the
person of Ferdinand Gotthold Max Eisenstein. 1 Gauss, in a
moment of enthusiasm, and without sufficiently weighing his
words, said of him : " There have been but three epoch-making
mathematicians, Archimedes, Newton, and Eisenstein." He
was brought up in poverty, showed no particular taste for
mathematics until he was nineteen, died at the age of twenty-
nine, and yet in ten years developed powers so remarkable as
to place him in the first rank of scholars. His most important
contributions were to the theory of ternary and quadratic
forms, the theory of numbers, and the theory of functions.
He has been spoken of by his countrymen as the real founder
of the theory of invariants.

Weierstrass. As a type of those great leaders who, towards
the close of the igth century, made Germany a great gathering
place for scholars there may be mentioned Karl Weierstrass. 2
He studied law and finance at Bonn (1834), taught in various
Gymnasien, went to Berlin in 1856 as a teacher in the Gewerbe-
institut, and became ordentlicher Professor of mathematics
in the University of Berlin in 1864. Here he became one of
the great leaders in the theory of elliptic and Abelian functions,
in the theory of functions in general, and in the development
of the theory of irrational numbers. The Berlin Akademie der
Wissenschaften began the publication of his collected works
in i894. 3

Dedekind, Cantor, and Fuchs. Julius Wilhelm Richard
Dedekind 4 stands out as one of the most prominent contrib-
utors of the i gth century to the theory of algebraic numbers.
He studied at Gottingen and in 1854 became a Dozent in the
university. In 1858 he went to the polytechnic school at

ifiorn at Berlin, April 16, 1823; died at Berlin, October n, 1852. F. Rudio,
"Eine Autobiographic von Gotthold Eisenstein," Abhandlungen, VII, 145.

2 Born at Ostenfelde, October 31, 1815; died at Berlin, February 19, 1897.

3 On his life, see Acta Mathematica, XXI, 79; XXII, r.

4 Born at Braunschweig, October 6, 1831 ; died at Braunschweig, February 12,
1916.

5io GERMANY

Zurich, and four years later became a professor in a similar
institution at Braunschweig. He wrote various important mem-
oirs on the binomial equation and on the theory of modular
and Abelian functions, but is best known for his treatises Was
sind und was sollen die Zahlen ? (1888) and Stctigkeit und irra-
tionale Zahlen (I872). 1 In the latter work he set forth his
idea of the Schnitt (cut) in relation to irrational numbers,
an idea which he had in mind as early as 1858.

Although Georg Cantor" was the son of a Danish merchant
and was born in Russia, he should properly be ranked among
the German mathematicians, having spent the greater part
of his life in German universities. He studied at Zurich, Got-
tingen, and Berlin, and became a Dozent at Halle in 1869 and a
professor three years later. He was a man of original ideas,
and the theory of assemblages is practically his creation. His
researches on this subject were first published in the Annalcn
in 1879.

Emmanuel Lazarus Fuchs 3 became professor of mathematics
at Greifswald in 1869 and afterwards occupied similar posi-
tions at Gottingen, Heidelberg, and Berlin. His earlier labors
were in the fields of higher geometry and the theory of num-
bers, but he attained his highest reputation in his work on linear
differential equations.

Other Writers. Among the other German writers of the
1 9th century not many were at the same time leaders in ad-
vanced research and contributors to elementary mathematics.
A few of the best-known names of those who extended the
boundaries of mathematics should, however, be mentioned.
These are Johann Friedrich Pfaff (1765-1825), professor at
Helmstadt (1788) and Halle (1810), known for his work in
astronomy, geometry, and analysis; Ludwig Otto Hesse
(1811-1874), one of the foremost writers on modern pure
geometry, analytic geometry, and determinants; Christoph
Gudermann (1798-1852), who wrote on hyperbolic functions;

1 English translations by Professor W. W. Beman, Chicago, 1901.
2 Born at Petrograd, March 3, 1845; died at Halle, January 6, 1918.
8 Born at Moschin, May 5, 1833; died at Berlin, April 26, 1902.

LATER WRITERS 511

Johann August Grunert (1797-1872), editor of the Archiv;
Ernst Ferdinand August (1795-1870), known for his work on
mathematical physics; Rudolff Friedrich Alfred Clebsch
(1833-1872), professor at Carlsruhe, Giessen, and Gottingen.
a contributor to modern geometry; Hermann Ludwig Fer-
dinand von Helmholtz (1821-1894), a contributor to many
fields of scientific research including that of non-Euclidean
geometry; Leopold Kronecker (1823-1891), a leading writer
on the theory of equations and on elliptic functions ; Friedrich
Wilhelm Bessel (1784-1846), one of the leading astronomers
of the century, a physicist, and well known for the functions
which bear his name; Paul Du Bois-Reymond (1831-1889),
known for his work on Fourier's series, the problem of conver-
gence, the calculus of variations, and integral equations;
Siegfried Heinrich Aronhqjld (1819-1884), professor in the
technical high school at Berlin, well known for his work on
invariants; and Karl Theodor Reye (1837-1919), whose Geo-
metric der Lage (3d ed. 1886-1892) is one of the best-known
textbooks on the subject.

5. ITALY

Nature of the Work. In the i8th and igth centuries Italy
produced a worthy line of mathematicians, but until recently
she has not made a serious effort to regain her earlier standing.
Her scholars seemed to work in isolation during much of this
period, a result due in some measure, no doubt, to the lack of
political homogeneity in Italy herself. A few names in the i8th
and the early i9th century, however, deserve our attention.

Ceva Brothers. About the middle of the i7th century there
were born two brothers, Giovanni Ceva 1 and Tommaso Ceva, 2
each of whom contributed to geometry and physics. The latter
was a teacher of mathematics in the Jesuit college at Milan,
while the former was in the service of the Duke of Mantua.
Tommaso wrote on the cycloid, the mechanical trisection of an

at Mantua, December, 1647; died May 13, 1736. These dates are
uncertain.

2 Born at Milan, December 20, 1648; died at Milan, February 3, 1737.

512 ITALY

angle, 1