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A HISTORY OF MATHEMATICS

BOOKS BY FLORIAN CAJORI

HISTORY OF MATHEMATICS
Revised and Enlarged Edition

HISTORY OF ELEMENTARY MATHEMATICS
Revised and Enlarged Edition

HISTORY OF PHYSICS

INTRODUCTION TO THE MODERN
THEORY OF EQUATIONS

A HISTORY OF

MATHEMATICS

BY
FLORIAN CAJORI, . PH. D.

PROFESSOR OF HISTORY OF MATHEMATICS IN THE
UNIVERSITY OF CALIFORNIA

" I am sure that no subject loses more than mathematics
by any attempt to dissociate it from its history." — J. W. L.
GLAISHER

SECOND EDITION, REVISED AND ENLARGED

,1U

THE MACMILLAN COMPANY

LONDON : MACMILLAN & CO., LTD.
1919

Set up and electrotyped January, 1894. Reprinted March,
1895; October, 1897; November, 1901; January, 1906; July, 1909;
February, ign.

Second Edition, Revised and Enlarged

BY THE MACMILLAN COMPANY
Set up and electrotyped. Published July, 1919

PREFACE TO THE SECOND EDITION

IN preparing this second edition the earlier portions of the book
have been partly re-written, while the chapters on recent mathematics
are greatly enlarged and almost wholly new. The desirability of
having a reliable one- volume history for the use of readers who cannot
devote themselves to an intensive study of the history of mathematics
is generally recognized. On the other hand, it is a difficult task to
give an adequate bird's-eye- view of the development of mathematics
from its earliest beginnings to the present time. In compiling this
history the endeavor has been to use only the most reliable sources.
Nevertheless, in covering such a wide territory, mistakes are sure to
have crept in. References to the sources used in the revision are
given as fully as the limitations of space would permit. These ref-
erences will assist the reader in following into greater detail the his-
tory of any special subject. Frequent use without acknowledgment
has been made of the following publications: Annuario Biografico del
Circolo Matematico di Palermo, 1914; Jahrbuch uber die Fortschritte der
Mathematik, Berlin; /. C. Poggendorffs Biographisch-Literarisches
Handworterbuch, Leipzig; Gedenktagebuch fur Mathematiker, von Felix
Miiller, 3. AufL, Leipzig und Berlin, 1912; Revue Semestrielle des Pub-
lications Mathematiques, Amsterdam.

The author is indebted to Miss Falka M. Gibson of Oakland, Cal.
for assistance in the reading of the proofs.

FLORIAN CAJORI.

University of California,
March, 1919.

TABLE OF CONTENTS

Page

INTRODUCTION i

THE BABYLONIANS 4

THE EGYPTIANS 9

THE GREEKS 15

Greek Geometry 15

The Ionic School . . .' 15

The School of Pythagoras 17

The Sophist School 20

The Platonic School 25

The First Alexandrian School 29

The Second Alexandrian School 45

Greek Arithmetic and Algebra 52

THE ROMANS 63

THE MAYA 69

THE CHINESE?/1 71

THE JAPANESE 78

THE HINDUS 83

THE ARABS 99

EUROPE DURING THE MIDDLE AGES 113

Introduction of Roman Mathematics 113

Translation of Arabic Manuscripts 118

The First Awakening and its Sequel 126

EUROPE DURING THE SIXTEENTH, SEVENTEENTH AND EIGHT-
EENTH CENTURIES , 130

The Renaissance 130

Vieta and Descartes 145

Descartes to Newton 173

Newton to Euler 190

Eider, Lagrange and Laplace 231

THE NINETEENTH AND TWENTIETH CENTURIES 278

Introduction 278

Definition of Mathematics 285

vii

viii CONTENTS

Page

Synthetic Geometry 286

Elementary Geometry of the Triangle and Circle 297

Link-motion 301

Parallel Lines, Non-Euclidean Geometry and Geometry of

n Dimensions 302

Analytic Geometry 309

Analysis Situs 323

Intrinsic Co-ordinates 324

Definition of a Curve 325

Fundamental Postulates 326

Geometric Models 328

Algebra 329

Theory of Equations and Theory of Groups 349

Solution of Numerical Equations 363

Magic Squares and Combinatory Analysis 316

Analysis 367

Calculus of Variations 369

Convergence of Series 373

Probability and Statistics 377

Differential Equations. Difference Equations 383

Integral Equations, Integro-differential Equations, General

Analysis, Functional Calculus 392

Theories of Irrationals and Theory of Aggregates 396

Mathematical Logic 407

Theory of Functions 411

Elliptic Functions 41 1

General Theory of Functions 1 419

Uniformization 433

Theory of Numbers 434

Fermat's "Last Theorem," Waring's Theorem 442

Other Recent Researches. Number Fields 444

Transcendental Numbers. The Infinite 446

Applied Mathematics 447

Celestial Mechanics 447

Problem of Three Bodies 452

General Mechanics 455

Fluid Motion 460

Sound. Elasticity 464

Light, Electricity, Heat, Potential 470

Relativity 479

Nomography 481

Mathematical Tables 482

Calculating Machines, Planimeters, Integraphs 485

Alphabetical Index 487

A HISTORY OF MATHEMATICS

INTRODUCTION

1 THE contemplation of the various steps by which mankind has
come into possession of the vast stock of mathematical knowledge
can hardly fail to interest the mathematician. He takes pride in the
fact that his science, more than any other, is an exact science, and
that hardly anything ever done in mathematics has proved to be use-
less. The chemist smiles at the childish efforts of alchemists, but the
mathematician finds the geometry of the Greeks and the arithmetic
of the Hindus as useful and admirable as any research of to-day. He
is pleased to notice that though, in course of its development, mathe-
matics has had periods of slow growth, yet in the main it has been
pre-eminently a progressive science.

The history of mathematics may be instructive as well as agreeable;
it may not only remind us of what we have, but may also teach us
how to increase our store. Says A. De Morgan, "The early history
of the mind of men with regard to mathematics leads us to point out
our own errors; and in this respect it is well to pay attention to the
history of mathematics." It warns us against hasty conclusions; it
points out the importance of a good notation upon the progress of the
science; it discourages excessive specialisation on the part of investi-
gators, by showing how apparently distinct branches have been found
to possess unexpected connecting links; it saves the student from
wasting time and energy upon problems which were, perhaps, solved
long since; it discourages him from attacking an unsolved problem by
the same method which has led other mathematicians to failure; it
teaches that fortifications can be taken in other ways than by direct
attack, that when repulsed from a direct assault it is well to recon-
noitre and occupy the surrounding ground and to discover the secret
paths by which the apparently unconquerable position can be taken.1
The importance of this strategic rule may be emphasised by citing a
case in which it has been violated. An untold amount of intellectual
energy has been expended on the quadrature of the circle, yet no con-
quest has been made by direct assault. The circle-squarers have,
existed in crowds ever since the period of Archimedes. After in-
numerable failures to solve the problem at a time, even, when in-

1 S. Giinther, Ziele und Resultate der neueren M athematisch-historischen Forschung.
Erlangen, 1876.

2 A HISTORY OF MATHEMATICS

vestigators possessed that most powerful tool, the differential calculus,
persons versed in mathematics dropped the subject, while those who
still persisted were completely ignorant of its history and generally
misunderstood the conditions of the problem. "Our problem," says
A. De Morgan, " is to square the circle with the old allowance of means:
Euclid's postulates and nothing more. We cannot remember an
instance in which a question to be solved by a definite method was
tried by the best heads, and answered at last, by that method, after
thousands of complete failures." But progress was made on this
problem by approaching it from a different direction and by newly
discovered paths. J. H. Lambert proved in 1761 that the ratio of the
circumference of a circle to its diameter is irrational. Some years
ago, F. Lindemann demonstrated that this ratio is also transcendental
and that the quadrature of the circle, by means of the ruler and com-
passes only, is impossible. He thus showed by actual proof that which
keen-minded mathematicians had long suspected; namely, that the
great army of circle-squarers have, for two thousand years, been
assaulting a fortification which is as indestructible as the firmament
of heaven.

Another reason for the desirability of historical study is the value
of historical knowledge to the teacher of mathematics. The interest
which pupils take in their studies may be greatly increased if the
solution of problems and the cold logic of geometrical demonstrations
are interspersed with historical remarks and anecdotes. A class in
arithmetic will be pleased to hear about the Babylonians and Hindus
and their invention of the "Arabic notation"; they will marvel at
the thousands of years which elapsed before people had even thought
of introducing into the numeral notation that Columbus-egg — the
zero; they will find it astounding that it should have taken so long
to invent a notation which they themselves can now learn in a month.
After the pupils have learned how to bisect a given angle, surprise
them by telling of the many futile attempts which have been made
to solve, by elementary geometry, the apparently very simple problem
of the trisection of an angle. When they know how to construct a
square whose area is double the area of a given square, tell them about
the duplication of the cube, of its mythical origin — how the wrath of
Apollo could be appeased only by the construction of a cubical altar
double the given altar, and how mathematicians long wrestled with
this problem. After the class have exhausted their energies on the
theorem of the right triangle, tell them the legend about its discov-
erer— how Pythagoras, jubilant over his great accomplishment,
sacrificed a hecatomb to the Muses who inspired him. When the
value of mathematical training is called in question, quote the in-
scription over the entrance into the academy of Plato, the philosopher:
"Let no one who is unacquainted with geometry enter here." Students
in analytical geometry should know something of Descartes, and,

INTRODUCTION 3

after taking up the differential and integral calculus, they should
become familiar with the parts that Newton, Leibniz, and Lagrange
played in creating that science. In his historical talk it is possible
for the teacher to make it plain to the student that mathematics is
not a dead science, but a living one in which steady progress is made.1

A similar point of view is taken by Henry S. White: 2 ''The ac-
cepted truths of to-day, even the commonplace truths of any science,
were the doubtful or the novel theories of yesterday. Some indeed
of prime importance were long esteemed of slight importance and
almost forgotten. The first effect of reading in the history of science
is a naive astonishment at the darkness of past centuries, but the
ultimate effect is a fervent admiration for the progress achieved by
former generations, for the triumphs of persistence and of genius^
The easy credulity with which a young student supposes that of
course every algebraic equation must have a root giv"es place finally
to a delight in the slow conquest of the realm of imaginary numbers,
and in the youthful genius of a Gauss who could demonstrate this
once obscure fundamental proposition."

The history of mathematics is important also as a valuable con-
tribution to the history of civilisation. Human progress is closely
identified with scientific thought. Mathematical and physical re-
searches are a reliable record of intellectual progress. The history
of mathematics is one of the large windows through which the philo-
sophic eye looks into past ages and traces the line of intellectual de-
velopment.

1 Cajori, F., The Teaching and History of Mathematics in the United States. Wash-
ington, 1890, p. 236.

. Am. Math. Soc., Vol. 15, 1909, p. 325.

THE BABYLONIANS

THE fertile valley of the Euphrates and Tigris was one of the
primeval seats of human society. Authentic history of the peoples
inhabiting this region begins only with the foundation, in Chaldaea
and Babylonia, of a united kingdom out of the previously disunited
tribes. Much light has been thrown on their history by the discovery
of the art of reading the cuneiform or wedge-shaped system of writing.

In the study of Babylonian mathematics we begin with the notation
of numbers. A vertical wedge )f stood for i, while the characters -^
and V>-» signified 10 and too respectively. Grotefend believes the
character for 10 originally to have been the picture of two hands, as
held in prayer, the palms being pressed together, the fingers close to
each other, but the thumbs thrust out. In the Babylonian notation
two principles were employed — the additive and multiplicative. Num-
bers below 100 were expressed by symbols whose respective values
had to be added. Thus, \ | stood for 2, \ ] f for 3, ^T for 4,<£*
for 23, ^ ^ ^ for 30. Here the symbols of higher order appear
always to the left of those of lower order. In writing the hun-
dreds, on the other hand, a smaller symbol was placed to the left of
100, and was, in that case, to be multiplied by 100. Thus, ^ y »~
signified 10 times 100, or 1000. But this symbol for 1000 was itself
taken for a new unit, which could take smaller coefficients to its left.
Thus, ^ ^ K a*- denoted, not 20 times 100, but 10 times 1000. Some
of the cuneiform numbers found on tablets in the ancient temple
library at Nippur exceed a million; moreover, some of these Nippur
tablets exhibit the subtracti-ue principle (20-1), similar to that shown
in the Roman notation, "XIX."

If, as is believed by most specialists, the early Sumerians were the
inventors of the cuneiform writing, then they were, in all probability,
also the inventors of the notation of numbers. Most surprising, in
this connection, is the fact that Sumerian inscriptions disclose the use,
not only of the above decimal system, but also of a sexagesimal one.
The latter was used chiefly in constructing tables for weights and
measures. It is full of historical interest. Its consequential develop-
ment, both for integers and fractions, reveals a high degree of
matical insight. We possess two Babylonian tablets which
its use. One of them, probably written between 2300 and 16
contains a table of square numbers up to 6o2. The numbers
16, 25, 36, 49, are given as the squares of the first seven inte

THE BABYLONIANS

lively. We have next i.4=82, 1.21 = g2, 1.40= io2, 2.1 = n2, et
remains unintelligible, unless we assume the sexagesimal scale,
ch makes 1.4 = 60-1-4, 1.21 = 60+21, 2.1 = 2.60+1. The second
let records the magnitude of the illuminated portion of the moon's
c for every day from new to full moon, the whole disc being assumed
consist of 240 parts. The illuminated parts during the first five
^ys are the series 5, io, 20, 40, 1.20 (=80), which is a geometrical
progression. From here on the series becomes an arithmetical progres-
sion, the numbers from the fifth to the fifteenth day being respectively
1.20, 1.36, 1.52, 2.8, 2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4. This table
not only exhibits the use of the sexagesimal system, but also indicates
the acquaintance of the Babylonians with progressions. Not to be
overlooked is the fact that in the sexagesimal notation of integers
the "principle of position" was employed. Thus, in 1.4 (=64), the
i is made to stand for 60, the unit of the second order, by virtue of
its position with respect to the 4. The introduction of this principle
at so early a date is the more remarkable, because in the decimal no-
tation it was not regularly introduced till about the ninth century
after Christ. The principle of position, in its general and systematic
application, requires a symbol for zero. We ask, Did the Babylonians
possess one? Had they already taken the gigantic step of representing
by a symbol the absence of units? Neither of the above tables answers
this question, for they happen to contain no number in which there
was occasion to use a zero. . Babylonian records of many centuries
later — of about 200 B. C. — give a symbol for zero which denoted the
absence of a figure but apparently was not used in calculation. It
consisted of two angular marks ^ one above the other, roughly re-
sembling two dots, hastily written. About 130 A. D., Ptolemy in
Alexandria used in his Almagest the Babylonian sexagesimal fractions,
and also the omicron o to represent blanks in the sexagesimal numbers.
This o was not used as a regular zero. It appears therefore that the
Babylonians had the principle of local value, and also a symbol for
zero, to indicate the absence of a figure, but did not use this zero in
computation. Their sexagesimal fractions were introduced into India
and with these fractions probably passed the principle of local value
and the restricted use of the zero.

The sexagesimal system was used also in fractions. Thus, in the
Babylonian inscriptions, | and | are designated by 30 and 20, the
reader being expected, in his mind, to supply the word "sixtieths/'
The astronomer Hipparchus, the geometer Hypsiclcs and the as-
tronomer Ptolemy borrowed the sexagesimal notation of fractions
from the Babylonians and introduced it into Greece. From that time
sexagesimal fractions held almost full sway in astronomical and mathe-
matical calculations until the sixteenth century, when they finally
yielded their place to the decimal fractions. It may be asked, What
led to the invention of the sexagesimal system? Why was it that 60

6 A HISTORY OF MATHEMATICS

parts were selected? To this we have no positive answer. 7Yj,s
chosen, in the decimal system, because it represents the numo{
fingers. But nothing of the human body could have suggeste^.
Did the system have an astronomical origin? It was supposed t
the early Babylonians reckoned the year at 360 days, that thiid
to the division of the circle into 360 degrees, each degree represeng
the daily amount of the supposed yearly revolution of the sun arod
the earth. Now they were, very probably, familiar with the fact
that the radius can be applied to its circumference as a chord 6 times,
and that each of these chords subtends an arc measuring exactly 60
degrees. Fixing their attention upon these degrees, the division into
60 parts may have suggested itself to them. Thus, when greater pre-
cision necessitated a subdivision of the degree, it was partitioned into
60 minutes. In this way the sexagesimal notation was at one time
supposed to have originated. But it now appears that the Babylonians
very early knew that the year exceeded 360 days. Moreover, it is
highly improbable that a higher unit of 360 was chosen first, and a
lower unit of 60 afterward. The normal development of a number
system is from lower to higher units. Another guess is that the
sexagesimal system arose as a mixture of two earlier systems of the
bases 6 and lo.1 Certain it is that the sexagesimal system became
closely interwoven with astronomical and geometrical science. The
division of the day into 24 hours, and of the hour into minutes and
seconds on the scale of 60, is attributed to the Babylonians. There is
strong evidence for the belief that they had also a division of the day
into 60 hours. The employment of a sexagesimal division in numeral
notation, in fractions, in angular as well as in time measurement, in-
dicated a beautiful harmony which was not disturbed for thousands
of years until Hindu and Arabic astronomers began to use sines and
cosines in place of parts of chords, thereby forcing the right angle to
the front as a new angular unit, which, for consistency, should* have
been subdivided sexagesimally, but was not actually so divided.

It appears that the people in the Tigro-Euphrates basin had made
very creditable advance in arithmetic. Their knowledge of arith-
metical and geometrical progressions has already been alluded to.
lamblichus attributes to them also a knowledge of proportion, and
even the invention of the so-called musical proportion. Though we

1 M. Cantor, Vorlesungen itber Geschichte der Mathematik, i. Bd., 3. Aufl., Leipzig,
1907, p. 37. This work has appeared in four large volumes and carries the history
down to 1799. The fourth volume (1008) was written with the cooperation of nine
scholars from Germany, Italy, Russia and the United States. Moritz Cantor (1829-
) ranks as the foremost general writer of the nineteenth century on the history
of mathematics. Born in Mannheim, a student at Heidelberg, at Gottingen under
Gauss and Weber, at Berlin under Dirichlet, he lectured at Heidelberg where in
1877 he became ordinary honorary professor. His first historical article was
brought out in 1856, but not until 1880 did the first volume of his well-known history
appear.

THE BABYLONIANS

possess no conclusive proof, we have nevertheless reason to believe
that in practical calculation they used the abacus. Among the races
of middle Asia, even as far as China, the abacus is as old as fable.
Now, Babylon was once a great commercial centre, — the metropolis of
many nations, — and it is, therefore, not unreasonable to suppose that
her merchants employed this most improved aid to calculation.

In 1889 H. V. Hilprecht began to make excavations at Nuffar (the
ancient Nippur) and found brick tablets containing multiplication and
division tables, tables of squares and square roots, a geometric progres-
sion and a few computations. He published an account of his findings
in 1906. l

The divisions in one tablet contain results like these: "6o4 divided
by 2 = 6,480,000 each," "6o4 divided by 3 = 4,320,000 each," and
so on, using the divisors 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18. The very
first division on the tablet is interpreted to be "6o4 divided by iJ/2 =
8,640,000." This strange appearance of f as a divisor is difficult to
explain. Perhaps there-is here a use of f corresponding to the Egyptian
use of | as found in the Ahmes papyrus at a, perhaps, contemporaneous
period. It is noteworthy that 6o4= 12,960,000, which Hilprecht found
in the Nippur brick text-books, is nothing less than the mystic " Platonic
number," the "lord of better and worse births," mentioned in Plato's
Republic. Most probably, Plato received the number from the
Pythagoreans, and the Pythagoreans from the Babylonians.2

In geometry the Babylonians accomplished little. Besides the divi-
sion of the circumference into 6 parts by its radius, and into 360 de-
grees, they had some knowledge of geometrical figures, such as the
triangle and quadrangle, which they used in their auguries. Like the
Hebrews (i Kin. 7:23), they took 7T = 3. Of geometrical demonstra-
tions there is, of course, no trace. "As a rule, in the Oriental mind
the intuitive powers eclipse the severely rational and logical."

Hilprecht concluded from his studies that the Babylonians pos-
sessed the rules for finding the areas of squares, rectangles, right tri-
angles, and trapezoids.

The astronomy of the Babylonians has attracted much attention.
They worshipped the heavenly bodies from the earliest historic times.
When Alexander the Great, after the battle of Arbela (331 B. c.), took
possession of Babylon, Callisthenes found there on burned brick as-
tronomical records reaching back as far as 2234 B. c. Porphyrius says
that these were sent to Aristotle. Ptolemy, the Alexandrian astrono-
mer, possessed a Babylonian record of eclipses going back to 747 B. c.

1 Mathematical, Metrological and Chronological Tablets from the Temple Library
of Nippur, by H. V. Hilprecht. Vol. XX, part I, Series A, Cuneiform Texts, pub-
lished by the Babylonian Expedition of the University of Pennsylvania, 1906. Con-
sult also D. E. Smith in Bull. Am. Math. Soc., Vol. 13, 1907, p. 392.

2 On the "Platonic number" consult P. Tannery in Revue philosophiquc, Vol. I,
1876, p. 170; Vol. XIII, 1881, p. 210; Vol. XV, 1883, p. 573. Also G. Loria in Le
scicnzc csaltc ncll'anlica grecia, 2 Ed., Milano, 1914, Appendice.

8 A HISTORY OF MATHEMATICS

Epping and Strassmaier 1 have thrown considerable light on Babylon-
ian chronology and astronomy by explaining two calendars of the
years 123 B. c. and in B. c., taken from cuneiform tablets coming,
presumably, from an old observatory. These scholars have succeeded
in giving an account of the Babylonian calculation of the new and
full moon, and have identified by calculations the Babylonian names
of the planets, and of the twelve zodiacal signs and twenty-eight
normal stars which correspond to some extent with the twenty-eight
nakshatras of the Hindus. We append part of an Assyrian astronomical
report, as translated by Oppert: —

"To the King, my lord, thy faithful servant, Mar-Istar."

"... On the first day, as the new moon's day of the month Thammuz
declined, the moon was again visible over the planet Mercury, as I had
already predicted to my master the King. I erred not."

1 Epping, J., Astronomisches CMS Babylon. Unter Mitwirkung von P. J. R. Strass-
maier. Freiburg, 1889.

THE EGYPTIANS

Though there is difference of opinion regarding the antiquity of
Egyptian civilisation, yet all authorities agree in the statement that,
however far back they go, they find no uncivilised state of society.
"Menes, the first king, changes the course of the Nile, makes a great
reservoir, and builds the temple of Phthah at Memphis." The Egyp-
tians built the pyramids at a very early period. Surely a people en-
gaging in enterprises of such magnitude must have known something
of mathematics — at least of practical mathematics.

All Greek writers are unanimous in ascribing, without envy, to
Egypt the priority of invention in the mathematical sciences. Plato
in Phcedrus says: "At the Egyptian city of Naucratis there was a
famous old god whose name was Theuth; the bird which is called the
Ibis was sacred to him, and he was the inventor of many arts, such
as arithmetic and calculation and geometry and astronomy and
draughts and dice, but his great discovery was the use of letters."

Aristotle says that mathematics had its birth in Egypt, because
there the priestly class had the leisure needful for the study of it.
Geometry, in particular, is said by Herodotus, Diodorus, Diogenes
Laertius, lamblichus, and other ancient writers to have originated in
Egypt.1 In Herodotus we find this (II. c. 109): "They said also that
this king [Sesostris] divided the land among all Egyptians so as to
give each one a quadrangle of equal size and to draw from each his
revenues, by imposing a tax to be levied yearly. But every one from
whose part the river tore away anything, had to go to him and notify
what had happened; he then sent the overseers, who had to measure
out by how much the land had become smaller, in order that the
owner might pay on what- was left, in proportion to the entire tax
imposed. In this way, it appears to me, geometry originated, which
passed thence to Hellas."

We abstain from introducing additional Greek opinion regarding
Egyptian mathematics, or from indulging in wild conjectures. We
rest our account on documentary evidence. A hieratic papyrus, in-
cluded in the Rhind collection of the British Museum, was deciphered
by Eisenlohr in 1877, and found to be a mathematical manual con-
taining problems in arithmetic and geometry. It was written by
Ahmes some time before 1700 B. c., and was founded on an older work
believed by Birch to date back as far as 3400 B. c. ! This curious

1 C. A. Bretschneider Die Geometric und die Geometer vor Euklides. Leipzig, 1870,
pp. 6-8. Carl Anton Bretschneider (1808-1878) was professor at the Realgymna-
sium at Gotha in Thuringia.

io A HISTORY OF MATHEMATICS

papyrus — the most ancient mathematical handbook known to us —
puts us at once in contact with the mathematical thought in Egypt of
three or five thousand years ago. It is entitled "Directions for ob-
taining the Knowledge of all Dark Things." We see from it that the
Egyptians cared but little for theoretical results. Theorems are not
found in it at all. It contains "hardly any general rules of procedure,
but chiefly mere statements of results intended possibly to be ex-
plained by a teacher to his pupils." * In geometry the forte of the
Egyptians lay in making constructions and determining areas. The
area of an isosceles triangle, of which the sides measure io khets (a
unit of length equal to 16.6 m. by one guess and about thrice that
amount by another guess 2) and the base 4 khets, was erroneously given
as 20 square khets, or half the product of the base by one side. The
area of an isosceles trapezoid is found, similarly, by multiplying half
the sum of the parallel sides by one of the non-parallel sides. The
area of a circle is found by deducting from the diameter ^ of its length
and squaring the remainder. Here TT is taken= (-196-)2 = 3.i6o4..., a
very fair approximation. The papyrus explains also such problems
as these, — To mark out in .the field a right triangle whose sides are
io and 4 units; or a trapezoid whose parallel sides are 6 and 4, and
the non-parallel sides each 20 units.

Some problems in this papyrus seem to imply a rudimentary knowl-
edge of proportion.

The base-lines of the pyramids run north and south, and east and
west, but probably only the lines running north and south were deter-
mined by astronomical observations. This, coupled with the fact
that the word harpedonaptce, applied to Egyptian geometers, means
"rope-stretchers," would point to the conclusion that the Egyptian,
like the Indian and Chinese geometers, constructed a right triangle
upon a given line, by stretching around three pegs a rope consisting
of three parts in the ratios 3 : 4 : 5, and thus forming a right triangle.3
If this explanation is correct, then the Egyptians were familiar, 2000
years B. c., with the well-known property of the right triangle, for
the special case at least when the sides are in the ratio 3: 4: 5.

On the walls of the celebrated temple of Horus at Edfu have been
found hieroglyphics, written about 100 B. c., which enumerate the
pieces of land owned by the priesthood, and give their areas. The
area of any quadrilateral, however irregular, is there found by the
formula -^-.-^"-. Thus, for a quadrangle whose opposite sides
are 5 and 8, 20 and 15, is given the area 113^ |.4 The incorrect for-

1 James Gow, A Short History of Greek Mathematics. Cambridge, 1884, p. 16.

* A. Eisenlohr, Ein mathematisches Handbuch der alien Aegypter, 2. Ausgabe, Leip-
zig, 1897, p. 103; F. L. Griffith in Proceedings of the Society of Biblical ArchcKology,
1891, 1894.

3 M. Cantor, op. cit. Vol. I, 3. Aufl., 1907, p. 105.

4 H. Hankel, Zur Gcschichtc dcr Mathemalik in Alterthum und Mittelaller, Leipzig,
1874, p. 86.

mulae of Ahmes of 3000 years B. c. yield generally closer approxima-
tions than those of the Edfu inscriptions, written 200 years after
Euclid!

The fact that the geometry of the Egyptians consists chiefly of
constructions, goes far to explain certain of its great defects. The
Egyptians failed in two essential points without which a science of
geometry, in the*true sense of the word, cannot exist. In the first
place, they failed to construct a rigorously logical system of geometry,
resting upon a few axioms and postulates. A great many of their
rules, especially those in solid geometry, had probably not been proved
at all, but were kno^n to be true merely from observation or as mat-
ters of fact. The second great defect was their inability to bring the
numerous special cases under a more general view, and thereby to
arrive at broader and more fundamental theorems. Some of the
simplest geometrical truths were divided into numberless special cases
of which each was supposed to require separate treatment.

Some particulars about Egyptian geometry can be mentioned more
advantageously in connection with the early Greek mathematicians
who came to the Egyptian priests for instruction.

An insight into Egyptian methods of numeration was obtained
through the ingenious deciphering of the hieroglyphics by Champol-
lion. Young, and th^ir successors. The symbols used were the fol-
lowing: [| for i, {\ for 10, (p for 100, § for 1000, (f for 10,000, ^>
for 100,000, ^ for 1,000,000, Q_ for 10,000,000.* The symbol for
i represents a vertical staff; that for 10,000 a pointing finger; that
for 100,000 a burbot; that for 1,000,000, a man in astonishment. The
significance of the remaining symbols is very doubtful. The writing
of numbers with these hieroglyphics was very cumbrous. The unit
symbol of each order was repeated as many times as there were units
in that order. The principle employed was the additive. Thus, 23
was written O H I I |

Besides the hieroglyphics, Egypt possesses the hieratic and demotic
writings, but for want of space we pass them by.

Herodotus makes an important statement concerning the mode of
computing among the Egyptians. He says that they "calculate with
pebbles by moving the hand from right to left, while the Hellenes
move it from left to right." Herein we recognise again that instru-
mental method of figuring so extensively used by peoples of antiquity.
The Egyptians used the decimal scale. Since, in figuring, they moved
their hands horizontally, it seems probable that they used ciphering-
boards with vertical columns. In each column there must have been
not more than nine pebbles, for ten pebbles would be equal to one
pebble in the column next to the left. J

1 M. Cantor, op. cit. Vol. I, 3. Aufl., 1907, p. 82.

12 A HISTORY OF MATHEMATICS

The Ahmes papyrus contains interesting information on the way
in which the Egyptians employed fractions. Their methods of opera-
tion were, of course, radically different from ours. Fractions were a
subject of very great difficulty with the ancients. Simultaneous
changes in both numerator and denominator were usually avoided.
In manipulating fractions the Babylonians kept the denominators (60)
constant. The Romans likewise kept them constant, but equal to 12.
The Egyptians and Greeks, on the other hand, kept the numerators
constant, and dealt with variable denominators. Ahmes used the
term "fraction" in a restricted sense, for he applied it only to unit-
fractions, or fractions having unity for the numerator. It was desig-
nated by writing the denominator and then placing over it a dot.
Fractional values which could not be expressed by any one unit-
fraction were expressed as the sum of two or more of them. Thus, he
wrote |- y^ in place of |. While Ahmes knows ^ to be equal to i- ^, he
curiously allows | to appear often among the unit-fractions and adopts
a special symbol for it. The first important problem naturally arising
was, how to represent any fractional value as the sum of unit-fractions.
This was solved by aid of a table, given in the papyrus, in which all

fractions of the form — -r— (where n designates successively all the

numbers up to 49) are reduced to the sum of unit-fractions. Thus,
2. = i^g-; ^7 = A*ri¥- When> by whom, and how this table was cal-

culated, we do not know. Probably it was compiled empirically at
different times, by different persons. It will be seen that by repeated
application of this table, a fraction whose numerator exceeds two can
be expressed in the desired form, provided that there is a fraction in
the table having the same denominator that it has. Take, for ex-
ample, the problem, to divide 5 by 21. In the first place", 5 = i-f- 2+ 2.
From the table we get ^ A A- Then ^ = ^+(A: A)+(i™) =
TT+ (A T?) = TT T TT = T TT = T Tf TV The papyrus contains prob-
lems in which it is required that fractions be raised by addition or multi-
plication to given whole numbers or to other fractions. For example,
it is required to increase i ^ ^ -^ -^ to i. The common denominator
taken appears to be 45, for the numbers are stated as u±, 5^ ^, 4*.,
i^, i. The sum of these is 23! i. £ forty-fifths. Add to this £ J^, and
the sum is f . Add -|, and we have i. Hence the quantity to be added
to the given fraction is ^ ^ -£$.

Ahmes gives the following example involving an arithmetical
progression: "Divide too loaves among 5 persons; \ of what the first
three get is what the last two get. What is the difference?" Ahmes
gives the solution: "Make the difference 5^; 23, 17^, 12, 6i, i.
Multiply by if; 38^, 29^, 20, io| |, i|." How did Ahmes come upon

THE EGYPTIANS 13

5^? Perhaps thus: l Let a and — d be the first term and the differ-
ence in the required arithmetical progression, then ^[a-\-(a — </)+
(a — 2d)] = (a — 3</)+(a — 4^), whence d=$%(a — ^d), i. e. the dif-
ference d is 5>^ times the last term. Assuming the last term i, he
gets his first progression. The sum is 60, but should be 100; hence
multiply by i|, for 6oXif = 100. We have here a method of solution
which appears again later among the Hindus, Arabs and modern
Europeans — the famous method of false position.

Ahmes speaks of a ladder consisting of the numbers 7, 49, 343,
2401, 16807. Adjacent to these powers of 7 are the words picture,
cat, mouse, barley, measure. What is the meaning of these mysterious
data? Upon the consideration of the problem given by Leonardo of
Pisa in his Liber abaci, 3000 years later: "7 old women go to Rome,
each woman has 7 mules, each mule carries 7 sacks, etc.", Moritz
Cantor offers the following solution to the Ahmes riddle: 7 persons
have each 7 cats, each cat eats 7 mice, each mouse eats 7 ears of
barley, from each ear 7 measures of corn may grow. How many
persons, cats, mice, ears of barley, and measures of corn, altogether?
Ahmes gives 19607 as the sum of the geometric progression. Thus,
the Ahmes papyrus discloses a knowledge of both arithmetical and
geometrical progression.

Ahmes proceeds to the solution of equations of one unknown quan-
tity. The unknown quantity is called 'hau' or heap. Thus the

oc
problem, "heap, its -J-, its whole, it makes 19," i. e. — \-x= 19. In

8x x

this case, the solution is as follows: — =19; ~~=2^^; *=I^i- But

in other problems, the solutions are effected by various other methods.
It thus appears that the beginnings of algebra are as ancient as those
of geometry.

That the period of Ahmes was a flowering time for Egyptian mathe-
matics appears from the fact that there exist other papyri (more re-
cently discovered) of the same period. They were found at Kahun,
south of the pyramid of Illahun. These documents bear close re-
semblance to Ahmes. They contain, moreover, examples of quadratic
equations, the earliest of which we have a record. One of them is: 2
A given surface of, say, 100 units of area, shall be represented as
the sum of two squares, whose sides are to each other as i : |.. In
modern symbols, the problem is, to find x and y, such that xz+y2=
100 and x:y=i:%. The solution rests upon the method of false
position. Try x= i and y=f-, then #2+:y2=ff and \/y| = |.. But
\/ioo= 10 and io-7-|-=8. The rest of the solution cannot be made

1 M. Cantor, op. cit., Vol. I, 3. Aufl., 1907, p. 78.

2 Cantor, op. cit. Vol. I, 1907, pp. 95, 96.

14 A HISTORY OF MATHEMATICS

out, but probably was #=8X i, y— 8X-| = 6. This solution leads to
the relation 62+82=io2. The symbol P was used to designate
square root.

In some ways similar to the Ahmes papyrus is also the Akhmim
papyrus,1 written over 2000 years later at Akhmim, a city on the
Nile in Upper Egypt. It is in Greek and is supposed to have been
written at some time between 500 and 800, A. D. It contains, besides
arithmetical examples, a table for finding "unit-fractions," like that
of Ahmes. Unlike Ahmes, it tells how the table was constructed. The

rule, expressed in modern symbols, is as follows :-^ = -r-^H — j~

<l~r P ~T~

For 2=2, this formula reproduces part of the table in Ahmes.

The principal defect of Egyptian arithmetic was the lack of a
simple, comprehensive symbolism — a defect which not even the Greeks
were able to remove. .i-u*5

The Ahmes papyrus and the other papyri of the same period repre-
sent the most advanced attainments of the Egyptians in arithmetic
and geometry. It is remarkable that they should have reached so
great proficiency in mathematics at so remote a period of antiquity.
But strange, indeed, is the fact that, during the next two thousand
years, they should have made no progress whatsoever in it. The con-
clusion forces itself upon us, that they resemble the Chinese in the
stationary character, not only of their government, but also of their
learning. All the knowledge of geometry which they possessed when
Greek scholars visited them, six centuries B. c., was doubtless known
to them two thousand years earlier, when they built those stupendous
and gigantic structures — the pyramids.

1 J. Baillet, "Le papyrus mathematique d'Akhmim," M6moire$ publics par les
membres de la mission archeologique franqaise an Caire, T. IX, ir fascicule, Paris,
1892, pp. 1-88. See also Cantor, op. cit. Vol. I, 1907, pp. 67, 504.

THE GREEKS

Greek Geometry

About the seventh century B. c. an active commercial intercourse
sprang up between Greece and Egypt. Naturally there arose an
interchange of ideas as well as of merchandise. Greeks, thirsting for
knowledge, sought the Egyptian priests for instruction. Thales,
Pythagoras, (Enopides, Plato, Democritus, Eudoxus, all visited the
land of the pyramids. Egyptian ideas were thus transplanted across
the sea and there stimulated Greek thought, directed it into new lines,
and gave to it a basis to work upon. Greek culture, therefore, is not
primitive. Not only in mathematics, but also in mythology and
art, Hellas owes a debt to older countries. To Egypt Greece is in-
debted, among other things, for its elementary geometry. But this
does not lessen our admiration for the Greek mind. From the mo-
ment that Hellenic philosophers applied themselves to the study of
Egyptian geometry, this science assumed a radically different aspect.
"Whatever we Greeks receive, we improve and perfect," says Plato.
The Egyptians carried geometry no further than was absolutely neces-
sary for their practical wants. The Greeks, on the other hand, had
within them a strong speculative tendency. They felt a craving tp
discover the reasons for things. They found pleasure in the con-
templation of ideal relations, and loved science as science.

Our sources of information on the history of Greek geometry before
Euclid consist merely of scattered notices in ancient writeis. The
early mathematicians, Thales and Pythagoras, left behind no written
records of their discoveries. A full history of Greek geometry and
astronomy during this period, written by Eudemus, a pupil of Aris-
totle, has been lost. It was well known to Prjacjus, who, in his com-
mentaries on Euclid, gives a brief account of it. This abstract con-
stitutes our most reliable information. We shall quote it frequently
under the name of Eudemian Summary.

The Ionic School

To Thales (640-546 B. c.), of Miletus, one of the "seven wise men,"
and the founder of the Ionic school, falls the honor of having intro-
duced the study of geometry into Greece. During middle life he
engaged in commercial pursuits, which took him to Egypt. He is
said to have resided there, and to have studied the physical sciences
and mathematics with the Egyptian priests. Plutarch declares that
Thales soon excelled his masters, and amazed King Amasis by measur-

16 A HISTORY OF MATHEMATICS

ing the heights of the pyramids from their shadows. According to
Plutarch, this was done by considering that the shadow cast by a
vertical staff of known length bears the same ratio to the shadow of
the pyramid as the height of the staff bears to the height of the pyra-
mid. This solution presupposes a knowledge of proportion, and the
Ahmes papyrus actually shows that the rudiments of proportion were
known to the Egyptians. According to Diogenes Laertius, the pyra-
mids were measured by Thales in a different way; viz. by finding the
length of the shadow of the pyramid at the moment when the shadow
of a staff was equal to its own length. Probably both methods were
used.

The Eudemian Summary ascribes to Thales the invention of the
theorems on the equality of vertical angles, the equality of the angles
at the base of an isosceles triangle, the bisection of a circle by any
diameter, and the congruence of two triangles having a side and the two
adjacent angles equal respectively. The last theorem, combined (we
have reason to suspect) with the theorem on similar triangles, he applied
to the measurement of the distances of ships from the shore. Thus
Thales was the first to apply theoretical geometry to practical uses.
The theorem that all angles inscribed in a semicircle are right angles
is attributed by some ancient writers to Thalesr by others to Pythag-
oras. Thales was doubtless laminar with other theorems, not re-
corded by the ancients. It has been inferred that he knew the sum
of the three angles of a triangle to be equal to two right angles, and
the sides of equiangular triangles to be proportional.1 The Egyptians
must have made use of the above theorems on the straight line, in
some of their constructions found in the Ahmes papyrus, but it was
left for the Greek philosopher to give these truths, which others saw,
but did not formulate into words, an explicit, abstract expression, and
to put into scientific language and subject to proof thart which others
merely felt to be true. Thales may be said to have created the geom-
etry of lines, essentially abstract in its character, while the Egyptians
studied only the geometry of surfaces and the rudiments of solid
geometry, empirical in their character.2

With Thales begins also the study of scientific astronomy. He
acquired great celebrity by the prediction of a solar eclipse in 585 B. c.
Whether he predicted the day of the occurrence, or simply the year,
is not known. It is told of him that while contemplating the stars
during an evening walk, he fell into a ditch. The good old woman
attending him exclaimed, "How canst thou know what is doing in
the heavens, when thou seest not what is at thy feet? "

The two most prominent pupils of Thales were Anaximander (b. 61 1

1 G. J. Allman, Greek Geometry from Thales to Euclid. Dublin, 1889, p. 10.
George Johnston Allman (1824-1904) was professor of mathematics at Queen's
College, Galway, Ireland.

2 G. J. Allman, op. cit., p. 15.

GREEK GEOMETRY 17

B. c.) and Anaximenes (b. 570 B. c.). They studied chiefly astronomy
and physical philosophy. Of Anaxagoras (500-428 B. c.), a pupil of
Anaximenes, and the last philosopher of the Ionic school, we know
little, except that, while in prison, he passed his time attempting to
square the circle. This is the first time, in the history of mathematics,
that we find mention of the famous problem of the quadrature of the
circle, that rock upon which so many reputations have been destroyed.
It turns upon the determination of the exact value of TT. Approxima-
tions to TT had been made by the Chinese, Babylonians, Hebrews, and
Egyptians. But the invention of a method to find its exact value, is
the knotty problem which has engaged the attention of many minds
from Jhe time of Anaxagoras down to our own. Anaxagoras did
not offer any solution of it, and seems to have luckily escaped par-
alogisms. The problem soon attracted popular attention, as appears
from the reference to it made in 414 B. c. by the comic poet Aris-
tophanes in his play, the "Birds." l

About the time of Anaxagoras, but isolated from the Ionic school,
flourished (Enopides of Chios. Proclus ascribes to him the solution
of the following problems: From a point without, to draw a per-
pendicular to a given line, and to draw an angle on a line equal to a
given angle. That a man could gain a reputation by solving problems
so elementary as these, indicates that geometry was still in its infancy,
and that the Greeks had not yet gotten far beyond the Egyptian con-
structions.

The Ionic school lasted over one hundred years. The progress of
mathematics during that period was slow, as compared with its
growth in a later epoch of Greek history. A new impetus to its prog-
ress was given by Pythagoras.

The School of Pythagoras

Pythagoras (5807-500? B. c.) was one of those figures which im-
pressed the imagination of succeeding times to such an extent that
their real histories have become difficult to be discerned through the
mythical haze that envelops them. The following account of Pythag-
oras excludes the most doubtful statements. He was a native of
Samos, and was drawn by the fame of Pherecydes to the island of
Syros. He then visited the ancient Thales, who incited him to study
in Egypt. He sojourned in Egypt many years, and may have visited
Babylon. On his return to Samos, he found it under the tyranny of
Polycrates. Failing in an attempt to found a school there, he quitted
home again and, following the current of civilisation, removed to
Magna Graecia in South Italy. He settled at Croton, and founded
the famous Pythagorean school. This was not merely an academy for

1 F. Rudio in Bibliotheca mathemalica, 3 S., Vol. 8, 1907-8, pp. 13-22.

i8 A HISTORY OF MATHEMATICS

the teaching of philosophy, mathematics, and natural science, but it
was a brotherhood, the members of which were united for life. This
brotherhood had observances approaching masonic peculiarity. They
were forbidden to divulge the discoveries and doctrines of their school.
Hence we are obliged to speak of the Pythagoreans as a body, and
find it difficult to determine to whom each particular discovery is to
be ascribed. The Pythagoreans themselves were in the habit of re-
ferring every discovery back to the great founder of the sect.

This school grew rapidly and gained considerable political ascend-
ency. But the mystic and secret observances, introduced in imitation
of Egyptian usages, and the aristocratic tendencies of the school,
caused it to become an object of suspicion. The democratic party in
Lower Italy revolted and destroyed the buildings of the Pythagorean
school. Pythagoras fled to Tarentum and thence to Metapontum,
where he was murdered.

Pythagoras has left behind no mathematical treatises, and our
sources of information are rather scanty. Certain it is that, in the
Pythagorean school, mathematics was the principal study. Pythag-
oras raised mathematics to the rank of a science. Arithmetic was
courted by him as fervently as ^geometry; In fact, arithmetic is the
foundation of his philosophic systenr-

The Eudemian Summary says that ''Pythagoras changed the study
of geometry into the form of a liberal education, for he examined its
principles to the bottom, and investigated its theorems in an imma-
terial and intellectual manner." His geometry was connected closely
with his arithmetic. He was especially fond of those geometrical
relations which admitted of arithmetical expression.

Like Egyptian geometry, the geometry of the Pythagoreans is much
concerned with areas. To Pythagoras is ascribed the important
theorem that the square on the hypotenuse of a right triangle is
equal to the sum of the squares on the other two sides. He had
probably learned from the Egyptians the truth of the theorem in the
special case when the sides are 3, 4, 5, respectively. The story goes,
that Pythagoras was so jubilant over this discovery that he sacrificed
a hecatomb. Its authenticity is doubted, because the Pythagoreans
believed in the transmigration of the soul and opposed the shedding
of blood. In the later traditions of the Neo-Pythagoreans this ob-
jection is removed by replacing this bloody sacrifice by that of "an
ox made of flour!" The proof of the law of three squares, given in
Euclid's Elements, I. 47, is due to Euclid himself, and not to the
Pythagoreans. What the Pythagorean method of proof was has
been a favorite topic for conjecture.

The theorem on the sum of the three angles of a triangle, presum-
ably known to Thales, was proved by the Pythagoreans after the
manner of Euclid. They demonstrated also that the plane about a
point is completely filled by six equilateral triangles, four squares, or

GREEK GEOMETRY 19

three regular hexagons, so that it is possible to divide up a plane into
figures of either kind.

From the equilateral triangle and the square arise the solids, namely,
the tetraedron, octaedron, icosaedron, and the cube. These solids
were, in all probability, known to the Egyptians, excepting, perhaps,
the icosaedron. In Pythagorean philosophy, they represent respec-
tively the four elements of the physical world; namely, fire, air, water,
and earth. Later another regular solid was discovered, namely, the
dodecaedron, which, in absence of a fifth element, was made to repre-
sent the universe itself. lamblichus states that Hippasus, a Pytha-
gorean, perished in the sea, because he boasted that he first divulged
" the sphere with the twelve pentagons." The same story of death at
sea is told of a Pythagorean who disclosed the theory of irrationals.
The star-shaped pentagram was used as a symbol of recognition by
the Pythagoreans, and was called by them Health.

Pythagoras called the sphere the most beautiful of all solids, and
the circle the most beautiful of all plane figures. The treatment of
the subjects of proportion and of irrational quantities by him and
his school will be taken up under the head of arithmetic.

According to Eudemus, the Pythagoreans invented the problems
concerning the application of areas, including the cases of defect and
excess, as in Euclid, VI. 28, ;><).

They were also familiar with the construction of a polygon equal,
in area to a given polygon and similar to another given polygqn. This
problem depends upon several important and somewhat advanced
theorems, and testifies to the fact that the Pythagoreans made no
mean progress in geometry.

Of the theorems generally ascribed to the Italian school, some
cannot be attributed to Pythagoras himself, nor to his earliest suc-
cessors. The progress from empirical to reasoned solutions must, of
necessity, have been slow. It is worth noticing that on the circle
no theorem of any importance was discovered by this school.

Though politics broke up the Pythagorean fraternity, yet the school
continued to exist at least two centuries longer. Among the later
Pythagoreans, Philolaus and Archytas are the most prominent.
Philolaus wrote a book on the Pythagorean doctrines. By him were
first given to the world the teachings of the Italian school, which had
been kept secret for a whole century. The brilliant Archytas (428-
347 B. c.) of Tarentum, known as a great statesman and general, and
universally admired for his virtues, was the only great geometer
among the Greeks when Plato opened his school. Archytas was the
first to apply geometry to mechanics and to treat the latter subject
methodically. He also found a very ingenious mechanical solution
to the problem of the duplication of the cube. His solution involves
clear notions on the generation of cones and cylinders. This probk-m
reduces itself to finding two mean proportionals between two given

20 A HISTORY OF MATHEMATICS

lines. These mean proportionals were obtained by Archytas from
the section of a half-cylinder. The doctrine of proportion was ad-
vanced through him.

There is every reason to believe that the later Pythagoreans exer-
cised a strong influence on the study and development of mathematics
at Athens. The Sophists acquired geometry from Pythagorean
sources. Plato bought the works of Philolaus, and had a warm friend
in Archytas.

The Sophist School

After the defeat of the Persians under Xerxes at the battle of
Salamis, 480 B. c., a league was formed among the Greeks to preserve
the freedom of the now liberated Greek cities on the islands and coast
of the jEgaean Sea. Of this league Athens soon became leader and
dictator. She caused the separate treasury of the league to be merged
into that of Athens, and then spent the money of her allies for her
own aggrandisement. Athens was also a great commercial centre.
Thus she became the richest and most beautiful city of antiquity.
All menial work was performed by slaves. The citizen of Athens was
well-to-do and enjoyed a large amount of leisure. The government
being purely democratic, every citizen was a politician. To make his
influence felt among his fellow-men he must, first of all, be educated.
Thus there arose a demand for teachers. The supply came principally
from Sicily, where Pythagorean doctrines had spread. These teachers
were called Sophists, or "wise men." Unlike the Pythagoreans, they
accepted pay for their teaching. Although rhetoric \vas the principal
feature of their instruction, they also taught geometry, astronomy,
and philosophy. Athens soon became the headquarters of Grecian
men of letters, and of mathematicians in particular. The home of
mathematics among the Greeks was first in the Ionian Islands, then
in Lower Italy, and during the time now under consideration, at
Athens.

The geometry of the circle, which had been entirely neglected by
the Pythagoreans, was taken up by the Sophists. Nearly all their
discoveries were made in connection with their innumerable attempts
to solve the following three famous problems: —

(1) To trisect an arc or an angle.

(2) To "double the cube," i. e., to find a cube whose volume is
double that of a given cube.

(3) To "square the circle," i. e. to find a square or some other
rectilinear figure exactly equal in area to a given circle.

These problems have probably been the subject of more discussion
and research than any other problems in mathematics. The bisection
of an angle was one of the easiest problems in geometry. The trisec-
tion of an angle, on the other hand, presented unexpected difficulties.
A right angle had been divided into three equal parts by the Pytha-

GREEK GEOMETRY 21

goreans. But the general construction, though easy in appearance,
cannot be effected by the aid only of ruler and compasses. Among
the first to wrestle with it was Hippias of Elis, a contemporary of
Socrates, and born about 460 B. c. Unable to reach a solution by
ruler and compasses only, he and other Greek geometers resorted to
the use of other means. Proclus mentions a man, Hippias, presum-
ably Hippias of Elis, as the inventor of a transcendental curve which
served to divide an angle not only into three, but into any number of
equal parts. This same curve was used later by Dinostratus and
others for the quadrature of the circle. On this account it is called
the qiiadratrix. The curve may be described thus: The side AH of the
square shown in the figure turns uniformly about A, the point B
moving along the circular arc BED. In the
same time, the side BC moves parallel to it- X
self and uniformly from the position of BC „
to that of AD. The locus of intersection of
AB and BC, when thus moving, is the
quadratrix BEG. Its equation we now write

7TX

y = xcot — . The ancients considered only
2r

the part of the curve that lies inside the ^ .

quadrant of the circle; they did not know G D

that x= ± 2r are asymptotes, nor that there

is an infinite number of branches. According to Pappus, Dinostratus

effected the quadrature by establishing the theorem that BED: AD

=AD:AG.

The Pythagoreans had shown that the diagonal of a square is the
side of another square having double the area of the original one.
This probably suggested the problem of the duplication of the cube,
i. e., to find the edge of a cube having double the volume of a given
cube. Eratosthenes ascribes to this problem a different origin. The
Delians were once suffering from a pestilence and were ordered by
the oracle to double a certain cubical altar. Thoughtless workmen
simply constructed a cube with edges twice as long, but brainless
work like that did not pacify the gods. The error being discovered,
Plato was consulted on the matter. He and his disciples searched
eagerly for a solution to this "Delian Problem." An important con-
tribution to this problem was made by Hippocrates of Chios (about
430 B. c.). He was a talented mathematician but, having been de-
frauded of his property, he was pronounced slow and stupid. It is
also said of him that he was the first to accept pay for the teaching of
mathematics. He showed that the Delian Problem could be reduced
to finding two mean proportionals between a given line and another
twice as long. For, in the proportion a : x=x ' y = y : 2a, since x* =
ay and y-=2ax and x*=a?y2, we have x* = ia*x and .v3 = 2a3. But,

22 A HISTORY OF MATHEMATICS

of course, he failed to find the two mean proportionals by geometric
construction with ruler and compasses. He made himself celebrated
by squaring certain lunes. According to Simplicius, Hippocrates
believed he actually succeeded in applying one of his lune-quadratures
to the quadrature" of the circle. That Hippocrates really committed
this fallacy is not generally accepted.

In the first lune which he squared, he took an isosceles triangle
ABC, right-angled at C, and drew a semi-circle on AB as a diameter,
, and passing through C. He drew also a semi-circle on AC as a diam-
eter and lying outside the .triangle ABC. The lunar area thus formed
is half the area of the triangle ABC. This is the first example of a
curvilinear area which admits of exact quadrature. Hippocrates
squared other lunes, hoping, no doubt, that he might be led to the
quadrature of the circle.1 In 1840 Th. Clausen found other quadrable
lunes, but in 1902 E. Landau of Gottingen pointed out that two of
the four lunes which Clausen supposed to be new, were known to
Hippocrates.2

In his study of the Quadrature and duplication-problems, Hip-
pocrates contributed mucnlo the geometry of the circle. He showed
that circles are to each other as the squares oLllieii-diajtneters, that
similar segments in a circle are as the squares of their chords and
contain equal angles, that in a segment less than a semi-circle the
angle is obtuse. Hippocrates contributed vastly to the logic of geom-
etry. His investigations are the oldest "reasoned geometrical proofs
in existence " (Gow) . For the purpose of describing geometrical figures
he used letters, a practice probably introduced by the Pythagoreans.

The subject of similar figures, as developed by Hippocrates, in-
volved the theory of proportion. Proportion had, thus iar, been used
by the Greeks only in numbers. They never succeeded in uniting
the notions of numbers and magnitudes. The term "number" was
used Ly them in a restricted sense. What we call irrational numbers
was not included under this notion. Not even rational fractions
were called numbers. They used the word in the same sense as we
use "positive integers." Hence numbers were conceived as discon-
tinuous, while magnitudes vfeie^^continuous . The iwo notions ap-
peared, therefore, entirely distinctT Tfie" chasm between them is ex-
posed to full view' in the statement of Euclid that "incommensurable
magnitudes do not have the same ratio as numbers." In Euclid's
Elements we find the theory of proportion of magnitudes developed
and treated independent of that of numbers. The transfer of the
theory of proportion from numbers to magnitudes (and to lengths in
particular) was a difficult and important step.

1 A full account is given by G. Loria in his Le scienze esatte nell'antica Grecia,
Milano, 2 edition, 1914, pp. 74-94. Loria gives also full bibliographical references
to the extensive literature on Hippocrates.

2E. W. Hobson, Squaring the Circle, Cambridge, 1913, p. 16.

GREEK GEOMETRY 23

Hippocrates added to his fame by writing a geometrical text-book,
called the Elements. This publication shows that the Pythagorean
habit of secrecy was being abandoned; secrecy was contrary to the
spirit of Athenian life.

The sophist Antiphon, a contemporary of Hippocrates, introduced
the process of exhaustion for the purpose of solving the problem of
the quadrature. He did himself credit by remarking that by inscrib-
ing in a circle a square or an equilateral triangle, and on its sides
erecting isosceles triangles with their vertices in the circumference,
and on the sides of these triangles erecting new triangles, etc., one
could obtain a succession of regular polygons, of which each approaches
nearer to the area of the circle than the previous one, until the circle
is finally exhausted. Thus is obtained an inscribed polygon whose
sides coincide with the circumference. Since there can be found
squares equal in area to any polygon, there also can be found a square
equal to the last polygon inscribed, and therefore equal to the circle-
itself. Bryson of Heraclea, a contemporary of Antiphon, advanced
the problem of the quadrature considerably by circumscribing poly-
gons at the same time that heinscribed jwlygons. He erred, however,
in assuming that the area of a circle was the arithmetical mean be-
tween circumscribed and inscribed polygons. Unlike Bryson and
the rest of Greek geometers, Antiphon seems to have believed it
possible, by continually doubling the sides of an inscribed polygon,
to obtain a polygon coinciding with the circle. This question gave
rise to lively disputes in Athens. If a polygon can coincide with the
circle, then, says Simplicius., we must put aside the notion that magni-
tudes are divisible ad infmitum. This difficult philosophical question
led to paradoxies that are difficult to explain and that deterred Greek
mathematicians from introducing ideas of infinity into their geometry;
rigor in geometric proofs demanded the exclusion of obscure concep-
tions. Famous are the arguments against the possibility of motion
that were advanced by Zeno of Elea, the great dialectician (early in
the 5th century B. c.). None of Zeno's writings have come down to
us. We know of his tenets only through his critics, Plato, Aristotle,
Simplicius. Aristotle, in his Physics, VI, 9, ascribes to Zeno four
arguments, called ''Zeno's paradoxies": (i) The "Dichotomy": You
cannot traverse an infinite number of points in a finite time; you
must traverse the half of a given distance before you traverse the
whole, and the half of that again before you can traverse the whole.
This goes on ad infinitum, so that (if space is made up of points) there
is an infinite number in any given space, and it cannot be traversed
in a finite time. (2) The "Achilles": Achilles cannot overtake a tor-
toise. For, Achilles must first reach the place from which the tortoise
started. By that time the tortoise will have moved on a little way.
Achilles must then traverse that, and still the tortojse will be ahead.
He is always nearer, yet never makes up to it. (3) The "Arrow":

24 A HISTORY OF MATHEMATICS

An arrow in any given moment of its flight must be at rest in som
particular point. (4) The "Stade": Suppose three parallel rows o
points in juxtaposition, as in Fig. i. One of these (B) is immovable

A .... «-A . . . .

B . . . . B ....

C . . . . C . . . .-*>

Fig. i Fig. 2

while A and C move in opposite directions with equal velocity, s<
as to come into the position in Fig. 2. The movement of C relative!]
to A will be double its movement relatively to B, or, in other words
any given point in C has passed twice as many points in A as it ha
in B. It cannot, therefore, be the case that an instant of time corre
spends to the passage from one point to another.

Plato says that Zeno's purpose was "to protect the arguments o
Parmenides against those who make fun of him"; Zeno argues tha
" there is no many" he " denies plurality." That Zeno's reasoning wa:
wrong has been the view universally held since the time of Aristotl<
down to the middle of the nineteenth century. More recently th<
opinion has been advanced that Zeno was incompletely and incor
rectly reported, that his arguments are serious efforts, conducted witl
logical rigor. This view has been advanced by Cousin, Grote and P
Tannery.1 Tannery claims that Zeno did not deny motion, bui
wanted to show that motion was impossible under the Pythagorear
conception of space as the sum of points, that the four arguments musl
be taken together as constituting a dialogue between Zeno and ar
adversary and that the arguments are in the form of a double dilemmE
into which Zeno forces his adversary. Zeno's arguments involve con-
cepts of continuity, of the infinite and infinitesimal ; they are as mucr.
the subjects of debate now as they were in the time of Aristotle
Aristotle did not successfully explain Zeno's paradoxes. He gave nc
reply to the query arising in the mind of the student, how is it pos-
sible for a variable to reach its limit? ' Aristotle's continuum was a
sensuous, physical one; he held that, since a line cannot be built up
of points, a line cannot actually be subdivided into points. "The
continued bisection of a quantity is unlimited, so that the unlimited
exists potentially, but is actually never reached." No satisfactory
explanation of Zeno's arguments was given before the creation oi
Georg Cantor's continuum and theory of aggregates.

The process of exhaustion due to Antiphon and Bryson gave rise
to the cumbrous but perfectly rigorous "method of exhaustion." Ir
determining the ratio of the areas between two curvilinear plane
figures, say two circles, geometers first inscribed or circumscribed
similar polygons, and then by increasing indefinitely the number oi

1 See F. Cajori, "The History of Zeno's Arguments on Motion" in the Americ.
Math. Monthly, Vol. 22, 1915, p. 3.

GREEK GEOMETRY 25

sides, nearly exhausted the spaces between the polygons and circum-
ferences. From the theorem that similar polygons inscribed in circles
are to each other as the squares on their diameters, geometers may
have divined the theorem attributed to Hipp'ocrates of Chios that the
circles, which differ but little from the last drawn polygons, must be
to each other as the squares on their diameters. But in order to ex-
clude all vagueness and possibility of doubt, later Greek geometers
applied reasoning like that in Euclid, XII, 2, as follows: Let C and c,
D and d be respectively the circles and diameters in question. Then
if the proportion D2 : d? = C : c is not true, suppose that Dz : d2 = C : c1.
If c1<c, then a polygon p can be inscribed in the circle c which comes
nearer to it in area than does c1. HP be the corresponding polygon
in C, then P : p = D2 : </2^TT :cl, and P : C = p : c1. Since p>cl, we
have P> C, which is absurd. Next they proved by this same method of
reductio ad absurdum the falsity of the supposition that cl>c. Since
c1 can be neither larger nor smaller than c, it must be equal to it,
Q.E.D. Hankel refers this Method of Exhaustion back to Hippocrates
of Chios, but the reasons for assigning it to this early writer, rather
than to Eudoxus, seem insufficient.

Though progress in geometry at this period is traceable only at
Athens, yet Ionia, Sicily, Abdera in Thrace, and Cyrene produced
mathematicians who made creditable contributions to the science. We
can mention,here only Democritus of Abdera (about 460-370 B. c.),
a pupil of Anaxagoras, a friend of Philolaus, and an admirer of the
Pythagoreans. He visited Egypt and perhaps even Persia. He was
a successful geometer and wrote on incommensurable lines, on geom-
etry, on numbers, and on perspective. None of these works are extant.
He used to boast that in the construction of plane figures with proof
no one had yet surpassed him, not even tl* so-called harpedonaptae
("rope-stretchers") of Egypt. By this asseftion he pays a flattering
compliment to the skill and ability of the E*rptians.

The Platonic School

During the Peloponnesian War (431-404 B. c.) the progress of geom-
etry was checked. After the war, Athens sank into the background
as a minor political power, but advanced more and more to the front
as the leader in philosophy, literature, and science. Plato was born
at Athens in 429 B. c., the year of the great plague, and died in 348.
He was a pupil and near friend of Socrates, but it was not from him
that he acquired his taste for mathematics. After the death of
Socrates, Plato travelled extensively. In Cyrene he studied mathe-
matics under Theodorus. He went to Egypt, then to Lower Italy
and Sicily, where he came in contact with the Pythagoreans. Archytas
of Tarentum and Timaeus of Locri became his intimate friends. On
his return to Athens, about 389 B. c., he founded his school in the

26 A HISTORY OF MATHEMATICS

groves of the Academia, and devoted the remainder of his life to teach
ing and writing.

Plato's physical philosophy is partly based on that of the Pytha
goreans. Like them, he sought in arithmetic and geometry the ke;
to the universe. When questioned about the occupation of the Deity
Plato answered that "He geometrises continually." Accordingly, ;
knowledge of geometry is a necessary preparation for the study o
philosophy. To show how great a value he put on mathematics an<
how necessary it is for higher speculation, Plato placed the inscrip
tion over his porch, "Let no one who is unacquainted with geometr
enter here." Xenocrates, a successor of Plato as teacher in th
Academy, followed in his master's footsteps, by declining to admit ,
pupil who had no mathematical training, with the remark, "Depart
for thou hast not the grip of philosophy." Plato observed that geom
etry trained the mind for correct and vigorous thinking. Hence i
was that the Eudemian Summary says, "He filled his writings wit]
mathematical discoveries, and exhibited on every occasion the re
markable connection between mathematics and philosophy."

With Plato as the head-master, we need not wonder that the Pla
tonic school produced so large a number of mathematicians. Plat
did little real original wTork, but he made valuable improvements ii
the logic and methods -employed in_ggiimetry. It is true that th
Sophist geometers of the previous century were fairly rigorous in thei
proofs, but as a rule they did not reflect on the inward nature of thei
methods. They used the axioms without giving them explicit ex
pression, and the geometrical concepts, such as the point, line, surface
etc., without assigning to them formal definitions.1 The Pythagorean
called a point "unity in position," but this is a statement of a philo
sophical theory rather than a definition. Plato objected to calling ;
point a "geometrical fiction." He defined a point as the "beginninj
of a line" or as "an indivisible line," and a line as "length withou
breadth." He called the point, line, surface, the "boundaries" o
th'eTine, surface, solid, respectively. Many of the definitions in Eucli<
are to be ascribed to the Platonic school. The same is probably tru
of Euclid's axioms. Aristotle refers to Plato the axiom that "equal
subtracted from equals leave equals."

One of the greatest achievements of Plato and his school is the in
vention of analysis as a method of proof. To be sure, this methoc
had been used unconsciously by Hippocrates and others; but Plato
like a true philosopher, turned the instinctive logic into a conscious
legitimate method.

1 "If any one scientific invention can claim pre-eminence over all others, I shoulc
be inclined myself to erect a monument to the unknown inventor of the mathe
matical point, as the supreme type of that process of abstraction which has beer
a necessary condition of scientific work from the very beginning." Horace Lamb'.
Address, Section A, Brit. Ass'n, 1904.

GREEK GEOMETRY 27

The terms synthesis and analysis are used in mathematics in a more
special sense than in logic. In ancient mathematics they had a 'dif-
ferent meaning from what they now have. The oldest definition of
mathematical analysis as opposed to synthesis is that given in Euclid,
XIII, 5, which in all probability was framed by Eudoxus: "Analysis is
the obtaining of the thing sought by assuming it and so reasoning up
to an admitted truth; synthesis is the obtaining of the thing sought
by reasoning up to the inference and proof of it." The analytic method
is not conclusive, unless all operations involved in it are known to
be reversible. To remove all doubt, the Greeks, as a rule, added to
the analytic process a synthetic one, consisting of a reversion of all
operations occurring in the analysis. Thus the aim of analysis was
to aid in the discovery of synthetic proofs or solutions.

Plato is said to have solved the problem of the duplication of the
cube. But the solution is open to the very same objection which he
made to the solutions by Archytas, Eudoxus, and Menaechmus. He
called their solutions not geometrical, but mechanical, for they re-
quired the use of other instruments than the ruler and compasses.
He said that thereby " the good of geometry is set aside and destroyed,
for we again reduce it to the world of sense, instead of elevating and
imbuing it with the eternal and incorporeal images of thought, even
as it is employed by God, for which reason He always is God." These
objections indicate either that the solution is wrongly attributed to
Plato or that he wished to show how easily non-geometric solutions
of that character can be found. It is now rigorously established that
the duplication problem, as well as the trisection and quadrature
problems, cannot be solved by means of the ruler and compasses
only.

Plato gave a healthful stimulus to the study of stereometry, which
until his time had been entirely neglected by the Greeks. The sphere
and the regular solids have been studied to some extent, but the prism,
pyramid, cylinder, and cone were hardly known to exist. All these
solids became the subjects of investigation by the^Platonic school.
One result of these inquiries was epoch-makingX'Menaechmus, an
associate of Plato and pupil of Eudoxus, invented the conic sections,
which, in course of only a century, raised geometry to the loftiest height
which it was destined to reach during antiquity. Menaechmus cut
three kinds of cones, the "right-angled," "acute-angled," and "obtuse-
angled," by planes at right angles to a side of the cones, and thus
obtained the three sections which we now call the parabola, ellipse,
and hyperbola. Judging from the two very elegant solutions of the
"Delian Problem" by means of intersections of these curves, Menaech-
mus must have succeeded well in investigating their properties. In
what manner he carried out the graphic construction of these curves
is not known.

Another great geometer was Dinostratus, the brother of Menaech-

2S A HISTORY OF MATHEMATICS

mus and pupil of Plato. Celebrated is his mechanical solution of th(
quadrature of the circle, by means of the quadratrix of Hippias.

Perhaps the "most brilliant mathematician of this period was
Eudoxus. He was born at Cnidus about 408 B. c., studied undei
Archytas, and later, for two months, under Plato. He was imbuec
with a true spirit of scientific inquiry, and has been called the fathei
of scientific astronomical observation. From the fragmentary notice;
of his astronomical researches, found in later writers, Ideler anc
Schiaparelli succeeded in reconstructing the system of Eudoxus wit!
its celebrated representation of planetary motions by "concentric
spheres." Eudoxus had a school at Cyzicus, went with his pupils tc
Athens, visiting Plato, and then returned to Cyzicus, where he diec
355 B. c. The fame of the academy of Plato is to a large extent du<
to Eudoxus's pupils of the school at Cyzicus, among whom are Men
sechmus, Dinostratus, Athenaeus, and Helicon. Diogenes Laertius de
scribes Eudoxus as astronomer, physician, legislator, as well as geom
eter. The Eudemian Summary says that Eudoxus "first increased th(
number of general theorems, added to the three proportions thm
more, and raised- to a considerable quantity the learning, begun b}
Plato, on the subject of the section, to which he applied the analytica
method." By this "section" is meant, no doubt, the "golden section'
(sectio aurea), which cuts a line in extreme and mean ratio. The first
five propositions in Euclid XIII relate to lines cut by this section, anc
are generally attributed to Eudoxus. Eudoxus added much to ttu
knowledge of solid geometry. He proved, says Archimedes, that a
pyramid is exactly one-third of a prism, and a cone one-third of a
cylinder, having equal base and altitude. The proof that spheres an
to each other as the cubes of their radii is probably due to him. Hf
made frequent and skilful use of the method of exhaustion, of which
he was in all probability the inventor. A scholiast on Euclid, though!
to be Proclus, says further that Eudoxus practically invented tht
whole of Euclid's fifth book. Eudoxus also found two mean propor-
tionals between two given lines, but the method of solution is not
known.

Plato has been called a maker of mathematicians. Besides the
pupils already named, the Eudemian Summary mentions the following:
Theaetetus of Athens, a man of great natural gifts, to whom, no doubt,
Euclid was greatly indebted in the composition of the loth book,1
treating of incommensurables and of the 13 th book; Leodamas oi
Thasos; Neocleides and his pupil Leon, who added much to the work
of their predecessors, for Leon wrote an Elements carefully designed,
both in number and utility of its proofs; Theudius of Magnesia, who
composed a very good book of Elements and generalised propositions,
which had been confined to particular cases; Hermotimus of Col-
ophon, who discovered many propositions of the Elements and com-
1 G. J. Allman, op. oil., p. 212.

GREEK GEOMETRY 29

posed some on loci; and, finally, the names of Amyclas of Heraclea,
Cyzicenus of Athens, and Philippus of Mende.

A skilful mathematician of whose life and works we have no details
is Aristseus, the elder, probably a senior contemporary of Euclid. The
fact that he wrote a work on conic sections tends to show that much
progress had been made in their study during the tune of Menaechmus.
Aristeus wrote also on regular solids and cultivated the analytic
method. His works contained probably a summary of the researches
of the Platonic school.1

Aristotle (384-322 B. c.), the systematiser of deductive logic, though
not a professed mathematician, promoted the science of geometry by
improving some of the most difficult definitions. His Physics contains
passages with suggestive hints of the principle of virtual velocities.
He gave the best discussion of continuity and-of Zeno's arguments
against motion, found in antiquity. About his time there appeared a
work called Mechanica, of which he is regarded by some as the author.
Mechanics was totally neglected by the Platonic school.

The First Alexandrian School

In the previous pages we have seen the birth of geometry in Egypt,
its transference to the Ionian Islands, thence to Lower Italy and to
Athens. We have witnessed its growth in Greece from feeble child-
hood to vigorous manhood, and now we shall see it return to the land
of its birth and there derive new vigor.

During her declining years, immediately following the Pelopon-
nesian War, Athens produced the greatest scientists and philosophers
of antiquity. It was the time of Plato and Aristotle. In 338 B. c., at
the battle of Chaeronea, Athens was beaten by Philip of Macedon,
and her power was broken forever. Soon after, Alexander the Great,
the son of Philip, started out to conquer the world. In eleven years
he built up a great empire which broke to pieces in a day. Egypt
fell to the lot of Ptolemy Soter. Alexander had founded the seaport
of Alexandria, which soon became the " noblest of all cities." Ptolemy
made Alexandria the capital. The history of Egypt during the next
three centuries is mainly the history of Alexandria. Literature,
philosophy, and art were diligently cultivated. Ptolemy created the
university of Alexandria. He founded the great Library and built
laboratories, museums, a zoological garden, and promenades. Alex-
andria soon became the great centre of learning.

Demetrius Phalereus was invited from Athens to take charge of the
Library, and it is probable, says Gow, that Euclid was invited with
him to open the mathematical school. According to the studies of
H. Vogt,2 Euclid was born about 365 B. c. and wrote his Elements

1 G. J. Allman, op. tit., p. 205.

2 Bibliotltcca inalhctnatlca, 3 S., Vol. 13, 1913, pp. 193-202.

3o A HISTORY OF .MATHEMATICS

between 330 and 320 B. c. Of the life of Euclid, little is known, except
what is added by Proclus to the Eudemian Summary. Euclid, says
Proclus, was younger than Plato and older than Eratosthenes and
Archimedes, the latter of whom mentions him. He was of the Platonic
sect, and well read in its doctrines. He collected the Elements, put
in order much that Eudoxus had prepared, completed many things of
Theaetetus, and was the first who reduced to unobjectionable demon-
stration the imperfect attempts of his predecessors. When Ptolemy
once asked him if geometry could not be mastered by an easier process
than by studying the Elements, Euclid returned the answer, "There
is no royal road to geometry." Pappus states that Euclid was distin-
guished by the fairness and kindness of his disposition, particularly
toward those who could do anything to advance the mathematical
sciences. Pappus is evidently making a contrast to Apollonius, of
whom he more than insinuates the opposite character.1 A pretty
little story is related by Stobaeus: 2 " A youth who had begun to read
geometry with Euclid, when he had learnt the first proposition, in-
quired, 'What" do I get by learning these things?' So Euclid called
his slave and said, 'Give him threepence, since he must make gain
out of what he learns.'" These are about all the personal details
preserved by Greek writers. Syrian and Arabian writers claim to
know much more, but they are unreliable. At one time Euclid of
Alexandria was universally confounded with Euclid of Megara, who
lived a century earlier.

The fame of Euclid has at all times rested mainly upon his book on
geometry, called the Elements. This book was so far superior to the
Elements written by Hippocrates, Leon, and Theudius, that the latter
works soon perished in the struggle for existence. The Greeks gave
Euclid the special title of "the author of the Elements." It is a re-
markable fact in the history of geometry, that the Elements of Euclid,
written over two thousand years ago, are still regarded by some as the
best introduction to the mathematical sciences. In England they
were used until the present century extensively as a text-book in
schools. Some editors of Euclid have, however, been inclined to credit
him with more than is his due. They would have us believe that a
finished and unassailable system of geometry sprang at once from the
brain of Euclid, "an armed Minerva from the head of Jupiter." They
fail to mention the earlier eminent mathematicians from whom Euclid
got his material. Comparatively few of the propositions and proofs
in the Elements are his own discoveries. In fact, the proof of the
"Theorem of Pythagoras" is the only one directly ascribed to him.
Allman conjectures that the substance of Books I, II, IV comes from
the Pythagoreans, that the substance of Book VI is due to the Pytha-

1 A. De Morgan, "Eucleides" in Smith's Dictionary of Greek and Roman Biography
and Mythology.

2 J. Gow, op. cit., p. 195.

GREEK GEOMETRY 31

goreans and Eudoxus, the latter contributing the doctrine of propor-
tion as applicable to incommensurables and also the Method of Ex-
haustions (Book XII), that Theastetus contributed much toward
Books X and XIII, that the principal part of the original work of
Euclid himself is to be found in Book X.1 Euclid was the greatest
systematiser of his time. By careful selection from the material before
him, and by logical arrangement of the propositions fse1ectedr \\e. built
up, from a few definitions and axioms, a proud andlpftystructure.
It would be erroneous to believe that he incorporated into his Elements
all the elementary theorems known at his time. Archimedes, Apol-
lonius, and even he himself refer to theorems not included in his Ele-
ments, as being well-known truths.

The text of the Elements that was commonly used in schools was
Theon's edition. Theon of Alexandria, the father of Hypatia, brought
out an edition, about 700 years after Euclid, with some alterations in
the text. As a consequence, later commentators, especially Robert
Simson, who labored under the idea that Euclid must be absolutely
perfect, made Theon the scapegoat for all the defects which they
thought they could discover in the text as they knew it. But among
the manuscripts sent by Napoleon I from the Vatican to Paris was
found a copy of the Elements believed to be anterior to Theon's recen-
sion. Many variations from Theon's version were noticed therein,
but they were not at all important, and showed that Theon generally
made only verbal changes. The defects in the Elements for which
Theon was blamed must, therefore, be due to Euclid himself. The
Elements used to be considered as offering models of scrupulously
rigorous demonstrations. It is certainly true that in point of rigor
it compares favorably with its modern rivals; but when examined
in the light of strict mathematical logic, it has been pronounced by
C. S. Peirce to be "riddled with fallacies." The results are correct
only because the writer's experience keeps him on his guard. In
many proofs Euclid relies partly upon intuition.

At the beginning of our editions of the Elements, under the head of
definitions, are given the assumptions of such notions as the point,
line, etc., and some verbal explanations^Thenfpllow three postulates
or demands, and twelve axioms. The term "axiom" was used by
Proclus, but not by EucIicT He speaks, instead, of "common no-
tions"— common either to all men or to all sciences. There has been
much Controversy among ancient and modern critics on the postulates
and axioms. An immense preponderance of manuscripts and the
testimony of Proclus place the "axioms" about _ right angles and
parallels among the postulates.2 This is indeed their proper place,

1 G. J. Allman, op. cit., p. 211.

2 A. De Morgan, loc. cit.; H. Hankel, Theorie dcr Complcxen Zahlcnsyslemc, Leip-
zig, 1867, p. 52. In the various editions of Euclid's Elements different numbers are
assigned to the axioms. Thus the parallel axiom is called by Robert Simson the

32 A HISTORY OF MATHEMATICS

for they are really assumptions, and not common notions or axioms.
The postulate about parallels plays an important role in the history
of non-Euclidean geometry. An important postulate which Euclid
missed was the one of superposition, according to which figures can
be moved about in space without any alteration in form or magnitude.
The Elements contains thirteen books by Euclid, and two, of which
it is supposed that Hypsicles and Damascius are the authors. The
first four books are on plane geometry. The fifth book treats of the
theory of proportion as applied to magnitudes in general. It has been
greatly admired because of its rigor of treatment. Beginners find the
book difficult. Expressed in modern symbols, Euclid's definition of
proportion is thus: Four magnitudes, a, b, c, d, are in proportion, when

for any integers m and n, we have simultaneously ma=nb, and me

nd. Says T. L. Heath,1 "certain it is that there is an exact corre-
spondence, almost coincidence, between Euclid's definition of equal
ratios and the modern theory of irrationals due to Dedekind. H. G.
Zeuthen finds a close resemblance between Euclid's definition and
Weierstrass' definition of equal numbers. The sixth book develops
the geometry of similar figures. Its 27th Proposition is the earliest
maximum theorem known to history. The seventh, eighth, ninth
books are on the theory of numbers, or on arithmetic. According to
P. Tannery, the knowledge of the existence of irrationals must have
greatly affected the mode of writing the Elements. The old naive
theory of proportion being recognized as untenable, proportions
are not used at all in the first four books. The rigorous theory of
Eudoxus was postponed as long as possible, because of its difficulty.
The interpolation of the arithmetical books VII-IX is explained
as a preparation for the fuller treatment of the irrational in book X.
Book VII explains the G. C. D. of two numbers by the process
of division (the so-called ''Euclidean method")- The theory of
proportion of (rational) numbers is then developed on the basis of
the definition, "Numbers are proportional when the first is the same
multiple, part, or parts of the second that the third is of the fourth."
This is believed to be the older, Pythagorean theory of proportion.2
The tenth treats of the theory of incommensurables. De Morgan con-
sidered this the most wonderful of all. We give a fuller account of it
under the head of Greek Arithmetic. The next three books are on

1 2th, by Bolyai the nth, by Clavius the i3th, by F. Peyrard the 5th. It is called
the sth postulate in old manuscripts, also by Heiberg and Menge in their annotated
edition of Euclid's works, in Greek and Latin, Leipzig, 1883, and by T. L. Heath
in his Thirteen Books of Euclid's Elements, Vols. I-III, Cambridge, 1908. Heath's
is the most recent translation into English and is very fully and ably annotated.

1 T. L. Heath, op. cit., Vol. II, p. 124.

2 Read H. B. Fine, "Ratio, Proportion and Measurement in the Elements of
Euclid," Annals of Mathematics, Vol. XIX, 1917, pp. 70-76.

GREEK GEOMETRY 33

stereometry. The eleventh contains its more elementary theorems;
the twelfth, the metrical relations of the pyramid, prism, cone, cylinder,
and sphere. The thirteenth treats of the regular polygons, especially
of the triangle and pentagon, and then uses them as faces of the five
regular solids; namely, the tetraedron, octaedron, icosaedron, cube,
and dodecaedron. The regular solids were studied so extensively by
the Platonists that they received the name of "Platonic figures." The
statement of Proclus that the whole aim of Euclid in writing the Ele-
ments was to arrive at the construction of the regular solids, is ob-
viously wrong. The fourteenth and fifteenth books, treating of solid
geometry, are apocryphal. It is interesting to see that to Euclid, and
to Greek mathematicians in general, the existence of areas was evident
from intuition. The notion of non-quadrable areas had not occurred
to them.

A remarkable feature of Euclid's, and of all Greek geometry before
Archimedes is that it eschews mensurajjon. Thus the theorem that
the area of a triangle equals naif the product of its base and its altitude
is foreign to Euclid.

Another extant book of Euclid is the Data. It seems to have been
written for those who, having completed the Elements, wish to acquire
the power of solving new problems proposed to them. The Data is
a course of practice in analysis. It contains little or nothing that an
intelligent student could not pick up from the Elements itself. Hence
it contributes little to the stock of scientific knowledge. The following
are the other works with texts more or less complete and generally
attributed to Euclid: Phenomena, a work on spherical geometry and
astronomy; Optics, which develops the hypothesis that light proceeds
from the eye, and not from the object seen; Catoptrica, containing
propositions on reflections from mirrors: De Divisionibus, a treatise on
the division of plane figures into parts having to one another a given
ratio; l Sectio Canonis, a work on musical intervals. His treatise on
Porisms is lost; but much learning has been expended by Robert Sim-
son and M. Chasles in restoring it from numerous notes found in the
writings of Pappus. The term "porism" is vague in meaning. Ac-
cording to Proclus, the aim of a porism is not to state some property
or truth, like a theorem, nor to effect a construction, like a problem,
but to find and bring to view a thing which necessarily exists with
given numbers or a given construction, as, to find the centre of a given
circle, or to find the G. C. D. of two given numbers. Porisms, ac-
cording to Chasles, are incomplete theorems, "expressing certain
relations between things variable according to a common law."
Euclid's other lost works are Fallacies, containing exercises in detec-
tion of fallacies; Conic Sections, in four books, which are the foundation
of a work on the same subject by Apollonius; and Loci on a Surface,

1 A careful restoration was brought out in 1915 by R. C. Archibald of Brown
University.

HISTORY OF MATHEMATICS

the meaning of which title is not understood. Heiberg believes it to
mean "loci which are surfaces."

The immediate successors of Euclid in the mathematical school at
Alexandria were probably Conon, Dositheus, and Zeuxippus, but
little is known of them.

Archimedes (2^^-212 B. c.), the greatest mathematician of an-
tiquity, was born in Syracuse. Plutarch calls him a relation of King
Hieron; but more reliable is the statement of Cicero, who tells us he
was of low birth. Diodorus says he visited Egypt, and, since he was
a great friend of Conon and Eratosthenes, it is highly probable that
he studied in Alexandria. This belief is strengthened by the fact that
he had the most thorough acquaintance with all the work previously
done in mathematics. He returned, however, to Syracuse, where he
made himself useful to his admiring friend and patron, King Hieron,
by applying his extraordinary inventive genius to the construction of
various war-engines, by which he inflicted much loss on the Romans
during the siege of Marcellus. The story that, by the use of mirrors
reflecting the sun's rays, he set on fire the Roman ships, when they
came within bow-shot of the walls, is probably a fiction. The city
was taken at length by the Romans, and Archimedes perished in the
indiscriminate slaughter which followed. According to tradition, he
was, at the time, studying the diagram to some problem drawn in the
sand. As a Roman soldier approached him, he called out, " Don't spoil
my circles." The soldier, feeling insulted, rushed upon him and killed
him. No blame attaches to the Roman general Marcellus, who ad-
mired his genius, and raised in his honor a tomb bearing the figure
of a sphere inscribed in a cylinder. When Cicero was in Syracuse,
he found the tomb buried under rubbish.

Archimedes was admired by his fellow-citizens chiefly for his me-
chanical inventions; he himself prized far more highly his discoveries
in pure science. He declared that "every kind of art which was con-
nected with daily needs was ignoble and vulgar." Some of his works
have been lost. The following are the extant books, arranged ap-
proximately in chronological order: i. Two books on Equiponderance
of Planes or Centres of Plane Gravities, between which is inserted his
treatise on the Quadrature of the Parabola; 2. The Method; 3. Two books
on the Sphere and Cylinder; 4. The Measurement of the Circle; 5. On
Spirals; 6. Conoids and Spheroids; 7. The Sand-Counter; 8. Two books
on Floating Bodies; 9. Fifteen Lemmas.

In the book on the Measurement of the Circle, Archimedes proves
first that the area of a circle is equal to that of a right triangle having
the length of the circumference for its base, and the radius for its
altitude. In this he assumes that there exists a straight line equal in
length to the circumference — an assumption objected to by some
ancient critics, on the ground that it is not evident that a straight
line can equal a curved one. The finding of such a line was the next

GREEK GEOMETRY 35

problem. He first finds an upper limit to the ratio of the circumfer-
ence to the diameter, or TT. To do this, he starts with an equilateral
triangle of which the base is a tangent and the vertex is the centre of
the circle. By successively bisecting the angle at the centre, by com-
paring ratios, and by taking the irrational square roots always a little
too small, he finally arrived at the conclusion that 7T<3|. Next he
finds a lower limit by inscribing in the circle regular polygons of 6, 12,
24, 48, 96 sides, finding for each successive polygon its perimeter,
which is, of course, always less than the circumference. Thus he
finally concludes that "the circumference of a circle exceeds three
times its diameter by a part which is less than ~ but more than ±~
of the diameter." This approximation is exact enough for most pur-
poses.

The Quadrature of the Parabola contains two solutions to the prob-
lem— one mechanical, the other geometrical. The method of ex-
haustion is used in both.

It is noteworthy that, perhaps through the influence of Zeno, in-
finitesimals (infinitely small constants) were not used in rigorous
demonstration. In fact, the great geometers of the period now under
consideration resorted to the radical measure of excluding them from
demonstrative geometry by a pos^ujate. This was done by Eudoxus,
Euclid, and Archimedes. In the preface to the Ouadrature of the Parab-
ola, occurs the so-called "Archimedean postulate," which Archimedes
himself attributes to Eudoxus: "When two spaces are unequal, it is
possible to add to itself the difference by which the lesser is surpassed
by the greater, so often that every finite space will be exceeded."
, Euclid (Elements V, 4) gives the postulate in the form of a definition :
^"Magnitudes are said to have a ratio to one another, when the less
lean be multiplied so as to exceed the other." Nevertheless, infinitesi-
Wals may have been used in tentative researches. That such was the
case with Archimedes is evident from his book, The Method, formerly
thought to be irretrievably lost, but fortunately discovered by Heiberg
in 1906 in Constantinople. The contents of this book shows that he
considered infinitesimals sufficiently scientific to suggest the truths of
theorems, but not to furnish rigorous proofs. In finding the areas of
parabolic segments, the volumes of spherical segments and other solids
of revolution, he uses a mechanical process, consisting of the weighing
of infinitesimal elements, which he calls straight lines or plane areas,
but which are really infinitely narrow strips or infinitely thin plane
laminae.1 The breadth or thickness is regarded as being the same in
the elements weighed at any one time. The Archimedean postulate
did not command the interest of mathematicians until the modern
arithmetic continuum was created. It was O. Stolz that showed that
it was a consequence of Dedekind's postulate relating to "sections."

1 T. L. Heath, Method of Architnedcs, Cambridge, 1912, p. 8.

36 A HISTORY OF MATHEMATICS

It would seem that, in his great researches, Archimedes' mode of
procedure was, to start with mechanics (centre of mass of surfaces and
solids) and by his infinitesimal-mechanical method to discover new
results for which later he deduced and published the rigorous proofs.
Archimedes knew the integral * fx3dx.

Archimedes studied also the ellipse and accomplished its quadrature,
but to the hyperbola he seems to have paid less attention. It is be-
lieved that he wrote a book on conic sections.

Of all his discoveries Archimedes prized most highly those in his
f Sphere and Cylinder. In it are proved the new theorems, that the
surface of a sphere is equal to four times a great circle ; that the surface;
segment of a sphere is equal to a circle whose radius is the straight
line drawn from the vertex of the segment to the circumference of its
basal circle; that the volume and the surface of a sphere are | of the
volume and surface, respectively, of the cylinder circumscribed about
the sphere. Archimedes desired that the figure to the last proposition
be inscribed on his tomb. This was ordered done by Marcellus.

The spiral now called the "spiral of Archimedes," and described in
the book On Spirals, was discovered by Archimedes, and not, as some
believe, by his friend Conon.2 His treatise thereon is, perhaps, the
most wonderful of all his works. Nowadays, subjects of this kind
are made easy by the use of the infinitesimal calculus. In its stead
the ancients used the method of exhaustion. Nowhere is the fertility
of his genius more grandly displayed than in his masterly use of this
method. With Euclid and his predecessors the method of exhaustion
was only the means of proving propositions which must have been
seen and believed before they were proved. But in the hands of
Archimedes this method, perhaps combined with his infinitesimal-
mechanical method, became an instrument of discovery.

By the word "conoid," in his book on Conoids and Spheroids, is
meant the solid produced by the revolution of a parabola or a hyper-
bola about its axis. Spheroids are produced by the revolution of an
ellipse, and are long or flat, according as the ellipse revolves around
the major or minor axis. The book leads up to the cubature of these

solids. A few constructions of geo-
metric figures were given by Archi-
medes and Appolonius which were
effected by "insertions." In the
following trisection of an angle, at-
tributed by the Arabs to Archi-
medes, the "insertion" is achieved by the aid of a graduated ruler.3
To trisect the angle CAB, draw the arc BCD. Then "insert" the

1 H. G. Zeuthen in Bibliotheca mathematica, 3 S., Vol. 7, 1906-7, p. 347.

2 M. Cantor, op. cit., Vol. I, 3 Aufl., 1907, p. 306.

3 F. Enriques, Fragen der Elementar geometric, deutsche Ausg. v. H. Fleischer, II,
Leipzig, 1907, p. 234.

GREEK GEOMETRY 37

distance FE, equal to AB, marked on an edge passing through C
and moved until the points E and F are located as shown in the
figure. The required angle is EFD.

His arithmetical treatise and problems will be considered later.
We shall now notice his works on mechanics. Archimedes is the
author of the first sound knowledge on this subject. Archytas, Aris-
totle, and others attempted to form the known mechanical truths into
a science, but failed. Aristotle knew the property of the lever, but
could not establish its true mathematical theory. The radical and
fatal defect in the speculations of the Greeks, in the opinion of Whewell,
was "that though they had in their possession facts and ideas, the
ideas were not distinct and appropriate to the fads." For instance,
Aristotle asserted that when a body at the end of a lever is moving,
it may be considered as having two motions; one in the direction of
the tangent and one in the direction of the radius; the former motion
is, he says, according to nature, the latter contrary to nature. These
inappropriate notions of "natural" and "unnatural" motions, to-
gether with the habits of thought which dictated these speculations,
made the perception of the true grounds of mechanical properties
impossible. It seems strange that even after Archimedes had en-
tered upon the right path, this science should have remained ab-
solutely stationary till the time of Galileo — a period of nearly two
thousand years.

The proof of the property of the lever, given in his Equiponderance
of Planes, holds its place in many text-books to this day. Mach 2
criticizes it. "From the mere assumption of the equilibrium of equal
weights at equal distances is derived the inverse proportionality of
weight and lever arm! How is that possible? " Archimedes' estimate
of the efficiency of the lever is expressed in the saying attributed to
him, " Give me a fulcrum on which to rest, and I will move the earth."

While the Equiponderance treats of solids, or the equilibrium of
solids, the book on Floating Bodies treats of hydrostatics. His atten-
tion was first drawn to the subject of specific gravity when King Hieron
asked him to test whether a crown, professed by the maker to be pure
gold, was not alloyed with silver. The story goes that our philosopher
was in a bath when the true method of solution flashed on his mind.
He immediately ran home, naked, shouting, "I have found it!" To
solve the problem, he took a piece of gold and a piece of silver, each
weighing the same as the crown. According to one author, he deter-
mined the volume of water displaced by the gold, silver, and crown
respectively, and calculated from that the amount of gold and silver

1 William Whewell, History of the Inductive Sciences, 3rd Ed., New York, 1858,
Vol. I, p. 87. William Whewell (1794-1866) was Master of Trinity College, Cam-
bridge.

2 E. Mach, The Science of Mechanics, tr. by T. McCgrmack, Chicago, 1907, p. 14.
Ernst Mach (1838-1916) was professor of the history and theory of the inductive
sciences at the university of Vienna.

38 A HISTORY OF MATHEMATICS

in the crown. According to another writer, he weighed separately
the gold, silver, and crown, while immersed in water, thereby deter-
mining their loss of weight in water. From these data he easily found
the solution. It is possible that Archimedes solved the problem by
both methods.

After examining the writings of Archimedes, one can well under-
stand how, in ancient times, an "Archimedean problem" came to
mean a problem too deep for ordinary minds to solve, and how an
"Archimedean proof" came to be the synonym for unquestionable
certainty. Archimedes wrote on a very wide range of subjects, and
displayed great profundity in each. He is the Newton of antiquity.

Eratosthenes, eleven years younger than Archimedes, was a native
of Gyrene. He was educated in Alexandria under Callimachus the
poet, whom he succeeded as custodian of the Alexandrian Library.
His many-sided activity may be inferred from his works. He wrote
on Good and Evil, Measurement of the Earth, Comedy, Geography,
Chronology, Constellations, and the Duplication of the Cube. He was
also a philologian and a poet. He measured the obliquity of the
ecliptic and invented a device for finding prime numbers, to be de-
scribed later. Of his geometrical writings we possess only a letter to
Ptolemy Euergetes, giving a history of the duplication problem and
also the description of a very ingenious mechanical contrivance of his
own to solve it. In his old age he lost his eyesight, and on that account
is said to have committed suicide by voluntary starvation.

About forty years after Archimedes flourished Apollonius of Perga,
whose genius nearly equalled that of his great predecessor. He incon-
testably occupies the second place in distinction among ancient mathe-
maticians. Apollonius was born in the reign of Ptolemy Euergetes
and died under Ptolemy Philopator, who reigned 222-205 B- c- He
studied at Alexandria under the successors of Euclid, and for some
time, also, at Pergamum, where he made the acquaintance of that
Eudemus to whom he dedicated the first three books of his Conic
Sections. The brilliancy of his great work brought him the title of the
"Great Geometer." This is all that is known of his life.

His Conic Sections were in eight books, of which the first four only
have come down to us in the original Greek. The next three books
were unknown in Europe till the middle of the seventeenth century,
when an Arabic translation, made about 1250, was discovered. The
eighth book has never been found. In 1710 E. Halley of Oxford pub-
lished the Greek text of the first four books and a Latin translation
of the remaining three, together with his conjectural restoration of
the eighth book, founded on the introductory lemmas of Pappus. The
first four books contain little more than the substance of what earlier
geometers had done. Eutocius tells us that Heraclides, in his life of
Archimedes, accused Appolonius of having appropriated, in his Conic
Sections, the unpublished discoveries of that great mathematician.

GREEK GEOMETRY 39

It is difficult to believe that this charge rests upon good foundation.
Eutocius quotes Geminus as replying that neither Archimedes nor
Apollonius claimed to have invented the conic sections, but that
Apollonius had introduced a real improvement. While the first three
or four books were founded on the works of Menaechmus, Aristaeus,
Euclid, and Archimedes, the remaining ones consisted almost entirely
of new matter. The first three books were sent to Eudemus at inter-
vals, the other books (after Eudemus's death) to one Attalus. The
preface of the second book is interesting as showing the mode in
which Greek books were "published" at this time. It reads thus:
"I have sent my son Apollonius to bring you (Eudemus) the second
book of my Conies. Read it carefully and communicate it to such
others as are worthy of it. If Philonides, the geometer, whom I intro-
duced to you at Ephesus, comes into the neighbourhood of Pergamum,
give it to him also." l

The first book, says Apollonius in his preface to it, "contains the
mode of producing the three sections and the conjugate hyperbolas
and their principal characteristics, more fully and generally, worked
out than in the writings of other authors." We remember that
Menaechmus, and all his successors down to Apollonius, considered only
sections of right cones by a plane perpendicular to their sides, and that
ths three sections were obtained each from a different cone. Apol-
lonius introduced an important generalisation. He produced all the
sections from one and the same cone, whether right or scalene, and
by sections which may or may not be perpendicular to its sides. The
old names for the three curves were now no longer applicable. Instead
of calling the three curves, sections of the "acute-angled," "right-
angled," and "obtuse-angled" cone, he called them ellipse, parabola,
and hyperbola, respectively. To be sure, we find the words "parabola "
and "ellipse" in the works of Archimedes, but they are probably only
interpolations. The word "ellipse" was applied because yz<px, p
being the parameter; the word "parabola" was introduced because
y2 = px, and the term "hyperbola" because y2>px.

The treatise of Apollonius rests on a unique property of conic sec-
tions, which is derived directly from the nature of the cone in which
these sections are found. How this property forms the key to the
system of the ancients is told in a masterly way by M. Chasles.2
"Conceive," says he, "an oblique cone on a circular base; the straight
line drawn from its summit to the centre of the circle forming its base
is called the axis of the cone. The plane passing through the axis,
perpendicular to its base, cuts the cone along two lines and determines
in the circle a diameter; the triangle having this diameter for its base

1 H. G. Zeuthen, Die Lclirc wit den Kcgckcluiiltai im Altcrthum, Kopenhagen,
1886, p. 502.

2 M. Chasles, Geschichle der Cicomrlric. Aus dem Franzosischcn ubertragen dun h,
Dr. L. A. Sohnckc, Halle, 1839, p. 15.

40 A HISTORY OF MATHEMATICS

and the two lines for its sides, is called the triangle through the <m:
In the formation of his conic sections, Apollonius supposed the cuttin
plane to be perpendicular to the plane of the triangle through th
axis. The points in which this plane meets the two sides of this tr
angle are the vertices of the curve; and the straight line which join
these two points is a diameter of it. Apollonius called this diamete
latus transversum. At one of the two vertices of the curve erect a pei
pendicular (latus rectum) to the plane of the triangle through th
axis, of a certain length, to be determined as we shall specify latei
and from the extremity of this perpendicular draw a straight line t
the other vertex of the curve; now, through any point whatever c
the diameter of the curve, draw at right angles an ordinate: the squar
of this ordinate, comprehended between the diameter and the curv<
will be equal to the rectangle constructed on the portion of the ordinat
comprised between the diameter and the straight line, and the pai
of the diameter comprised between the first vertex and the foot of th
ordinate. Such is the characteristic property which Apollonius reco£
nises in his conic sections and which he uses for the purpose of ir
ferring from it, by adroit transformations and deductions, nearly a
the rest. It plays, as we shall see, in his hands, almost the same rol
as the equation of the second degree with two variables (abscissa an
ordinate) in the system of analytic geometry of Descartes." Apo
lonius made use of co-ordinates as did Menaechmus before him
Chasles continues:

"It will be observed from this that the diameter of the curve an
the perpendicular erected at one of its extremities suffice to construe
the curve. These are the two elements which the ancients used, wit
which to establish their theory of conies. The perpendicular in que<
tion was called by them latus erectum; the moderns changed this nam
first to that of latus rectum, and afterwards to that of parameter."

The first book of the Conic Sections of Apollonius is almost wholl
devoted to the generation of the three principal conic sections.

The second book treats mainly of asymptotes, axes, and diameter;

The third book treats of the equality or proportionality of triangles
rectangles, or squares, of which the component parts are determine
by portions of transversals, chords,' asymptotes, or tangents, whic
are frequently subject to a great number of conditions. It also touche
the subject of foci of the ellipse and hyperbola.

In the fourth book, Apollonius discusses the harmonic division c
straight lines. He also examines a system of two conies, and show
that they cannot cut each other in more than four points. He inves
tigates the various possible relative positions of two conies, as, fc
instance, when they have one or two points of contact with each othei

The fifth book reveals better than any other the giant intellect c
its author. Difficult questions of maxima and minima, of which fe^
1T. L. Heath, Apollonius of Perga, Cambridge, 1896, p. CXV.

GREEK GEOMETRY 41

examples are found in earlier works, are here treated most exhaustively.
The subject investigated is, to find the longest and shortest lines that
can be drawn from a given point to a conic. Here are also found the
germs of the subject of evolutes and centres of osculation.

The sixth book is on the similarity of conies.

The seventh book is on conjugate diameters.

The eighth book, as restored by Halley, continues the subject of
conjugate diameters.

It is worthy of notice that Apollonius nowhere introduces the
notion of directrix for a conic, and that, though he incidentally dis-
covered the focus of an ellipse and hyperbola, he did not discover the
focus of a parabola.1 Conspicuous in his geometry is also the absence
of technical terms and symbols, which renders the proofs long and
cumbrous. R. C. Archibald claims that Apollonius was familiar with
the centres of similitude of circles, usually attributed to Monge.
T. L. Heath2 comments thus: "The principal machinery used by
Apollonius as well as by the earlier geometers comes under the head
of what has been not inappropriately called a geometrical algebra."

The discoveries of Archimedes and Apollonius, says M. Chasles,
marked the most brilliant epoch of ancient geometry. Two questions
which have occupied geometers of all periods may be regarded as
having originated with them. The first of these is the quadrature of
curvilinear figures, which gave birth to the infinitesimal calculus. The
second is the theory of conic sections, which was the prelude to the
theory of geometrical curves of all degrees, and to that portion of
geometry which considers only the forms and situations of figures
and uses only the intersection of lines and surfaces and the ratios of
rectilineal distances. These two great divisions of geometry may be
designated by the names of Geometry of Measurements and Geometry
of Forms and Situations, or, Geometry of Archimedes and of Apol-
lonius.

Besides the Conic Sections, Pappus ascribes to Apollonius the fol-
lowing works: On Contacts, Plane Loci, Inclinations, Section of an Area,
Determinate Section, and gives lemmas from which attempts have
been made to restore the lost originals. Two books on De Sectione
Rationis have been found in the Arabic. The book on Contacts, as
restored by F. Vieta, contains the so-called "Apollonian Problem":
Given three circles, to find a fourth which shall touch the three.

Euclid, Archimedes, and Apollonius brought geometry to as high
a state of perfection as it perhaps could be brought without first in-
troducing some more general and more powerful method than the old
method of exhaustion. A briefer symbolism, a Cartesian geometry,
an infinitesimal calculus, were needed. The Greek mind was not

1 J. Gow, op. ell., p. 252.

2 T. L. Heath, Apollonius of Perga, edited in modern notation. Sambridge,

p. ci.

42 A HISTORY OF MATHEMATICS

adapted to the invention of general methods. Instead of a climb t<
still loftier heights we observe, therefore, on the part of later Creel
geometers, a descent, during which they paused here and there to lool
around for details which had been passed by in the hasty ascent.1

Among the earliest successors of Apollonius was Nicomedes. Noth
ing definite is known of him, except that he invented the conchou
("mussel-like"), a curve of the fourth order. He devised a littl
machine by which the curve could be easily described. With aid o
the conchoid he duplicated the cube. The curve can also be used fo
trisecting angles in a manner resembling that in the eighth lemma o
Archimedes. Proclus ascribes this mode of trisection to Nicomedes
but Pappus, on the other hand, claims it as his own. The conchoi<
was used by Newton in constructing curves of the third degree.

About the time of Nicomedes (say, 180 B. c.), flourished alsi
Diocles, the inventor of the cissoid ("ivy-like"). This curve he use<
for finding two mean proportionals between two given straight line?
The Greeks did not consider the companion-curve to the cissoid
in fact, they considered only the part of the cissoid proper whid
lies inside the circle used in constructing the curve. The part of th
area of the circle left over when the two circular areas on the concav
sides of the branches of the curve are removed, looks somewhat lik
an ivy-leaf. Hence, probably, the name of the curve. That the tw
branches extend to infinity appears to have been noticed first by G. P
de Roberal in 1640 and then by R. de Slusc.2

About the life of Perseus we know as little as about that of Nico
medes and Diocles. He lived some time between 200 and 100 B. c
From Heron and Geminus we learn that he wrote a work on the spirt
a sort of anchor-ring surface described by Heron as being produced b;
the revolution of a circle around one of its chords as an axis. Th
sections of this surface yield peculiar curves called spiral sections
which, according to Geminus, were thought out by Perseus. Thes
curves appear to be the same as the Hippopede of Eudoxus.

Probably somewhat later than Perseus lived Zenodorus. He wrot
an interesting treatise on a new subject; namely, isoperimetrical figures
Fourteen propositions are preserved by Pappus and Theon. Her
are a few of them: Of isoperimetrical, regular polygons, the one havin:
the largest number of angles has the greatest area; the circle has
greater area than any regular polygon of equal periphery; of all isc
perimentrical polygons of n sides, the regular is the greatest; c
all solids having surfaces equal in area, the sphere has the greates
volume.

Hypsicles (between 200 and 100 B. c.) was supposed to be th
author of both the fourteenth and fifteenth books of Euclid, but recen
critics are of opinion that the fifteenth book was written by an autho

1 M. Cantor, op. cit., Vol. I, 3 AufL, 1907, p. 350.

2 G. Loria, Ebenc Ciirvcn, transl. by F. Schiitte, I, 1910, p. 37.

who lived several centuries after Christ. The fourteenth book con-
tains seven elegant theorems on regular solids. A treatise of Hypsicles
on Risings is of interest because it gives the division of the circum-
ference into 360 degrees after the fashion of the Babylonians.

Hipparchus of Nicaea in Bithynia was the greatest astronomer of
antiquity. He took astronomical observations between 161 and 127
B. c. He established inductively the famous theory of epicycles and
eccentrics. As might be expected, he was interested in mathematics,
not per se, but only as an aid to astronomical inquiry. No mathe-
matical writings of his are extant, but Theon of Alexandria informs us
that Hipparchus originated the science of trigonometry, and that he
calculated a "table of chords" in twelve books. Such calculations
must have required a ready knowledge of arithmetical and algebraical
operations. He possessed arithmetical and also graphical devices for
solving geometrical problems in* a plane and on a sphere. He gives
indication of having seized the idea of co-ordinate representation, found
earlier in Apollonius.

About ico B. c. flourished Heron the Elder of Alexandria. He was
the pupil of Ctesibius, who was celebrated for his ingenious mechanical
inventions, such as the hydraulic organ, the water-clock, and catapult.
It is believed by some that Heron was a son of Ctesibius. He ex-
hibited talent of the same order as did his master by the invention of
the eolipile and a curious mechanism known as " Heron's fountain."
Great uncertainty exists concerning his writings. Most authorities
believe him to be the author of an important Treatise on the Dioptra,
of which there exist three manuscript copies, quite dissimilar. But
M. Marie x thinks that the Dioptra is the work of Heron the Younger,
who lived in the seventh or eighth century after Christ, and that
Geodesy, another book supposed to be by Heron, is only a corrupt and
defective copy of the former work. Dioptra contains the important
formula for finding the area of a triangle expressed in terms of its
sides; its derivation is quite laborious and yet exceedingly ingenious.
"It seems to me difficult to believe," says Chasles, "that so beautiful
a theorem should be found in a work so ancient as that of Heron the
Elder, without that some Greek geometer should have thought to
cite it." Marie lays great stress on this silence of the ancient writers,
and argues from it that the true author must be Heron the Younger
or some writer much more recent than Heron the Elder. But no re-
liable evidence has been found that there actually existed a second
mathematician by the name of Heron. P. Tannery has shown that,
in applying this formula, Heron found the irrational square roots by

the approximation, \/A~z(a-}-—), where a2 is the square nearest to

1 Maximilien Marie, Histoirr drs .sr/V;/<r.y matktHUiiyutS ft physiques. Par!-;,
Tunic I, 1883, p. 178.

44 A HISTORY OF MATHEMATICS

in the place of a in the above formula. Apparently, Heron som<
times found square and cube roots also by the method of "doub
false position."

"Dioptra," says Venturi, were instruments which had great n
semblance to our modern theodolites. The book Dioptra is a treatis
on geodesy containing solutions, with aid of these instruments, of
large number of questions in geometry, such as to find the distam
between two points, of which one only is accessible, or between t\\
points which are visible but both inaccessible; from a given point 1
draw a perpendicular to a line which cannot be approached; to fin
the difference of level between two points; to measure the area of
field without entering it.

Heron was a practical surveyor. This may account for the fa<
that his writings bear so little resemblance to those of the Gree
authors, who considered it degrading the science to apply geometry 1
surveying. The character of his geometry is not Grecian, but di
cidedly Egyptian. This fact is the more surprising when we considi
that Heron demonstrated his familiarity with Euclid by writing a con
mentary on the Elements. Some of Heron's formulas point to an ol
Egyptian origin. Thus, besides the above exact formula for the an

of a triangle in terms of its sides, Heron gives the formula - —X
which bears a striking likeness to the formula — - -X — - ~ f<

2 2

finding the area of a quadrangle, found in the Edfu inscription
There are, moreover, points of resemblance between Heron's writing
and the ancient Ahmes papyrus. Thus Ahmes used unit-fractior
exclusively (except the fraction |.) ; Heron uses them of tener than oth<
fractions. Like Ahmes and the priests at Edfu, Heron divides con
plicated figures into simpler ones by drawing auxiliary lines; like then
he shows, throughout, a special fondness for the isosceles trapezoid.

The writings of Heron satisfied a practical want, and for that reasc
were borrowed extensively by other peoples. We find traces of thei
in Rome, in the Occident during the Middle Ages, and even in Indi;

The works attributed to Heron, including the newly discovere
Metrica published in 1903, have been edited by J. H. Heiberj
H. Schone and W. Schmidt.

Geminus of Rhodes (about 70 B. c.) published an astronomical wor
still extant. He wrote also a book, now lost, on the Arrangement <
Mathematics, which contained many valuable notices of the earl
history of Greek mathematics. Proclus and Eutocius quote it fr<
quently. Theodosius is the author of a book of little merit on th

GREEK GEOMETRY 45

geometry of the sphere. Investigations due to P. Tannery and A. A.
Bjornbo J seem to indicate that the mathematician Theodosius was
not Theodosius of Tripolis, as formerly supposed, but was a resident
of Bithynia and contemporary of Hipparchus. Dionysodorus of
Amisus in Pontus applied the intersection of a parabola and hyperbola
to the solution of a problem which Archimedes, in his Sphere and
Cylinder, had left incomplete. The problem is "to cut a sphere so
that its segments shall be in a given ratio."

We have now sketched the progress of geometry down to the time
of Christ. Unfortunately, very little is known of the history of geom-
etry between the time of Apollonius and the beginning of the Christian
era. The names of quite a number of geometers have been mentioned,
but very few of their works are now extant. It is certain, however,
that there were no mathematicians of real genius from Apollonius to
Ptolemy, excepting Hipparchus and perhaps Heron.

The Second Alexandrian School

The close of the dynasty of the Lagides which ruled Egypt from the
time of Ptolemy Soter, the builder of Alexandria, for 300 years; the
absorption of Egypt into the Roman Empire; the closer commercial
relations between peoples of the East and of the West; the gradual
decline of paganism and spread of Christianity, — these events were
of far-reaching influence on the progress of the sciences, which then
had their home in Alexandria. Alexandria became a commercial and
intellectual emporium. Traders of all nations met in her busy streets,
and in her magnificent Library, museums, lecture-halls, scholars from
the East mingled with those of the West; Greeks began to study older
literatures and to compare them with their own. In consequence of
this interchange of ideas the Greek philosophy became fused with
Oriental philosophy. Neo-Pythagoreanism and Neo-Platonism were
the names of the modified systems. These stood, for a time, in op-
position to Christianity. The study of Platonism and Pythagorean
mysticism led to the revival of the theory of numbers. Perhaps the
dispersion of the Jews and their introduction to Greek learning helped
in bringing about this revival. The th£Qiy__of numbers became a
favorite study. This new line of mathematical inquiry ushered in
what we may call a new school. There is no doubt that even now
geometry continued to be one of the most important studies in the
Alexandrian course. This Second Alexandrian School may be said to
begin with the Christian era. It was made famous by the names of
Claudius Ptolemaeus, Diophantus, Pappus, Theon of Smyrna, Theon
of Alexandria, lamblichus, Porphyrius, and others.

By the side of these we may place Serenus of Antinceia, as having

1 Axel Anthon Bjornbo (1874-1911) of Copenhagen was a historian of mathe-
matics. See Blbllolheca mathemalica, 3 S., Vol. 12, 1911-12, pp. 337~344-

46 A HISTORY OF MATHEMATICS

been connected more or less with this new school. He wrote on sec-
tions of the cone and cylinder, in two books, one of which treated
only of the triangular section of the cone through the apex. He solved
the problem, "given a cone (cylinder), to find a cylinder (cone), so
that the section of both by the same plane gives similar ellipses." Of
particular interest is the following theorem, which is the foundation
of the modern theory of harmonics: If from D we draw DF, cutting
the triangle ABC, and choose H on it, so that DE : DF — ER : HF,
and if we draw the line AH, then every transversal through D, such
as DG, will be divided by AH so that DK : DG= KJ : JG. Menelaus
of Alexandria (about 98 A. D.) was the author of Sphcerica, a work
extant in Hebrew and Arabic, but not in Greek. In it he proves the

theorems on the congruence of
spherical triangles, and describes
their properties in much the same
way as Euclid treats plane tri-
angles. In it are also found the
theorems that the sum of the three
sides of a spherical triangle is less
than a great circle, and that the
sum of the three angles exceeds two right angles. Celebrated are two
theorems of his on plane and spherical triangles. The one on plane tri-
angles is that, " if the three sides be cut by a straight line, the product of
the three segments which have no common extremity is equal to the.
product of the other three." L. N. M. Carnot makes this proposition,
known as the "lemma of Menelaus," the base of his theory of trans-
versals. The corresponding theorem for spherical triangles, the so-
called "regula sex quantitatum," is obtained from the above by
reading "chords of three segments doubled," in place of "three seg-
ments."

Claudius Ptolemy, a celebrated astronomer, was a native of Egypt.
Nothing is known of his personal history except that he flourished in
Alexandria in 139 A. D. and that he made the earliest astronomical
observations recorded in his works, in 125 A. D., the latest in 151 A. D.
The chief of his works are the Syntaxis Mathematica (or the Almagest,
as the Arabs call it) and the Geographica, both of which are extant.
The former work is based partly on his own researches, but mainly
on those of Hipparchus. Ptolemy seems to have been not so much of
an independent investigator, as a corrector and improver of the work
of his great predecessors. The Almagest l forms the foundation of
all astronomical science down to N. Copernicus. The fundamental
idea of his system, the "Ptolemaic System," is that the earth is in the
centre of the universe, and that the sun and planets revolve around
the earth. Ptolemy did considerable for mathematics. He created,

1 On the importance of the Almagest in the history of astronomy, consult P.
Tannery, Recherches sur I'histoire de I' astronomic, Paris, 1893.

GREEK GEOMETRY 47

for astronomical use, a trigonometry remarkably perfect in form. The
foundation of this science was laid by the illustrious Hipparchus.

The Almagest is in 13 books. Chapter 9 of the first book shows how
to calculate tables of chords. The circle is divided into 360 degrees,
each of which is halved. The diameter is divided into 120 divisions;
each of these into 60 parts, which are again subdivided into 60 smaller
parts. In Latin, these parts were called paries minuta primes and
paries minuta secundcr. Hence our names, "minutes" and "seconds."
The sexagesimal method of dividing the circle is of Babylonian origin,
and was known to Geminus and Hipparchus. But Ptolemy's method
of calculating chords seems original with him. He first proved the
proposition, now appended to Euclid VI (/)), that "the rectangle
contained by the diagonals of a quadrilateral figure inscribed in a
circle is equal to both the rectangles contained by its opposite sides."
He then shows how to find from the chords of two arcs the chords of
their sum and difference, and from the chord of any arc that of its
half. These theorems he applied to the calculation of his tables of
chords. The proofs of these theorems are very pretty. Ptolemy's
construction of sides of a regular inscribed pentagon and decagon was
given later by C. Clavius and L. Mascheroni, and now is used much
by engineers. Let the radius BD be _L to AC,
DE=EC. Make EF = EB, then BF is the side of
the pentagon and DF is the side of the decagon.

Another chapter of the first book in the Alma-
gest is devoted to trigonometry, and to spherical
trigonometry in particular. Ptolemy proved the
"lemma of Menelaus," and also the "regula sex quantitatum."
Upon these propositions he built up his trigonometry. In trigono-
metric computations, the Greeks did not use, as*did the Hindus, half
the chord of twice the arc (the "sine"); the Greeks used instead
the whole chord of double the arc. Only in graphic constructions,
referred to again later, did Ptolemy and his predecessors use half the
chord of double the arc. The fundamental theorem of plane trigo-
nometry, that two sides of a triangle are to each other as the chords
of double the arcs measuring the angles opposite the two sides, was
not stated explicitly by Ptolemy, but was contained implicitly in other
theorems. More complete are the propositions in spherical trigo-
nometry.

The fact that trigonometry was cultivated not for its own sake, but
to aid astronomical inquiry, explains the rather startling fact that
spherical trigonometry came to exist in a developed state earlier than
plane trigonometry.

The remaining books of the Almagest are on astronomy. Ptolemy
has written other works which have little or no bearing on mathe-
matics, except one on geometry. Extracts from this book, made by
Proclus, indicate that Ptolemy did not regard the parallel-axiom of

48 A HISTORY OF MATHEMATICS

Euclid as self-evident, and that Ptolemy was the first of the long line
of geometers from ancient time down to our own who toiled in the vain
attempt to prove it. The untenable part of his demonstration is the
assertion that, in case of parallelism, the sum of the interior angles on
one side of a transversal must be the same as their sum on the other
side of the transversal. Before Ptolemy an attempt to improve the
theory of parallels was made by Posidonius (first cent. B. c.) who de-
fined parallel lines as lines that are coplanar and equidistant. From
an Arabic writer, Al-Nirizi (ninth cent.) it appears that Simplicius
brought forward a proof of the 5th postulate, based upon this def-
inition, and due to his friend Aganis (Geminus?).1
• In the making of maps of the earth's surface and of the celestial
sphere, Ptolemy (following Hipparchus) used stereographic projection.
The eye is imagined to be at one of the poles, the projection being
thrown upon the equatorial plane. He devised an instrument, a form
of astrolabe planisphere, which is a stereographic projection of the
celestial sphere.2 Ptolemy wrote a monograph on the analemma which
was a figure involving orthographic projections of the celestial sphere
upon three mutually perpendicular planes (the horizontal, meridian
and vertical circles) . The analemma was used in determining positions
of the sun, the rising and setting of the stars. The procedure was
probably known to Hipparchus and the older astronomers. It fur-
nished a graphic method for the solution of spherical triangles and was
used subsequently by the Hindus, the Arabs, and Europeans as late
as the seventeenth century.3

Two prominent mathematicians of this time wrere Nicomachus and
Theon of Smyrna. Their favorite study was theory of numbers.
The investigations in this science culminated later in the algebra of
Diophantus. But no important geometer appeared after Ptolemy
for 150 years. An occupant of this long gap was Sextus Julius
Africanus, who wrote an unimportant work on geometry applied
to the art of war, entitled Cestes. Another was the sceptic, Sextus
Empiricus (200 A. D.); he endeavored to elucidate Zeno's "Arrow"
by stating another argument equally paradoxical and therefore far
from illuminating: Men never die, for if a man die, it must either
be at a time when he is alive, or at a time when he is not alive;
hence he never dies. Sextus Empiricus advanced also the paradox,
that, when a line rotating in a plane about one of its ends describes
a circle with each of its points, these concentric circles are of un-
equal area, yet each circle must be equal to the neighbouring circle
which it touches.1

1 R. Bonola, Non-Euclidean Geometry, trans, by H. S. Carslaw, Chicago, 1912,
pp. 3-8. Robert Bonola (1875-1911) was professor in Rome.

2 See M. Latham, " The Astrolabe," Am. Math. Monthly, Vol. 24, 1917, p. 162.

3 See A. v. Braunmiihl, Geschichte dcr Trigonometric, Leipzig, I, 1900, p. n.
Alexander von Braunmiihl (1853-1908) was .prof essor at the technical high school
in Munich.

GREEK GEOMETRY 49

Pappus, probably born about 340 A. D., in Alexandria, was the
last great mathematician of the Alexandrian school. His genius was
inferior to that of Archimedes, Apollonius, and Euclid, who flourished
over 500 years earlier. But living, as he did, at a period when interest
in geometry was declining, he towered above his contemporaries "like
the peak of Teneriffa above the Atlantic." He is the author of a
Commentary on the Almagest, a Commentary on Euclid's Elements, a
Commentary on the Analemma of Diodorus, — a writer of whom nothing
is known. All these works are lost. Proclus, probably quoting from
the Commentary on Euclid, says that Pappus objected to the state-
ment that an angle equal to a right angle is always itself a right
angle.

The only work of Pappus still extant is his Mathematical Collections.
This was originally in eight books, but the first and portions of the
second are now missing. The Mathematical Collections seems to have
been written by Pappus to supply the geometers of his time with a
succinct analysis of the most difficult mathematical works and to
facilitate the study of them by explanatory lemmas. Hut these
lemmas are selected very freely, and frequently have little or no con-
nection with the subject on hand. However, he gives very accurate
summaries of the works of which he 'treats. The Mathematical Col-
lections is invaluable to us on account of the rich information it gives
on various treatises by the foremost Greek mathematicians, which
are now lost. Mathematicians of the last century considered it pos-
sible to restore lost work from the resume by Pappus alone.

We shall now cite the more important of those theorems in the
Mathematical Collections which are supposed to be original with
Pappus. First of all ranks the elegant theorem re-discovered by P.
Guldin, over 1000 years later, that the volume generated by the
revolution of a plane curve which lies wholly on one side of the axis,
equals the area of the curve multiplied by the circumference de-
scribed by its center of gravity. Pappus proved also that the centre
of gravity of a triangle is that of another triangle whose vertices lie
upon the sides of the first and divide its three sides in the same ratio.
In the fourth book are new and brilliant propositions on the quadra-
trix which indicate an intimate acquaintance with curved surfaces.
He generates the quadratrix as follows: Let a spiral line be drawn
upon a right circular cylinder; then the perpendiculars to the axis
of the cylinder drawn from each point of the spiral line form the
surface of a screw. A plane passed through one of these perpendicu-
lars, making any convenient angle with the base of the cylinder, cuts
the screw-surface in a curve, the orthogonal projection of which upon
the base is the quadratrix. A second mode of generation is no less
admirable: If we make the spiral of Archimedes the base of a right

JSee K. Lasswitz, Gescklchte der Atomisiik, I, Hamburg und Leipzig, 1890,
p. 148.

50 A HISTORY OF MATHEMATICS

cylinder, and imagine a cone of revolution having for its axis the side
of the cylinder passing through the initial point of the spiral, then
this cone cuts the cylinder in a curve of double curvature. The per-
pendiculars to the axis drawn through every point in this curve form
the surface of a sqrew which Pappus here calls the plectoidal surface.
A plane passed through one of the perpendiculars at any convenient
angle cuts that surface in a curve whose orthogonal projection upon
the plane of the spiral is the required quadratrix. Pappus considers
curves of double curvature still further. He produces a spherical
spiral by a point moving uniformly along the circumference of a
great circle of a sphere, while the great circle itself revolves uniformly
around its diameter. He then finds the area of that portion of the
surface of the sphere determined by the spherical spiral, "a complana-
tion which claims the more lively admiration, if we consider that,
although the entire surface of the sphere was known since Archimedes'
time, to measure portions thereof, such as spherical triangles, was
then and for a long time afterwards an unsolved problem." A
question which was brought into prominence by Descartes and Newton
is the "problem of Pappus." Given several straight lines in a plane,
to find the locus of a point such that when perpendiculars (or, more
generally, straight lines at given angles) are drawn from it to the
given lines, the product of certain ones of them shall be in a given
ratio to the product of the remaining ones. It is worth noticing that
it was Pappus who first found the focus of the -parabola and pro-
pounded the theory of the involution of points. He used the directrix
and was the first to put in definite form the definition of the conic
sections as loci of those points whose distances from a fixed point
and from a fixed line are in a constant ratio. He solved the problem
to draw through three points lying in the same straight line, three
straight lines which shall form a triangle inscribed in a given circle.
From the Mathematical Collections many more equally difficult the-
orems might be quoted which are original with Pappus as far as we
know. It ought to be remarked, however, that he has been charged
in three instances with copying theorems without giving due ciedit,
and that he may have done the same thing in other cases in which
we have no data by which to ascertain the real discoverer.2

About the time of Pappus lived Theon of Alexandria. He brought
out an edition of Euclid's Elements with notes, which he probably
used as a text-book in his classes. His commentary on the Almagest
is valuable for the many historical notices, and especially for the
specimens of Greek arithmetic which it contains. Theon's daughter
Hypatia, a woman celebrated for her beauty and modesty, was the
last Alexandrian teacher of reputation, and is said to have been an

1 M. Cantor, op. cit., Vol. I, 3 Aufl., 1907, p. 451.

2 For a defence of Pappus against these charges, see J. H. Weaver in Bull. Am.
Math. Soc., Vol. 23, 1916, pp. 131-133.

GREEK GEOMETRY 51

abler philosopher and mathematician than her father. Her notes on
the works of Diophantus and Apollonius have been lost. Her tragic
death in 415 A. D. is vividly described in Kingsley's Hypatia.

From now on, mathematics ceased to be cultivated in Alexandria.
The leading subject of men's thoughts was Christian theology.
Paganism disappeared, and with it pagan learning. The Neo-Platonic
school at Athens struggled on a century longer. Proclus, Isidorus, and
others kept up the "golden chain of Platonic succession." Proclus,
the successor of Syrianus, at the Athenian school, wrote a commentary
on Euclid's Elements. We possess only that on the first book, which
is valuable for the information it contains on the history of geometry.
Damascius of Damascus, the pupil of Isidorus, is now believed to be
the author of the fifteenth book of Euclid. Another pupil of Isidorus
was Eutocius of Ascalon, the commentator of Apollonius and Archi-
medes. Simplicius wrote a commentary on Aristotle's De Casio.
Simplicius reports Zeno as saying: "That which, being added to
another, does not make it greater, and being taken away from another
does not make it less, is nothing." According to this, the denial of
the existence of the infinitesimal goes back to Zeno. This momentous
question presented itself centuries later to Leibniz, who gave different
answers. The report made by Simplicius of the quadratures of Anti-
phon and Hippocrates of Chios is one of the best sources of historical
information on this point.1 In the year 529, Justinian, disapproving
heathen learning, finally closed by imperial edict the schools at
Athens.

As a rule, the geometers of the last 500 years showed a lack of
creative power. They were commentators rather than discoverers.

The principal characteristics of ancient geometry are: —

(1) A wonderful clearness and definiteness of its__concepts and an
almost perfect logical rigor of its conclusions.

(2) A complete want of general principles and methods. Ancient
geometry is decidedly special. Thus the Greeks possessed no general
method of drawing tangents. "The determination of the tangents
to the three conic. sections did not furnish any rational assistance for
drawing the tangent to any other new curve, such as the conchoid,
the cissoid, etc." In the demonstration of a theorem, there were, for
the ancient geometers, as many different cases requiring separate
proof as there were different positions for^he lines. The greatest
geometers considered it necessary to treat all possible cases inde-
pendently of each other, and to prove each with equal fulness. To
devise methods by which the various cases could all be disposed of
by one stroke, was beyond the power of the ancients. "If we com-
pare a mathematical problem with a huge rock, into the interior of
which we desire to penetrate, then the work of the Greek mathe-

1 See F. Rudio in Bibliothcca mathcmatica, 3 S., Vol. 3, 1902, pp. 7-62.

52 A HISTORY OF MATHEMATICS

maticians appears to us like that of a vigorous stonecutter who, with
chisel and hammer, begins with indefatigable perseverance, from with-
out, to crumble the rock slowly into fragments; the modern mathe-
matician appears like an excellent miner, who first bores through the
rock some few passages, from which he then bursts it into pieces
with one powerful blast, and brings to light the treasures within." *

Greek Arithmetic and Algebra

Greek mathematicians were in the habit of discriminating between
the science of numbers and the art of calculation. The former they
called arithmetica, the latter logistica. The drawing of this distinction
between -the- two was very natural and proper. The difference be-
tween them is as marked as that between theory and practice. Among
the Sophists the art of calculation was a favorite study. Plato, on
the other hand, gave considerable attention to philosophical arith-
metic, but pronounced calculation a vulgar and childish art.

In sketching the history of Greek calculation, we shall first give a
brief account of the Greek mode of counting and of writing numbers.
Like the Egyptians and Eastern nations, the earliest Greeks counted
on their fingers or with pebbles. In case of large numbers, the pebbles
were probably arranged in parallel vertical lines. Pebbles on the
first line represented units, those on the second tens, those on the
third hundreds, and so on. Later, frames came into use, in which
strings or wires took the place of lines. According to tradition,
Pythagoras, who travelled in Egypt and, perhaps, in India, first
introduced this valuable instrument into Greece. The abacus, as it
is called, existed among different peoples and at different times, in
various stages of perfection. An abacus is still employed by the
Chinese under the name of Swan-pan. We possess no specific informa-
tion as to how the Greek abacus looked or how it was used. Boethius
says that the Pythagoreans used with the abacus certain nine signs
called apices, which resembled in form the nine "Arabic numerals."
But the correctness of this assertion is subject to grave doubts.

The oldest Grecian numerical symbols were the so-called Herodianic
signs (after Herodianus, a Byzantine grammarian of about 200 A. D.,
who describes them). These signs occur frequently in Athenian in-
scriptions and are, on that account, now generally called Attic. For
some unknown reason these symbols were afterwards replaced by the
alphabetic numerals, in which the letters of the Greek alphabet were
used, together with three strange and antique letters £, 9 , and "P),
and the symbol M. This change was decidedly for the worse, for the
old Attic numerals were less burdensome on the memory, inasmuch

1 H. Hankel, Die Entwickelung der Mathematik in den lelzten Jahrhundertcn.
Tubingen, 1884, p. 16.

as they contained fewer symbols and were better adapted to show
forth analogies in numerical operations. The following table shows
the Greek alphabetic numerals and their respective values: —

afty8es£-i)0i K A /* v £ o IT <?

i 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90

po-r v $ x ^ w ~TD <a *P >y etc-

zoo 200 300 400 500 600 700 800 900 1000 2000 3ooo

ft y

M M M etc.

10,000 20,000 30,000

It will be noticed that at 1000, the alphabet is begun over again,
but, to prevent confusion, a stroke is now placed before the letter
and generally somewhat below it. A horizontal line drawn over a
number served to distinguish it more readily from words. The co-
efficient for M was sometimes placed before or behind instead of over
the M. Thus 43,678 was written SM/yx01?- It is to be observed that
the Greeks had no zero.

Fractions were denoted by first writing the numerator marked with
an accent, then the denominator marked with two accents and written
twice. Thus, iy'/<0"K0" = |-|. In case of fractions having unity for
the numerator, the a' was omitted and the denominator was written
only once. Thus fi8" = ^.

The Greeks had the name epimorion for the ratio - . •. Archytas
proved the theorem that if an epimorion » is reduced to its lowest

terms -, then v=/n-f-i. This theorem is found later in the musical

writings of Euclid and of the Roman Boethius. The Euclidean form
of arithmetic, without perhaps the representation of numbers by lines,
existed as early as the time of Archytas.1

Greek writers seldom refer to calculation with alphabetic numerals.
Addition, subtraction, and even multiplication were probably per-
formed on the abacus. Expert mathematicians may have used the
symbols. Thus Eutocius, a commentator of the sixth century after
Christ, gives a great many multiplications of which the following is
a specimen: 2 —

1 P. Tannery in Bibliotheca mathemalica, 3 S., Vol. VI, 1905, p. 228.

2 J. Gow, op. cit., p. 50.

54 A HISTORY OF MATHEMATICS

— T— The operation is explained suf-

; " 5 ficiently by the modern numerals

°"£e 265 appended. In case of mixed

~5~~o numbers, the process was still

MM/^a 40000, 12000, looo more clumsy. Divisions are found

a in Theon of Alexandria's com-

M^/rx1" 12000, 3600, 300 mentary on the Almagest. As

.ar/ce iooo, 300, 25 might be expected, the process is

-> — long and tedious.

M O-KC 70225 We have seen in geometry that

the more advanced mathematicians

frequently had occasion to extract the square root. Thus Archimedes
in his Mensuration of the Circle gives a large number of square roots.
He states, for instance, that -v/3<-VW- an^ V^3>TTF ' ^ut ^e g^ves no
clue to the method by which he obtained these approximations. It
is not improbable that the earlier Greek mathematicians found the
square root by trial only. Eutocius says that the method of extracting
it was given by Heron, Pappus, Theon, and other commentators on
the Almagest. Theon's is the only one of these methods known to us.
It is the same as the one used nowadays, except that sexagesimal
fractions are employed in place of our decimals. What the mode of
procedure actually was when sexagesimal fractions were not used, has
been the subject of conjecture on the part of numerous modern writers.

Of interest, in connection with arithmetical symbolism, is the Sand-
Counter (Arenarius), an essay addressed by Archimedes to Gelon,
king of Syracuse. In it Archimedes shows that people are in error who
think the sand cannot be counted, or that if it can be counted, the
number cannot be expressed by arithmetical symbols. He shows that
the number of grains in a heap of sand not only as large as the whole
earth, but as large as the entire universe, can be arithmetically ex-
pressed. Assuming that 10,000 grains of sand suffice to make a little
solid of the magnitude of a poppy-seed, and that the diameter of a
poppy-seed be not smaller than ~ part of a finger's breadth; assuming
further, that the diameter of the universe (supposed to extend to the
sun) be less than 10,000 diameters of the earth, and that the latter
be less than 1,000,000 stadia, Archimedes finds a number which would
exceed the number of grains of sand in the sphere of the universe.
He goes on even further. Supposing the universe to reach out to the
fixed stars, he finds that the sphere, having the distance from the
earth's centre to the fixed stars for its radius, would contain a number
of grains of sand less than iooo myriads of the eighth octad. In our
notation, this number would be io63 or i with 63 ciphers after it. It
can hardly be doubted that one object which Archimedes had in view
in making this calculation was the improvement of the Greek sym-
bolism. It is not known whether he invejited some short notation by
which to represent the above number or not.

GREEK ARITHMETIC AND ALGEBRA 55

We judge from fragments in the second book of Pappus that Apol-
lonius proposed an improvement in the Greek method of writing
numbers, but its nature we do not know. Thus we see that the Greeks
never possessed the boon of a clear, comprehensive symbolism. The
honor of giving such to the world was reserved by the irony of fate
for a nameless Indian of an unknown time, and we know not whom to
thank for an invention of such importance to the general progress of
intelligence.1

Passing from the subject of logistica to that of arithmetica, our at-
tention is first drawn to the scienclfoTnumbers of Pythagoras. Before
founding his school, Pythagoras studied for many years under the
Egyptian priests and familiarised himself with Egyptian mathematics
and mysticism. If he ever was in Babylon, as some authorities claim,
he may have learned the sexagesimal notation in use there; he may
have picked up considerable knowledge on the theory of proportion,
and may have found a large number of interesting astronomical
observations. Saturated with that speculative spirit then pervading
the Greek mind, he endeavored to discover some principle, oi-bQmo-
geneitv in the, universe. Before him, the philosophers of the Ionic
school had sought iT* m the matter of things; Pythagoras looked for
it in the structure of things. He observed various numerical relations
or analogies between numbers and the phenomena of the universe.
Being convinced that it was in numbers and their relations that he
was to find the foundation to true philosophy, he proceeded to trace
the origin of all things to numbers. Thus he observed that musical
strings of equal length stretched by weights having the proportion of
i> •§> f> produced intervals which were an octave, a fifth, and a fourth.
Harmony, therefore, depends on musical, proportion; it is nothing but
a mysterious numerical relation. Where harmony is, there are
numbers. . Hence the order and beauty of the universe have their
origin in numbers. There are seven intervals in the musical scale,
and also seven planets crossing the heavens. The same numerical
relations which underlie the former must underlie the latter. But
where numbers are, there is harmony. Hence his spiritual ear dis-
cerned in the planetary motions a wonderful ''harmony of the spheres."
The Pythagoreans invested particular numbers with extraordinary
attributes. Thus one is the essence of things; it is an absolute number;
hence the origin of all numbers and so of all things. Four is the most
perfect number, and was in some mystic way conceived to correspond
to the human soul. Philolaus believed that 5 is the cause of color, 6 of
cold, 7 of mind and health and light, 8 of love and friendship.2 In
Plato's works are evidences of a similar belief in religious relations of
numbers. Even Aristotle referred the virtues to numbers.

Enough has been said about these mystic speculations to show
what lively interest in mathematics they must have created and

1 J. Gow, op. cit., p. 63. - J. Cow, op. ciL, p. 69.

56 A HISTORY OF MATHEMATICS

maintained. Avenues of mathematical inquiry were opened up by
them which otherwise would probably have remained closed at that
time.

The Pythagoreans classified numbers into odd and even. They
observed that the sum of the series of odd numbers from i to 2n + 1
was always a complete square, and that by addition of the even num-
bers arises the series 2, 6, 12, 20, in which every number can be de-
composed into two factors differing from each other by unity. Thus,
6=2.3, 12 = 3.4, etc. These latter numbers were considered of
sufficient importance to receive the separate name of heteromecic (not

4? ('Yl ~L- T )

equilateral). Numbers of the form were called triangular,

•

because they could always be arranged thus, ,\V. Numbers which
were equal to the sum of all their possible factors, such as 6, 28, 496,
were called perfect; those exceeding that sum, excessive; and those
which were less, defective. Amicable numbers were those of which
each was the sum of the factors in the other. Much attention was
paid by the Pythagoreans to the subject of proportion. The quan-
tities a, b, c, d were said to be in arithmetical proportion when a — b =
c—d; in geometrical proportion, when a:b = c:d; in harmonic propor-
tion, when a—b:b—c=a:c. It is probable that the Pythagoreans

were also familiar with the musical proportion a: = -'.b.

2 a+b

lamblichus says that Pythagoras introduced it from Babylon.

In connection with arithmetic, Pythagoras made extensive investi-
gations into geometry. He believed that an arithmetical fact had
its analogue in geometry, and vice versa. In connection with his
theorem on the right triangle he devised a rule by which integral
numbers could be found, such that the sum of the squares of two of
them equalled the square of the third. Thus, take for one side an odd

(2/2-{-i)2— i
number (2^+1); then =2»2+2W=the other side, and

(2^2+ zn-\- 1) = hypotenuse. If in-\- 1 = 9, then the other two numbers
are 40 and 41. But this rule only applies to cases in which the hy-
potenuse differs from one of the sides by i. In the study of the right
triangle there doubtless arose questions of puzzling subtlety. Thus,
given a number equal to the side of an isosceles right triangle, to find
the number which the hypotenuse is equal to. The side may have
been taken equal to i, 2, |, |-, or any other number, yet in every in-
stance all efforts to find a number exactly equal to the hypotenuse
must have remained fruitless. The problem may have been attacked
again and again, until finally " some rare genius, to whom it is granted,
during some happy moments, to soar with eagle's flight above the
level of human thinking," grasped the happy thought that this prob-

GREEK ARITHMETIC AND ALGEBRA 57

lem cannot be solved. In some such manner probably arose the theory
of irrational quantities, which is attributed by Eudemus to the Pytha-
goreans. It was indeed a thought of extraordinary boldness, to as-
sume that straight lines could exist, differing from one another not
only in length, — that is, in quantity, — but also in a quality, which,
though real, was absolutely invisible.1 Need we wonder that the
Pythagoreans saw in irrationals a deep mystery, a symbol of the un-
speakable? We are told that the one who first divulged the theory of
irrationals, which the Pythagoreans kept secret, perished in conse-
quence in a shipwreck, "for the unspeakable and invisible should
always be kept secret." Its discovery is ascribed to Pythagoras, but
we must remember that all important Pythagorean discoveries were,
according to Pythagorean custom, referred back to him. The first
incommensurable ratio known seems to have been that of the side
of a square to its diagonal, as i :\/2- Theodoras of Gyrene added to
this the fact that the sides of squares represented in length by A/3,
A/5, etc., up to A/i7, and Theaetetus, that the sides of any square,
represented by a surd, are incommensurable with the linear unit.
Euclid (about 300 B. c.), in his, Elements, X, 9, generalised still further:
Two magnitudes whose squares are (or are not) to one another as a
square number to a square number are commensurable (or incom-
mensurable), and conversely. In the tenth book, he treats of incom-
mensurable quantities at length. He investigates every possible

variety of lines which can be represented by A/Va^ Vb, a and b
representing two commensurable lines, and obtains 25 species. Every
individual of every species is incommensurable with all the individuals
of every other species. "This book," says De Morgan, "has a com-
pleteness which none of the others (not even the fifth) can boast of;
and we could almost suspect that Euclid, having arranged his ma-
terials in his own mind, and having completely elaborated the tenth
book, wrote the preceding books after it, and did not live to revise
them thoroughly." 2 The theory of incommensurables remained
where Euclid left it, till the fifteenth centilry.

If it be recalled that the early Egyptians had some familiarity with
quadratic equations, it is not surprising if similar knowledge is dis-
played by Greek Biters in the time of Pythagoras. Hippocrates, in
the fifth century B.»C., when working on the areas of lunes, assumes
the geometrical equivalent of the solution of the quadratic equation
#2-|-A/f ax=a2. The complete geometrical solution was given by
Euclid in his Elements, VI, 27-29. He solves certain types of quad-
ratic equations geometrically in Book II, 5, 6, n.

1 H. Hankel, Zur Geschichte der Malhematik in Mitldalter und Alterthum, 1874,
p. 102.

- A. I)e Morgan, "Eucleides" in Smith's Dictionary of Greek and Roman Biog.

and Myth.

58 A HISTORY OF MATHEMATICS

Euclid devotes the seventh, eighth, and ninth books of his Elements
to arithmetic. Exactly how much contained in these books is Euclid's
own invention, and how much is borrowed from his predecessors, we
have no means of knowing. Without doubt, much is original with
Euclid. The seventh book begins with twenty-one definitions. All
except that for "prime" numbers are known to have been given by
the Pythagoreans. Next follows a process for finding the G. C. D.
of two or more numbers. The eighth book deals with numbers in con-
tinued proportion, and with the mutual relations of squares, cubes,
and plane numbers. Thus, XXII, if three numbers are in continued
proportion, and the first is a square, so is the third. In the ninth book,
the same subject is continued. It contains the proposition that the
number of primes is greater than any given number.

After the death of Euclid, the theory of numbers remained almost
stationary for 400 years. Geometry monopolised the attention of all
Greek mathematicians. Only two are known to have done work in
arithmetic worthy of mention. Eratosthenes (275-194 B. c.) invented
a " sieve" for finding prime numbers. All composite numbers are
"sifted" out in the following manner: Write down the odd numbers
from 3 up, in succession. By striking out every third number after
the 3, we remove all multiples of 3. By striking out every fifth num-
ber after the 5, we remove all multiples of 5. In this way, by rejecting
multiples of 7, n, 13, etc., we have left prime numbers only. Hyp-
sicles (between 200 and 100 B. c.) worked at the subjects of polygonal
numbers and arithmetical progressions, which Euclid entirely neg-
lected. In his work on "risings of the stars," he showed (i) that in
an arithmetical series of 2» terms, the sum of the last n terms exceeds
the sum of the first n by a multiple of w2; (2) that in such a series of
2n-\- 1 terms, the sum of the series is the number of terms multiplied
by the middle term; (3) that in such a series of in terms, the sum is
half the number of terms multiplied by the two middle terms.1

For two centuries after the time of Hypsicles, arithmetic disappears
from history. It is brought to light again about 100 A. D. by Ni-
comachus, a Neo-Pythagorean, who inaugurated the final era of Greek
mathematics. From now on, arithmetic was a favorite study, while
geometry was neglected. Nicomachus wrote a work entitled In-
troductio Arithmetica, which was very famous in its day. The great
number of commentators it has received vouch for its popularity.
Boethius translated it into Latin. Lucian could pay no higher com-
pliment to a calculator than this: "You reckon like Nicomachus of
Gerasa." The Introductio Arithmetica was the first exhaustive work
in which arithmetic was treated quite independently of geometry.
Instead of drawing lines, like Euclid, he illustrates things by real
numbers. To be sure, in his book the old geometrical nomenclature is
retained, but the method is inductive instead of deductive. "Its sole
1 J. Gow, op. cit., p. 87.

GREEK ARITHMETIC AND ALGEBRA 59

business is classification, and all its classes are derived from, and
exhibited by, actual numbers." The work contains few results that
are really original. We mention one important proposition which is
probably the author's own. He states that cubical numbers are al-
ways equal to the sum of successive odd numbers. Thus, 8=23 =
3+5. 27 = 33=7+9+n, 64= 43=13+15+ 17+ 19, and so on. This
theorem was used later for finding the sum of the cubical numbers
themselves. Theon of Smyrna is the author of a treatise on " the
mathematical rules necessary for the study of Plato." The work is
ill arranged and of little merit. Of interest is the theorem, that every
square number, or that number minus i, is divisible by 3 or 4 or both.
A remarkable discovery is a proposition given by lamblichus in his
treatise on Pythagorean philosophy. It is founded on the observation
that the Pythagoreans called i, 10, 100, 1000, units of the first, second,
third, fourth "course" respectively. The theorem is this: If we add
any three consecutive numbers, of which the highest is divisible by 3,
then add the digits of that sum, then, again, the digits of that sum,
and so on, the final sum will be 6. Thus, 61+62+63= I^6, 1+8+6 =
I5t IH~5 = 6. This discovery was the more remarkable, because the
ordinary Greek numerical symbolism was much less likely to suggest
any such property of numbers than our "Arabic" notation would
have been.

Hippolytus, who appears to have been bishop at Portus Romae in
Italy in the early part of the third century, must be mentioned for the
giving of "proofs" by casting out the Q'S and the 7's.

The works of Nicomachus, Theon of Smyrna, Thymaridas, and
others contain at times investigations of subjects which are really
algebraic iri their nature. Thymaridas in one place uses the Greek,
word meaning "unknown quantity" in a way which would lead one
to believe that algebra was not far distant. Of interest in tracing the
invention of algebra are the arithmetical epigrams in the Palatine
Anthology, which contain about fifty problems leading to linear equa-
tions. Before the introduction of algebra these problems were pro-
pounded as puzzles. A riddle attributed to Euclid and contained in
the Anthology is to this effect: A mule and a donkey were walking
along, laden with corn. The mule says to the donkey, "If you gave
me one measure, I should carry twice as much as you. If I gave you
one, we should both carry equal burdens. Tell me their burdens, O
most learned master of geometry." *

It will be allowed, says Gow, that this problem, if authentic, was
not beyond Euclid, and the appeal to geometry smacks of antiquity.
A far more difficult puzzle was the famous "cattle-problem," which
Archimedes propounded to the Alexandrian mathematicians. The
problem is indeterminate, for from only seven equations, eight un-
known quantities in integral numbers are to be found. It may be
1 J. Gow, op. dt., p. 99.

60 A HISTORY OF MATHEMATICS

stated thus: The sun had a herd of bulls and cows, of different colors.
(i) Of Bulls, the white (W) were, in number, (^-f-^) of the blue (B)
and yellow (Y): the B were (£+£) of the Y and piebald (P): the P
were (£-f-y) of the W and Y. (2) Of Cows, which had the same colors
(w, b, y, p},

(JF-Fw).

Find the number of bulls and cows.1 This leads to high numbers,
but, to add to its complexity, the conditions are superadded that
W-|-B = a square, and P-f-Y=a triangular number, leading to an in-
determinate equation of the second degree. Another problem in the
Anthology is quite familiar to school-boys: "Of four pipes, one fills the
cistern in one day, the next in two days, the third in three days, the
fourth in four days: if all run together, how soon will they fill the
cistern?" A great many of these problems, puzzling to an arith-
metician, would have been solved easily by an algebraist. They be-
came very popular about the time of Diophantus, and doubtless acted
as a powerful stimulus on his mind.

Diophantus was one of the last and most fertile mathematicians of
the second Alexandrian school. He flourished about 250 A. D. His
age was eighty-four, as is known from an epitaph to this effect: Dio-
phantus passed ^ of his life in childhood, -^ in youth, and \ more as
a bachelor; five years after his marriage was born a son who died four
years before his father, at half his father's age. The place of nativity
and parentage of Diophantus are unknown. If his works were not
written in Greek, no one would think for a moment that they were
the product of Greek mind. There is nothing in his works that
reminds us of the classic period of Greek mathematics. His were al-
most entirely new ideas on a new subject. In the circle of Greek
mathematicians he stands alone in his specialty. Except for him,
we should be constrained to say that among the Greeks algebra was
almost an unknown science.

Of his works we have lost the Porisms, but possess a fragment of
Polygonal Numbers, and seven books of his great work on Arithmetica,
said to have been written in 13 books. Recent editions of the Arith-
metica were brought out by the indefatigable historians, P. Tannery
and T. L. Heath, and by G. Wertheim.

If we except the Ahmes papyrus, which contains the first sugges-
tions of algebraic notation, and of the solution of equations, then his
Arithmetica is the earliest treatise on algebra now extant. In this work
is introduced the idea of an algebraic equation expressed in algebraic
symbols. His treatment is purely analytical and completely divorced
from geometrical methods. He states that "a number to be sub-

1 J. Gow, op. tit., p. 99.

GREEK ARITHMETIC AND ALGEBRA 61

tracted, multiplied by a number to be subtracted, gives a number to
be added." This is applied to the multiplication of differences, such
as (x— i) (x—2). It must be remarked, that Diophantus had no
notion whatever of negative numbers standing by themselves. All
he knew were differences, such as (2^—10), in which 2x could not be
smaller than 10 without leading to an absurdity. He appears to be
the first who could perform such operations as (x— i)x(x— 2) without
reference to geometry. Such identities as (a +b)2 = a2 + 2ab +b2, which
with Euclid appear in the elevated rank of geometric theorems, are
with Diophantus the simplest consequences of the algebraic laws of
operation. His sign for subtraction was 1*, for equality i. For un-
known quantities he had only one symbol, s. He had no sign for
addition except juxtaposition. Diophantus used but few symbols,
and sometimes ignored even these by describing an operation in words
when the symbol would have answered just as well.

In the solution of simultaneous equations Diophantus adroitly
managed with only one symbol for the unknown quantities and ar-
rived at answers, most commonly, by the method of tentative assump-
tion, which consists in assigning to some of the unknown quantities
preliminary values, that satisfy only one or two of the conditions.
These values lead to expressions palpably wrong, but which generally
suggest some stratagem by which values can be secured satisfying
all the conditions of the problem.

Diophantus also solved determinate equations of the second degree.
Such equations were solved geometrically by Euclid and Hippocrates.
Algebraic solutions appear to have been found by Heron of Alexandria,
who gives 8| as an approximate answer to the equation 144.^(14 — 0!;) =
6720. In the Geometry, doubtfully attributed to Heron, the solution of
the equation ^x2-\-'2^-x =212 is practically stated in the form x =

\/(i54X 212 +841) - 20

— — -^ -. Diophantus nowhere goes through with the

whole process of solving quadratic equations; he merely states the
result. Thus, tl84x2+jx =7, whence x is found =£." From partial
explanations found here and there it appears that the quadratic equa-
tion was so written that all terms were positive. Hence, from the point
of view of Diophantus, there were three cases of equations with a
positive root: axz+bx =c, ax2 =bx+c, ax*+c =bx, each case requiring
a rule slightly different from the other two. Notice he gives only one
root. His failure to observe that a quadratic equation has two roots,
even when both roots are positive, rather surprises us. It must be
remembered, however, that this same inability to perceive more than
one out of the several solutions to which a problem may point is com-
mon to all Greek mathematicians. Another point to be observed
.is that he never accepts as an answer a quantity which is negative
Cor irrational.

62 A HISTORY OF MATHEMATICS

Diophantus devotes only the first book of his Arilhmetica to the
solution of determinate equations. The remaining books extant
treat mainly of indeterminate quadratic equations of the form Ax2 +
Bx+C =y2, or of two simultaneous equations of the same form. He
considers several but not all the possible cases which may arise in
these equations. The opinion of Nesselmann on the method of Dio-
phantus, as stated by Gow, is as follows: " (i) Indeterminate equations
of the second degree are treated completely only when the quadratic
or the absolute term is wanting: his solution of the equations Ax2 +
C =y2 and Ax2+Bx+C—y2 is in many respects cramped. (2) For
the ' double equation ' of the second degree he has a definite rule only
when the quadratic term is wanting in both expressions: even then
his solution is not general. More complicated expressions occur only
under specially favourable circumstances." Thus, he solves Bx+C2
=y2, £iX+Ci2=;yi2.

The extraordinary ability of Diophantus lies rather in another di-
rection, namely, in his wonderful ingenuity to reduce all sorts of
equations to particular forms which he knows how to solve. Very
great is the variety of problems considered. The 130 problems found
in the great work of Diophantus contain over 50 different classes of
problems, which are strung together without any attempt at classi-
fication. But still more multifarious than the problems are the solu-
tions. General methods are almost unknown to Dipohantus. Each
problem has its own distinct method, which is often useless for the
most closely related problems. "It is, therefore, difficult for a modern,
after studying 100 Diophantine solutions, to solve the loist." This
statement, due to Hankel, is somewhat overdrawn, as is shown by
Heath, i

That which robs his work of much of its scientific value is the
fact that he always feels satisfied with one solution, though his equa-
tion may admit of an indefinite number of values. Another great
defect is the absence of general methods. Modern mathematicians,
such as L. Euler, J. Lagrange, K. F. Gauss, had to begin the study of
indeterminate analysis anew and received no direct aid from Dio-
phantus in the formulation of methods. In spite of these defects
we cannot fail to admire the work for the wonderful ingenuity ex-
hibited therein in the solution of particular equations.

1 T. L. Heath, Diophantus of Alexandria, 2 Ed., Cambridge, 1910, pp. 54-97.

THE ROMANS

Nowhere is the contrast between the Greek and Roman minds
shown forth more distinctly than jn_their attitude toward the nmthe-
matical science. The sway of the Greek was a flowering time for
mathematics, but that of ihe_Rjoman .a periojLoLsterility. In philos-
ophy, poetry, and art the Roman was an imitator. BiiLJn mathe-
matics he did not even rise to the desire for imitaiian The mathe-
matical fruits of Greek genius lay before him untasted. In him a
science which had no direct bearing on practical life could awake no
interest. As a consequence, not only the higher geometry of Archi-
medes and Apollonius, but even the Elements of Euclid, were neglected.
What ]jttle mathematics the Romans possessed did not come altogether
from the Greeks, buLcame in parti rom more ancient sources. Exactly
_where and how some of it originated is a matter of doubt. It seems
most probable that the 1' Roman notation," as well as the early
practical geometry ..of the Romans, came from the old Etruscans,
w_ho^at the earliest period to which our knowledge of them extends,
inhabited the district between the Arno and Tiber.

Livy tells us that the Eimscans were in the habit of representing

the_number of years elapsed, by driving yearly a nail into the sancr.

tuary of Minerva^ and that the Romans continued this practice. ^A

less primitive mode of designating numbers, presumably of Etruscan

^origin, was a notation resembling the present "Roman notation."

This system is noteworthy from the fact that a principle is involved

in it which is rarely met with in others, namely, the principle of sub-

Jraction. If a letter be placed before another of greater value, its

"value is not to be added to, but subtracted from, that of the greater.

In the designation of large numbers a horizontal bar placed over a

letter was made to increase its value one thousand fold. In fractions

the Romans used the duodecimal system.

Of arithmetical calculations, the Romans employed three different
kinds: Reckoning on the fingers, upon the abacus, and by tables pre-
pared for the purpose.1 Finger-symbolism was known as early as the
time of King Numa, for he had erected, says Pliny, a statue of the
double-faced Janus, of which the fingers indicated 365 (355?), the
number of days in a year. Many other passages from Roman authors
point out the use of the fingers as aids to calculation. In fact, a finger-
symbolism of practically the same form was in use not only in Rome,
but also in Greece and throughout,, the East, certainly as early as the
beginning of the Christian era, and continued to be used in Europe

1 M. Cantor, op. cit., Vol. I, 3 Aufl., 1907, p. 526.
63

A HISTORY OF MATHEMATICS

during the Middle Ages. We possess no knowledge as to where or when
it was invented. The second mode of calculation, by the abacus, was
a subject of elementary instruction in Rome. Passages in Roman
writers indicate that the kind of abacus most commonly used was
covered with dust and then divided into columns by drawing straight
lines. Each column was supplied with pebbles (calculi, whence "cal-
culare" and "calculate") which served for calculation.

The Romans used also another kind of abacus, consisting of a
metallic plate having grooves with movable buttons. By its- use all
integers between i and 9,999,999, as well as some fractions, could be
represented. In the two adjoining figures l the lines represent grooves

C X T c X I •£

C X T C X I

and the circles buttons. The Roman numerals indicate the value of
each button in the corresponding groove below, the button in the
shorter groove above having a fivefold value. Thus If = 1,000,000;
hence each button in the long left-hand groove, when in use, stands
for 1,000,000, and the button in the short upper groove stands for
5,000,000. The same holds for the other grooves labelled by Roman
numerals. The eighth long groove from the left (having 5 buttons)
represents duodecimal fractions, each button indicating JL, while the
button above the dot means — •. In the ninth column the upper
button represents ~^, the middle ~, and two lower each T^. Our
first figure represents the positions of the buttons before the operation
begins; our second figure stands for the number 852 | -±±. The eye
has here to distinguish the buttons in use and those left idle. Those
counted are one button above c (=500), and three buttons below
c ( =300) ; one button above x ( = 50) ; two buttons below I ( = 2) ; four
buttons indicating duodecimals ( = -5-); and the button for ~^-

Suppose now that 10,318 | i J^ is to be added to 852 ^ -^. The
operator could begin with the highest units, or the lowest units, as he
pleased. Naturally the hardest part is the addition of the fractions.

1 G. Friedlein, Die Zahlzeichen und das elementare Rechnen der Griechen und R'omer,
Erlangen, 1869, Fig. 21. Gottfried Friedlein (1828-1875) was "Rektor der Kgl.
Studienanstalt zu Hof " in Bavaria.

THE ROMANS 65

In this case the button for ^, the button above the dot and three
buttons below the dot were used to indicate the sum | -£$. The addi-
tion of 8 would bring all the buttons above and below i into play,
making 10 units. Hence, move them all back and move up one button
in the groove below x. Add 10 by moving up another of the buttons
below x; add 300 to 800 by moving back all buttons above and below
c, except one button below, and moving up one button below I; add
10,000 by moving up one button below x. In subtraction the operation
was similar.

Multiplication could be carried out in several ways. In case of
3** 2 TT times 25 ^, the abacus may have shown successively the follow-
ing values: 600 (=30.20), 760 (=600+20.8), 770 (=760 + *. 20),
77°}£ (=770+^.20), 920^ (=77°rlr+30-5), 96o £ (=920 !» +
8 -5), 963 I (=96o^+s-5), 9631 A (=963 HA -5), 973 ^^
(=963 I 3^+i-3°), 976 A 2T (=973 \ A +8-4), 976 | A (=976

A^t"*' 97«i A: A (=976 | A+4-^)-1

In division the abacus was used to represent the remainder resulting
from the subtraction from the dividend of the divisor or of a con-
venient multiple of the divisor. The process was complicated and
difficult. These methods of abacal computation show clearly how
multiplication or division can be carried out by a series of successive
additions or subtractions. In this connection we suspect that recourse
was had to mental operations and to the multiplication table. Pos-
sibly finger-multiplication may also have been used. But the multi-
plication of large numbers must, by either method, have been beyond
the power of the ordinary arithmetician. To obviate this difficulty,
the arithmetical tables mentioned above were used, from which the
desired products could be copied at once. Tables of this kind were
prepared by Victorius of Aquitania. His tables contain a peculiar
notation for fractions, which continued in use throughout the Middle
Ages. Victorius is best known for his canon paschalis, a rule for find-
ing the correct date for Easter, which he published in 457 A. D.

Payments of interest and problems in interest were very old among
the Romans. The Roman laws of inheritance gave rise to numerous
arithmetical examples. Especially unique is the following: A dying
man wills that, if his wife, being with child, gives birth to a son, the
son shall receive § and she ^ of his estates; but if a daughter is born,
she shall receive ^ and his wife f . It happens that twins are born, a
boy and a girl. How shall the estates be divided so as to satisfy the
will? The celebrated Roman jurist, Salvianus Julianus, decided that
the estates shall be divided into seven equal parts of which the son
receives four, the wife two, the daughter one.

We next consider Roman geometry. He who expects to find in
1 Friedlein, op. oil., p. 89.

66 A HISTORY OF MATHEMATICS

Rome a science of geometry, with definitions, axioms, theorems, and
proofs arranged in logical order, will be disappointed. The only
geometry known was a practical geometry, which, like the old Egyp-
tian, consisted only of empirical rules. This practical geometry was
employed in surveying. Treatises thereon have come down to us.
compiled by the Roman surveyors, called agrimensores or gromatici.
One would naturally expect rules to be clearly formulated. But no;
they are left to be abstracted by the reader from a mass of numerical
examples. "The total impression is as though the Roman gromatic
were thousands of years older than Greek geometry, and as though
a deluge were lying between the two." Some of their rules were prob-
ably inherited from the Etruscans, but others are identical with those
of Heron. Among the latter is that for finding the area of a triangle
from its sides and the approximate formula, ^-§a2, for the area of
equilateral triangles (a being one of the sides). But the latter area
was also calculated by the formulas 5 (a2 -fa) and ^a2, the first of
which was unknown to Heron. Probably the expression ^a2 was de-
rived from the Egyptian formula . for the determination of

2 2

the surface of a quadrilateral. This Egyptian formula was used by
the Romans for finding the area, not only of rectangles, but of any
quadrilaterals whatever. Indeed, the gromatici considered it even
sufficiently accurate to determine the areas of cities, laid out irregu-
larly, simply by measuring their circumferences.1 Whatever Egyptian
geometry the Romans possessed was transplanted across the Mediter-
ranean at the time of Julius Ccesar, who ordered a survey of the whole
empire to secure an equitable mode of taxation. Caesar also reformed
the calendar, and, for that purpose, drew from Egyptian learning.
He secured the services of the Alexandrian astronomer, Sosigenes.

Two Roman philosophical writers deserve our attention. The
philosophical poet, Titus Lucretius (g6?-55 B. c.), in his De rerum
natura, entertains conceptions of an infinite multitude and of an in-
finite magnitude which accord with the modern definitions of those
terms as being not variables but constants. However, the Lucretian
infinites are not composed of abstract things, but of material particles.
His infinite multitude is of the denumerable variety; he made use
of the whole-part property of infinite multitudes.2

Cognate topics are discussed several centuries later by the cele-
brated father of the Latin church, St. Augustine (354-430 A. D.), in
his references to Zeno of Elea. In a dialogue on the question, whether
or not the mind of man moves when the body moves, and travels with
the body, he is led to a. definition of motion, in which he displays some
levity. It has been said of scholasticism that it has no sense of humor.

1 H. Hankel, op. cit., p. 297.

2 C. J. Keyser in Bull. Am. Math. Soc., Vol. 24, 1918, p. 268, 321.

THE ROMANS 67

Hardly does this apply to St. Augustine. He says: "When this dis-
course was concluded, a boy came running from the house to call us
to dinner. I then remarked that this boy compels us not only to
define motion, but to see it before our very eyes. So let us go, and
pass from this place to another; for that is, if I am not mistaken,
nothing else than motion." St. Augustine deserves the credit of
having accepted the existence of the actually infinite and to have
recognized it as being, not a variable, but a constant. He recognized
all finite positive integers as an infinity of that type. On this point
he occupied a radically different position than his forerunner, the
Greek father of the church, Origen of Alexandria. Origen's arguments
against the actually infinite have been pronounced by Georg Cantor
the profoundest ever advanced against the actually infinite.

In the fifth century, the Western Roman Empire was fast falling
to pieces. Three great branches — Spain, Gaul, and the province of
Africa — broke off from the decaying trunk. In 476, the Western
Empire passed away, and the Visigothic chief, Odoacer, became king.
Soon after, Italy was conquered by the Ostrogoths under Theodoric.
It is remarkable that this very period of political humiliation should
be the one during which Greek science was studied in Italy most
zealously. School-books began to be compiled from the elements of
Greek authors. These compilations are very deficient, but are of
absorbing interest, from the fact that, down to the twelfth century,
they were the only sources of mathematical knowledge in the Occident.
Foremost among these writers is Boethius (died 524). At first he
was a great favorite of King Theodoric, but later, being charged by
envious courtiers with treason, he was imprisoned, and at last decapi-
tated. While in prison he wrote On the Consolations of Philosophy. As
a mathematician, Boethius was a Brobdingnagian among Roman
Scholars, but a Liliputian by the side of Greek masters. He wrote
an Institutis Arithmetica, which is essentially a translation of the arith-
metic of Nicomachus, and a Geometry in several books. Some of the
most beautiful results of Nicomachus are omitted in Boethius' arith-
metic. The first book on geometry is an extract from Euclid's Ele-
ments, which contains, in addition to definitions, postulates, and
axioms, the theorems in the first three books, without proofs. How
can this omission of proofs be accounted for? It has been argued by
some that Boethius possessed an incomplete Greek copy of the Ele-
ments; by others, that he had Theon's edition before him, and be-
lieved that only the theorems came from Euclid, while the proofs were
supplied by Theon. The second book, as also other books on geometry
attributed to Boethius, teaches, from numerical examples, the men-
suration of plane figures after the fashion of the agrimensores.

A celebrated portion in the geometry of Boethius is that pertaining
to an abacus, which he attributes to the Pythagoreans. A consider-
able improvement on the old abacus is there introduced. Pebbles

68 A HISTORY OF MATHEMATICS

are discarded, and apices (probably small cones) are used. Upon each
of these apices is drawn a numeral giving it some value below 10.
The names of these numerals are pure Arabic, or nearly so, but are
added, apparently, by a later hand. The o is not mentioned by
Boethius in the text. These numerals bear striking resemblance to
the Gubar-numerals of the West-Arabs, which are admittedly of
Indian origin. These facts have given rise to an endless controversy.
Some contended that Pythagoras was in India, and from there brought
the nine numerals to Greece, where the Pythagoreans used them
secretly. This hypothesis has been generally abandoned, for it is
not certain that Pythagoras or any disciple of his ever was in India,
nor is there any evidence in any Greek author, that the apices were
known to the Greeks, or that numeral signs of any sort were used by
them with the abacus. It is improbable, moreover, that the Indian
signs, from which the apices are derived, are so old as the time of
Pythagoras. A second theory is that the Geometry attributed to
Boethius is a forgery; that it is not older than the tenth, or possibly
the ninth, century, and that the apices are derived from the Arabs.
But there is an Encyclopaedia written by Cassiodorius (died about
585) in which both the arithmetic and geometry of Boethius are men-
tioned. Some doubt exists as to the proper interpretation of this
passage in the Encyclopaedia. At present the weight of evidence is
that the geometry of Boethius, or at least the part mentioning the
numerals, is spurious.1 A third theory (Woepcke's) is that the
Alexandrians either directly or indirectly obtained the nine numerals
from the Hindus, about the second century A. D., and gave them to
the Romans on the one hand, and to the Western Arabs on the other.
This explanation is the most plausible.

It is worthy of note that Cassiodorius was the first writer to use
the terms "rational" and "irrational" in the sense now current in
arithmetic and algebra.2

1 A. good discussion of this so-called "Boethius question," which has been de-
bated for two centuries, is given by D. E. Smith and L. C. Karpinski in their Hindu-
Arabic Numerals, 1911, Chap. V.

2 Encyclopedic des sciences mathematiques, Tome I, Vol. 2, 1907, p. 2. An il-
luminating article on ancient finger-symbolism is L. J. Richardson's " Digital
Reckoning Among the Ancients " in the Am. Math. Monthly, Vol 23, 1816,
PP- 7-13-

THE MAYA

The Maya of Central America and Southern Mexico developed
hieroglyphic writing, as found in inscriptions and codices dating ap-
parently from about the beginning of the Christian era, that ranks
"probably as the foremost intellectual achievement of pre-Columbian
times in the New World." Maya number systems and chronology
are remarkable for the extent of their early development. Perhaps
five or six centuries before the Hindus gave a systematic exposition
of their decimal number system with its zero and principle of local
value, the Maya in the flatlands of Central America had evolved
systematically a vigesimal number system employing a zero and the
principle of local value. In the Maya number system found in the
codices the ratio of increase of successive units was not 10, as in the
Hindu system; it was 20 in all positions except the third. That is,
20 units of the lowest order (kins, or days) make one unit of the next
higher order (uinals, or 20 days), i& uinals make one unit of the third
order (tun, or 360 days), 20 tuns make one unit of the fourth order
(katun, or 7200 days), 20 katuns make one unit of the fifth order
(cycle, or 144,000 days) and finally, 20 cycles make i great cycle of
2,880,000 days. In Maya codices we find symbols for i to 19, ex-
pressed by bars and dots. Each bar stands for 5 units, each dot for
i unit. For instance,

2 4 5 7 ii 19

The zero is represented by a symbol that looks roughly like a half-
closed eye. In writing 20 the principle of local value enters. It is
expressed by "a dot placed over the symbol for zero. The numbers
are written vertically, the lowest order being assigned the lowest
position. Accordingly, 37 was expressed by the symbols for 17 (three
bars and two dots) in the kin place, and one dot representing 20,
placed above 17 in the uinal place. To write 360 the Maya drew
two zeros, one above the other, with one dot higher up, in third place
(1X18x20+0+0=360). The highest number found in the codices
is in our decimal notation 12,489,781.

A second numeral system is found on Maya inscriptions. It em-
ploys the zero, but not the principle of local value. Special symbols
are employed to designate the different units. It is as if we were to
write 203 as " 2 hundreds, o tens, 3 ones." 1

1 For an account of the Maya number-systems and chronology, see S. G. Morley
A n Introduction to the Study of the Maya Hierogliphs, Government Printing Office,
Washington, 1915.

7o A HISTORY OF MATHEMATICS

The Maya had a sacred year of 260 days, an official year of 360
days and a solar year of 365+ days. The fact that 18x20=360
seems to account for the break in the vigesimal system, making 18
(instead of 20) uinals equal to i tun. The lowest common multiple
of 260 and 365, or 18980, was taken by the Maya as the "calendar
round," a period of 52 years, which is "the most important period in
Maya chronology."

We may add here that the number systems of Indian tribes in North
America, while disclosing no use of the zero nor of the principle of
local value, are of interest as exhibiting not only quinary, decimal, and
vigesimal system^, but also ternary, quarternary, and octonary sys-
tems.1

1 See W. C. Eells, "Number Systems of the North American Indians" in Amer-
ican Math. Monthly, Vol. 20, 1913, pp. 263-272, 293-299; also Bibliolkcca mallic-
malica, 3 S., Vol. 13, 1913, pp. 218-222.

THE CHINESE 1

The oldest extant Chinese work of mathematical interest is an
anonymous publication, called Chou-pei and written before the
second century, A. D., perhaps long^beforeT In one of the dialogues the
Chou-pei is believed to reveal the state of mathematics and astronomy
in China as early as 1 100 B. c. The Pythagorean theorem of the right
triangle appears to have been known at that early date.

Next to the Chou-pei in age is the Chiu-chang Suan-shu ("Arith-
metic in Nine Sections"), commonly called the Chiu-chang, the most
celebrated Chinese Text on arithmetic. Neither its authorship nor
the time of its composition is known definitely. By an edict of the
despotic emperor Shih Hoang-ti of the Ch'in Dynasty "all books were
burned and all scholars were buried in the year 213 B. c." After the
death of this emperor, learning revived again. We are told that a
scholar named CHANG T'SANG found some old writings, upon which
he based this famous treatise^_the Chiu-chang. About a century later
a revision of it was made by Ching ClT'ou^ch'ang; commentaries on
this classic text were made by Liu Hui in 263 A. D. and by Li Ch'un-
feng in the seventh century. How much of the ''Arithmetic in Nine
Sections," as it exists to-day, is due to the old records ante-dating
213 B. c., how much to Chang T'sang and how much to Ching Ch'ou-
ch'ang, it has not yet been found possible to determine.

The "Arithmetic in Nine Sections" begins with mensuration; it
gives the area of a triangle as | b h, of a trapezoid as \ (b +b')h, of a
circle variously as \c .%d, \cd, \d- and ^c~, where c is the circumference
and d is the diameter. Here TT is taken equal to 3. The area of a
segment of a circle is given as ^(ca+a2), where c is the chord and a
the altitude. Then follow fractions, commercial arithmetic including
percentage and proportion, partnership, and square and cube root of
numbers. Certain parts exhibit a partiality for unit-fractions. Divi-
sion by a fraction is effected by inverting the fraction and multiplying.
The rules of operation are usually stated in obscure language. There
are given rules for finding the volumes of the prism, cylinder, pyramid,
truncated pyramid and cone, tetrahedron and wedge. Then follow
problems in alligation. There are indications of the use of positive
and negative numbers. Of interest is the following problem because
centuries later it is found in a work of the Hindu Brahmagupta:

1 All our information on Chinese mathematics is drawn from Yoshio Mikami's
The Development of Mathematics in China and Japan, Leipzig, 1912, and from Da\ ill
KiiKimc Smith and Yoshio Mikami's History of Japanese Mathematics, Chicago,

101 \.

72 A HISTORY OF MATHEMATICS

There is a bamboo 10 ft. high, the upper end of which is broken anc
reaches to the ground 3 ft. from the stem.. What is the height of th<

break? In the solution the height of the break is taken — TT —

2x10

Here is another: A square town has a gate at the mid-point of ead
side. Twenty paces north of the north gate there is a tree whjfi
is visible from a point reached by walking from the south gate i^
paces south and then 1775 paces west. Find the side of the square
The problem leads to the quadratic equation x2+(2o+i4)x — 2Xio>
1775 =o. The derivation and solution of this equation are not mad<
clear in the text. There is an obscure statement to the effect tha
the answer is obtained by evolving the root of an expression whicl
is not monomial but has an additional term [the term of the firs
degree (20 + 14)2;]. It has been surmised that the process here re
ferred to was evolved more fully later and led to the method close!}
resembling Horner's process of approximating to the roots, and tha
the process was carried out by the use of calculating boards. Anothe;
problem leads to a quadratic equation, the rule for the solution o
which fits the solution of literal quadratic equations.

We come next to the Sun-Tsu Suan-ching ("Arithmetical Classi<
of Sun-Tsu"), which belongs to the first century, A. D. The author
SuN-Tsu, says: "In making calculations we must first know position:
of numbers. Unity is vertical and ten horizontal; the hundred stand:
while the thousand lies; and the thousand and the ten look equally
and so also the ten thousand and the hundred." This is evidently £
reference to abacal computation, practiced from time immemorial ir
China, and carried on by the use of computing rods. These rods
made of small bamboo or of wood, were in Sun-Tsu's time much longer
The later rods were about if inches long, red and black in color
representing respectively positive and negative numbers. According
to Sun-Tsu, units are represented by vertical rods, tens by horizonta
rods, hundreds by vertical, and so on; for 5 a single rod suffices. The
numbers 1-9 are represented by rods thus: |, \\, |||, ||||, |||H, |,j|, ]j|7 |U|;
the numbers in the tens column, 10, 20, . . ., 90 are written thus

— , =, =, =, ==, I , JL, =, =. The number 6728 is designated
by _j_ ~f|" = Iff . The rods were placed on a board ruled in columns
and were rearranged as the computation advanced. The successive
steps in the multiplication of 321 by 46 must have been about as
follows:

321 321 321

138 1472 14766

46 46 46

The product was placed between the multiplicand and multiplier,
The 46 is multiplied first by 3, then by 2, and last by i, the 46 being

THE CHINESE" "

moved to the right one place at each step. Sun-Tsu does not
division, except when the divisor consists of one digit. Square root
is explained more clearly than in the "Arithmetic in Nine Sections. "£
Algebra is involved in the problem suggested by the reply made by a
woman washing dishes at a river: "I don't know how many guests
there were; but every two used a dish for rice between them; every
three a dish for broth; every four a dish for meat; and there were 65
dishes in all. — Rule: Arrange the 65 dishes, and multiply by 12, when
we get 780. Divide by 13, and thus we obtain the answer."

An indeterminate equation is involved in the following: "There are **
certain things whose number is unknown. Repeatedly divide by 3,
— the remainder is 2 ; by 5 the remainder is 3 ; and by 7 the remainder is ^
2. What will be the number?" Only one solution is given, viz. 23.

The Hai-tao Suan-ching ("Sea-island Arithmetical Classic") was *
written by Liu Hui, the commentator on the "Arithmetic in Nine "^
Sections," during the war-period in the third century, A. D. He gives ^
complicated problems indicating marked proficiency in algebraic
manipulation. The first problem calls for the determination of the
distance of an island and the height of a peak on the island, when two
rods 30' high and 1000' apart are in line with the peak, the top of the
peak being in line with the top of the nearer (more remote) rod, when
seen from a point on the level ground 123' (127') behind this nearer ., <^^.
(more remote) rod. The rules given for solving the problem are
equivalent to the expressions obtained from proportions arising from V
the similar triangles.

Of the treatises brought forth during the next centuries only a few
are extant. We mention the "Arithmetical Classic of Chang Ch'iu- v
chien" of the sixth century which gives problems on proportion, arith-
metical progression and mensuration. He proposes the "problem of
100 hens" which is given again by later Chinese authors: "A cock V^&
costs 5 pieces of money, a hen 3 pieces, and 3 chickens i piece. If
then we buy with 100 pieces 100 of them, what will be their respective
numbers?"

The early values of TT used in China were 3 and \/io- Liu Hui
calculated the perimeters of regular inscribed polygons of 12, 24, 48,
96, 192 sides and arrived at TT =3.14+. Tsu Ch'ung-chih in the fifth
century took the diameter io8 and obtained as upper and lower limits
for TT 3.1415927 and 3.1415926, and from these the "accurate" and
"inaccurate" values 355/113, 22/7. The value 22/7 is the upper limit
given by Archimedes and is found here for the first time in Chinese
history. The ratio 355/113 became known to the Japanese, but in
the West it was not known until Adriaen Anthonisz, the father of
Adriaen Metius, derived it anew, sometime between 1585 and 1625.
However, M. Curtze's researches would seem to show that it was
known to Valentin Otto as early as 1573.*

1 Bibliotheca malhemalica, 3 S., Vol. 13, 1913, p. 264. A neat geometric construe-

74 A HISTORY OF MATHEMATICS

In the first half of the seventh century WANG Hs' IAO-T'UNG brought
forth a work, the Ch'i-ku Suan-ching, in which numerical cubic equa-
tions appear for the first time in Chinese mathematics. This took
place seven or eight centuries after the first Chinese treatment of
quadratics. Wang Hs'iao-t'ung gives several problems leading to
cubics: "There is a right triangle, the product of whose two sides is
706 -j^, and whose hypotenuse is greater than the first side by 30 ^.
It is required to know the lengths of the three sides." He gives the
answer 14 ^_, 49 |, 51 1-, and the rule: "The Product (P) being
squared and being divided by twice the Surplus (S),make the result
shih or the constant class. Halve the surplus and make it the lien-fa
or the second degree class. A$d carry out the operation of evolution
according to the extraction oif cube root. The result gives the first
side. Adding the surplus^to it, one gfets the hypotenuse. Divide the
product with the first side and the^jufotient is the second side." This
rule leads to the cubic equation x3 +S/2X2-— =o. The mode of solu-
tion is similar to the process of extracting cube roots, but details of
the process are not revealed.

In 1247 CH'IN CHIU-SHAO wrote the Su-shu Chiu-chang ("Nine
Sections of Mathematics") which makes a decided advance on the
solution of numerical equations. At first Ch'in Chiu-shao led a mili-
tary life; he lived at the time of the Mongolian invasion. For ten
years stricken with disease, he recovered and then devoted himself to
study. The following problem led him to an equation of the tenth
degree: There is a circular castle of unknown diameter, having 4
gates. Three miles north of the north gate is a tree which is visible
from a point 9 miles east of the south gate. The unknown diameter
is found to be 9. He passes beyond Sun-Tsu in his ability to solve
indeterminate equations arising for a number which will give the
residues r\, r%, . ., ra when divided by m\, tn^, ., mn, respectively.

Ch'in Chiu-shao solves the equation — x4 +763 aoo.r2 — 40642560000
= o by a process almost identical with Homer's method. /However,
the computations were very probably carried out on a computing
board, divided into columns, and by the use of computing rods.
Hence the arrangement of the work must have been different from
that of Horner. But the operations performed were the same. The
first digit in the root being 8, (8 hundreds), a transformation is ef-
fected which yields #4— 3 2oox3 — 3O768oox2 — 826880000^+3820544-
oooo = o, the same equation that is obtained by Horner's process.
Then, taking 4 as the second figure in the root, the absolute term
vanishes in the operation, giving the root 840. Thus the Chinese had

tion of the fraction ^ || =3 +42 -r- (72+82) is given anonymously in Grunerfs Archiv.
Vol. 12, 1849, p. 98. Using f f|, T. M. P. Hughes gives in Nature, Vol. 93,
1914, p. no, a method of constructing a triangle that gives the area of a given
circle with great accuracy.

THE CHINESE 75

invented Homer's method of solving numerical equations more than
five centuries before Ruffini and Horner. This solution of higher
numerical equations is given later in the writings of Li Yeh and others.
Ch'in Chiu-shao marks an advance over Sun-Tsu in the use of o as a
symbol for zero. Most likely this symbol is an importation from
India. Positive and negative numbers were distinguished by the use
of red and black computing rods. This author gives for the first time
a problem which later became a favorite one among the Chinese; it
involved the trisection of a trapezoidal field under certain restrictions
in the mode of selection of boundaries.

We have already mentioned a contemporary of Ch'in Chiu-shao,
namely, Li YEH; he lived far apartjn^a rival monarchy and worked
independently^ He was the author of Tse-yuan Ilai-ching ("Sea-
"Mirror of thlFCircle-Measurements"), 1248, and of the I-ku Yen-tuan,
1259. He used the symbol o for zero. On account of the inconven-
ience of writing and printing positive and negative numbers in dif-
ferent colors, he designated negative numbers by drawing a cancella-
tion mark across the symbol. Thus J_o stood for 60, Jio stood for
— 60. The unknown quantity was represented by unity which was
probably represented on the counting board by a rod easily distin-
guished from the other rods. The terms of an equation were written,
not in a horizontal, but in a vertical line. In Li Yeh's work of 1259,
as also in the work of Ch'in Chiu-shao, the absolute term is put in the
top line; in Li Yeh's work of 1248 the order of the terms is reversed,
so that the absolute term is in the bottom line and the highest power
of the unknown in the top line. In the thirteenth century Chinese
algebra reached a much higher development than formerly. This
science, with its remarkable method (our Horner's) of solving numer-
ical equations, was designated by the Chinese "the celestial element
method."

A third prominent thirteenth century mathematician was YANG
Hui, of whom several books are, still extant. They deal with the
summation of arithmetical progresslongppf tne series i +3 +6 + . . +
(1+2 + . . +«) =n(«+i)(»+2)-5-6, i?+22+. . +n2=£»(»+$)(n+i),
also with proportion, simultaneous linear equations^ quadratic and
quartic equations.

Half a century later, Chinese algebra reached its height in the
treatise Suan-hsiao Chi-mtng ("Introduction to Mathematical
Studies"), 1299, and the Szu-yuen Yu-chien ("The Precious Mirror
of the Four Elements"), 1303, which came from the pen of CHU
Sinn-CiiiEii. The first work contains no new results, but exerted a
great stimulus on Japanese mathematics in tin- seventeenth century.
At one time the book was lost in China, but in iS^) it was restored
by the discovery of a copy of a Korean reprint, made in 1660. The
"Precious Mirror" is a more original work. It treats fully of the
"celestial element method." He gives as an "ancient method" a

A HISTORY OF MATHEMATICS

triangle (known in the West as Pascal's arithmetical triangle), dis-
playing the binomial coefficients, which were known to the Arabs in
the eleventh century and were probably imported into China. Chu
shih-Chieh's algebraic notation was altogether different from our
modern notation. Thus, a +b +c +d was written

i
1*1

202
O *20
22 O 2

as shown on the left, except that, in the central position, we employ
an asterisk in place of the Chinese character t'ai (great extreme, ab-
solute term) and that we use the modern numerals in place of the
sangi forms. The square of a +b +c +d, namely, a2 +b2 +c2 +d2 +2ab
+2ac+2ad+2bc+2bd+2cd, is represented as shown on the right.
In further illustration of the Chinese notation, at the time of Chu
Shih-Chieh, we give *

= — 22

= XZ

= 2yz

In the fourteenth century astronomy and the calendar were studied.
They involved the rudiments of geometry and spherical trigonometry.
In this field importations from the Arabs are disclosed.

After the noteworthy achievements of the thirteenth century,
Chinese mathematics for several centuries was in a period of decline.
The famous "celestial element method" in the solution of higher
equations was abandoned and forgotten. Mention must be made,
however, of CH'ENG TAI-WEI, who in 1593 issued his Suan-fa T'ung-
tsung ("A Systematised Treatise on Arithmetic"), which is the oldest
work now extant that contains a diagram of the form of the abacus,
called suan-pan, and the explanation of its use. The instrument was
known in China in the twelfth century. Resembling the old Roman
abacus, it contained balls, movable along rods held by a wooden
frame. The suan-pan replaced the old computing rods. The "Sys-
tematised Treatise on Arithmetic" is famous also for containing some
magic squares and magic circles. Little is known of the early history

1In the symbol for "xz" notice that the "i" is one space down,(x) and one
space to the right (z) of *, and is made to stand for the product xz. In the symbol
for " zyz" the three o's indicate the absence of the terms y, x, xy; the small "2"
means twice the product of the two letters in the same row, respectively one space
to the right and to the left of *, i. e., 2 yz. The limitations of this notation are ob-
vious.

THE CHINESE

of magic squares. Myth tells us that, in early times, the sage Yii,
the enlightened emperor, saw on the calamitous Yellow River a divine
tortoise, whose back was decorated with the figure made up of the
numbers from i to 9, arranged in form of a magic square or lo-shu.

<\>ooooooooo/X

**. J- ^.

\J IV *J

The lo-shu.

The numerals are indicated by knots in strings: black knots repre-
sent even numbers (symbolizing imperfection), white knots repre-
sent odd numbers (perfection).

Christian missionaries entered China in the sixteenth century.
The Italian Jesuit Matteo Ricci (1552-1610) introduced European
astronomy and mathematics. With the aid of a Chinese scholar
named Hsu, he brought out in 1607 a translation of the first six books
of Euclid. Soon after followed a sequel to Euclid and a treatise on
surveying. The missionary Mu Ni-ko sometime before 1660 intro-
duced logarithms. In 1713 Adrian Vlack's logarithmic tables to n
places were reprinted. Ferdinand Verbiest 1 of West Flanders, a
noted Jesuit missionary and astronomer, was in 1669 made vice-
president of the Chinese astronomical board and in 1673 its president.
European algebra found its way into China. Mei Ku-cWeng noticed
that the European algebra was essentially of the same principles as
the Chinese "celestial element method" of former days which had
been forgotten. Through him there came a revival of their own
algebraic method, without, however, displacing European science.
Later Chinese studies touched mainly three subjects: The determina-
tion of TT by geometry and by infinite series, the solution of numerical
equations, and the theory of logarithms.

We shall see later that Chinese mathematics stimulated the growth
of mathematics in Japan and India. We have seen that, in a small
way, there was a taking as well as a giving. Before the influx of
recent European science, China was influenced somewhat by Hindu
and Arabic mathematics. The Chinese achievements which stand
out most conspicuously are the solution of numerical equations and
the origination of magic squares and magic circles.

1 Consult H. Bosnians, Ferdinand Verbiest, Louvain, 1912. Extract from Revue
des Questions scientifiques, January-April, 1912.

THE JAPANESE l

According to tradition, there existed in Japan in remote times a
system of numeration which extended to high powers of ten and re-
sembled somewhat the sand counter of Archimedes. About 552 A. D.
Buddhism was introduced into Japan. This new movement was
fostered by Prince Shotoku Taishi who was deeply interested in all
learning. Mathematics engaged his attention to such a degree that
he came to be called the father of Japanese mathematics. A little
later the Chinese system of weights and measures was adopted. In
701 a university system was established in which mathematics figured
prominently. Chinese science was imported, special mention being
made in the official Japanese records of nine Chinese texts on mathe-
matics, which include the Chou-pei, the Suan-ching written by Sun-
Tsu and the great arithmetical work, the Chiu-chang. But this eighth
century interest in mathematics was of short duration ; the Chiu-chang
was forgotten and the dark ages returned. Calendar reckoning and
the rudiments of computation are the only signs of mathematical
activity until about the seventeenth century of our era. On account
of the crude numeral systems, devoid of the principal of local value
and of a symbol for zero, mechanical aids of computation became a
necessity. These consisted in Japan, as in China, of some forms of
the abacus. In China there came to be developed an instrument,
called the suan-pan, in Japan it was called the soroban. The importa-
tion of the suan-pan into Japan is usually supposed to have occurred
before the close of the sixteenth century. Bamboo computing rods
were used in Japan in the seventh century. These round pieces were
replaced later by the square prisms (sangi pieces). Numbers were
represented by these rods in the manner practiced by the Chinese.
The numerals were placed inside the squares of a surface ruled like a
chess board. The soroban was simply a more highly developed form
of abacal instrument.

The years 1600 to 1675 mark a period of great mathematical ac-
tivity. It was inaugurated by MORI KAMBEI SHIGEYOSHI, who popu-
larized the use of the soroban. ' His pupil, YOSHIDA SHICHIBEI KOYU,
is the author of Jinko-ki, 1627, which attained wide popularity and
is the oldest Japanese mathematical work now extant. It explains
operations on the soroban, including square and cube root. In one of

1This account is compiled from David Eugene Smith and Yoshio Mikami's
History of Japanese Mathematics, Chicago, 1914, from Yoshio Mikami's Develop-
ment of Mathematics in China and Japan, Leipzig, 1912, and from T. Hayashi's
A Brief History of the Japanese Mathematics, Overgedrukt uit het Nieuw Archief
voor Wiskunde VI, pp. 296-361; VII, pp. 105-161.

THE JAPANESE 79

his later editions Yoshida appended a number of advanced problems
to be solved by competitors. This procedure started among the
Japanese the practice of issuing problems, which was kept up until
1813 and helped to stimulate mathematical activity.

Another pupil of Mori was IMAMURA CHISHO who, in 1639, pub-
lished a treatise entitled Jugairoku, written in classical Chinese. He
took up the mensuration of the circle, sphere and cone. Another
author, ISOMURA KITTOKU, in his Ketsugisho, 1660 (second edition
1684), when considering problems on mensuration, makes a crude
approach to integration. He gives magic squares, both odd and even
celled, and also magic circles. Such squares and circles became favor-
ite topics among the Japanese. In the 1684 edition, Isomura gives
also magic wheels. TANAKA KISSHIN arranges the integers 1-96 in
six 42-celled magic squares, such that the sum in each row and column
are 194; placing the six squares upon a cube, he obtains his "magic
cube." Tanaka formed also "magic rectangles." l MURAMATSU in
1663 gives a magic square containing as many as ig2 cells and a magic
circle involving 129 numbers. Muramatsu gives also the famous
" Josephus Problem" in the following form: 'Once upon a time there
lived a wealthy farmer who had thirty children, half being of his first
wife and half of his second one. The latter wished a favorite son to
inherit all the property, and accordingly she asked him one day, say-
ing: Would it not be well to arrange our 30 children on a circle, calling
one of them the first and counting out every tenth one until there
should remain only one, who should be called the heir. The hus-
band assenting, the wife arranged the children . . ; the counting . .
resulted in the elimination of 14 step-children at once, leaving only
one. Thereupon the wife, feeling confident of her success, said, . .
let us reverse the order. . The husband agreed again, and the
counting proceeded in the reverse order, with the unexpected result
that all of the second wife's children were stricken out and there re-
mained only the step-child, and accordingly he inherited the property."
The origin of this problem is not known. It is found much earlier in
the Codex Einsidelensis (Einsideln, Switzerland) of the tenth century,
while a Latin work of Roman times attributes it to Flavius Josephus.
It commonly appears as a problem relating to Turks and Christians,
half of whom must be sacrificed to save a sinking ship. It was very
common in early printed European books on arithmetic and in books
on mathematical recreations.

In 1666 SATO SEIKO wrote his Kongenki which, in common with
other works of his day, considers the computation of TT( =3.14).
He is the first Japanese to take up the Chinese "celestial elenu-nt
method " in algebra. He applies it to equations of as high a degree a3
the sixth. His successor, SAWAGUCIII, and a contemporary NOZAWA,
give a crude calculus resembling that of Cavalieri. Sawaguchi rises
1 Y. Mikami in Archiv der Mathemalik u. Physik, Vol. 20, pp. 183-186.

8o A HISTORY OF MATHEMATICS

above the Chinese masters in recognising the plurality of roots, but
he declares problems which yield them to be erroneous in their nature.
Another evidence of a continued Chinese influence is seen in the
Chinese value of TT, |4f > which was made known in Japan by IKEDA.

We come now to SEKI KOWA (1642-1708) whom the Japanese con-
sider the greatest mathematician that their country has produced.
The year of his birth was the year in which Galileo died and Newton
was born. Seki was a great teacher who attracted many gifted pupils.
Like Pythagoras, he discouraged divulgence of mathematical dis-
coveries made by himself and his school. For that reason it is difficult
to determine with certainty the exact origin and nature of some of the
discoveries attributed to him. He is said to have left hundreds of
manuscripts; the transcripts of a few of them still remain. He pub-
lished only one book, the Hatsubi Sampb, 1674, in which he solved
15 problems issued by a contemporary writer. Seki's explanations
are quite incomplete and obscure. Takebe, one of his pupils, lays
stress upon Seki's clearness. The inference is that Seki gave his ex-
planations orally, probably using the computing rods or sangi, as he
proceeded. Noteworthy among his mathematical achievements are
the tenzan method and the yendan method. Both of these refer to
improvements in algebra. The tenzan method is an improvement of
the Chinese "celestial element" method, and has reference to nota-
tion, while the yendan refers to explanations or method of analysis.
The exact nature and value of these two methods are not altogether
clear. By the Chinese "celestial element" method the roots of equa-
tions were computed one digit at a time. Seki removed this limita-
tion. Building on results of his predecessors, Seki gives also rules for
writing down magic squares of (2n + i)2 cells. In the case of the more
troublesome even celled squares, Seki first gives a rule for the con-
struction of a magic square of 42 cells, then of ^(n + i)2 and 16 w2 cells.
He simplified also the treatment of magic circles. Perhaps the most
original and important work of Seki is the invention of determinants,
sometime before 1683. Leibniz, to whom the first idea of determinants
is usually assigned, made his discovery in 1693 when he stated that
three linear equations in x and y can have the same ratio only when
the determinant arising from the coefficients vanishes. Seki took n
equations and gave a more general treatment. Seki knew that a
determinant of the nth order, when expanded, has n\ terms and that
rows and columns are interchangeable.1 Usually attributed to Seki
is the invention of \h& yenri or "circle-principle" which, it is claimed,
accomplishes somewhat the same things as the differential and in-
tegral calculus. Neither the exact nature nor the origin of the yenri
is well understood. Doubt exists whether Seki was its discoverer.
TAKEBE, a pupil of Seki, used the yenri and may be the chief originator

1 For details consult Y. Mikami, "On the Japanese theory of determinants" in
Isis, Vol. II, 1914, PP- 9-36-

THE JAPANESE 81

of it, but his explanations are incomplete and obscure. Seki, Takebe
and their co-workers dealt with infinite series, especially in the study
of the circle and of IT. Probably some knowledge of European mathe-
matics found its way into Japan in the seventeenth century. A
Japanese, under the name of Petrus Hartsingius, is known to have
studied at Leyden under Van Schooten, but there is no clear evidence
that he returned to Japan. In 1650 a Portuguese astronomer, whose
real name is not known and whose adopted name was Sawano Chuan,
translated a European astronomical work into Japanese.1

In the eighteenth century the followers of Seki were in control.
Their efforts were expended upon problem-solving, the mensuration
of the circle and the study of infinite series. Of KURUSHIMA GITA,
who died in 1757, fragmentary manuscripts remain, which show a
"magic cube" composed of four 42-celled magic squares in which the
sums of rows and columns is 130, and the sums of corresponding cells
of the four squares is likewise 130. This "magic cube" is evidently a
different thing from Tanaka's "Magic cube." Near the close of the
eighteenth century, during the waning days of the Seki school, there
arose a bitter controversy between FUJITA SADASUKE, then the head
of the Seki school, and AIDA AMMEI. Of the two, Aida was the younger
and more gifted — an insurgent against the old and involved methods
of exposition. Aida worked on the approximate solution of numerical
equations.2 The most noted work of the time was done by a man
living in peaceful seclusion, AJIMA CHOKUYEN of Yedo, who died in
1798. He worked on Diophantine analysis and on a problem known
in the West as "the Malfatti problem," to inscribe three circles in a
triangle, each tangent to the other two. This problem appeared in
Japan in 1781. Malfatti's publication on it appeared in 1803, but
the special case of the isosceles triangle had been considered by Jakob
Bernoulli before 1744. Ajima treats special cases of the problem to
determine the number of figures in the repetend in circulating decimals.
He improved the yenri and placed mathematics on the highest plane
that it reached in Japan during the eighteenth century.

In the early part of the nineteenth century there was greater in-
filtration of European mathematics. There was considerable activity,
but no noteworthy names appeared, except WADA NEI (1787-1840)
who perfected the yenri still further, developing an integral calculus
that served the ordinary purposes of mensuration, and giving reasons
where his predecessors ordinarily gave only rules. He worked par-
ticularly on maxima and minima, and on roulettes. Japanese re-
searches of his day relate to groups of ellipses and other figures which

1 Y. Mikami in Annaes da academia polyt. do Porto, Vol. VIII, 1913.

2 Y. Mikami, "On Aida Ammei's solution of an equation" in Annaes da Academia
Polyt. do Porto, Vol. VIII, 1913. This article gives details on the solution of equa-
tions in China and Japan. See also Mikami's article on Miyai Antai in the Tohoku
Math. Journal, Vol. 5, Nos. 3, 4, 1914.

82 A HISTORY OF MATHEMATICS

can be drawn upon a folding fan. Here mathematics finds application
to artistic design.

After the middle of the nineteenth century the native mathematics
of Japan yielded to a strong influx of Western mathematics. The
movement in Japan became a part of the great international advance.
In 1911 there was started the Tdhoku Mathematical Journal, under
the editorship of T. Hayashi. It is devoted to advanced mathematics,
contains articles in many of our leading modern languages and is quite
international in character.1

Looking back we see that Japan produced some able mathemati-
cians, but on account of her isolation, geographically and socially,
her scientific output did not affect or contribute to the progress of
mathematics in the West. The Babylonians, Hindus, Arabs, and to
some extent even the Chinese through their influence on the Hindus,
contributed to the onward march of mathematics in the West. But
the Japanese stand out in complete isolation.

1 G. A. Miller, Historical Introduction to Mathematical Literature, 1916, p. 24.

THE HINDUS

After the time of the ancient Greeks, the first people whose re-
searches wielded a wide influence in the world march of mathematics,
belonged, like the Greeks, to the Aryan race. It was, however, not a
European, but an Asiatic nation, and had its seat in far-off India.

Unlike the Greek, Indian society was fixed into castes. The only
castes enjoying the privilege and leisure for advanced study and
thinking were the Brahmins, whose prime business was religion and
philosophy, and the Kshatriyas, who attended to war and government.

Of the development of Hindu mathematics we know but little. A
few manuscripts bear testimony that the Indians had climbed to a
lofty height, but their path of ascent is no longer traceable. It would
seem that Greek mathematics grew up under more favorable condi-
tions than the Hindu, for in Greece it attained an independent exist-
ence, and was studied for its own sake, while Hindu mathematics
always remained merely a servant to astronomy. Furthermore, in
Greece mathematics was a science of the people, free to be cultivated
by all who had a liking for it; in India, as in Egypt,- it was in the
hands chiefly of the priests. Again, the Indians were in the habit of
putting into verse all mathematical results they obtained, and of
clothing them in obscure and mystic language, which, though well
adapted to aid the memory of him who already understood the subject,
was often unintelligible to the uninitiated. Although the great Hindu
mathematicians doubtless reasoned out most or all of their discoveries,
yet they were not in the habit of preserving, the proofs, so that the
naked theorems and processes of operation are all that have come
down to our time. Very different in these respects were the Greeks.
Obscurity of language was generally avoided, and proofs belonged
to the stock of knowledge quite as much as the theorems themselves.
Very striking was the difference in the bent of mind of the Hindu and
Greek; for, while _the Greek mind, was preeminently geometrical, the
Indian was first of all prithme.tir.al. The *j[jpdy ffcfl-lfr- Vth ""mber, the
Greek with farm. Numerical symbolism, the science ol numbers, "
and algebra attained in India far greater perfection than they had
previously reached in Greece. On the other hand, Hindu geometry
was merely mensuration, unaccompanied by demonstration. Hindu
trigonometry is meritorious, but rests on arithmetic more than on
geometry.

An interesting but difficult task is the tracing of the relation be-
tween Hindu and Greek mathematics. It is well known that more
or less trade was carried on between Greece and India from early

84 A HISTORY OF MATHEMATICS

times. After Egypt had become a Roman province, a more lively
commercial intercourse sprang up between Rome and India, by way
of Alexandria. A priori, it does not seem improbable, that with the
traffic of merchandise there should also be an interchange of ideas.
That communications of thought from the Hindus to the Alexandrians
actually did take place, is evident from the fact that certain philo-
sophic and theologic teachings of the Manicheans, Neo-Platonists,
Gnostics, show unmistakable likeness to Indian tenets. Scientific
facts passed also from Alexandria to India. This is shown plainly
by the Greek origin of some of the technical terms used by the Hindus.
Hindu astronomy was influenced by Greek astronomy. A part of
the geometrical knowledge which they possessed is traceable to Alex-
andria, and to the writings of Heron in particular. In algebra there
was, probably, a mutual giving and receiving.

There is evidence also of an intimate connection between Indian
and Chinese mathematics. In the fourth and succeeding centuries of
our era Indian embassies to China and Chinese visits to India are
recorded by Chinese authorities.1 We shall see that undoubtedly
there was an influx of Chinese mathematics into India.

The history of Hindu mathematics may be resolved into two
periods: First the S'ulvastitra period which terminates not later than
200 A. D., second, the astronomical and mathematical period, extending
from about 400 to 1200 A. D.

The term S'ulvasutra means "the rules of the cord"; it is the name
given to the supplements of the Kalpasutras which explain the con-
struction of sacrificial altars.2 The S'ulvasutras were composed some-
time between 800 B. c. and 200 A. D. They are known to modern
scholars through three quite modern manuscripts. Their aim is
primarily not mathematical, but religious. The mathematical parts
relate to the construction of squares and rectangles. Strange to say,
none of these geometrical constructions occur in later Hindu works;
later Indian mathematics ignores the S'ulvasutras!

The second period of Hindu mathematics probably originated
with an influx from Alexandria of western astronomy. To the fifth
century of our era belongs an anonymous Hindu astronomical work,
called the Surya Siddhdnta ("Knowledge from the sun") which came
to be regarded a standard work. In the sixth century A. D., Varaha
Mihira wrote his Pancha Siddhdntikd which gives a summary of the
Surya Siddhdnta and four other astronomical works then in use; it
contains matters of mathematical interest.

In 1881 there was found at Bakhshdll, in northwest India, buried
in the earth, an anonymous arithmetic, supposed, from the peculiar-

. * G. R. Kaye, Indian Mathematics, Calcutta & Simla, 1915, p. 38. We are draw-
ing heavily upon this book which embodies the results of recent studies of Hindu
mathematics.

2 G. R. Kaye, op. cit., p. 3.

THE HINDUS 85

ities of its verses, to date from the third or fourth century after Christ.
The document that was found is of birch bark, and is an incomplete
copy, prepared probably about the eighth century, of an older manu-
script. * It contains arithmetical computation.

The noted Hindu astronomer Aryabhata was born 476 A. D. at
Pataliputja, on the upper Ganges. His celebrity rests on a work
entitled Aryabhatiya, of which the third chapter is devoted to mathe-
matics. About one hundred years later, mathematics in India reached
the highest mark. At that time flourished Brahmagupta (born 598).
In 628 he wrote his Brahma-sphula-siddhanta ("The Revised System
of Brahma"), of which the twelfth and eighteenth chapters belong
to mathematics.

Probably to the ninth century belongs Mahavlra, a Hindu author
on elementary mathematics, whose writings have only recently been
brought to the attention of historians. He is the author of the Ganita-
Sara-Sangraha which throws light upon Hindu geometry and arith-
metic. The following centuries produced only two names of impor-
tance; namely, S'ndhara, who wrote a Ganita-sara ("Quintessence
of Calculation"), and Padmanabha, the author of an algebra. The
science seems to have made but little progress at this time; for a
work entitled Siddhanta S'iromani ("Diadem of an Astronomical
System"), written by Bhaskara in 1150, stands little higher than that
of Brahmagupta, written over 500 years earlier. The two most im-
portant mathematical chapters in this work are the IMavatl ( ="the
beautiful," i. e. the noble science) and Vlja-ganita (=" root-extrac-
tion"), devoted to arithmetic and algebra. From now on, the Hindus
in the Brahmin schools seemed to content themselves with studying
the masterpieces of their predecessors. Scientific intelligence de-
creases continually, and in more modern times a very deficient Arabic
work of the sixteenth century has been held in great authority.

The mathematical chapters of the Brahma-siddhanta and Siddhanta
S'iromani were translated into English by H. T. Colebrooke, London,
1817. The Surya-siddhanta was translated by E. Burgess, and anno-
tated by W. D. Whitney, New Haven, Conn., 1860. Mahavira's
Ganita-Sara-Sangraha was published in 1912 in Madras by M. Ranga-
carya.

We begin with geometry, the field in which the Hindus were least
proficient. The S'ulvasutras indicate that the Hindus, perhaps as
early as 800 B. c., applied geometry in the construction of altars.
Kaye2 states that the mathematical rules found in the S'ulvasutras
" relate to (i) the construction of squares and rectangles, (2) the rela-
tion of the diagonal to the sides, (3) equivalent rectangles and squares,
(4) equivalent circles and squares." A knowledge of the Pythagorean

1 The Bakhshali Manuscript, edited by Rudolf Hoernly in the Indian Antiquary,
xv"> 33-48 and 275-279, Bombay, 1888.

2 G. R. Kaye, op. cit., p. 4.

86 A HISTORY OF MATHEMATICS

theorem is disclosed in such relations as 32+42=52, i22+i62=2o2,
i52+362 =3Q2. There is no evidence that these expressions were
obtained from any general rule. It will be remembered that special
cases of the Pythagorean theorem were known as early as 1000 B. c.
in China and as early as 2000 B. c. in Egypt. A curious expression
for the relation of the diagonal to a square, namely,

v 2 = I + 3T"3T1 — 3.4-34 ;

is explained by Kaye as being "an expression of a direct measure-
ment" which may be obtained by the use of a scale of the kind named
in one of the S'ulvasutra manuscripts, and based upon the change
ratios 3, 4, 34. It is noteworthy that the fractions used are all unit
fractions and that the expression yields a result correct to five decimal
places. The S'ulvasutra rules yield, by the aid of the Pythagorean
theorem, constructions for finding a square equal to the sum or dif-
ference o£ two squares; they yield a rectangle equal to a given square,
with aV2 and ^aVJ as the sides of the rectangle; they yield by
geometrical construction a square equal to a given rectangle, and
satisfying the relation ab =(b+[a— b]/2)2— \(a-b}z, corresponding
to Euclid II, 5. In the S'ulvasutras the altar building ritual explains
the construction of a square equal to a circle. Let a be the side of
a square and d the diameter of an equivalent circle, then the given
rules may be expressed thus: l d =a+(aV2-a)/3, a=d—2dfi$, a =
^(i~¥+8^T~8.219.6 +8~.29.6.8)- This third expression may_t)e ob-
tained from the first by the aid of the approximation for V 2, given
above. Strange to say, none of these geometrical constructions occur
in later Hindu works; the latter completely ignore the mathematical
contents of the S'ulvasutras. _

During the six centuries from the time of Aryabhata to that of
Bhaskara, Hindu geometry deals mainly with mensuration. The
Hindu gave no definitions, no postulates, no axioms, no logical chain
of reasoning. His knowledge of mensuration was largely borrowed
from the Mediterranean and from China, through imperfect channels of
communication. Aryabhata gives a rule for the area of triangle which
holds only for the isosceles triangle. Brahmagupta distinguishes
between approximate and exact areas, and gives Heron of Alexan-
dria's famous formula for the triangular area, V.y(s— a)(s — b)(s — c),
Heron's formula is given also by Mahavira2 who advanced be-
yond his predecessors in giving the area of an equilateral triangle as
<?' v3/4- Brahmagupta and Mahavira make a remarkable extension
of Heron's formula by giving V(s— a)(s— b)(s — c)(s — d) as the area
of a quadrilateral whose sides are a, b, c, d, and whose semiperimeter
is s. That this formula is true only for quadrilaterals that can be in-

1 G. R. Kaye, op. tit., p. 7.

2 D. E. Smith, in I sis, Vol. i, 1913, pp. 199, 200.

THE HINDUS 87

scribed in a circle was recognized by Brahmagupta, according to Can-
tor's 1 and Kaye's 2 interpretation of Brahmagupta's obscure ex-
position, but Hindu commentators did not understand the limitation
and Bhaskara finally pronounced the formula unsound. Remarkable
is "Brahmagupta's theorem" on cyclic quadrilaterals, x2=(ad-{-b)c.
(ac-\-bd)l(ab-\-cd) and y2=(ab+cd] (ac-\-bd}l(ad-\-bc}J where x and
y are the diagonals and a, b, c, d, the lengths of the sides; also the the-
orem that, if a?-\-b2 = c2 and A2-\-B2 = C2, then the quadrilateral
("Brahmagupta's trapezium"), (aC, cB, bC, cA) is cyclic and has its
diagonals at right angles. Kaye says: From the triangles (3, 4, 5)
and (5, 12, 13) a commentator obtains the quadrilateral (39, 60, 52,
25), with diagonals 63 and 56, etc. Brahmagupta (says Kaye) also in-
troduces a proof of Ptolemy's theorem and in doing this follows Dio-
phantus (III, 19) in constructing from right triangles (a, b, c) and
(a, (3, y) new right triangles (ay, by, cy) and (ac, @c, yc) and
uses the actual examples given by Diophantus, namely (39, 52, 65)
and (25, 60, 65). Parallelisms of this sort show unmistakably that the
Hindus drew from Greek sources.

_ In the mensuration of solids remarkable inaccuracies occur in
Aryabhata. He gives the volume of a pyramid as half the product of
the base and the height; the volume of a sphere as TT* r3. Aryab-
hata gives in one place an extremely accurate value for TT, viz. 3^^
( = 3.1416), but he himself never utilized it, nor did any other Hindu
mathematician before the twelfth century. A frequent Indian prac-
tice was to take 7T = 3, or Vio. Bhaskara gives two values, — the
above mentioned 'accurate,' f-ff-j> and the 'inaccurate,' Archimedean
value, — . A commentator on Lilavatl says that these values were
calculated by beginning with a regular inscribed hexagon, and apply-

ing repeatedly the formula AD=-\/2 — ^l^-AB2, wherein AB is the
side of the given polygon, and AD that of one with double the number
of sides. In this way were obtained the perimeters of the inscribed
polygons of 12, 24, 48, 96, 192, 384 sides. Taking the radius = 100, the
perimeter of the last one gives the value which Aryabhata used for TT.
The empirical nature of Hindu geometry is illustrated by Bhaskara's
proof of the Pythagorean theorem.
He draws the right triangle four
times in the square of the hypote-
nuse, so that in the middle there re-
mains a square whose side equals
the difference between the two sides

of the right triangle. Arranging this square and the four triangles in
a different way, they are seen, together, to make up the sum of the
square of the two sides. "Behold!" says Bhaskara, without adding

1 Cantor, op. cit., Vol. I, 3rd Ed., 1907, pp. 649-653.
1G. R. Kaye, op. cit., pp. 20-22.

88 A HISTORY OF MATHEMATICS

another word of explanation. Bretschneider conjectures that the
Pythagorean proof was substantially the same as this. Recently it
has been shown that this interesting proof is not of Hindu origin, but
was given much earlier (early in the Christian era) by the Chinese
writer Chang Chun-Ch'ing, in his commentary upon the ancient treat-
ise, the Chou-pei.1 In another place, Bhaskara gives a second dem-
onstration of this theorem by drawing from the vertex of the right
angle a perpendicular to the hypotenuse, and comparing the two tri-
angles thus obtained with the given triangle to which they are similar.
This proof was unknown in Europe till Wallis rediscovered it. The
only Indian work that touches the subject of the conic sections is Ma-
havira's book, which gives an inaccurate treatment of the ellipse. It
is readily seen that the Hindus cared little for geometry. Brahma-
gupta's cyclic quadrilaterals constitute the only gem in their geom-
etry.

The grandest achievement of the Hindus and the one which, of all
mathematical inventions, has contributed most toJihe progress of
intelligence, is the perfecting of the so-called " Arabic Notation."
That this notation did not originate with the Arabs is now admitted
by every one. Until recently the preponderance of authority favored
the hypothesis that our numeral system, with^its concept of local
value and our symbol for zero, was wholly of HmcTu origin. Now it
appears that the principal ol local value was Used in the sexagesimal
system found on Babylonian tablets dating from 1600 to 2300 B. c.
and that Babylonian records from the centuries immediately preced-
ing the Christian era contain a symbol for zero which, however, was
not used in computation. These .sexagesimal fractions appear in
Ptolemy's Almagest in 130 A. D., where the omicron o is made to des-
ignate blanks in the sexagesimal numbers, but was not used as a reg-
ular zero. The Babylonian origin of the sexagesimal fractions used by
Hindu astronomers is denied by no one. .The earliest form of the In-
dian symbol for zero was that of a dot which, according to Buhler,2
was "commonly used in inscriptions and manuscripts in order to
mark a blank." This restricted early use of the symbol for zero re-
sembles somewhat the still earlier use made of it by the Babylonians
and by Ptolemy. It is therefore probable that an imperfect notation
involving the principle of local value and the use of the zero was im-
ported into India, that it was there transferred from the sexagesimal
to the decimal scale and ttien, in the course of centuries, brought to
final perfection. ~"Ti the'SU vitws di'c fuund IjyTurther research to be
correct7~thcn the name " Babylonic-Hindu " notation will be more
appropriate than either "Arabic" or "Hindu- Arabic." It appears

1 Yoshio Mikami, "The Pythagorean Theorem" in Archiv d. Math. u. Physik,
3. S., Vol. 22, 1912, pp. 1-4.

2 Quoted by D. E. Smith and L. C. Karpinski in their Hindu-Arabic Numerals,
Boston and London, 1911, p. 53.

THE HINDUS 89

that in India various numeral forms were used long before the prin-
ciple of local value and the zero came to be used. Early Hindu numer-
als have been classified under three great groups. Numeral forms of
one of these groups date from the third century B. c.1 and are believed
to be the forms from which our present system developed. That the
nine figures were introduced quite early and that the principle of lo-
cal value and the zero were incorporated later is a belief which re-
ceives support from the fact that on the island of Ceylon a notation
resembling the Hindu, but without the zero has been preserved. We
know that Buddhism and Indian culture were transplanted to Ceylon
about the third century after Christ, and that this culture remained
stationary there, while it made progress on the continent. It seems
highly probable, then, that the numerals of Ceylon are the old, im-
perfect numerals of India. In Ceylon, nine figures were used for the
units, nine others for the tens, one for 100, and also one for 1000.
These 20 characters enabled them to write all the numbers up to
9999. Thus, 8725 would have been written with six signs, represent-
ing the following numbers: 8, 1000, 7, 100, 20, 5. These Singhalesian
signs, like the old Hindu numerals, are supposed originally to have
been the initial letters of the corresponding numeral adjectives. There
is a marked resemblance between the notation of Ceylon and the one
used by Aryabhata in the first chapter of his work, and there only.
Although the zero and the principle of position were unknown to the
scholars of Ceylon, they were probably known to Aryabhata; for, in
the second chapter, he gives directions for extracting the square and
cube roots, which seem to indicate a knowledge of them. The sym-
bol for zero was called sunya (the void). It is found in the form of a
dot in the Bakhshall arithmetic, the date of which is uncertain. The
earliest undoubted occurrence of our zero in India is in 876 A. D.2
The earliest known mention of Hindu numerals outside of India was
made in 662 A. D. by the Syrian writer, Severus Sebokht. He speaks
of Hindu computations "which excel the spoken word and . . . are
done with nine symbols." 3

There appear to have been several notations in use in different parts
of India, which differed, not in principle, but merely in the forms of
the signs employed. Of interest is also a symbolical system of position,
in which the figures generally were not expressed by numerical adjec-
tives, but by objects suggesting the particular numbers in question.
Thus, for i were used the words moon, Brahma, Creator, or form; for
4, the words Veda, (because it is divided into four parts) or ocean, etc.
The following example, taken from the Surya Siddhanta, illustrates
the idea. The number 1,577,917,828 is expressed from right to left as

1 D. E. Smith and L. C. Karpinski, op. cit., p. 22.

2 Ibid, p. 52.

3 M. F. Nau in Journal Asiatique, S. 10, Vol. 16, 1910; D. E. Smith in Bull. Am.
Math. Soc., Vol. 23, 1917, p. 366.

90 A HISTORY OF MATHEMATICS

follows: Vasu (a class of 8 gods)+ two+eight+ mountains (the 7 moun-
tain-chains^ form-f- digits (the 9 digits)+seven+mountains+ lunar
days (half of which equal 15). The use of such notations made it pos-
sible to represent a number in several different ways. This greatly
facilitated the framing of verses containing arithmetical rules or sci-
entific constants, which could thus be more easily remembered.

At an early period the Hindus exhibited great skill in calculating,
even with large numbers. Thus, they tell us of an examination to
which Buddha, the reformer of the Indian religion, had to submit,
when a youth, in order to win the maiden he loved. In arithmetic,
after having astonished his examiners by naming all the periods of
numbers up to the 53d, he was asked whether he could determine the
number of primary atoms which, when placed one against the other,
would form a line one mile in length. Buddha found the required an-
swer in this way: 7 primary atoms make a very minute grain of dust,

7 of these make a minute grain of dust, 7 of these a grain of dust whirled
up by the wind, and so on. Thus he proceeded, step by step, until he
finally reached the length of a mile. The multiplication of all the fac-
tors gave for the multitude of primary atoms in a mile a number con-
sisting of 15 digits. This problem reminds one of the " Sand-Counter"
of Archimedes.

After the numerical symbolism had been perfected, figuring was
made much easier. Many of the Indian modes of operation differ
from ours. The Hindus were generally inclined to follow the motion
from left to right, as in writing. Thus, they added the left-hand col-
umns first, and made the necessary corrections as they proceeded.
For instance, they would have added 254 and 663 thus: 2+6 = 8,
5+6=11, which changes 8 into 9, 4+3 = 7. Hence the sum 917. In
subtraction they had two methods. Thus in 821 — 348 they would say,

8 from 11 = 3,4 from 11 = 7, 3 from 7 = 4. Or they would say, 8 from
11 = 3, 5 from 12 = 7, 4 from 8 = 4. In multiplication of a number by
another of only one digit, say 569 by 5, they generally said, 5.5 = 25,
5.6 = 30, which changes 25 into 28, 5.9 = 45, hence the o must be in-
creased by 4. The product is 2845. In the multiplication with each
other of many-figured numbers, they first multiplied, in the manner
just indicated, with the left-hand digit of the multiplier, which was
written above the multiplicand, and placed the product above the
multiplier. On multiplying with the next digit of the multiplier, the
product was not placed hi a new row, as with us, but the first product
obtained was corrected, as the process continued, by erasing, when-
ever necessary, the old digits, and replacing them by new ones, until
finally the whole product was obtained. We who possess the modern
luxuries of pencil and paper, would not be likely to fall in love with
this Hindu method. But the Indians wrote "with a cane-pen upon
a small blackboard with a white, thinly liquid paint which made marks
that could be easily erased, or upon a white tablet, less than a foot

THE HINDUS 91

square, strewn with red flour, on which they wrote the figures with a
small stick, so that the figures appeared white on a red ground." l
Since the digits had to be quite large to be distinctly legible, and
since the boards were small, it was desirable to have a method which
would not require much space. Such a one was the above method
of multiplication. Figures could be easily erased and replaced by
others without sacrificing neatness. But the Hindus had also other
ways of multiplying, of which we mention the following: The tablet
was divided into squares like a chess-board. Diagonals were also
drawn, as seen in the figure. The multiplication of 12X735 = 8820 is
exhibited in the adjoining diagram.2 According to Kaye,3 this mode
of multiplying was not of Hindu origin and was
known earlier to the Arabs. The manuscripts
extant give no information of how divisions were
executed.

Hindu mathematicians of the twelfth century
test the correctness of arithmetical computations 8830
by "casting out nines," but this process is not of Hindu origin;
it was known to the Roman bishop Hippolytos in the third cen-
tury.

In the Bakhshdll arithmetic a knowledge of the processes of com-
putation is presupposed. In fractions, the numerator is written above
the denominator without a dividing line. Integers are written as
fractions with the denominator i. In mixed expressions the integral

part is written above the fraction. Thus, i = if . In place of our =

they used the word phalam, abbreviated into pha. Addition was in-
dicated by yu, abbreviated from yuta. Numbers to be combined

were often enclosed in a rectangle. Thus, pha 1 2 j \ \yu\ means £+ £ =

12. An unknown quantity is sunya, and is designated thus . by a
heavy dot. The word sunya means "empty," and is the word for
zero, which is here likewise represented by a dot. This double use of
the word and dot rested upon the idea that a position is "empty" if
not filled out. It is also to be considered " empty " so long as the num-
ber to be placed there has not been ascertained.4

The Bakhshali arithmetic contains problems of which some are
solved by reduction to unity or by a sort of false position. Example:
B gives twice as much as A, C three times as much as B, D four times as
much as C; together they give 132 ; how much did A give? Take i for
the unknown (sunya), thenA=i, B = 2, C = 6, D = 24, their sum =
33. Divide 132 by 33, and the quotient 4 is what A gave.

The method of false position we have encountered before among

1 H. Hankel, op. tit., 1874, p. 186.

2 M. Cantor, op. cit., Vol. I, 3 Aufl., 1907, p. 611.
* G. R. Kaye, op. cit., p. 34.

4 Cantor, I, 3 Ed., 1907, pp. 613-618.

92 A HISTORY OF MATHEMATICS

the early Egyptians. With them it was an instinctive procedure;
with the Hindus it had risen to a conscious method. Bhaskara uses
it, but while the Bakhshall document preferably assumes i as the
unknown, Bhaskara is partial to 3. Thus, if a certain number is
taken five-fold, ^ of the product be subtracted, the remainder divided
by 10, and |, | and j of the original number added, then 68 is ob-
tained. What is the number? Choose 3, then you get 15, 10, i, and

= ¥•

Then (684-\7-)3 = 48, the answer.

We shall now proceed to the consideration of some arithmetical
problems and the Indian modes of solution. A favorite method was
that of inversion. With laconic brevity, Aryabhata describes it thus:
"Multiplication becomes division, division becomes multiplication;
what was gain becomes loss, what loss, gain; inversion." Quite different
from this quotation in style is the following problem from Aryabhata,
which illustrates the method: "Beautiful maiden with beaming eyes,
tell me, as thou understandst the right method of inversion, which
is the number which multiplied by 3, then increased by f of the prod-
uct, divided by 7, diminished by f of the quotient, multiplied by it-
self, diminished by 52, the square root extracted, addition of 8, and
division by 10, gives the number 2?" The process consists in begin-
ning with 2 and working backwards. Thus, (2.10— 8)2+52= 196,
\/i96= 14, and 14.1.7.^4-3 = 28, the answer.

Here is another example taken from Lilavafi, a chapter in Bhas-
kara's great work: "The square root of half the number of bees in a
swarm has flown out upon a jessamine-bush, f of the whole swarm
has remained behind; one female bee flies about a male that is buzz-
ing within a lotus-flower into which he was allured in the night by its
sweet odor, but is now imprisoned in it. Tell me the number of bees."
Answer, 72. The pleasing poetic garb in which all arithmetical prob-
lems are clothed is due to the Indian practice of writing all school-books
in verse, and especially to the fact that these problems, propounded
as puzzles, were a favorite social amusement. Says Brahmagupta:
"These problems are proposed simply for pleasure; the wise man can
invent a thousand others, or he can solve the problems of others by
the rules given here. As the stfn eclipses the stars by his brilliancy,
so the man of knowledge will eclipse the fame of others in assemblies
of the people if he proposes algebraic problems, and still more if he
solves them."

The Hindus solved problems in interest, discount, partnership,
alligation, summation of arithmetical and geometric series, and de-
vised rules for determining the numbers of combinations and permu-
tations. It may here be added that chess, the profoundest of all
games, had its origin in India. The invention of magic squares is
sometimes erroneously attributed to the Hindus. Among the Chi-
nese and Arabs magic squares appear much earlier. The first occur-
rence of a magic square among the Hindus is at Dudhai, Ihansi, hi

THE HINDUS 93

northern India. It is engraved upon a stone found in the ruins of
a temple assigned to the eleventh century, A. D.1 After the time of
Bhaskara magic squares are mentioned by Hindu writers.

The Hindus made frequent use of the " rule of three." Their method
of "false position," is almost identical with that of the "tentative
assumption" of Diophantus. These and other rules were applied to
a large number of problems.

Passing now to algebra, we shall first take up the symbols of opera-
tion. Addition was indicated simply by juxtaposition as in Diophan-
tine algebra; subtraction, by placing a dot over the subtrahend; mul-
tiplication, by putting after the factors, bha, the abbreviation of the
word bhavita, "the product"; division, by placing the divisor beneath
the dividend; square- root, by writing ka, from the word karana (irra-
tional), before the quantity. The unknown quantity was called by
Brahmagupta ydvattdvat (quantum tantuni). When several unknown
quantities occurred, he gave, unlike Diophantus, to each a distinct
name and symbol. The first unknown was designated by the general
term "unknown quantity." The rest were distinguished by names
of colors, as the black, blue, yellow, red, or green unknown. The ini-
tial syllable of each word constituted the symbol for the respective
unknown quantity. Thus yd meant x; kd (from kdlaka = black) meant
y; yd kd bha, "x times y"; ka 15 ka 10, " A/iS — \/io."

The Indians were the first to recognize the existence of absolutely
negative quantities. They brought out the difference between posi-
tive and negative quantities by attaching to the one the idea of "pos-
session," to the other that of "debts." The conception also of oppo-
site directions on a line, as an interpretation of + and — quantities,
was not foreign to them. They advanced beyond Diophantus in ob-
serving that a quadratic has always two roots. Thus Bhaskara gives
x=$o and .v= — 5 for the roots of x2 — 45*= 250. "But," says he,
"the second value is in this case not to be taken, for it is inadequate;
people do not approve of negative roots." Commentators speak of
this as if negative roots were seen, but not admitted.

Another important generalization, says Hankel, was this, that since
the time of Bhaskara the Hindus never confined their arithmetical
operations to rational numbers. For instance, Bhaskara showed how,

by the formula vVf-VT=-\ a+Va2~6+-N a~Va2~6 the square

2 2

root of the sum of rational and irrational numbers could be found.
The Hindus never discerned the dividing line between numbers and
magnitudes, set up by the Greeks, which, though the product of a
scientific spirit, greatly retarded the progress of mathematics. They
passed from magnitudes to numbers and from numbers to magnitudes
without anticipating that gap which to a sharply discriminating mind
1 Bull. Am. Math. Soc., Vol. 24, 1917, p. 106.

94 A HISTORY OF MATHEMATICS

exists between the continuous and discontinuous. Yet by doing so
the Indians greatly aided the general progress of mathematics. "In-
deed, if one understands by algebra the application of arithmetical
operations to complex magnitudes of all sorts, whether rational or irra-
tional numbers or space-magnitudes, then the learned Brahmins of
Hindostan are the real inventors of algebra." i

Let us now examine more closely the Indian algebra. In extract-
ing the square and cube roots they used the formulas (<H-&)2j=a2+
2ab-\-b2 and (a-\-b}z=az-\-$aib-\-T)ab'2-\-bz. In this connection Aryab-
hata speaks of dividing a number into periods of two and three digits.
From this we infer that the principle of position and the zero in the
numerical notation were already known to him. In figuring with
zeros, a statement of Bhaskara is interesting. A fraction whose de-
nominator is zero, says he, admits of no alteration, though much be
added or subtracted. Indeed, in the same way, no change takes place
in the infinite and immutable Deity when worlds are destroyed or
created, even though numerous orders of beings be taken up or brought
forth. Though in this he apparently evinces clear mathematical no-
tions, yet in other places he makes a complete failure in figuring with
fractions of zero denominator.

In the Hindu solutions of determinate equations, Cantor thinks
he can see traces of Diophantine methods. Some technical terms be-
tray their Greek origin. Even if it be true that the Indians borrowed
from the Greeks, they deserve credit for improving the solutions of
linear and quadratic equations. Recognizing the existence of neg-
ative numbers, Brahmagupta was able to unify the treatment of
the three forms of quadratic equations considered by Diophantus,
viz., ax*-{-bx=c, bx-\-c = ax,2 ax'2-\-c = bx, (a, b and c being pos-
itive numbers), by bringing the three under the one general case,
pxz-\-qx-\-r=o. To S'rldhara is attributed the "Hindu method"
of completing the square which begins by multiplying both sides
of the equation by 4/>. Bhaskara advances beyond the Greeks
and even beyond Brahmagupta when he says that "the square of
a positive, as also of a negative number, is positive; that the square
root of a positive number is twofold, positive and negative. There is
no square root of a negative number, for it is not a square." Kaye
points out, however, that the Hindus were not the first to give double
solutions of quadratic equations.2 The Arab Al-Khowarizmi of the
ninth century gave both solutions of x~-{- 2i = iox. Of equations of
higher degrees, the Indians succeeded in solving only some special
cases in which both sides of the equation could be made perfect powers
by the addition of certain terms to each.

Incomparably greater progress than in the solution of determinate
equations was made by the Hindus in the treatment of indeterminate
equations. Indeterminate analysis was a subject to which the Hindu
1 H. Hankel, op. «*/., p. 195. * G. R. Kaye, op. cit., p. 34.

THE HINDUS 95

mind showed a happy adaptation. We have seen that this very sub-
ject was a favorite with Diophantus, and that his ingenuity was al-
most inexhaustible in devising solutions for particular cases. But the
glory of having invented general methods in this most subtle branch
of mathematics belongs to the Indians. The Hindu indeterminate
analysis differs from the Greek not only in method, but also in aim.
The object of the former was to find all possible integral solutions.
Greek analysis, on the other hand, demanded not necessarily integral,
but simply rational answers. Diophantus was content with a_single
solution; the Hindus endeavored to find all solutions possible. Aryab-
hata gives solutions in integers to linear equations of the form ax=±
by = c, where a, b, c are integers. The rule employed is called the pul-
verizer. For this, as for most other rules, the Indians give no proof.
Their solution is essentially the same as the one of Euler. Euler's

process of reducing r to a continued fraction amounts to the same as

the Hindu process of finding the greatest common divisor of a and b
by division. This is frequently called the Diophantine method. Han-
kel protests against this name, on the ground that Diophantus not
only never knew the method, but did not even aim at solutions purely
integral.1 These equations probably grew out of problems in astron-
omy. They were applied, for instance, to determine the time when
a certain constellation of the planets would occur in the heavens.

Passing by the subject of linear equations with more than two un-
known quantities, we come to indeterminate quadratic equations. In
the solution of xy=ax-{-by-{-c, they applied the method re-invented
later by Euler, of decomposing (ab-}-c) into the product of two integers
m.n and of placing x = m-}-b and y = n-\-a.

Remarkable is the Hindu solution of the quadratic equation cy- =
ax2-\-b. With great keenness of intellect they recognized in the special
case y2=axz-}-i a fundamentarproblem in indeterminate quadratics.
They solved it by the cyclic method. "It consists," says De Morgan,
"in a rule for finding an indefinite number of solutions of y2 = ax2-\-i
(a being an integer which is not a square), by means of one solution
given or found, and of feeling for one solution by making a solution
of y2 = ax2-\-b give a solution of y2 = ax2-}-b2. It amounts to the fol-
lowing theorem: If p and q be one set of values of x and y in y2 = ax'2-\-b
and p' and q' the same or another set, then qP+pq' and app'-}-qq'
are values of x and y in y2 = ax2-\-b2. From this it is obvious that one
solution of y2 = ax2-}-i may be made to give any number, and that if,
taking b at pleasure, y2 = ax2-{-b2 can be solved so that x and y
are divisible by b, then one preliminary solution of y2 = ax2-{-i can be
be found. Another mode of trying for solutions is a combination of
the preceding with the cuttaca (pulverizer)." These calculations were
used in astronomy.

1 H. Hankel, op. cit., p. 196.

96 A HISTORY OF MATHEMATICS

Doubtless this "cyclic method" constitutes the greatest invention
in the theory of numbers before the time of Lagrange. The perver-
sity of fate has willed it, that the equation y2 = ax2+i should now be
called Pell's equation; the first incisive work on it is due to Brahmin
scholarship, reinforced, perhaps, by Greek research. It is a problem
that has exercised the highest faculties of some of our greatest modern
analysts. By them the work of the Greeks and Hindus was done over
again; for, unfortunately, only a small portion of the Hindu algebra and
the Hindu manuscripts, which we now possess, were known in the
Occident. Hankel attributed the invention of the "cyclic method"
entirely to the Hindus, but later historians, P. Tannery, M. Cantor,
T. Heath, G. R. Kaye favor the hypothesis of ultimate Greek origin.
If the missing parts of Diophantus are ever found, light will probably
be thrown upon this question.

Greater taste than for geometry was shown by the Hindus for trig-
onometry. Interesting passages are found in Varaha Mihira's Pancha
Siddhantika of the sixth century A. D.,1 which, in our notation for
unit radius, gives 7T=Vio, sin 30° =|, sin 6o°=Vi — \, sin2'y =
(sin 27) 2/4+ ( i — sin (90°— 27) ) 2/4. This is followed by a table of
24 sines, the angles increasing by intervals of 3°45' (the eighth part
.of 30°), obviously taken from Ptolemy's table of chords. However,
instead of dividing the radius into 60 parts in the manner of Ptolemy,
the Hindu astronomer divides it into 120 parts, which device enabled
him to convert Ptolemy's table of. "chords into a table of sines without
changing the numerical values. Aryabhata took a still different value
for the radius, namely, 3438, obtained apparently from the relation
2X 3. 14167 = 21,600. The Hindus followed the Greeks and Babylo-
nians in the practice of dividing the circle into quadrants, each quad-
rant into 90 degrees and 5400 minutes — thus dividing the whole circle
into 21,600 equal parts. Each quadrant was divided also into 24 equal
parts, so that each part embraced 225 units of the whole circumference,
and corresponded to 3! degrees. Notable is the fact that the Indians
never reckoned, like the Greeks, with the whole chord of double the arc,
but always with the sine (joa) and versed sine. Their mode of calcula-
ting tables was theoretically very simple. The sine of 90° was equal to
the radius, or 3438; the sine of 30° was evidently half that, or 1719.

F
Applying the formula sin2a+ cos2a = r2, they obtained sin 45° = r~ =

243 1. Substituting for cos a its equal sin (90 — a), and making a = 60°,

V~TP
they obtained sin 60° = — — = 2978. With the sines of 90, 60, 45, and 30

as starting-points, they reckoned the sines of half the angles by the
formula ver sin 2 a=2 sin2a, thus obtaining the sines of 22° 30', 15°,

1 G. R. Kaye, op. cit., p. 10.

THE HINDUS 97

11° 15', 7° 30', 3° 45'. They now figured out the sines of the comple-.
ments of these angles, namely, the sines of 86° 15', 82° 30', 78° 45',
75°> 67° 30'; then they calculated the sines of half these angles; then
of their complements; then, again, of half their complements; and so
on. By this very simple process they got the sines of angles at inter-
vals of 3° 45'. In this table they discovered the unique law that if
a, b, c be three successive arcs such that a—b = b—c=s° 45', then

sin a — sin b = (sin b — sin c) . This formula was afterwards used

225

whenever a re-calculation of tables had to be made. No Indian trig-
onometrical treatise on the triangle is extant. In astronomy they
solved plane and spherical triangles.1

Now that we have a fairly complete history of Chinese mathematics,
Kaye has been able to point out parallelisms between Hindu and
Chinese mathematics which indicate that India is indebted to China.
The 2 Chiu-chang Suan-shu ("Arithmetic in Nine Sections") was com-
posed in China at least as early as 200 B. c.; the Chinese writer Chang
T'sang wrote a commentary on it in 263 A. D. The "Nine Sections"
gives the approximate area of a segment of a circle = | (c-\-a)a, where
c is the chord and a is the perpendicular. This rule occurs in the work
of the later Hindu author Mahavlra. Again, the Chinese problem of
the bamboo 10 ft. high, the upper end of which being broken reaches the
ground three feet from the stem; to determine the height of the break —
occurs in all Hindu books after the sixth century. The Chinese arith-
metical treatise, Sun-Tsu Suan-ching, of about the first century A.
D. has an example asking for a number which, divided by 3 yields the
remainder 2, by 5 the remainder 3, and by 7 the remainder 2. Exam-
ples of this type occur in Indian works of the seventh and ninth cen-
turies, particularly in Brahmagupta and Mahavlra. On a preceding
page we called attention to the fact that Bhaskara's dissection proof
of the Pythagorean theorem is found much earlier in China. Kaye
gives several other examples of Chinese origin that are found later in
Hindu books.

Notwithstanding the Hindu indebtedness to other nations, it is
remarkable to what extent Indian mathematics enters into the
science of our time. Both the form and the spirit of the arithmetic
and algebra of modern times are essentially Indian. Think of our
notation of numbers, brought to perfection by the Hindus, think of
the Indian arithmetical operations nearly as perfect as our own, think
of their elegant algebraical methods, and then judge whether the
Brahmins on the banks of the Ganges are not entitled to some credit.
Unfortunately, some of the most brilliant results in indeterminate an-
alysis, found in Hindu works, reached Europe too late to exert the in-

1 A. Arneth, Geschichle der reinen Malhcmatik. Stuttgart, 1852, p. 174.

2 G. R. Kaye, op. cit., pp. 38-41.

98 A HISTORY OF MATHEMATICS

fluence they would have exerted, had they come two or three centu-
ries earlier.

At the beginning of the twentieth century, mathematical activity
along modern lines sprang up in India. In the year 1907 there was
founded the Indian Mathematical Society; in 1909 there was started
at Madras the Journal of the Indian Mathematical Society.1

1 (Three recent writers have advanced arguments tending to disprove the Hindu
origin of our numerals. We refer (i) to G. R. Kaye's articles in Scientia, Vol. 24,
1918, pp. 53-55; in Journal Asiatic Soc. Bengal, III, 1907, pp. 475-508, also VII,
1911, pp. 801-816: in Indian Antiquary, 1911, pp. 50-56; (2) to Carra de Vaux's
article in Scientia, Vol. 21, 1917, pp. 273-282; (3) to a Russian book brought out
by Nikol. Bubnow in 1908 and translated into German in 1914 by Jos. Lezius.
Kaye claims to show that the proofs of the Hindu origin of our numerals are
largely legendary, that the question has been clouded by a confusion between the
words Hindi- (Indian) and hindasi (measure geometrical), that the symbols are
not modified letters of the alphabet. We must hold our minds in suspense on
this difficult question and await further evidence.)

THE ARABS

After the flight of Mohammed from Mecca to Medina in 622 A. D.,
an obscure people of Semitic race began to play an important part in
the drama of history. Before the lapse of ten years, the scattered
tribes of the Arabian peninsula were fused by the furnace blast of
religious enthusiasm into a powerful nation. With sword in hand the
united Arabs subdued Syria and Mesopotamia. Distant Persia and
the lands beyond, even unto India, were added to the dominions of
the Saracens. They conquered Northern Africa, and nearly the
whole Spanish peninsula, but were finally checked from further prog-
ress in Western Europe by the firm hand of Charles Martel (732 A. D.).
The Moslem dominion extended now from India to Spain; but a war
of succession to the caliphate ensued, and in 755 the Mohammedan
empire was divided, — one caliph reigning at Bagdad, the other at Cor-
dova in Spain. Astounding as was the grand march of conquest by
the Arabs, still more so was the ease with which they put aside their
former nomadic life, adopted a higher civilization, and assumed the
sovereignty over cultivated peoples. Arabic was made the written
language throughout the conquered lands. With the rule of the Abba-
sides in the East began a new period in the history of learning. The
capital, Bagdad, situated on the Euphrates, lay half-way between
two old centres of scientific thought, — India in the East, and Greece
in the West. The Arabs were destined to be the custodians of the
torch of Greek science, to keep it ablaze during the period of confu-
sion and chaos in the Occident, and afterwards to pass it over to the
Europeans. This remark applies in part also to Hindu science. Thus
science passed from Aryan to Semitic races, and then back again to
the Aryan. Formerly it was held that the Arabs added but little to
the knowledge of mathematics; recent studies indicate that they must
be credited with novelties once thought to be of later origin.

The Abbasides at Bagdad encouraged the introduction of the
sciences by inviting able specialists to their court, irrespective of na*
tionality or religious belief. Medicine and astronomy were their fa-
vorite sciences. Thus Harun-al-Rashid, the most distinguished Sara-
cen ruler, drew Indian physicians to Bagdad. In the year 772 there
came to the court of Caliph Almansur a Hindu astronomer with as-
tronomical tables which were ordered to be translated into Arabic.
These tables, known by the Arabs as the Sindhind, and probably taken
from the Brahma-sphuta-siddhanta of Brahmagupta, stood in great
authority. They contained the important Hindu table of sines.

Doubtless at this time, and along with these astronomical tables,

ioo A HISTORY OF MATHEMATICS

the Hindu numerals, with the zero and the principle of position, were \
introduced among the Saracens. Before the time of Mohammed the
Arabs had no numerals. Numbers were written out in words. Later,
the numerous computations connected with the financial administra-
tion over the conquered lands made a short symbolism indispensable.
In some localities, the numerals of the more civilized conquered na-
tions were used for a time. Thus, in Syria, the Greek notation was
retained; in Egypt, the Coptic. In some cases, the numeral adjec-
tives may have been abbreviated in writing. The Diwani-numerals,
found in an Arabic-Persian dictionary, are supposed to be such ab-
breviations. Gradually it became the practice to employ the 28 Ara-
bic letters of the alphabet for numerals, in analogy to the Greek sys-
tem. This notation was in turn superseded by the Hindu notation,
which quite early was adopted by merchants, and also by writers on
arithmetic. Its superiority was generally recognized, except in as-
tronomy, where the alphabetic notation continued to be used. Here
the alphabetic notation offered no great disadvantage, since in the
sexagesimal arithmetic, taken from the Almagest, numbers of gen- /
erally only one or two places had to be written. 1

As regards the form of the so-called Arabic numerals, the state-
ment of the Arabic writer A l-Biruni (died 1039), who spent many
years in India, is of interest. He says that the shape of the numer-
als, as also of the letters in India, differed in different localities, and
that the Arabs selected from the various forms the most suitable. An
Arabian astronomer says there was among people much difference in
the use of symbols, especially of those for 5, 6, 7, and 8. The symbols
used by the Arabs can be traced back to the tenth century. We find
material differences between those used by the Saracens in the East
and those used in the West. But most surprising is the fact that the
symbols of both the East and of the West Arabs deviate so extraordi-
narily from the Hindu Devanagari numerals ( = divine numerals) of
to-day, and that they resemble much more closely the apices of the
Roman writer Boethius. This strange similarity on the one hand,
and dissimilarity on the other, is difficult to explain. The most plau-
sible theory is the one of Woepcke: (i) that about the second cen-
tury after Christ, before the zero had been invented, the Indian nu-
merals were brought to Alexandria, whence they spread to Rome
and also toWesFTtftica; (2) that in the eighth century, after the no-
tation in India had been already much modified and perfected by the
invention of the zero, the Arabs at Bagdad got it from the Hindus;
(3) that the Arabs of the West borrowed the Columbus-egg, the zero,
from those in the East, but retained the old forms of the nine numer-
als, if for no other reason, simply to be contrary to their political ene-
mies of the East; (4) that the old forms were remembered by the West-
Arabs to be of Indian origin, and were hence called Gubar-numerals
1 H. Hankel, op. cit., p. 255.

THE ARABS 101

( = dust-numerals, in memory of the Brahmin practice of reckoning
on tablets strewn with dust or sand; (5) that, since the eighth cen-
tury, the numerals in India underwent further changes, and assumed
the greatly modified forms of the modern Devanagari-numerals. 1 This
is rather a bold theory, but, whether true or not, it explains better
than any other yet propounded, the relations between the apices, the
Gubar, the East-Arabic, and Devanagari numerals.

It has been mentioned that in 772 the Indian Siddhanta was brought
to Bagdad and there translated into Arabic. There is no evidence that
any intercourse existed between Arabic and Indian astronomers either
before or after this time, excepting the travels of Al-Binmi. But
we should be very slow to deny the probability that more extended
communications actually did take place.

Better informed are we regarding the way in which Greek science,
in successive waves, dashed upon and penetrated Arabic soil. In
Syria the sciences, especially philosophy and medicine, were culti-
vated by Greek Christians. Celebrated were the schools at Antioch
and Emesa, and, first of all, the flourishing Nestorian school at Edessa.
From Syria, Greek physicians and scholars were called to Bagdad.
Translations of works from the Greek began to be made. A large
number of Greek manuscripts were secured by Caliph Al-Mamun (813-
833) from the emperor in Constantinople and were turned over to
Syria. The successors of Al-Mamun continued the work so auspi-
ciously begun, until, at the beginning of the tenth century, the more
important philosophic, medical, mathematical, and astronomical
works of the Greeks could all be read in the Arabic tongue. The trans-
lations of mathematical works must have been very deficient at first,
as it was evidently difficult to secure translators who were masters of
both the Greek and Arabic and at the same time proficient in mathe-
matics. The translations had to be revised again and again before
they were satisfactory. The first Greek authors made to speak in
Arabic were Euclid and Ptolemy. This was accomplished during the
reign of the famous Harun-al-Rashid. A revised translation of Eu-
clid's Elements was ordered by Al-Mamun. As this revision still con-
tained numerous errors, a new translation was made, either by the
learned Hunain ibn Ishak, or by his son, Ishak ibn Hunain. The
word " ibn " means " son." To the thirteen books of the Elements were
added the fourteenth, written by Hypsicles, and the fifteenth attrib-
uted by some to Damascius. But it remained for Tabit ibn Korra to
bring forth an Arabic Euclid satisfying every need. Still greater dif-
ficulty was experienced in securing an intelligent translation of the
Almagest. Among other important translations into Arabic were the
works of Apollonius, Archimedes, Heron, and Diophantus. Thus we
see that in the course of one century the Arabs gained access to the
vast treasures of Greek science.

1 M. Cantor, op. cit., Vol. I, 1907, p. 711.

102 A HISTORY OF MATHEMATICS

In astronomy great activity in original research existed as early as
the ninth century. The religious observances demanded by Moham-
medanism presented to astronomers several practical problems. The
Moslem dominions being of such enormous extent, it remained in
some localities for the astronomer to determine which way the " Be-
liever " must turn during prayer that he may be facing Mecca. The
prayers and ablutions had to take place at definite hours during the
day and night. This led to more accurate determinations of time. To
fix the exact date for the Mohammedan feasts it became neces-
sary to observe more closely the motions of the moon. In addition to
all this, the old Oriental superstition that extraordinary occurrences
in the heavens in some mysterious way affect the progress of human
affairs added increased interest to the prediction of eclipses.1

For these reasons considerable progress was made. Astronomical
tables and instruments were perfected, observatories erected, and a
connected series of observations instituted. This intense love for as-
tronomy and astrology continued during the whole Arabic scientific
period. As in India, so here, we hardly ever find a man exclusively
devoted to pure mathematics. Most of the so-called mathematicians
were first of all astronomers.

The first notable author of mathematical books was Mohammed
ibn Musa Al-Khowarizmi, who lived during the reign of Caliph Al-
Mamun (813-833). Our chief source of information about Al-Khow-
rizmi is the book of chronicles, entitled Kitab Al-Fihrist, written by
Al-Nadim, about 987 A. D., and containing biographies of learned
men. Al-Khowarizmi was engaged by the caliph in making extracts
from the Sindhind, in revising the tablets of Ptolemy, in taking ob-
servations at Bagdad and Damascus, and in measuring a degree of
the earth's meridian. Important to us is his work on algebra and
arithmetic. The portion on arithmetic is not extant in the original,
and it was not till 1857 that a Latin translation of it was found. It
begins thus: "Spoken has Algoritmi. Let us give deserved praise to
God, our leader and defender." Here the name of the author, Al-
Khowarizmi has passed into Algorilmi, from which come our modern
word algorithm, signifying the art of computing in any particular
way, and the obsolete form augrim, used by Chaucer.2 The arith-
metic of Khowarizmi, being based on the principle of position and
the Hindu method of calculation, "excels," says an Arabic writer,
"all others in brevity and easiness, and exhibits the Hindu intellect
and sagacity in the grandest inventions." This book was followed
by a large number of arithmetics by later authors, which differed
from the earlier ones chiefly in the greater variety of methods. Ara-
bian arithmetics generally contained the four operations with inte-

1 H. Hankel, op. cit., pp. 226-228.

2 See L. C. Karpinski, "Augrimstones" in Modern Language Notes, Vol. 27,
1912, pp. 206-209.

THE ARABS 103

gers and fractions, modelled after the Indian processes. They ex-
plained the operation of casting out the Q'S, also the regula falsa and
the regula duorum falsorum, sometimes called the rules of "false po-
sition" and of "double position" or "double false position," by which
algebraical examples could be solved without algebra. The regula
falsa or falsa positio was the assigning of an assumed value to the
unknown quantity, which value, if wrong, was corrected by some
process like the "rule of three." It was known to the Hindus and to
the Egyptian Ahmes. Diophantus used a method almost identical
with this. The regula duorum falsorum was as follows:1 To solve an
equation f(x) = V, assume, for the moment, two values for x; namely,
x = a and x = b. Then form f(a) =A and /(&) = £, and determine the

7 77 T?

errors V— A = Ea and V — B = Eb, then the required x = — =? — =•- is

Ea—Eb

generally a close approximation, but is absolutely accurate whenever
f(x) is a linear function of x.

We now return to Khowarizmi, and consider the other part of his
work,— the algebra. This is the first book known to contain this word
itself as title. Really the title consists of two words, al-jebr w'almu-
qabala, the nearest English translation of which is "restoration and
reduction." By "restoration" was meant the transposing of negative
terms to the other side of the equation; by "reduction," the uniting of
similar terms. Thus, x2 — 2x = $x-}-6 passes by al-jebr into £2=5.t-{-
2^+6; and this, by almuqabala, into x2 =jx-\- 6. The work on alge-
bra, like the arithmetic, by the same author, contains little that is
original. It explains the elementary operations and the solutions of
linear and quadratic equations. From whom did the author borrow
his knowledge of algebra? That it came entirely from Indian sources
is impossible, for the Hindus had no rules like the "restoration" and
"reduction." They were, for instance, never in the habit of making
all terms in an equation positive, as is done by the process of " restora-
tion." Diophantus gives two rules which resemble somewhat those
of our Arabic author, but the probability that the Arab got all his al-
gebra from Diophantus is lessened by the considerations that he rec-
ognized both roots of a quadratic, while Diophantus noticed only one;
and that the Greek algebraist, unlike the Arab, habitually rejected
irrational solutions. It would seem, therefore, that the algebra of
Al-Khowarizmi was neither purely Indian nor purely Greek. Al-
Khowarizimi's fame among the Arabs was great. He gave the ex-
amples .t2+iox = 39, x2-\-2i = IQX, 3.r+4 = .r2 which are used by later
authors, for instance, by the poet and mathematician Omar Khayyam.
"The equation ar-f- 1 0^ = 39 runs like a thread of gold through the
algebras of several centuries" (L. C. Karpinski). It appears in the
algebra of Abu Kamil who drew extensively upon the work of Al-

1 H. Hankel, op. oil., p. 259.

io4 A HISTORY OF MATHEMATICS

Khowarizmi. Abu Kamil, in turn, was the source largely drawn upon
by the Italian, Leonardo of Pisa, in his book of 1202.

The algebra of Al-Khowarizmi contains also a few meagre frag-
ments on geometry. He gives the theorem of the right triangle, but
proves it after Hindu fashion and only for the simplest case, when the
right triangle is isosceles. He then calculates the areas of the tri-
angle, parallelogram, and circle. For ?r he uses the value 3^, and also
the two Indian, TT = V io and TT = -f ft® fft. Strange to say, the last value
was afterwards forgotten by the Arabs, and replaced by others less
accurate. Al-Khowarizmi prepared astronomical tables, which, about
1000 A. D., were revised by Maslama al-Majrifi, and are of importance
as containing not only the sine function, but also the tangent function.1
The former is evidently of Hindu origin, the latter may be an addi-
tion made by Maslama and was formerly attributed to Abu'l Wefa.

Next to be noticed are the three sons of Musa Sakir, who lived in
Bagdad at the court of the Caliph Al-Mamun. They wrote several
works, of which we mention a geometry containing the well-known
formula for the area of a triangle expressed in terms of its sides. We
are told that one of the sons travelled to Greece, probably to collect
astronomical and mathematical manuscripts, and that on his way back
he made acquaintance with Tabit ibn Korra. Recognizing in him a
talented and learned astronomer, Mohammed procured for him a place
among the astronomers at the court in Bagdad. Tabit ibn Korra
(836-901) was born at Harran in Mesopotamia. He was proficient
not only in astronomy and mathematics, but also in the Greek, Arabic,
and Syrian languages. His translations of Apollonius, Archimedes,
Euclid, Ptolemy, Theodosius, rank among the best. His dissertation
on amicable numbers (of which each is the sum of the factors of the
other) is the first known specimen of original work in mathematics on
Arabic soil. It shows that he was familiar with the Pythagorean the-
ory of numbers. Tabit invented the following rule for finding amicable
numbers, which is related to Euclid's rule for perfect numbers: If
P = $.2n — i, 5 = 3. 2W~1 — i, r=9.22n~~1— i (n being a whole number)
are three primes, then a= 2npq, b=2nr are a pair of amicable numbers.
Thus, if n=2, then p=n, q = 5, r=7i, and 0=220, 6=284. Tabit
also trisected an angle.

Tabit ibn Korra is the earliest writer outside of China to discuss
magic squares. Other Arabic tracts on this subject are due to Ibn
Al-Haitam and later writers.2

Foremost among the astronomers of the ninth century ranked Al-

1 See H. Suter, "Die astronomischen Tafeln des Muhammed ibn Musa Al-
Khwarizmf in der Bearbeitung des Maslama ibn Ahmed Al-Madjnti und der Latein.
Uebersetzung des Athelhard von Bath," in Memoires de V Academic R. des Sciences
et des Lettres de Danemark, Copenhague, 7me S., Section des Lettres, t. Ill, no. i,
1914.

2 See H. Suter, Die M athematiker u. Astronomen der Araber u. ihre Werke, 1900,
PP- 36, 93, 136, 139, 140, 146, 218.

THE ARABS 105

Battani, called Albategnius by the Latins. Battan in Syria was his
birthplace. His observations were celebrated for great precission.
His work, De scientia stellarum, was translated into Latin by Plato
Tiburtinus, in the twelfth century. Out of this translation sprang the
word "sinus," as the name of a trigonometric function. The Arabic
word for "sine," jiba was derived from the Sanskrit jiva, and resem-
bled the Arabic word jaib, meaning an indentation or gulf. Hence
the Latin "sinus." l Al-Battani was a close student of Ptolemy, but
did not follow him altogether. He took an important step for the
better, when he introduced the Indian "sine" or half the chord, in
place of the whole chord of Ptolemy. He was the first to prepare a
tsib\e_oi_cpjangents. He dealt with horizontal and also vertical sun
dials, and accordingly considered a horizontal shadow (umbra extensa
in Latin translation) and vertical shadow (umbra versa}. These de-
noted, respectively, the "cotangent" and "tangent"; the former
came to be called umbra recta by Latin writers. Al-Battani probably
knew the law of sines; that this law was known to Al-Biruni is certain.
Another improvement on Greek trigonometry made by the Arabs
points likewise to Indian influences. Propositions and operations
which were treated by the Greeks geometrically are expressed by the
Arabs algebraically. Thus, Al-Battani at once gets from an equation

-n = D. the value of 6 by means of sin 6 = . — r—p-i, , — a process
# Vi+Z>2

cos

unknown to the ancients. He knows all the formulas for spherical
triangles given in the Almagest, but goes further, and adds an impor-
tant one of his own for oblique-angled triangles; namely, cos a = cos b.
cos c-\- sin b sin c cos A .

At the beginning or the tenth century political troubles arose in the
East, and as a result the house of the Abbasides lost power. One prov-
ince after another was taken, till, in 945, all possessions were wrested
from them. Fortunately, the new rulers at Bagdad, the Persian Buy-
ides, were as much interested in astronomy as their predecessors. The
progress of the sciences was not only unchecked, but the conditions
for it became even more favorable. The Emir Adud-ed-daula (978-
983) gloried in having studied astronomy himself. His son Saraf-ed-
daula erected an observatory in the garden of his palace, and called
thither a whole group of scholars.2 Among them were Abu'l-Wefa,
Al-Kuhi, Al-Sagani.

Abu'l Wefa (940-998) was born at Buzshan in Chorassan, a region
among the Persian mountains, which has brought forth many Arabic
astronomers. He made the brilliant discovery of the variation of the
moon, an inequality usually supposed to have been first discovered by
Tycho Brahe. Abu'1-Wefa translated Diophantus. He is one of the

1 M. Cantor, op. cil., Vol. I, 3 Aufl., 1907, p. 737.

2 H. Hankel, op. ell., p. 242.

106 A HISTORY OF MATHEMATICS

last Arabic translators and commentators of Greek authors. The fact
that he esteemed the algebra of Mohammed ibn Musa Al-Khowarizimi
worthy of his commentary indicates that thus far algebra had made
little or no progress on Arabic soil. Abu'1-Wefa invented a method for
computing tables of sines which gives the sine of half a degree correct
to nine decimal places. He used the tangent and calculated a table of
tangents. In considering the shadow-triangle of sun-dials he intro-
duced also the secant and cosecant. Unfortunately, these new trigo-
nometric functions and the discovery of the moon's variation ex-
cited apparently no notice among his contemporaries and followers.
A treatise by Abu'1-Wefa on "geometric constructions" indicates that
efforts were being made at that time to improve draughting. It con-
tains a neat construction of the corners of the regular polyedrons on
the circumscribed sphere. Here, for the first time, appears the con-
dition which afterwards became very famous in the Occident, that
the construction be effected with a single opening of the compasses.

Al-Kuhi, the second astronomer at the observatory of the emir at
Bagdad, was a close student of Archimedes and Apollonius. He solved
the problem, to construct a segment of a sphere equal in volume to
a given segment and having a curved surface equal in area to that of
another given segment. He, Al-Sagani, and Al-Biruni made a study
of the trisection of angles. Abu'l Jud, an able geometer, solved the
problem by the intersection of a parabola with an equilateral hyper-
bola.

The Arabs had already discovered the theorem that the sum of two
cubes can never be a cube. This is a special case of the " last theorem
of Fermat." Abu Mohammed Al-Khojandi of Chorassan thought he
had proved this. His proof, now lost, is said to have been defective.
Several centuries later Beha-Eddin declared the impossibility of
xz-\-yz = zz. Creditable work in theory of numbers and algebra was
done by Al-Karkhi of Bagdad, who lived at the beginning of the elev-
enth century. His treatise on algebra is the greatest algebraic work
of the Arabs. In it he appears as a disciple of Diophantus. He was
the first to operate with higher roots and to solve equations of the
form x-n-\-axn = b. For the solution of quadratic equations he gives
both arithmetical and geometrical proofs. He was the first Arabic
author to give and prove the theorems on the summation of the se-
ries: —

Al-Karkhi also busied himself with indeterminate analysis. He
showed skill in handling the methods of Diophantus, but added no-
thing whatever to the stock of knowledge already on hand. Rather
surprising is the fact that Al-Karkhi's algebra shows no traces what-

THE ARABS 107

ever of Hindu indeterminate analysis. But most astonishing it is,
that an arithmetic by the same author completely excludes the Hindu
numerals. It is constructed wholly after Greek pattern. Abu'1-Wefa,
also, in the second half of the tenth century, wrote an arithmetic in
which Hindu numerals find no place. This practice is the very oppo-
site to that of other Arabian authors. The question, why the Hindu
numerals were ignored by so eminent authors, is certainly a puzzle.
Cantor suggests that at one time there may have been rival schools,
of which one followed almost exclusively Greek mathematics, the
other Indian.

The Arabs were familiar with geometric solutions of quadratic equa-
tions. Attempts were now made to solve cubic equations geometri-
cally. They were led to such solutions by the study of questions like
the Archimedean problem, demanding the section of a sphere by a
plane so that the two segments shall be in a prescribed ratio. The
first to state this problem in form of a cubic equation was Al-Mahani
of Bagdad, while Abu Ja'far Alchazin was the first Arab to solve the
equation by conic sections. Solutions were given also by Al-Kuhi,
Al-Hasan ibn Al-Haitam, and others. Another difficult problem, to
determine the side of a regular heptagon, required the construction of
the side from the equation x3— x2 — 2x-\-i = o. It was attempted by
many and at last solved by Abu'l Jud.

The one who did most to elevate to a method the solution of alge-
braic equations by intersecting conies, was the poet Omar Khayyam
of Chorassan (about 1045-1123). He divides cubics into two classes,
the trinomial and quadrinomial, and each class into families and spe-
cies. Each species is treated separately but according to a general
plan. He believed that cubics could not be solved by calculation, nor
bi-quadratics by geometry. He rejected negative roots and often
failed to discover all the positive ones. Attempts at bi-quadratic
equations were made by Abu'l- Wefa,1 who solved geometrically xA = a
and x*-\-ax3 = b.

The solution of cubic equations by intersecting conies was the great-
est achievement of the Arabs in algebra. The foundation to this work
had been laid by the Greeks, for it was Menaechmus who first con-
structed the roots of x3 — c = o or x3— 2a3 = o. It was not his aim to
find the number corresponding to x, but simply to determine the side
x of a cube double another cube of side a. The Arabs, on the other
hand, had another object in view: to find the roots of given numerical
equations. In the Occident, the Arabic solutions of cubics remained
unknown until quite recently. Descartes and Thomas Baker invented
these constructions anew. The works of Al-Khayyam, Al-Karkhi,
Abu'l Jud, show how the Arabs departed further and further from

1 L. Matthiessen, Orundziige der Antiken und Modernen Algebra der Litter alcn
Gleichungen, Leipzig, 1878, p. 923. Ludwig Matthiessen (1830-1906) was professor
of physics at Rostock.

io8 A HISTORY OF MATHEMATICS

the Indian methods, and placed themselves more immediately under
Greek influences.

With Al-Karkhi and Omar Khayyam, mathematics among the
Arabs of the East reached flood-mark, and now it begins to ebb. Be-
tween uoo and 1300 A. D. come the crusades with war and bloodshed,
during which European Christians profited much by their contact with
Arabian culture, then far superior to their own. The crusaders were
not the only adversaries of the Arabs. During the first half of the
thirteenth century, they had to encounter the wild Mongolian hordes,
and, in 1256, were conquered by them under the leadership of Hulagu.
The caliphate at Bagdad now ceased to exist. At the close of the four-
teenth century still another empire was formed by Timur or Tamer-
lane, the Tartar. During such sweeping turmoil, it is not surprising
that science declined. Indeed, it is a marvel that it existed at all.
During the supremacy of Hulagu, lived Nasir-Eddin (1201-1274),
a man of broad culture and an able astronomer. He persuaded Hu-
lagu to build him and his associates a large observatory at Maraga.
Treatises on algebra, geometry, arithmetic, and a translation of Eu-
clid's Elements, were prepared by him. He for the first time elabo-
rated trigonometry independently of astronomy and to such great
perfection that, had his work been known, Europeans of the fifteenth
century might have spared their labors.1 He tried his skill at a proof
of the parallel-postulate. His proof assumes that if AB is perpendic-
ular to CD at C, 'and if another straight line EDF makes an angle
EDC acute, then the perpendiculars to AB, comprehended between
AB and EF, and drawn on the side of CD toward E, are shorter and
shorter, the further they are from CD. His proof, in Latin translation,
was published by Wallis in 165 1.2 Even at the court of Tamerlane in
v D Samarkand, the sciences were by

no means neglected. A group of
astronomers was drawn to this
B court. Uleg Beg (1393-1449), a

C grandson of Tamerlane, was him-

self an astronomer. Most prominent at this tune was Al-Kashi, the
author of an arithmetic. Thus, during intervals of peace, science
continued to be cultivated in the East for several centuries. The
last Oriental writer was Beha-Eddin (1547-1622). His Essence of
Arithmetic stands on about the same level as the work of Mohammed
ibn Musa Khowarizmi, written nearly 800 years before.

"Wonderful is the expansive power of Oriental peoples, with which
upon the wings of the wind they conquer half the world, but more
wonderful the energy with which, in less than two generations, they
raise themselves from the lowest stages of cultivation to scientific

1 Bibliotheca mathematica (2), 7, 1893, P- 6.

2 R. Bonola, Nan-Euclidean Geometry, transl. by H. S. Carslaw, Chicago, 1917,
pp. 10-12.

THE ARABS 109

efforts." During all these centuries, astronomy and mathematics in
the Orient greatly excel these sciences in the Occident.

Thus far we have spoken only of the Arabs in the East. Between
the Arabs of the East and of the West, which were under separate gov-
ernments, there generally existed considerable political animosity.
In consequence of this, and of the enormous distance between the two
great centres of learning, Bagdad and Cordova, there was less scien-
tific intercourse among them than might be expected to exist between
people* having the same religion and written language. Thus the
course of science in Spain was quite independent of that in Persia.
While wending our way westward to Cordova, we must stop in Egypt
long enough to observe that there, too, scientific activity was re-
kindled. Not Alexandria, but Cairo with its library and observatory,
was now the home of learning. Foremost among her scientists ranked
Ibn Junos (died 1008), a contemporary of Abu'1-Wefa. He solved
some difficult problems in spherical trigonometry. Another Egyptian
astronomer was Ibn Al-Haitam (died 1038), who computed the vol-
umes of paraboloids formed by revolving a parabola about any diam-
eter or any ordinate; he used the method of exhaustion and gave the
four summation formulas for the first four powers of the natural num-
bers.1 Travelling westward, we meet in Morocco Abu'l Hasan Ali,
whose treatise "on astronomical instruments" discloses a thorough
knowledge of the Conies of Apollonius. Arriving finally in Spain
at the capital, Cordova, we are struck by the magnificent splendor of
her architecture. At this renowned seat of learning, schools and li-
braries were founded during the tenth century.

Little is known of the progress of mathematics in Spain. The ear-
liest name that has come down to us is Al-Majriti (died 1007), the
author of a mystic paper on "amicable numbers." His pupils founded
schools at Cordova, Dania, and Granada. But the only great astron-
omer among the Saracens in Spain is Jabir ibn Aflah of Se villa, fre-
quently called Geber. He lived in the second half of the eleventh cen-
tury. It was formerly believed that he was the inventor of algebra,
and that the word algebra came from "Jabir" or "Geber." He ranks
among the most eminent astronomers of this time, but, like so many
of his contemporaries, his writings contain a great deal of mysticism.
His chief work is an astronomy in nine books, of which the first is de-
voted to trigonometry. In his treatment of spherical trigonometry,
he exercises great independence of thought. He makes war against
the time-honored procedure adopted by Ptolemy of applying "the
rule of six quantities," and gives a new way of his own, based on the
" rule of four quantities." This is: If PP\ and QQ\ be two arcs of great
circles intersecting in A, and if PQ and P\Q\ be arcs of great circles
drawn perpendicular to QQ\, then we have the proportion

sin AP : sin PQ= sin AP\ : sin PiQi.
1 H. Suter in Bibliotheca tnathcnuitlca, 3. S., Vol. 12, 1911-12, pp. 320-322.

no A HISTORY OF MATHEMATICS

From this he derives the formulas for spherical right triangles. This
sine-formula was probably known before this to Tabit ibn Korra and
others.1 To the four fundamental formulas already given by Ptolemy,
he added a fifth, discovered by himself. If a, b, c, be the sides, and
A, B, C, the angles of a spherical triangle, right-angled at A, then
cos 5 = cos b sin C. This is frequently called "Geber's Theorem."
Radical and bold as were his innovations in spherical trigonometry,
in plane trigonometry he followed slavishly the old beaten path of
the Greeks. Not even did he adopt the Indian "sine" and "cosine,"
but still used the Greek "chord of double the angle." So painful was
the departure from old ideas, even to an independent Arab!

It is a remarkable fact that among the early Arabs no trace what-
ever of the use of the abacus can be discovered. At the close of the
thirteenth century, for the first time, do we find an Arabic writer, Ibn
Albanna, who uses processes which are a mixture of abacal and Hindu
computation. Ibn Albanna lived in Bugia, an African seaport, and it
is plain that he came under European influences and thence got a
knowledge of the abacus. Ibn Albanna and Abraham ibn Esra be-
fore him, solved equations of the first degree by the rule of "double
false position." After Ibn Albanna we find it used by Al-Kalsadi
and Beha-Eddin (i547-i622).2 If ax-\-b = o, let m and n be any two
numbers ("double false position"), let also am-\-b = M, an-\-b = N,

Of interest is the approximate solution of the cubic x3-\-Q = Px,
which grew out of the computation of # = sin T°. The method is
shown only in this one numerical example. It is given in Miram
Chelebi in 1498, in his annotations of certain Arabic astronomical
tables. The solution is attributed to Atabeddin Jamshid3 Write
x=(Q-{-x3)-z-P. If Q+P = a-{-R-i-P, then a is the first approxima-
tion, x being snail. We have Q = aP-\-R, and consequently x — a-\-
(R-\-a3)-^-P = a-\-b-}-S-i-P, say. Then a-f-i is the second approxima-
tion. We have R = bP+S-a3 and Q=(a+b)P+S-a3. Hence
x=a+b+(S-a3+(a+b)3+P=a+b+c+T+P, say. Here a+b
-\-c is the third approximation, and so on. In general, the amount of
computation is considerable, though for finding #=sin i° the method
answered very well. This example is the only known approximate
arithmetical solution of an affected equation due to Arabic writers.
Nearly three centuries before this, the Italian, Leonardo of Pisa,
carried the solution of a cubic to a high degree of approximation, but
without disclosing his method.

The latest prominent Spanish-Arabic scholar was Al-Kalsadi of
Granada, who died in 1486. He wrote the Raising of the Veil of the
Science of Gubar. The word "gubar" meant originally "dust" and

1 See Bibliotheca mathematica, 2 S., Vol. 7, 1893, P- 7-

* L. Matthiessen, Gnindziige d. Antiken u. modcrnen Algebra, Leipzig, 1878, p. 275.

3 See Cantor, op. cit. Vol. I, 3rd Ed., 1907, p. 782.

THE ARABS in

stands here for written arithmetic with numerals, in contrast to men-
tal arithmetic. In addition, subtraction and multiplication, the
result is written above the other figures. The square root was indi-
cated by the initial Arabic letter of the word " jidre," meaning "root,"
particularly "square root." He had symbols for the unknown and
had, in fact, a considerable amount of algebraic symbolism. His
approximation for the square root Va2+6, namely (4a3+ 306) / (4024~
b), is believed by S. Giinther to disclose a method of continued frac-
tions, without our modern notation, since (4a3+30£)/(4#2+£) =
a-\-bl(2a-\-blza). Al-Kalsadi's work excels other Arabic works in
the amount of algebraic symbolism used. Arabic algebra before him
contained much less symbolism then Hindu algebra. With Nessel-
mann1, we divide algebras, with respect to notation, into three classes:
(i) R/tetorical algebras, in which no symbols are used, everything
being written out in words, (2) Syncopated algebras, in which, as in
the first class, everything is written out in words, except that abbrevia-
tions are used for certain frequently recurring operations and ideas,
(3) Symbolic algebras, in which all forms and operations are repre-
sented by a fully developed algebraic symbolism, as for example,
x^+iox+j. According to this classification, Arabic works (excepting
those of the later western Arabs), the Greek works of lamblichus and
Thymaridas, and the works of the early Italian writers and of Regio-
montanus are rhetorical in form; the works of the later western Arabs,
of Diophantus and of the later European writers down to about the
middle of the seventeenth century (excepting Vieta's and Oughtred's)
are syncopated in form; the Hindu works and those of Vieta and
Oughtred, and of the Europeans since the middle of the seventeenth
century, are symbolic in form. It is thus seen that the western Arabs
took an advanced position in matters of algebraic notation, and were
inferior to none of their predecessors or contemporaries, except the
Hindus.

In the year in which Columbus discovered America, the Moors
lost their last foot-hold on Spanish soil; the productive period of
Arabic science was passed.

We have witnessed a laudable intellectual activity among the
Arabs. They had the good fortune to possess rulers who, by their
munificence, furthered scientific research. At the courts of the ca-
liphs, scientists were supplied with libraries and observatories. A
large number of astronomical and mathematical works were written
by Arabic authors. It has been said that the Arabs were learned,
but not original, With our present knowledge of their work, this
dictum needs revision; they have to their credit several substantial
accomplishments. They solved cubic equations by geometric con-
struction, perfected trigonometry to a marked degree and made nu-

1 G. H. F. Nesselmann, Die Algebra der Griechen, Berlin, 1842, pp. 301-306.

ii2 A HISTORY OF MATHEMATICS

merous smaller advances all along the line of mathematics, physics
and astronomy. Not least of their services to science consists in this,
that they adopted the learning of Greece and India, and kept what
they received with care. When the love for science began to grow
in the Occident, they transmitted to the Europeans the valuable treas-
ures of antiquity. Thus a Semitic race was, during the Dark Ages,
the custodian of the Aryan intellectual possessions.

EUROPE DURING THE MIDDLE AGES

With the third century after Christ begins an era of migration of
nations in Europe. The powerful Goths quit their swamps and forests
in the North and sweep onward in steady southwestern current, dis-
lodging the Vandals, Sueves, and Burgundians, crossing the Roman
territory, and stopping and recoiling only when reaching the shores
of the Mediterranean. From the Ural Mountains wild hordes sweep
down on the Danube. The Roman Empire falls to pieces, and the
Dark Ages begin. But dark though they seem, they are the germinat-
ing season of the institutions and nations of modern Europe. The
Teutonic element, partly pure, partly intermixed with the Celtic and
Latin, produces that strong and luxuriant growth, the modern civili-
zation of Europe. Almost all the various nations of Europe belong
to the Aryan stock. As the Greeks and the Hindus — both Aryan races
— were the great thinkers of antiquity, so the nations north of the Alps
and Italy became the great intellectual leaders of modern times.

Introduction of Roman Mathematics

We shall now consider how these as yet barbaric nations of the
North gradually came in possession of the intellectual treasures of
antiquity. With the spread of Christianity the Latin language was
introduced not only in ecclesiastical but also in scientific and all im-
portant worldly transactions. Naturally the science of the Middle
Ages was drawn largely from Latin sources. In fact, during the earlier
of these ages Roman authors were the only ones read in the Occident.
Though Greek was not wholly unknown, yet before the thirteenth
century not a single Greek scientific work had been read or translated
into Latin. Meagre indeed was the science which could be gotten
from Roman writers, and we must wait several centuries before any
substantial progress is made in mathematics.

After the time of Boethius and Cassiodorius mathematical activity
in Italy died out. The first slender blossom of science among tribes
that came from the North was an encyclopaedia entitled Origenes,
written by Isidorus (died 636 as bishop of Seville). This work is
modelled after the Roman encyclopaedias of Martianus Capella of
Carthage and of Cassiodorius. Part of it is devoted to the quadrivium,
arithmetic, music, geometry, and astronomy. He gives definitions
and grammatical explications of technical terms, but does not de-
scribe the modes of computation then in vogue. After Isidorus there
follows a century of darkness which is at last dissipated by the appear-

ii4 A HISTORY OF MATHEMATICS

ance of Bede the Venerable (672-735), the most learned man of his
time. He was a native of Wearmouth, in England. His works con-
tain treatises on the Computus, or the computation of Easter-time,
and on finger-reckoning. It appears that a finger-symbolism was then
widely used for calculation. The correct determination of the time
of Easter was a problem which in those days greatly agitated the
Church. It became desirable to have at least one monk at each mon-
astery who could determine the day of religious festivals and could
compute the calendar. Such determinations required some knowledge
of arithmetic. Hence we find that the art of calculating always found
some little corner in the curriculum for the education of monks.

The year in which Bede died is also the year in which Alcuin (735-
804) was born. Alcuin was educated in Ireland, and was called to the
court of Charlemagne to direct the progress of education in the great
Prankish Empire. Charlemagne was a great patron of learning and
of learned men. In the great sees and monasteries he founded schools
in which were taught the psalms, writing, singing, computation (com-
putus), and grammar. By computus was here meant, probably, not
merely the determination of Easter-time, but the art of computation
in general. Exactly what modes of reckoning were then employed
we have no means of knowing. It is not likely that Alcuin was familiar
with the apices of Boethius or with the Roman method of reckoning
on the abacus. He belongs to that long list of scholars who dragged
the theory of numbers into theology. Thus the number of beings
created by God, who created all things well, is 6, because 6 is a perfect
number (the sum of its divisors being 1+2+3 = 6); 8, on the other
hand, is an imperfect number (i+2+4<8); hence the second origin
of mankind emanated from the number 8, which is the number of souls
said to have been in Noah's ark.

There is a collection of "Problems for Quickening the Mind" (prop-
ositiones ad acuendos iuvenes), which are certainly as old as 1000 A. D.
and possibly older. Cantor is of the opinion that they were written
much earlier and by Alcuin. The following is a specimen of these
"Problems": A dog chasing a rabbit, which has a start of 150 feet,
jumps 9 feet every time the rabbit jumps 7. In order to determine in
how many leaps the dog overtakes the rabbit, 150 is to be divided by 2.
In this collection of problems, the areas of triangular and quadrangular
pieces of land are found by the same formulas of approximation as
those used by the Egyptians and given by Boethius in his geometry.
An old problem is the "cistern-problem" (given the time in which
several pipes can fill a cistern singly, to find the time in which they
fill it jointly), which has been found previously in Heron, in the Greek
Anthology, and in Hindu works. Many of the problems show that
the collection was compiled chiefly from Roman sources. The prob-
lem which, on account of its uniqueness, gives the most positive testi-
mony regarding the Roman origin is that on the interpretation of a

INTRODUCTION OF ROMAN MATHEMATICS 115

will in a case where twins are born. The problem is identical with the
Roman, except that different ratios are chosen. Of the exercises for
recreation, we mention the one of the wolf, goat, and cabbage, to be
rowed across a river in a boat holding only one besides the ferry-man.
Query: How must he carry them across so that the goat shall not eat
the cabbage, nor the wolf the goat? l The solutions of the "problems
for quickening the mind" require no further knowledge than the recol-
lection of some few formulas used in surveying, the ability to solve
linear equations and to perform the four fundamental operations with
integers. Extraction of roots was nowhere demanded; fractions hardly
ever occur. 2

The great empire of Charlemagne tottered and fell almost imme-
diately after his death. War and confusion ensued. Scientific pur-
suits were abandoned, not to be resumed until the close of the tenth
century, when under Saxon rule in Germany and Capetian in France,
more peaceful times began. The thick gloom of ignorance commenced
to disappear. The zeal with which the study of mathematics was now
taken up by the monks is due principally to the energy and influence
of one man, — Gerbert. He was born in Aurillac in Auvergne. After
receiving a monastic education, he engaged in study, chiefly of mathe-
matics, in Spain. On his return he taught school at Rheims for ten
years and became distinguished for his profound scholarship. By
King Otto I, and his successors Gerbert was held in highest esteem.
He was elected bishop of Rheims, then of Ravenna, and finally was
made Pope under the name of Sylvester II, by his former Emperor
Otho III. He died in 1003, after a life intricately involved in many
political and ecclesiastical quarrels. Such was the career of the great-
est mathematician of the tenth century in Europe. By his contem-
poraries his mathematical knowledge was considered wonderful.
Many even accused him of criminal intercourse with evil spirits.

Gerbert enlarged the stock of his knowledge by procuring copies
of rare books. Thus in Mantua he found the geometry of Boethius.
Though this is of small scientific value, yet it is of great importance
in history. It was at that time the principal book from which Euro-
pean scholars could learn the elements of geometry. Gerbert studied
it with zeal, and is generally believed himself to be the author of a ge-
ometry. H. Weissenborn denied his authorship, and claimed that the
book in question consists of three parts which cannot come from one
and the same author. More recent study favors the conclusion that
Gerbert is the author and that he compiled it from different sources.3
This geometry contains little more than the one of Boethius, but the
fact that occasional errors in the latter are herein corrected shows that

1 S. Giinther, Gcschichtc des mathem. Unterrichts im deulschen Mittdalter. Berlin,
1887, p. ,?-'.

• M. C;mtor, op. cit., Vol. I, 3. Aufl., 1907, p. 839.

3 S. Giinther, Geschichle der Mathematik, i. Teil, Leipzig, 1908, p. 249.

n6 A HISTORY OF MATHEMATICS

the author had mastered the subject. "The first mathematical paper
of the Middle Ages which deserves this name," says Hankel, "is a
letter of Gerbert to Adalbold, bishop of Utrecht," in which is explained
the reason why the area of a triangle, obtained "geometrically" by
taking the product of the base by half its altitude, differs from the
area calculated "arithmetically," according to the formula \a(a-\-i),
used by surveyors, where a stands for a side of an equilateral triangle.
He gives the correct explanation that in the latter formula all the
small squares, in which the triangle is supposed to be divided, are
counted in wholly, even though parts of them project beyond it.
D. E. Smith l calls attention to a great medieval number game, called
rithmomachia, claimed by some to be of Greek origin. It was played
as late as the sixteenth century. It called for considerable arithmeti-
cal ability, and was known to Gerbert, Oronce Fine, Thomas Brad-
wardine and others. A board resembling a chess board was used. Re-
lations like 81 = 72+1 of 72, 42 = 36+ g of 36 were involved.

Gerbert made a careful study of the arithmetical works of Boethius.
He himself published the first, perhaps both, of the following two
works, — A Small Book on the Division of Numbers, and Rule of Compu-
tation on the Abacus.' They give an insight into the methods of calcu-
lation practised in Europe before the introduction of the Hindu nu-
merals. Gerbert used the abacus, which was probably unknown to
Alcuin. Bernelinus, a pupil of Gerbert, describes it as consisting of
a smooth board upon which geometricians were accustomed to strew
blue sand, and then to draw their diagrams. For arithmetical pur-
poses the board was divided into 30 columns, of which 3 were reserved
for fractions, while the remaining 27 were divided into groups with
3 columns in each. In every group the columns were marked respec-
tively by the letters C (centum), D (decem), and S (singularis) or
M (monas). Bernelinus gives the nine numerals used, which are the
apices of Boethius, and then remarks that the Greek letters may be
used in their place. By the use of these columns any number can be
written without introducing a zero, and all operations in arithmetic
can be performed in the same way as we execute ours without the col-
umns, but with the symbol for zero. Indeed, the methods of adding,
subtracting, and multiplying in vogue among the abacists agree sub-
stantially with those of to-day. But in a division there is very great
difference. The early rules for division appear to have been framed
'to satisfy the following three conditions: (i) The use of the multipli-
cation table shall be restricted as far as possible; at least, it shall never
be required to multiply mentally a figure of two digits by another of
one digit. (2) Subtractions shall be avoided as much as possible and
replaced by additions. (3) The operation shall proceed in a purely
mechanical way, without requiring trials.2 That it should be neces-
sary to make such conditions seems strange to us; but it must be re_

1 Am. Math. Monthly, Vol. 28, 1911, pp. 73-80. 2 H. Hankel, op. cil., p. 318.

INTRODUCTION OF ROMAN MATHEMATICS .117

remembered that the monks of the Middle Ages did not attend school
during childhood and learn the multiplication table while the memory
was fresh. Gerbert's rules for division are the oldest extant. They
are so brief as to be very obscure to the uninitiated. They were prob-
ably intended simply to aid the memory by calling to mind the suc-
cessive steps in the work. In later manuscripts they are stated more
fully. In dividing any number by another of one digit, say 668 by 6,
the divisor was first increased to 10 by adding 4. The process is ex-
hibited in the adjoining figure. l As it continues, we must imagine the
digits which are crossed out, to be erased and then replaced by the
ones beneath. It is as follows: 6oo-j- 10 = 60, but, to rectify the error,
4X60, or 240, must be added; 200-7-10=20, but 4X20, or 80, must
be added. We now write for 60+40+80, its sum 180, and continue
thus: IOO-T-IO=IO; the correction necessary is 4X10, or 40, which,
added to 80, gives 120. Now IOO-T- 10= 10, and the correction 4X 10,
together with the 20, gives 60. Proceeding as before, 60-7-10 = 6; the
correction is 4X 6 = 24. Now 20-7-10= 2, the correction being 4X 2 = 8.
In the column of units we have now 8+4+8, or 20. As before, 20-7-
10= 2; the correction is 2X4 = 8, which is not divisible by 10, but only
by 6, giving the quotient i and the remainder 2. All the partial quo-
tients taken together give 60+20+10+10+6+2+2+1 = 111, and
the remainder 2.

Similar but more complicated, is the process when the divisor con-
tains two or more digits. Were the divisor 27, then the next higher
multiple of 10, or 30, would be taken for the divisor, but corrections
would be required for the 3. He who has the patience to carry such
a division through to the end, will understand why it has been said of
Gerbert that "Regulas dedit, qua; a sudantibus abacistis vix intelli-
guntur." He will also perceive why the Arabic method of division,
when first introduced, was called the divisio aurea, but the one on the
abacus, the divisio ferrea.

In his book on the abacus, Bernelinus devotes a chapter to fractions.
These are, of course, the duodecimals, first used by the Romans. For
want of a suitable notation, calculation with them was exceedingly
difficult. It would be so even to us, were we accustomed, like the
early abacists, to express them, not by a numerator or denominator,
but by the application of names, such as uncia for j1^, quincunx for ^
dodrans for T\.

In the tenth century, Gerbert was the central figure among the
learned. In his time the Occident came into secure possession of all
mathematical knowledge of the Romans. During the eleventh cen-
tury it was studied assiduously. Though numerous works were
written on arithmetic and geometry, mathematical knowledge in the
Occident was still very insignificant. Scanty indeed were the mathe-
matical treasures obtained from Roman sources.

1 M. Cantor, op. clt., Vol. I, 3. Aufl., 1907, p. 882.

n8 A HISTORY OF MATHEMATICS

Translation of Arabic Manuscripts

By his great erudition and phenomenal activity, Gerbert infused
new life into the study not only of mathematics, but also of philosophy.
Pupils from France, Germany, and Italy gathered at Rheims to enjoy
his instruction. When they themselves became teachers, they taught
of course not only the use of the abacus and geometry, but also what
they had learned of the philosophy of Aristotle. His philosophy was
known, at first, only through the writings of Boethius. But the grow-
ing enthusiasm for it created a demand for his complete works. Greek
texts were wanting. But the Latins heard that the Arabs, too, were
great admirers of Peripatetism, and that they possessed translations
of Aristotle's works and commentaries thereon. This led them finally
to search for and translate Arabic manuscripts. During this search,
mathematical works also came to their notice, and were translated
into Latin. . Though some few unimportant works may have been
translated earlier, yet the period of greatest activity began about 1 100.
The zeal displayed in acquiring the Mohammedan treasures of knowl-
edge excelled even that of the Arabs themselves, when, in the eighth
century, they plundered the rich coffers of Greek and Hindu science.

Among the earliest scholars engaged in translating manuscripts into
Latin was Athelard of Bath. The period of his activity is the first
quarter of the twelfth century. He travelled extensively in Asia
Minor, Egypt, perhaps also in Spain, and braved a thousand perils,
that he might acquire the language and science of the Mohammedans.
He made one of the earliest translations, from the Arabic, of Euclid's
Elements. He translated the astronomical tables of Al-Khowarizmi.
In 1857, a manuscript was found in the library at Cambridge, which
proved to be the arithmetic by Al-Khowarizmi in Latin. This trans-
lation also is very probably due to Athelard.

At about the same time flourished Plato of Tiwli or Plato Tiburtinus.
He effected a translation of the astronomy of Al-Battani and of the
Sphcerica of Theodosius.

About the middle of the twelfth century there was a group of Chris-
tian scholars busily at work at Toledo, under the leadership of Ray-
mond, then archbishop of Toledo. Among those who worked under
his direction, John of Seville was most prominent. He translated
works chiefly on Aristotelian philosophy. Of importance to us is a
liber alghoarismi, compiled by him from Arabic authors. The rule for

the division of one fraction by another is proved as follows: 3-^3 =

d b ad a a

—-:--—:=—. This same explanation is given by the thirteenth cen-
bd bd be

tury German writer, Jordanus Nemorarius. On comparing works
like this with those of the abacists, we notice at once the most striking
difference, which shows that the two parties drew from independent

TRANSLATION OF ARABIC MANUSCRIPTS 119

sources. It is argued by some that Gerbert got his apices and his arith-
metical knowledge, not from Boethius, but from the Arabs in Spain,
and that part or the whole of the geometry of Boethius is a forgery,
dating from the time of Gerbert. If this were the case, then the writ-
ings of Gerbert would betray Arabic sources, as do those of John of
Seville. But no points of resemblance are found. Gerbert could not
have learned from the Arabs the use of the abacus, because all evidence
we have goes to show that they did not employ it. Nor is it probable
that he borrowed from the Arabs the apices, because they were never
used in Europe except on the abacus. In illustrating an example in
division, mathematicians of the tenth and eleventh centuries state
an example in Roman numerals, then draw an abacus and insert in it
the necessary numbers with the apices. Hence it seems probable that
the abacus and apices were borrowed from the same source. The
contrast between authors like John of Seville, drawing from Arabic
works, and the abacists, consists in this, that, unlike the latter, the
former mention the Hindus, use the term algorism, calculate with the
zero, and do not employ the abacus. The former teach the extraction
of roots, the abacists do not; they teach the sexagesimal fractions used
by the Arabs, while the abacists employ the duodecimals of the Ro-
mans.1

A little later than John of Seville flourished Gerard of Cremona in
Lombardy. Being desirous to gain possession of the Almagest, he
went to Toledo, and there, in 1175, translated this great work of Ptol-
emy. Inspired by the richness of Mohammedan literature, he gave
himself up to its study. He translated into Latin over 70 Arabic works.
Of mathematical treatises, there were among these, besides the Al-
magest, the 15 books of Euclid, the Sphcerica of Theodosius, a work of
Menelaus, the algebra of Al-Khowarizmi, the astronomy of Jabir ibn
Aflah, and others less important. Through Gerard of Cremona the
term sinus was introduced into trigonometry. Al-Khawarizmi's al-
gebra was translated also by Robert of Chester; his translation prob-
ably antedated Cremona's.

In the thirteenth century, the zeal for the acquisition of Arabic
learning continued. Foremost among the patrons of science at this
time ranked Emperor Frederick II of Hohenstaufen (died 1250).
Through frequent contact with Mohammedan scholars, he became
familiar with Arabic science. He employed a number of scholars in
translating Arabic manuscripts, and it was through him that we came
in possession of a new translation of the Almagest. Another royal
head deserving mention as a zealous promoter of Arabic science was
Alfonso X of Castile (died 1284). He gathered around him a number
of Jewish and Christian scholars, who translated and compiled astro-
nomical works from Arabic sources. Astronomical tables prepared
by two Jews spread rapidly in the Occident, and constituted the basis
1 M. Cantor, op. oil., Vol. I, 3. Aufl., 1907, p. 879, chapter 40.

120 A HISTORY OF MATHEMATICS

of all astronomical calculation till the sixteenth century. The num-
ber of scholars who aided in transplanting Arabic science upon Chris-
tian soil was large. But we mention only one, Giovanni Campano of
Novara (about 1260), who brought out a new translation of Euclid,
which drove the earlier ones from the field, and which formed the
basis of the printed editions. 1

At the middle of the twelfth century, the Occident was in possession
of the so-called Arabic notation. At the close of the century, the
Hindu methods of calculation began to supersede the cumbrous meth-
ods inherited from Rome. Algebra, with its rules for solving linear
and quadratic equations, had been made accessible to the Latins. The
geometry of Euclid, the Sphwrica of Theodosius, the astronomy of
Ptolemy, and other works were now accessible in the Latin tongue.
Thus a great amount of new scientific material had come into the
hands of the Christians. The talent necessary to digest this hetero-
geneous mass of knowledge was not wanting. The figure of Leonardo
of Pisa adorns the vestibule of the thirteenth century.

It is important to notice that no work either on mathematics or
astronomy was translated directly from the Greek previous to the
fifteenth century.

The First Awakening and its Sequel

Thus far, France and the British Isles have been the headquarters
of mathematics in Christian Europe. But at the beginning of the
thirteenth century the talent and activity of one man was sufficient to
assign the mathematical science a new home in Italy. This man was
not a monk, like Bede, Alcuin, or Gerbert, but a layman who found
time for scientific study. Leonardo of Pisa is the man to whom we
owe the first renaissance of mathematics on Christian soil. He is also
called Fibonacci, i.e. son of Bonaccio. His father was secretary at one of
the numerous factories erected on the south and east coast of the Med-
iterranean by the enterprising merchants of Pisa. He made Leonardo,
when a boy, learn the use of the abacus. The boy acquired a strong
taste for mathematics, and, in later years, during extensive travels in
Egypt, Syria, Greece, and Sicily, collected from the various peoples
all the knowledge he could get on this subject. Of all the methods of
calculation, he found the Hindu to be unquestionably the best. Re-
turning to Pisa, he published, in 1202, his great work, the Liber Abaci.
A revised edition of this appeared in 1228. This work contains the
knowledge the Arabs possessed in arithmetic and algebra, and treats
the subject in a free and independent way. This, together. with the
other books of Leonardo, shows that he was not merely a compiler,
nor, like other writers of the Middle Ages, a slavish imitator of the
form in which the subject had been previously presented. The extent

1 H. Hankel, op. cit., pp. 338, 339.

THE FIRST AWAKENING AND ITS SEQUEL 121

of his originality is not definitely known, since the sources from which
he drew have not all been ascertained. Karpinski has shown that
Leonardo drew extensively from Abu Kamil's algebra. Leonardo's
Practica geometries is partly drawn from the Liber embadorum of Sav-
asorda, a learned Jew of Barcelona and a co-worker of Plato of
Tivoli.

Leonardo was the first great mathematician to advocate the adop- |
tion of the " Arabic_notatioflJ,' The calculation with the zero was the
portion of ""Arabic matEematics earliest adopted by the Christians.
The minds of men had been prepared for the reception of this by the
use of the abacus and the apices. The reckoning with columns was
gradually abandoned, and the very word abacus changed its meaning
and became a synonym for algorism. For the zero, the Latins adopted
the name zephirum, from the Arabic sifr (sifra = empty); hence our
English word cipher. The new notation was accepted readily by the
enlightened masses, but, at first, rejected by the learned circles. The
merchants of Italy used it as early as the thirteenth century, while
the monks in the monasteries adhered to the old forms. In 1299,
nearly 100 years after the publication of Leonardo's Liber Abaci, the
Florentine merchants were forbidden the use of the Arabic numeral
in book-keeping, and ordered either to employ the Roman numerals
or to write the numeral adjectives out in full. This decree is probably
due to the variety of forms of certain digits and the consequent am-
biguity, misunderstanding and fraud. Some interest attaches to the
earliest dates indicating the use of Hindu-Arabic numerals in the Oc-
cident. Many erroneous or doubtful early dates have been given by
writers inexperienced in the reading of manuscripts and inscriptions.
The numerals are first found in manuscripts of the tenth century, but
they were not well known until the beginning of the thirteenth cen-
tury.1 About 1275 they began to be widely used. The earliest Arabic
manuscripts containing the numerals are of 874 and 888 A. D. They
appear in a work written at Shiraz in Persia in 970 A. D. A church-
pillar not far from the Jeremias Monastery in Egypt has the date 349
A. H. ( = 961 A. D.) The oldest definitely dated European manuscript
known to contain the numerals is the Codex Vigilanus, written in the
Albelda Cloister in Spain in 976 A. D. The nine characters without
the zero are given, as an addition, in a Spanish copy of the Origines
by Isidorus of Seville, 992 A. D. A tenth century manuscript with
forms differing materially from those in the Codex Vigilanus was found
in the St. Gall manuscript now in the University Library at Zurich.
The numerals are contained in a Vatican manuscript of 1077, a Sicilian
coin of 1138, a Regensburg (Bavaria) chronicle of 1197. The earliest
manuscript in French giving the numerals dates about 1275. In the

1 G. F. Hill, The Development of Arabic Numerals in Europe, Oxford, 1915, p. n.
Our dates are taken from this book and from D. E. Smith and L. C. Karpinski's
Hindu-Arabic Numeral^, Boston and London, 1911, pp. 133-146.

122 A HISTORY OF MATHEMATICS

British Museum one English manuscript is of about 1230-50, another
is of 1246. The earliest undoubted Arabic numerals on a gravestone
are at Pforzheim in Baden of 1371 and one at Ulm of 1388. The
earliest coins dated in the Arabic numerals are as follows: Swiss 1424,
Austrian 1484, French 1485, German 1489, Scotch 1539, English 1551.
The earliest calendar with Arabic figures is that of Kobel, 1518. The
forms of the numerals varied considerably. The 5 was the most
freakish. An upright 7 was rare in the earlier centuries.

In the fifteenth century the abacus with its counters ceased to be
used in Spain and Italy. In France it was used later, and it did not
disappear in England and Germany before the middle of the seven-
teenth century.1 The method of abacal computation is found in the
English exchequer for the last time in 1676. In the reign of Henry I
the exchequer was distinctly organized as a court of law, but the finan-
cial business of the crown was also carried on there. The term " ex-
chequer" is derived from the chequered cloth which covered the table
at which the accounts were made up. Suppose the sheriff was sum-
moned to answer for the full annual dues "in money or in tallies."
"The liabilities and the actual payments of the sheriff were balanced
by means of counters placed upon the squares of the chequered table,
those on the one side of the table representing the value of the tallies,
warrants and specie presented by the sheriff, and those on the other
the amount for which he was liable," so that it was easy to see whether
the sheriff had met his obligations or not. In Tudor times "pen and
ink dots" took the place of counters. These dots were used as late as
i676.2 The "tally" upon which accounts were kept was a peeled
wooden rod split in such a way as to divide certain notches previously
cut in it. One piece of the tally was given to the payer; the other piece
was kept by the exchequer. The transaction could be verified easily
by fitting the two halves together and noticing whether the notches
" tallied" or nor. Such tallies remained in use as late as 1783.

In the Winter's Tale (IV. 3), Shakespeare lets the clown be embar-
rassed by a problem which he could not do without counters. lago (in
Othello, i, i) expresses his contempt for Michael Cassio, "forsooth a
great mathematician," by calling him a "counter-caster." 3 So gen-
eral, indeed, says Peacock, appears to have been the practice of this
species of arithmetic, that its rules and principles form an essential
part of the arithmetical treatises of that day. The real fact seems to
be that the old methods were used long after the Hindu numerals were

George Peacock, "Arithmetic" in the Encyclopedia of Pure Mathematics,
London, 1847, p. 408.

2 Article "Exchequer" in Palgrave's Dictionary of Political Economy, London,
1894.

3 For additional information, consult F. P. Barnard, The Casting-Counter and
the Counting-Board, Oxford, 1916. He gives a list of 159 extracts from English
inventories referring to counting boards and also photographs of reckoning tables
at Basel and Niirnberg, of reckoning cloths at Munich, etc.

THE FIRST AWAKENING AND ITS SEQUEL 123

in common and general use. With such dogged persistency does man
cling to the old !

The Liber Abaci was, for centuries, one of the storehouses from
which authors got material for works on arithmetic and algebra. In
it are set forth the most perfect methods of calculation with integers
and fractions, known at that time; the square and cube root are ex-
plained, cube root no^having been considered in the Christian Occi-
dent before; equations of the first and second degree leading to prob-
lems, either determinate or indeterminate, are solved by the methods
of " single " or "double position," and also by real algebra. He recog-
nized that the quadratic x*+c=bx may be satisfied by two values of x.
He took no cognizance of negative and imaginary roots. The book
contains a large number of problems. The following was proposed to
Leonardo of Pisa by a magister in Constantinople, as a difficult prob-
lem: If A gets from B 7 denare, then A's sum is five-fold B's; if B gets
from A 5 denare, then B's sum is seven-fold A's. How much has each?
The Liber Abaci contains another problem, which is of historical in-
terest, because it was given with some variations by Ahmes, 3000 years
earlier: 7 old women go to Rome; each woman has 7 mules, each mule
carries 7 sacks, each sack contains 7 loaves, with each loaf are 7 knives,
each knife is put up in 7 sheaths. What is the sum total of all named?
Ans. 137, 256. l Following the practice of Arabic and of Greek and
Egyptian writers, Leonardo frequently uses unit fractions. This was
done also by other European writers of the Middle Ages. He ex-
plained how to resolve a fraction into the sum of unit fractions. He
was one of the first to separate the numerator from the denominator
by a fractional line. Before his time, when fractions were written in
Hindu-Arabic numerals, the denominator was written beneath the
numerator, without any sign of separation.

In 1220, Leonardo of Pisa published his Practica Geometries, which
contains all the knowledge of geometry and trigonometry transmitted
to him. The writings of Euclid and of some other Greek masters were
known to him, either from Arabic manuscripts directly or from the
translations made by his countrymen, Gerard of Cremona and Plato
of Tivoli. As previously stated, a principal source of his geometrical
knowledge was Plata of Tivolis' translation in 1116, from the Hebrew
into Latin, of the Liber embadorum of Abraham Savasorda.2 Leo-
nardo's Geometry contains an elegant geometrical demonstration of
Heron's formula for the area of a triangle, as a function of its three
sides; the proof resembles Heron's. Leonardo treats the rich material
before him with skill, some originality and Euclidean rigor.

Of still greater interest than the preceding works are those contain-

1 M. Cantor, op. cit., Vol. IT, 2. Aufl., 1900, p. 26. See a problem in the Ahmes
papyrus believed to be of the same type as this.

2 See M. Curtze, Urkutiden zur GeschlclUe der Mathcmatik, I Theil, Leipzig, 1902,
P- 5-

i24 A HISTORY OF MATHEMATICS

ing Fibonacci's more original investigations. We must here preface
that after the publication of the Liber Abaci, Leonardo was presented
by the astronomer Dominicus to Emperor Frederick II of Hohen-
staufen. On that occasion, John of Palermo, an imperial notary,
proposed several problems, which Leonardo solved promptly. The
first (probably an old familiar problem to him) was to find a number x,
such that #2-r-5 and x2— 5 are each square numbers. The answer is

* = 3i\; f°r (3i\)M-5=(4-iV)2> ($&)*- 5 -(»&)*• His masterly so-
lution of this is given in his liber quadratorum, a manuscript which was
not printed, but to which reference is made in the second edition of
his Liber Abaci. The problem was not original with John of Palermo,
since the Arabs had already solved similar ones. Some parts of Leo-
nardo's solution may have been borrowed from the Arabs, but the
method which he employed of building squares by the summation of
odd numbers is original with him.

The second problem proposed to Leonardo at the famous scientific
tournament which accompanied the presentation of this celebrated al-
gebraist to that great patron of learning, Emperor Frederick II, was
the solving of the equation x3-\- 2x2-\- iox= 20. As yet cubic equations
had not been solved algebraically. Instead of brooding stubbornly
over this knotty problem, and after many failures still entertaining
new hopes of success, he changed his method of inquiry and showed
by clear and rigorous demonstration that the roots of this equation
could not be represented by the Euclidean irrational quantities, or, in
other words, that they could not be constructed with the ruler and
compass only. He contented himself with finding a very close ap-
proximation to the required root. His work on this cubic is found in
the Flos, together with the solution of the following third problem
given him by John of Palermo: Three men possess in common an un-
known sum of money /; the share of the first is -; that of the second, -;

/ 2

that of the third, -. Desirous of depositing the sum at a safer place,

each takes at hazard a certain amount; the first takes x, but deposits

X "V

only - ; the second carries y, but deposits only -; the third takes 2, and

deposits -. Of the amount deposited each one must receive exactly f ,

in order to possess his share of the whole sum. Find x, y, z. Leonardo
shows the problem to be indeterminate. Assuming 7 for the sum
drawn by each from the deposit, he finds / = 47, £=33, y=i3, z=i.
One would have thought that after so brilliant a beginning, the
sciences transplanted from Mohammedan to Christian soil would
have enjoyed a steady and vigorous development. But this was not
the case. During the fourteenth and fifteenth centuries, the mathe-

THE FIRST AWAKENING AND ITS SEQUEL 125

matical science was almost stationary. Long wars absorbed the ener-
gies of the people and thereby kept back the growth of the sciences.
The death of Frederick II in 1254 was followed by a period of con-
fusion in Germany. The German emperors and the popes were con-
tinually quarrelling, and Italy was inevitably drawn into the struggles
between the Guelphs and the Ghibellines. France and England were
engaged in the Hundred Years' War (1338-1453). Then followed in
England the Wars of the Roses. The growth of science was retarded
not only by war, but also by the injurious influence of scholastic phi-
losophy. The intellectual leaders of those times quarrelled over subtle
subjects in metaphysics and theology. Frivolous questions, such as
"How many angels can stand on the point of a needle? " were discussed
with great interest. Indistinctness and confusion of ideas charac-
terized the reasoning during this period. The writers on mathematics
during this period were not few in number, but their scientific efforts
were vitiated by the method of scholastic thinking. Though they
possessed the Elements of Euclid, yet the true nature of a mathematical
proof was so little understood, that Hankel believes it no exaggeration
to say that "since Fibonacci, not a single proof, not borrowed from
Euclid, can be found in the whole literature of these ages, which fulfils
all necessary conditions."

The only noticeable advance is a simplification of numerical opera-
tions and a more extended application of them. Among the Italians
are evidences of an early maturity of arithmetic. Peacock J says:
The Tuscans generally, and the Florentines in particular, whose city
was the cradle of the literature and arts of the thirteenth and four-
teenth centuries, were celebrated for their knowledge of arithmetic
and book-keeping, which were so necessary for their extensive com-
merce; the Italians were in familiar possession of commercial arith-
metic long before the other nations of Europe; to them we are indebted
for the formal introduction into books of arithmetic, under distinct
heads, of questions in the single and double rule of three, loss and gain,
fellowship, exchange, simple and compound interest, discount, and
so on.

There was also a slow improvement in the algebraic notation. The
Hindu algebra possessed a tolerable symbolic notation, which was,
however, completely ignored by the Mohammedans. In this respect,
Arabic algebra approached much more closely to that of Diophantus,
which can scarcely be said to employ symbols in a systematic way.
Leonardo of Pisa possessed no algebraic symbolism. Like the early
Arabs, he expressed the relations of magnitudes to each other by lines
or in words. But in the mathematical writings of Chuquet (1484), of
Widmann (1489) and of the monk Luca Pacioli (also called Lucas de
Burgo sepulchri) symbols began to appear. Pascioli's consisted merely

1 G. Peacock, op. cil., 1847, p. 429.

126 A HISTORY OF MATHEMATICS

in abbreviations of Italian words, such as p for piu (more), m for meno
(less), co for cosa (the unknown x), ce for censo (x2), cece for censocenso
(x?), "Our present notation has arisen by almost insensible degrees
as convenience suggested different marks of abbreviation to different
authors; and that perfect symbolic language which addresses itself
solely to the eye, and enables us to take in at a glance the most com-
plicated relations of quantity, is the result of a large series of small im-
provements." 1

We shall now mention a few authors who lived during the thirteenth
and fourteenth and the first half of the fifteenth centuries.

We begin with the philosophic writings of Thomas Aquinas (1225-
1274), the great Italian philosopher of the Middle Ages, who gave in
the completes! form the ideas of Origen on infinity. Aquinas' notion
of a continuum, particularly a linear continuum, made it potentially
divisible to infinity, since practically the divisions could not be carried
out to infinity. There was, therefore, no minimum line. On the other
hand, the point is not a constituent part of the line, since it does not
possess the property of infinite divisibility that parts of a line possess,
nor can the continuum be constructed out of points. However, a
point by its motion has the capacity of generating a line.2 This con-
tinuum held a firm ascendancy over the ancient atomistic doctrine
which assumed matter to be composed of very small, indivisible par-
ticles. No continuum superior to this was created before the nine-
teenth century. Aquinas explains Zeno's arguments against motion,
as they are given by Aristotle, but hardly presents any new point of
view. The Englishman, Roger Bacon [i2i4(?)-i294J likewise argued
against a continuum of indivisible parts different from points. Re-
newing arguments presented by the Greeks and early Arabs, he held
that the doctrine of indivisible parts of uniform size would make the
diagonal of a square commensurable with a side. Likewise, if through
the ends of an indivisible arc of a circle radii are drawn, these radii
intercept an' arc on a concentric circle of smaller radius; from this it
would follow that the inner circle is of the same length as the outer
circle, which is impossible. Bacon argued against infinity. If time
were infinite, the absurdity would follow that the part is equal to the
whole. Bacon's views were made known more widely through Duns
Scotus (1265-1308), the theological and philosophical opponent of
Thomas Aquinas. However, both argued against the existence of
indivisible parts (points). Duns Scotus wrote on Zeno's paradoxies,
but without reaching new points of view. His commentaries were
annotated later by the Italian theologian, Franciscus de Pitigianis,
who expressed himself in favor of the admission of the actual infinity
to explain the "Dichotomy" and the "Achilles," but fails to ade-
quately elaborate the subject. Scholastic ideas on infinity and the

1 J. F. W. Herschel, "Mathematics" in Edinburgh Encyclopaedia.

2 C. R. Wallner, in Bibliotheca mathematica, 3. F., Bd. IV, 1903, pp. 29, 30.

THE FIRST AWAKENING AND ITS SEQUEL 127

continuum find expression in the writings of Bradwardine, the Eng-
lish doctor profundis.1

About the time of Leonardo of Pisa (1200 A. D.), lived the German
monk Jordanus Nemorarius (£-1237), who wrote a once famous work
on the properties of numbers, printed in 1496 and modelled after the
arithmetic of Boethius. The most trifling numeral properties are
treated with nauseating pedantry and prolixity. A practical arith-
metic based on the Hindu notation was also written, by him. John
Halifax (Sacro Bosco, died 1256) taught in Paris and made an extract
from the Almagest containing only the most elementary parts of that
work. This extract was for nearly 400 years a work of great popular-
ity and standard authority, as was also his arithmetical work, the
Tractatus de arte numerandi. Other prominent writers are Albertus
Magnus (1193?-! 280) and Georg Peurbach (1423-1461) in Ger-
many. It appears that here and there some of our modern ideas were
anticipated by writers of the Middle Ages. Thus, Nicole Oresme
(about 1323-1382), a bishop in Normandy, first conceived the notion of
fractional powers, afterwards rediscovered by Stevin, and suggested a
notation. Since 43 = 64, and 64^ = 8, Oresme concluded that 4^ = 8.

In his notation, 4^ is expressed, ip.^ 4, or j^ 4. Some of the

mathematicians of the Middle Ages possessed some idea of a function.
Oresme even attempted a graphic representation. But of a numeric
dependance of one quantity upon another, as found in Descartes,
there is no trace among them.2

In an unpublished manuscript Oresme found the sum of the infinite
series ^+|+l+i4G+ 3%~h • • in inf. Such recurrent infinite series were
formerly supposed to have made their first appearance in the eight-
eenth century. The use of infinite series is explained also' in the Liber
de triplici molu, by the Portuguese mathematician Alvarus Thomas*
in 1509. He gives the division of a line-segment into parts represent-
ing the terms of a convergent geometric series; that is, a segment AB
is divided in to parts such that AB :P1B = P1B :P2B = . . =P\B :Pi+l
B= . . Such a division of a line-segment occurs later in Napier's
kinematical discussion of logarithms.

Thomas Bradwardine (about 1290-1349), archbishop of Canter-
bury, studied star-polygons. The first appearance of such polygons- was
with Pythagoras and his school. We next meet with such polygons
in the geometry of Boethius and also in the translation of Euclid from
the Arabic by Athelard of Bath. To England falls the honor of hav-
ing produced the earliest European writers on trigonometry. The

1 F. Cajori, Amcric. Math. Monthly, Vol. 22, IQIS, pp. 45-47.

2H. Wieleitner in Bibliolhcca mathcmalica, 3. S., Vol. 13, 1913, pp. 115-145.

3 See Etudes stir Leonard da Vinci, Vol. Ill, Paris, 1913, pp. 393, 540, 541, by
Pierre Duhem (1861-1916) of the University of Bordeaux; see also Wieleitner in
Bibliotheca mathcmalica, Vol. 14, 1914, pp. 150-168.

128 A HISTORY OF MATHEMATICS

writings of Bradwardine, of Richard of Wallingford, and John Maud-
ith, both professors at Oxford, and of Simon Bredon of Winchecombe,
contain trigonometry drawn from Arabic sources.

The works of the Greek monk Maximus Planudes (about 1260-
1310), are of interest only as showing that the Hindu numerals were
then known in Greece. A writer belonging, like Planudes, to the By-
zantine school, wras Manuel Moschopulus who lived in Constantino-
ple in the early part of the fourteenth century. To him appears to be
due the introduction into Europe of magic squares. He wrote a treatise
on this subject. Magic squares were known before this to the Arabs
and Japanese; they originated with the Chinese. Mediaeval astrol-
ogers and physicians believed them to possess mystical properties and
to be a charm against plague, when engraved on silver plate.

Recently there has been printed a Hebrew arithmetical work by
the French Jew, Levi ben Gerson, written in 13 2 1,1 and handed down
in several manuscripts. It contains formulas for the number of per-
mutations and combinations of n things taken k at a time. It is worthy
of note that the earliest practical arithmetic known to have been
brought out in print appeared anonymously in Treviso, Italy, in 1478,
and is referred to as the "Treviso arithmetic." Four years later, in
1482, came out at Bamberg the first printed German arithmetic. It
is by Ulrich Wagner, a teacher of arithmetic at Niirnberg. It was
printed on parchment, but only fragments of one copy are now extant. 2

According to Enestrom, Ph. Calandri's De ariihmetrica opusculum,
Florence, 1491, is the first printed treatise containing the word "zero";
it is found in some fourteenth century manuscripts.

In 1494 was printed the Summa de Arithmetica, Geometria, Propor-
tione et Proportionality, written by the Tuscan monk Luca Pacioli
(1445-1514?), who, as we remarked, introduced several symbols in
algebra. This contains all the knowledge of his day on arithmetic,
algebra, and trigonometry, and is the first comprehensive work which
appeared after the Liber Abaci of Fibonacci. It contains little of im-
portance which cannot be found in Fibonacci's great wrork, published
three centuries earlier. Pacioli came in personal touch with two ar-
tists who were also mathematicians, Leonardo da Vinci 3 (1452-1519)
and Pier delta Francesca (1416-1492). Da Vinci inscribed regular
polygons in circles, but did not distinguish between accurate and ap-
proximate constructions. It is interesting to note that da Vinci was
familiar with the Greek text of Archimedes on the measurement of
the circle. Pier della Francesca advanced the theory of perspective,
and left a manuscript on regular solids which was published by

1 BibHotheca mathemalica, 3. S., Vol. 14, 1916, p. 261.

2 See D. E. Smith, Rara arithmetica, Boston and London, 1908, pp. 3, 12, 15;
F. linger, Mettwdik der Praktischen Arithmetik in Historischer Entwickelung, Leip-
zig, 1888, p. 39.

3 Consult P. Duhem's Etudes sur Leonard de Vinci, Paris, 1909.

129

Pacioli in 1509 as his own work, in a book entitled, Divina pro-
portione.

Perhaps the greatest result of the influx of Arabic learning was the
establishment of universities. What was their attitude toward mathe-
matics? The University of Paris, so famous at the beginning of the
twelfth century under the teachings of Abelard paid but little atten-
tion to this science during the Middle Ages. Geometry was neglected,
and Aristotle's logic was the favorite study. In 1336, a rule was in-
troduced that no student should take a degree without attending lec-
tures on mathematics, and from a commentary on the first six books
of Euclid, dated 1536, it appears that candidates for the degree of
A. M. had to give an oath that they had attended lectures on these
books.1 Examinations, when held at all, probably did not extend be-
yond the first book, as is shown by the nickname "magister mathe-
seos," applied to the Theorem of Pythagoras, the last in the first book.
More attention was paid to mathematics at the University of Prague,
founded 1384. For the Baccalaureate degree, students were required
to take lectures on Sacro Bosco's famous work on astronomy. Of can-
didates for the A.M. were required not' only the six books of Euclid,
but an additional knowledge of applied mathematics. Lectures were
given on the Almagest. At the University of Leipzig, the daughter of
Prague, and at Cologne, less work was required, and, as late as the
sixteenth century, the same requirements were made at these as at
Prague in the fourteenth. The universities of Bologna, Padua, Pisa,
occupied similar positions to the ones in Germany, only that purely
astrological lectures were given in place of lectures on the Almagest.
At Oxford, in the middle of the fifteenth century, the first two books
of Euclid were read.2

Thus it will be seen that the study of mathematics was maintained
at the universities only in a half-hearted manner. No great mathe-
matician and teacher appeared, to inspire the students. The best
energies of the schoolmen were expended upon the stupid subtleties of
their philosophy. The genius of Leonardo of Pisa left no permanent
impress upon the age, and another Renaissance of mathematics was
wanted.

1 H. Hankel, op. cit., p. 355. a J. Gow, op. tit., p. 207.

EUROPE DURING THE SIXTEENTH, SEVENTEENTH
AND EIGHTEENTH CENTURIES

We find it convenient to choose the time of the capture of Constan-
tinople by the Turks as the date at which the Middle Ages ended and
Modern Times began. In 1453, the Turks battered the walls of this
celebrated metropolis with cannon, and finally captured the city; the
Byzantine Empire fell, to rise no more. Calamitous as was this event
to the East, it acted favorably upon the progress of learning in the
West. A great number of learned Greeks fled into Italy, bringing with
them precious manuscripts of Greek literature. This contributed
vastly to the reviving of classic learning. Up to this time, Greek mas-
ters were known only through the often very corrupt Arabic manu-
scripts, but now they began to be studied from original sources and
in their own language. The first English translation of Euclid was
made in 1570 from the Greek by Sir Henry Billingsley, assisted by
John Dee.1 About the middle of the fifteenth century, printing was
invented; books became cheap and plentiful; the printing-press trans-
formed Europe into an audience-room. Near the close of the fifteenth
century, America was discovered, and, soon after, the earth was cir-
cumnavigated. The pulse and pace of the world began to quicken.
Men's minds became less servile; they became clearer and stronger.
The indistinctness of thought, which was the characteristic feature of
mediaeval learning, began to be remedied chiefly by the steady cultiva-
tion of Pure Mathematics and Astronomy. Dogmatism was attacked;
there arose a long struggle with the authority of the Church and the
established schools of philosophy. The Copernican System was set
up in opposition to the time-honored Ptolemaic System. The long
and eager contest between the two culminated in a crisis at the time
of Galileo, and resulted in the victory of the new system. Thus, by
slow degrees, the minds of men were cut adrift from their old scholastic
moorings and sent forth on the wide sea of scientific inquiry, to dis-
cover new islands and continents of truth.

The Renaissance

With the sixteenth century began a period of increased intellectual
activity. The human mind made a vast effort to achieve its freedom.
Attempts at its emancipation from Church authority had been made
before, but they were stifled and rendered abortive. The first great
and successful revolt against ecclesiastical authority was made in

1 G. B. Halsted in Am. Jour, of Math., Vol. II, 1879.
130

THE RENAISSANCE 131

Germany. The new desire for judging freely and independently in
matters of religion was preceded and accompanied by a growing spirit
of scientific inquiry. Thus it was that, for a time, Germany led the
van in science. She produced Regiomontanus , Copernicus, RJiaticus
and Kepler, at a period when France and England had7as yet, brought
forth hardly any great scientific thinkers. This remarkable scientific
productiveness was no doubt due, to a great extent, to the commercial
prosperity of Germany. Material prosperity is an essential condition
for the progress of knowledge. As long as every individual is obliged
to collect the necessaries for his subsistence, there can be no leisure
for higher pursuits. At this time, Germany had accumulated con-
siderable wealth. The Hanseatic League commanded the trade of
the North. Close commercial relations existed between Germany and
Italy. Italy, too, excelled in commercial activity and enterprise.
We need only mention Venice, whose glory began with the crusades,
and Florence, with her bankers and her manufacturers of silk and wool.
These two cities became great intellectual centres. Thus, Italy, too,
produced men in art, literature, and science, who shone forth in fullest
splendor. In fact, Italy was the fatherland of what is termed the Re-
naissance.

For the first great contributions to the mathematical sciences we
must, therefore, look to Italy and Germany. In Italy brilliant acces-
sions were made to algebra, in Germany progress was made in astron-
omy and trigonometry.

On the threshold of this new era we meet in Germany with the figure
of John Mueller, more generally called Regiomontanus (1436-1476).
Chiefly to him we owe the revival of trigonometry. He studied as-
tronomy and trigonometry at Vienna under the celebrated George
Peurbach. The latter perceived that the existing Latin translations
of the Almagest were full of errors, and that Arabic authors had not
remained true to the Greek original. Peurbach therefore began to
make a translation directly from the Greek. But he did not live to
finish it. His work was continued by Regiomontanus, who went be-
yond his master. Regiomontanus learned the Greek language from
Cardinal Bessarion, whom he followed to Italy, where he remained
eight years collecting manuscripts from Greeks who had fled thither
from the Turks. In addition to the translation of and the commen-
tary on the Almagest, he prepared translations of the Conies of Apol-
lonius, of Archimedes, and of the mechanical works of Heron. Regio-
montanus and Peurbach adopted the Hindu sine in place of the Greek
chord of double the arc. The Greeks and afterwards the Arabs divided
the radius into 60 equal parts, and each of these again into 60 smaller
ones. The Hindu expressed the length of the radius by parts of the
circumference, saying that of the 21,600 equal divisions of the latter,
it took 3438 to measure the radius. Regiomontanus, to secure greater
precision, constructed one table of sines on a radius divided into

132 A HISTORY OF MATHEMATICS

600,000 parts, and another on a radius divided decimally into
10,000,000 divisions. He emphasized the use of the tangent in trigo-
nometry. Following out some ideas of his master, he calculated a
table of tangents. German mathematicians were not the first Euro-
peans to use this function. In England it was known a century earlier
to Bradwardine, who speaks of tangent (umbra versa) and cotangent
(umbra recta), and to John Maudith. Even earlier, in the twelfth
century, the umbra versa and umbra recta are used in a translation from
Arabic into Latin, effected by Gerard of Cremona, of the Toledian
Tables of Al-Zarkali, who lived in Toledo about 1080. Regiomontanus
was the author of an arithmetic and also of a complete treatise on
trigonometry, containing solutions of both plane and spherical tri-
angles. Some innovations in trigonometry, formerly attributed to
Regiomontanus, are now known to have been introduced by the Arabs
before him. Nevertheless, much credit is due to him. His complete
mastery of astronomy and mathematics, and his enthusiasm for them,
were of far-reaching influence throughout Germany. So great was his
reputation, that Pope Sixtus IV called him to Italy to improve the
calendar. Regiomontanus left his beloved city of Nurnberg for Rome,
where he died in the following year.

After the time of Peurbach and Regiomontanus, trigonometry and
especially the calculation of tables continued to occupy German schol-
ars. More refined astronomical instruments were made, which gave
observations of greater precision; but these would have been useless
without trigonometrical tables of corresponding accuracy. Of the sev-
eral tables calculated, that by Georg Joachim of Feldkirch in Tyrol, gen-
erally called Rhaeticus (1514-1567) deserves special mention. He cal-
culated a table of sines with the radius =10,000,000,000 and from 10"
to 10"; and, later on, another with the radius =1,000,000,000,000,000,
and proceeding from 10" to 10". He began also the construction of
tables of tangents and secants, to be carried to the same degree of
accuracy; but he died before finishing them. For twelve years he had
had in continual employment several calculators. The work was com-
pleted in 1596 by his pupil, Valentine Otho (iS5o?-i6o5). This was
indeed a gigantic work, — a monument of German diligence and inde-
fatigable perseverance. The tables were republished in 1613 by Bar-
tholomaus Pitiscus (1561-1613) of Heidelberg, who spared no pains
•to free them of errors. Pitiscus was perhaps the first to use the word
"trigonometry." Astronomical tables of so great a degree of accu-
racy had never been dreamed of by the Greeks, Hindus, or Arabs.
That Rhaeticus was not a ready calculator only, is indicated by his
views on trigonometrical lines. Up to his time, the trigonometric
functions had been considered always with relation to the arc; he was
the first to construct the right triangle and to make them depend di-
rectly upon its angles. It was from the right triangle that Rhrcticus
got his idea of calculating the hypotenuse; i. e. he was the first to plan

THE RENAISSANCE 133

a table of secants. Good work in trigonometry was done also by Vieta
and Romanus.

We shall now leave the subject of trigonometry to witness the prog-
ress in the solution of algebraical equations. To do so, we must quit
Germany for Italy. The first comprehensive algebra printed was that
of Luca Pacioli. He closes his book by saying that the solution of the
equations xs-{-mx = n, x3-\-n=mx is as impossible at the present state
of science as the quadrature of the circle. This remark doubtless stim-
ulated thought. The first step in the algebraic solution of cubics was
taken by Scipione del Ferro (1465-1526), a professor of mathematics
at Bologna, who solved the equation x3-\-mx=n. He imparted it to
his pupil, Floridas, in 1505, but did not publish it. It was the practice
in those days and for two centuries afterwards to keep discoveries
secret, in order to secure by that means an advantage over rivals by
proposing problems beyond their reach. This practice gave rise to
numberless disputes regarding the priority of inventions. A second
solution of cubics was given by Nicolo of Brescia [i499(?)-i557].
When a boy of six, Nicolo was so badly cut by a French soldier that
he never again gained the free use of his tongue. Hence he was called
Tartaglia, i. e. the stammerer. His widowed mother being too poor to
pay his tuition in school, he learned to read and picked up a knowledge
of Latin, Greek, and mathematics by himself. Possessing a mind of
extraordinary power, he was able to appear as teacher of mathematics
at an early age. He taught in Venice, then in Brescia, and later again
in Venice. In 1530, one Colla proposed him several problems, one
leading to the equation x3-\-px2 = q. Tartaglia found an imperfect
method for solving this, but kept it secret. He spoke about his secret
in public and thus* caused Del Ferro's pupil, Floridas, to proclaim his
own knowledge of the form x*-\-mx = n. Tartaglia, believing him to
be a mediocrist and braggart, challenged him to a public discussion, to
take place on the 22d of February, 1535. Hearing, meanwhile, that
his rival had gotten the method from a deceased master, and fearing
that he would be beaten in the contest, Tartaglia put in all the zeal,
industry, and skill to find the rule for the equations, and he succeeded
in it ten days before the appointed date, as he himself modestly says. *
The most difficult step was, no doubt, the passing from quadratic ir-1
rationals, used in operating from time of old, to cubic irrationals.!
Placing x=\/~t— \/u, Tartaglia perceived that the irrationals dis-
appeared from the equation x3 = mx— n, making « = /—«. But this
last equality, together with (\m}z = tu, gives at once

This is Tartaglia's solution of xs-{-mx = n. On the i3th of February,

he found a similar solution for x? = mx-\-n. The contest began on the

1 H. Hankel, op. cit., p. 362.

134 A HISTORY OF MATHEMATICS

22<d. Each contestant proposed thirty problems. The one who could
solve the greatest number within fifty days should be the victor. Tar-
taglia solved the thirty problems proposed by Floridas in two hours;
Floridas could not solve any of Tartaglia's. From now on, Tartaglia
studied cubic equations with a will. In 1541 he discovered a general
solution for the cubic .T3± px2 = =±= q, by transforming it into the form
x3^=mx==±n. The news of Tartaglia's victory spread all over Italy.
Tartaglia was entreated to make known his method, but he declined
to do so, saying that after his completion of the translation from the
Greek of Euclid and Archimedes, he would publish a large algebra
containing his method. But a scholar from Milan, named Hieronimo
Cardano (1501-1576), after many solicitations, and after giving the
most solemn and sacred promises of secrecy, succeeded in obtaining
from Tartaglia a knowledge of his rules. Cardan was a singular mix-
ture of genius, folly, self-conceit and mysticism. He was successively
professor of mathematics and medicine at Milan, Pavia and Bologna,
In 1570 he was imprisoned for debt. Later he went to Rome, was
admitted to the college of physicians and was pensioned by the pope.
At this time Cardan was writing his Ars Magna, and he knew no
better way to crown his work than by inserting the much sought for
rules for solving cubics. Thus Cardan broke his most solemn vows,
and published in 1545 in his Ars Magna Tartaglia's solution of cubics.
However, Cardan did credit "his friend Tartaglia" with the discovery
of the rule. Nevertheless, Tartaglia became desperate. His most
cherished hope, of giving to the world an immortal work which should
be the monument of his deep learning and power for original research,
was suddenly destroyed; for the crown intended for his work had
been snatched away. His first step was to write a history of his in-
vention; but, to completely annihilate his enemies, he challenged
Cardan and his pupil Lodovico Ferrari to a contest: each party
should propose thirty-one questions to be solved by the other within
fifteen days. Tartaglia solved most questions in seven days, but the
other party did not send in their solutions before the expiration of the
fifth month; moreover, all their solutions except one were wrong. A
replication and a rejoinder followed. Endless were the problems pro-
posed and solved on both sides. The dispute produced much chagrin
and heart-burnings to the parties, and to Tartaglia especially, who
met with many other disappointments. After having recovered him-
self again, Tartaglia began, in 1556, the publication of the work which
he had had in his mind for so long; but he died before he reached the
consideration of cubic equations. Thus the fondest wish of his life re-
mained unfulfilled. How much credit for the algebraic solution of the
general cubic is due to Tartaglia and how much to Del Ferro it is now
impossible to ascertain definitely. Del Ferro 's researches were never
published and were lost. We know of them only through the remarks
of Cardan and his pupil L. Ferrari who say that Del Ferro's and Tar-

THE RENAISSANCE 135

taglia's methods were alike. Certain it is that the customary desig-
nation, "Cardan's solution of the cubic" ascribes to Cardan what
belongs to one or the other of his predecessors.

Remarkable is the great interest that the solution of cubics excited
throughout Italy. It is but natural that after this great conquest
mathematicians should attack bi-quadratic equations. As in the case
of cubics, so here, the first impulse was given by Colla, who, in 1540,
proposed for solution the equation x*-\- 6^+36 = 6ox. To be sure,
Cardan had studied particular cases as early as 1539. Thus he solved
the equation i3£2 = #4+2£3+2#+i by a process similar to that em-
ployed by Diophantus and the Hindus; namely, by adding to both
sides 3#2 and thereby rendering both numbers complete squares. But
Cardan failed to find a general solution; it remained for his pupil
Lodovico Ferrari (1522-1565) of Bologna to make the brilliant dis-
covery of the general solution of bi-quadratic equations. Ferrari re-
duced Colla's equation to the form (x2-\-6~)2=6ox-}-6x2. In order to
give also the right member the form of a complete square he added to
both members the expression 2(x2-}-6}y-^-y3, containing a new un-
known quantity y. This gave him (.-r+6-f-;y)2= (6-\-2y)x2-)r6ox-\-
(i2y+;y2). The condition that the right member be a complete square
is expressed by the cubic equation (2^+6) (i2y-\-y~} = goo. Extract-
ing the square root of the bi-quadratic, he got x--{-6-\-y=x^J 2y-\-6

-\ — ===== . Solving the cubic for y and substituting, it remained

only to determine x from the resulting quadratic. L. Ferrari pursued
a similar method with other numerical bi-quadratic equations. 1 Car-
dan had the pleasure of publishing this discovery in his Ars Magna
in 1545. Ferrari's solution is sometimes ascribed to R. Bombelli, but
he is no more the discoverer of it than Cardan is of the solution called
by his name.

To Cardan algebra is much indebted. In his Ars Magna he takes
notice of negative roots of an equation, calling them fictitious, while
the positive roots are called real. He paid some attention to compu-
tations involving the square root of negative numbers, but failed
to recognize imaginary roots. Cardan also observed the difficulty
in the irreducible case in the cubics, which, like the quadrature of the
circle, has since "so much tormented the perverse ingenuity of mathe-
maticians." But he did not understand its nature. It remained for
Raphael Bombelli of Bologna, who published in 1572 an algebra of
great merit, to point out the reality of the apparently imaginary ex-
pression which a root assumes, also to assign its value, when rational,
and thus to lay the foundation of a more intimate knowledge of imagi-
nary quantities. Cardan was an inveterate gambler. In 1663 there
was published posthumously his gambler's manual, De ludo alece,
1 H. Hankel, op. cit., p. 368.

136 A HISTORY OF MATHEMATICS

which contains discussions relating to the chances favorable for throw-
ing a particular number with two dice and also with three dice. Car-
dan considered another problem in probabilities. Stated in general
terms, the problem is: What is the proper division of a stake between
two players, if the game is interrupted and one player has taken si
points, the other 52 points, s points being required to win.1 Cardan
gives the ratio (1+2+ . . -\-[s—sz])l(i-]-2-\- . . -\-[s—si]), Tartaglia
gives (s-hsi— sz)!(s-\-5z— si). Both of these answers are wrong. Car-
dan considered also what later became known as the "Petersburg
problem."

After the brilliant success in solving equations of the third and
fourth degrees, there was probably no one who doubted, that with
aid of irrationals of higher degrees, the solution of equations of any
degree whatever could be found. But all attempts at the algebraic
solution of the quintic were fruitless, and, finally, Abel demonstrated
that all hopes of finding algebraic solutions to equations of higher
than the fourth degree were purely Utopian.

Since no solution by radicals of equations of higher degrees could
be found, there remained nothing else to be done than the devising of
processes by which the real roots of numerical equations could be
found by approximation. The Chinese method used by them as early
as the thirteenth century was unknown in the Occident. We have
seen that in the early part of the thirteenth century Leonardo of Pisa
solved a cubic to a high degree of approximation, but we are ignorant
of his method. The earliest known process in the Occident of ap-
proaching to a root of an affected numerical equation was invented by
Nicolas Chuquet, who, in 1484 at Lyons, wrote a work of high rank,
entitled Le triparty en la science des nombres. It was not printed until

i88o.2 If — >x<-;, then Chuquet takes the intermediate value — : — .
c d c+d

as a closer approximation to the root x. He finds a series of successive
intermediate values. We stated earlier that in 1498 the Arabic writer
Miram Chelebi gave a method of solving x3-{-Q = Px which he attrib-
utes to Atabeddin Jamshid. This cubic arose in the computation of
# = sin i°.

The earliest printed method of approximation to the roots of af-
fected equations is that of Cardan, who gave it in the Ars Magna,
1545, under the title of regula aurea. It is a skilful application of
the rule of "false position," and is applicable to equations of any de-
gree. This mode of approximation was exceedingly rough, yet this
fact hardly explains why Clavius, Stevin and Vieta did not refer to it.

1 M. Cantor, IT, 2 Aufl., 1900, pp. 501, 520, 537.

2 Printed in the Bullelino Boncompagni, T xiii, 1880; see pp. 653-654. See also
F. Cajori, "A History of the Arithmetical Methods of Approximation to the
Roots of Numerical Equations of one Unknown Quantity" in Colorado College
Publication, General Series Nos. 51 and 52, 1910.

THE RENAISSANCE 137

Processes of approximation were given by the Frenchman J. Peletier
(1554), the Italian R. Bombelli (1572), the German R. Ursus (1601),
the Swiss Joost Biirgi, the German Pitiscus (1612), and the Belgian
Simon Stevin. But far more important than the processes of these
men was that of the Frenchman, Francis Vieta (1540-1603), which
initiates a new era. It is contained in a work published at Paris in
1600 by Marino Ghetaldi as editor, with Vieta's consent, under the
title: De numerosa protestatum pur arum atque adfectarum ad exegesin
resolutione tractatus. His method is not of the nature of the rule of
"double false position," used by Cardan and Biirgi, but resembles
the method of ordinary root-extraction. Taking f(x) = k, where k is
taken positive, Vieta separates the required root from the rest, then
substitutes an approximate value for it and shows that another figure
of the root can be obtained by division. A repetition of this process
gives the next figure, and so on. Thus, in xb— $x*-\-$oox =7905504,
he takes r=2o, then computes 7905504 -r5+S^ — SOOT and divides
the result by a value which in our modern notation takes the form
\(f(r+sl) —f(r))\ —Sin, where n is the degree of the equation and s\ is
a unit of the denomination of the digit next to be found. Thus, if the
required root is 243, and r has been taken to be 200, then s\ is 10; but
if r is taken as 240, then s\ is i. In our example, where r= 20, the
divisor is 878295, and the quotient yields the next digit of the root
equal to 4. We obtain ^=20+4=24, the required root. Vieta's
procedure was greatly admired by his contemporaries, particularly
the Englishmen, T. Harriot, W. Oughtred and J. Wallis, each of whom
introduced some minor improvements.

We pause a moment to sketch the life of Vieta, the most eminent
French mathematician of the sixteenth century. He was born in
Poitou and died at Paris. He was employed throughout life in the
service of the state, under Henry III. and Henry IV. He was, there-
fore, not a mathematician by profession, but his love for the science
was so great that he remained in his chamber studying, sometimes
several days in succession, without eating and sleeping more than was
necessary to sustain himself. So great devotion to abstract science
is the more remarkable, because he lived at a time of incessant po-
litical and religious turmoil. During the war against Spain, Vieta
rendered service to Henry IV by deciphering intercepted letters writ-
ten in a species of cipher, and addressed by the Spanish Court to their
governor of Netherlands. The Spaniards attributed the discovery of
the key to magic.

In 1579 Vieta published his Canon mathematicus seu ad triangula
cum appendicibus, which contains very remarkable contributions to
trigonometry. It gives the first systematic elaboration in the Occi-
dent of the methods of computing plane and spherical triangles by
the aid of the six trigonometric functions.1 He paid special attention
1 A. v. Braunmiihl, Geschichlc dcr Trigonometry, I, Leipzig, 1900, p. 160.

138 A HISTORY OF MATHEMATICS

also to goniometry, developing such relations as sin a = sin (6o°+ a)

— sin (60° — a), csca-\-ctna = cin— . — ctna+csca = ta,n— , with the

2 2

aid of which he could compute from the functions of angles below
30° or 45°, the functions of the remaining angles below 90°, essentially
by addition and subtraction alone. Vieta is the first to apply alge-
braic transformation to trigonometry, particularly to the multisection
of angles. Letting 2 cosa=#, he expresses cos »a as a function of x
for all integers «<n; letting 2 sina=:r and 2 sin 2a=y, he expresses
2.rn~2sin no. in terms of x and y. Vieta exclaims: "Thus the analysis
of angular sections involves geometric and arithmetic secrets which
hitherto have been penetrated by no one."

An ambassador from Netherlands once told Henry IV that France
did not possess a single geometer capable of solving a problem pro-
pounded to geometers by a Belgian mathematician, Adrianus Ro-
manus. It was the solution of the equation of the forty-fifth degree: —

4S:V-3795:y3+95634/- • • • +945/1-45/3+/5=C1.
Henry IV called Vieta, who, having already pursued similar investi-
gations, saw at once that this awe-inspiring problem was simply the
equation by which C=2 sin 0 was expressed in terms of y=2 sin^g- </>;
that, since 45 = 3.3.5, it was necessary only to divide an angle once
into 5 equal parts, and then twice into 3, — a division which could be
effected by corresponding equations of the fifth and third degrees.
Brilliant was the discovery by Vieta of 23 roots to this equation, in-
stead of only one. The reason why he did not find 45 solutions, is
that the remaining ones involve negative sines, which were unintel-
ligible to him. Detailed investigations on the famous old problem
of the section of an angle into an odd number of equal parts, led Vieta
to the discovery of a trigonometrical solution of Cardan's irreducible

case in cubics. He applied the equation (2 cos f $)3— 3^ 2 cos-0] =

2 cos<£ to the solution of x3-3a?x=a?b, when a>%b, by placing x=
2a cos|<£, and determining 0from b = 2a cos<£.

The main principle employed by him in the solution of equations
is that of reduction. He solves the quadratic by making a suitable
substitution which will remove the term containing x to the first de-
gree. Like Cardan, he reduces the general expression of the cubic to
the form x3+mx+n=o; then, assuming x=(\a— z^-t-z and substi-
tuting, he gets g6— 6z3— ^a3=o. Putting z3=y, he has a quadratic.
In the solution of bi-quadratics, Vieta still remains true to his principle
of reduction. This gives him the well-known cubic resolvent. He
thus adheres throughout to his favorite principle, and thereby in-
troduces into algebra a uniformity of method which claims our lively
admiration. In Vieta's algebra we discover a partial knowledge of
the relations existing between the coefficients and the roots of an equa-

THE RENAISSANCE 139

tion. He shows that if the coefficient of the second term in an equa-
tion of the second degree is minus the sum of two numbers whose
product is the third term, then the two numbers are roots of the equa-
tion. Vieta rejected all except positive roots; hence it was impossible
for him to fully perceive the relations in question.

The most epoch-making innovation in algebra due to Vieta is the
denoting of general or indefinite quantities by letters of the alphabet.
To be sure, Regiomontanus and Stifel in Germany, and Cardan in
Italy, used letters before him, but Vieta extended the idea and first
made it an essential part of algebra. The new algebra was called by
him logistica speciosa in distinction to the old logistica numerosa.
Vieta's formalism differed considerably from that of to-day. The
equation a3+^a'2b+^ab2+b3=(a+b)3 was written by him "a cubus
+b in a quadr. 3+0 in b quadr. 3+6 cubo sequalia a+b cubo." In
numerical equations the unknown quantity was denoted by N, its
square by Q, and its cube by C. Thus the equation x3 — Sx2+ 16^=40
was written i C — 8Q+i6 N (equal. 40. Vieta used the term "co-
efficient," but it was little used before the close of the seventeenth
century.1 Sometimes he uses also the term "polynomial." Observe
that exponents and our symbol (=) for equality were not yet in use;
but that Vieta employed the Maltese cross (+) as the short-hand
symbol for addition, and the ( — ) for subtraction. These two char-
acters had not been in very general use before the time of Vieta. "It
is very singular," says Hallafn, "that discoveries of the greatest con-
venience, and, apparently, not above the ingenuity of a village school-
master, should have been overlooked by men of extraordinary acute-
ness like Tartaglia, Cardan, and L. Ferrari; and, hardly less so that, by
dint of that acuteness, they dispensed with the aid of these contriv-
ances in which we suppose that so much of the utility of algebraic ex-
pression consists." Even after improvements in notation were once
proposed, it was with extreme slowness that they were admitted into
general use. They were made oftener by accident than design, and
their authors had little notion of the effect of the change which they
were making. The introduction of the + and — symbols seems to be
due to the Germans, who, although they did not enrich algebra dur-
ing the Renaissance with great inventions, as did the Italians, still cul-
tivated it with great zeal. The arithmetic of John Widmann, brought
out in 1489 in Leipzig, is the earliest printed book in which the + and
— symbols have been found. The + sign/ is not restricted by him to
ordinary addition; it has the more general meaning "et" or "and"
as in the heading, "regula augment! + decrementi." The — sign- is
used to indicate subtraction, but not regularly so. The word "plus"
does not occur in Widmann's text; the word "minus" is used only two
or three times. The symbols + and — are used regularly for addi-

1 Encyclopedic dca sciences mathimaliqiies, Tome I, Vol. 2, 1907, p. 2.

i4o A HISTORY OF MATHEMATICS

tion and subtraction, in 152 1,1 in the arithmetic of Grammateus,
(Heinrich Schreiber, died 1525) a teacher at the University of Vienna.
His pupil, Christoff Rudolff, the writer of the first text-book on algebra
in the German language (printed in 1525), employs these symbols also.
So did Stifel, who brought out a second edition of Rudolll's Coss in
1553. Thus, by slow degrees, their adoption became universal. Sev-
eral independent paleographic studies of Latin manuscripts of the
fourteenth and fifteenth centuries make it almost certain that the
sign + conies from the Latin et, as it was cursively written in manu-
scripts just before the time of the invention of printing.2 The ori-
gin of the sign — is still uncertain. There is another short-hand
symbol of which we owe the origin to the Germans. In a manu-
script published sometime in the fifteenth century, a dot placed
before a number is made to signify the extraction of a root of
that number. This dot is the embryo of our present symbol for the
square root. Christoff Rudolff, in his algebra, remarks that <4the
radix quadrata is, for brevity, designated in his algorithm with the
character V, as ^4." Here the dot has grown into a symbol much
like our own. This same symbol was used by Michael Stifel. Our
sign of equality is due to Robert Recorde (1510-1558), the author of
The Whetstone of Witte (1557), which is the first English treatise on
algebra. He selected this symbol because no two things could be
more equal than two parallel lines =. The sign -r- for division was
first used by Johann Heinrich Rahn, a Swiss, in his Teutsche Algebra,
Zurich, 1659, and was introduced in England through Thomas
Brancker's translation of Rahn's book, London, 1668.

Michael Stifel (i486?-i567), the greatest German algebraist of the
sixteenth century, was born in Esslingen, and died in Jena. He was
educated in the monastery of his native place, and afterwards be-
came Protestant minister. The study of the significance of mystic
numbers in Revelation and in Daniel drew him to mathematics. He
studied German and Italian works, and published in' 1544, in Latin,
a book entitled Arithmetica Integra. Melanchthon wrote a preface to
it. Its three parts treat respectively of rational numbers, irrational
numbers, and algebra. Stifel gives a table containing the numerical
values of the binomial coefficients for powers below the i8th. He ob-
serves an advantage in letting a geometric progression correspond to
an arithmetical progression, and arrives at the designation of integral
powers by numbers. Here are the germs of the theory of exponents
and of logarithms. In 1545 Stifel published an arithmetic in German.
His edition of Rudolff's Coss contains rules for solving cubic equations,
derived from the writings of Cardan.

1 G. Enestrom in Bibliotheca mathematica, 3. S., Vol. 9, 1908-09, pp. 155-157;
Vol. 14, 1914, p. 278.

2 For references see M. Cantor, op. cit., Vol. II, 2. Ed., 1900, p. 231; J. Tropfke,
op. cit., Vol. I, 1902, pp. 133, 134.

THE RENAISSANCE 141

We remarked above that Vieta discarded negative roots of equa-
tions. Indeed, we find few algebraists before and during the Renais-
sance who understood the significance even of negative quantities.
Fibonacci seldom uses them. Pacioli states the rule that "minus times
minus gives plus," but applies it really only to the development of the
product of (a— b) (c—d); purely negative quantities do not appear
in his work. The German "Cossist" (algebraist), Michael Stifel,
speaks as early as 1544 of numbers which are "absurd" or "fictitious
below zero," and which arise when "real numbers above zero" are
subtracted from zero. Cardan, at last, speaks of a "pure minus";
"but these ideas," says H. Hankel, "remained sparsely, and until
the beginning of the seventeenth century, mathematicians dealt ex-
clusively with absolute positive quantities." One of the first alge-
braists who occasionally place a purely negative quantity by itself on
one side of an equation, is T. Harriot in England. As regards the rec-
ognition of negative roots, Cardan and Bombelli were far in advance
of all writers of the Renaissance, including Vieta. Yet even they
mentioned these so-called false or fictitious roots only in passing, and
without grasping their real significance and importance. On this
subject Cardan and Bombelli had advanced to about the same point
as had the Hindu Bhaskara, who saw negative roots, but did not ap-
prove of them. The generalization of the conception of quantity so
as to include the negative, was an exceedingly slow and difficult process
in the development of algebra.

We shall now consider the history of geometry during the Renais-
sance. Unlike algebra, it made hardly any progress. The greatest
gain was a more intimate knowledge of Greek geometry. No essen-
tial progress was made before the time of Descartes. Regiomontanus,
Xylander (Wilhelm Holzmann, 1532-1576) of Augsburg, Tartaglia,
Federigo Commandino (1509-1575) of Urbino in Italy, Maurolycus
and others, made translations of geometrical works from the Greek.
The description and instrumental construction of a new curve, the
epicycloid, is explained by Albrecht Diirer (1471-1528), the celebrated
painter and sculptor of Niirnberg, in a book, Underweysung der Mes-
sung mil dem Zyrkel und rychtscheyd, 1525. The idea of such a curve
goes back at least as far as Hipparchus who used it in his astronomical
theory of epicycles. The epicycloid does not again appear in history
until the time of G. Desargues and P. La Hire. Diirer is the earliest
writer in the Occident to call attention to magic squares. A simple
magic square appears in his celebrated painting called "Melancholia."

Johannes Werner (1468-1528) of Niirnberg published in 1522 the
first work on conies which appeared in Christian Europe. Unlike the
geometers of old, he studied the sections in relation with the cone, and
derived their properties directly from it. This mode of studying the
conies was followed by Franciscus Maurolycus (1494-1575) of Mes-
sina. The latter is, doubtless, the greatest geometer of the sixteenth

142 A HISTORY OF MATHEMATICS

century. From the notes of Pappus, he attempted to restore the miss-
ing fifth book of Apollonius on maxima and minima. His chief work is
his masterly and original treatment of the conic sections, wherein he
discusses tangents and asymptotes more fully than Apollonius had
done, and applies them to various physical and astronomical problems.
To Maurolycus has been ascribed also the discovery of the inference
by mathematical induction.1 It occurs in his introduction to his Opus-
cula mathematica^ Venice, 1575. Later, mathematical induction was
used by Pascal in his Traite du triangle arithmetique (1662). Processes
akin to mathematical induction, some of which would yield the mod-
ern mathematical induction by introducing some slight change in the
mode of presentation or in the point of view, were given before Mau-
rolycus. Giovanni Campano (latinized form, Campanus) of Novara
in Italy, in his edition of Euclid (1260), proves the irrationality of the
golden section by a recurrent mode of inference resulting in a reductio
ad absurdum. But he does not descend by a regular progression from n
to n— i, n— 2, etc., but leaps irregularly over, perhaps, several integers.
Campano's process was used later by Fermat. A recurrent mode of
inference is found in Bhaskara's "cyclic method" of solving inde-
terminate equations, in Theon of Smyrna (about 130 A. D.) and in
Proclus's process for finding numbers representing the sides and di-
agonals of squares; it is found in Euclid's proof (Elements IX, 20) that
the number of primes is infinite.

The foremost geometrician of Portugal was Pedro Nunes2 (1502-
1578) or Nonius. He showed that a ship sailing so as to make equal
angles with the meridians does not travel in a straight line, nor usually
along the arc of a great circle, but describes a path called the loxo-
dromic curve. Nunes invented the "nonius" and described it in
his De crepusculis, Lisbon, 1542. It consists in the juxtaposition of
equal arcs, one arc divided into m equal parts and the other into m+i
equal parts. Nonius took m=8g. The instrument is also called
a "vernier," after the Frenchman Pierre Vernier, who re-invented it
in 1631. The foremost French mathematician before Vieta was Peter
Ramus (1515-1572), who perished in the massacre of St. Bartholomew.
Vieta possessed great familiarity with ancient geometry. The new
form which he gave to algebra, by representing general quantities by
letters, enabled him to point out more easily how the construction of
the roots of cubics depended upon the celebrated ancient problems of
the duplication of the cube and the trisection of an angle. He reached
the interesting conclusion that the former problem includes the solu-
tions of all cubics in which the radical in Tartaglia's formula is real,
but that the latter problem includes only those leading to the irredu-
cible case.

1 G. Vacca in Bulletin Am. ~Math. Society, 2. S., Vol. 16, 1909, p. 70. See also
F. Cajori in Vol. 15, pp. 407-409.

2 See R. Guimaraes, Pedro Nunes, Coimpre, 1915.

THE RENAISSANCE 143

The problem of the quadrature of the circle was revived in this age,
and was zealously studied even by men of eminence and mathematical
ability. The army of circle-squarers became most formidable during
the seventeenth century. Among the first to revive this problem was
the German Cardinal Nicolaus Cusanus (1401-1464), who had the
reputation of being a great logician. His fallacies were exposed to
full view by Regiomontanus. As in this case, so in others, every quad-
rator of note raised up an opposing mathematician: Oronce Fine was
met by Jean Buteo (c. 1492-1572) and P. Nunes; Joseph Scaliger by
Vieta, Adrianus Romanus, and Clavius; a Quercu by Adriaen An-
thonisz (1527-1607). Two mathematicians of Netherlands, Adrianus
Romanus (1561-1615) and Ludolph van Ceulen (1540-1610), occu-
pied themselves with approximating to the ratio between the circumfer-
ence and the diameter. The former carried the value TT to 15, the lat-
ter to 35, places. The value of TT is therefore often named "Ludolph's
number." His performance was considered so extraordinary, that the
numbers were cut on his tomb-stone (now lost) in St. Peter's church-
yard, at Leyden. These men had used the Archimedian method of
in- and circum-scribed polygons, a method refined in 1621 by Wille-
brord Snellius (1580-1626) who showed how narrower limits may be
obtained for TT without increasing the number of sides of the poly-
gons. Snellius used two theorems equivalent to \ (2 sin 6 tan 6} Z. dZ.
3/(2 csc0+cot0). The greatest refinements in the use of the geo-
metrical method of Archimedes were reached by C. Huyghens in his
De circuli magnitudine inventa, 1654, and by James Gregory (1638-
1675), professor at St. Andrews and Edinburgh, in his Exercitationes
geometric^, 1668, and Vera circuli el hyperbolae quadratures, 1667.
Gregory gave several formulas for approximating to TT and in the
second of these publications boldly attempted to prove by the Ar-
chimedean algorithm that the quadrature of the circle is impossible.
Huyghens showed that Gregory's proof is not conclusive, although
he himself believed that the quadrature is impossible. Other attempts
to prove this impossibility were made by Thomas Fautat De Lagny
(1660-1734) of Paris, in 1727, Joseph Saurin (1659-1737) in 1720,
Isaac Newton in his Principia I, 6, lemma 28, E. Waring, L. Euler,
1771.

That these proofs would lack rigor was almost to be expected, as
long as no distinction was made between algebraical and transcen-
dental numbers.

The earliest explicit expression for TT by an infinite number of op-
erations was found by Vieta. Considering regular polygons of 4, 8,
16, . . . sides, inscribed in a circle of unit radius, he found that the
area of the circle is

144 A HISTORY OF MATHEMATICS

from which we obtain

Z= — _ * — , which may be derived from Euler's formula 1

2 ViVjhTvT^

8= — z— — , ( 0^ TT), by taking 0=7T/2 .

COS0/2 COS0/4 COS0/8 ...

As mentioned earlier,it was Adrianus Romanus (1561-1615) of Lou-
vain who propounded for solution that equation of the forty-fifth degree
solved by Vieta. On receiving Vieta's solution, he at once departed for
Paris, to make his acquaintance with so great a master. Vieta proposed
to him the Apollonian problem, to draw a circle touching three given
circles. " Adrianus Romanus solved the problem by the intersection of
two hyperbolas; but this solution did not possess the rigor of the ancient
geometry. Vieta caused him to see this, and then, in his turn, pre-
sented a solution which had all the rigor desirable." 2 Romanus
did much toward simplifying spherical trigonometry by reducing, by
means of certain projections, the 28 cases in triangles then considered
to only six.

Mention must here be made of the improvements of the Julian
calendar. The yearly determination of the movable feasts had for
a long time been connected with an untold amount of confusion. The
rapid progress of astronomy led to the consideration of this subject,
and many new calendars were proposed. Pope Gregory XIII con-
voked a large number of mathematicians, astronomers, and prelates,
who decided upon the adoption of the calendar proposed by the Jesuit
Christophorus Clavius (1537-1612) of Rome. To rectify the errors of
the Julian calendar it was agreed to write in the new calendar the i5th
of October immediately after the 4th of October of the year 1582.
The Gregorian calendar met with a great deal of opposition both
among scientists and among Protestants. Clavius, who ranked high
as a geometer, met the objections of the former most ably and effec-
tively; the prejudices of the latter passed away with time.

The passion for the study of mystical properties of numbers de-
scended from the ancients to the moderns. Much was written on
numerical mysticism even by such eminent men as Pacioli and Stifel.
The Numerorum Mysteria of Peter Bungus covered 700 quarto pages.
He worked with great industry and satisfaction on 666, which is the
number of the beast in Revelation (xiii, 18), the symbol of Antichrist.
He reduced the name of the "impious" Martin Luther to a form which
may express this formidable number. Placing a=i, b = 2, etc., k = io,
1= 20, etc., he finds, after misspelling the name, that M(30)A(i)R(80)T(ioo>
I(9)N(40)L(2o)V(20o)T(ioo)E(5)R(80)A(i) constitutes the number required.
These attacks on the great reformer were not unprovoked, for his

1 E. W. Hobson, Squaring the Circle, Cambridge, 1913, pp. 26, 27, 31.

2 A. Quetelet, Histoire des Sciences mathematiques et physiques chez Us Beiges.
Bruxelles, 1864, p. 137.

•

VIETA TO DESCARTES 145

friend, Michael Stifel, the most acute and original of the early mathe-
maticians of Germany, exercised an equal ingenuity in showing that
the above number referred to Pope Leo X, — a demonstration which
gave Stifel unspeakable comfort.1

Astrology also was still a favorite study. It is well known that Car-
dan, Maurolycus, Regiomontanus, and many other eminent scientists
who lived at a period even later than this, engaged in deep astrological
study; but it is not so generally known that besides the occult sciences
already named, men engaged in the mystic study of star-polygons
and magic squares. "The pentagramma gives you pain," says Faust
to Mephistopheles. It is of deep psychological interest to see scientists,
like the great Kepler, demonstrate on one page a theorem on star-
polygons, with strict geometric rigor, while on the next page, perhaps,
he explains their use as amulets or in conjurations. Playfair, speaking
of Cardan as an astrologer, calls him "a melancholy proof that there
is no folly or weakness too great to be united to high intellectual at-
tainments." ' Let our judgment not be too harsh. The period under
consideration is too near the Middle Ages to admit of complete eman-
cipation from mysticism even among scientists. Scholars like Kepler,
Napier, Albrecht Diirer, while in the van of progress and planting
one foot upon the firm ground of truly scientific inquiry, were still
resting with the other foot upon the scholastic ideas of preceding ages.

Vieta to Descartes

The ecclesiastical power, which in the ignorant ages was an unmixed
benefit, in more enlightened ages became a serious evil. Thus, in
France, during the reigns preceding that of Henry IV, the theological
spirit predominated. This is painfully shown by the massacres of
Vassy and of St. Bartholomew. Being engaged in religious disputes,
people had no leisure for science and for secular literature. Hence,
down to the time of Henry IV, the French "had not put forth a single
work, the destruction of which would now be a loss to Europe." In
England, on the other hand, no religious wars were waged. The people
were comparatively indifferent about religious strifes; they concen-
trated their ability upon secular matters, and acquired, in the six-
teenth century, a literature which is immortalized by the genius of
Shakespeare and Spenser. This great literary age in England was
followed by a great scientific age. At the close of the sixteenth cen-
tury, the shackles of ecclesiastical authority were thrown off by France.
The ascension of Henry IV to the throne was followed in 1598 by the
Edict of Nantes, granting freedom of worship to the Huguenots, and
thereby terminating religious wars. The genius of the French nation

1 G. Peacock, op. cit., p. 424.

2 John Playfair, "Progress of the Mathematical and Physical Sciences" in En-
cyclopedia Britannica, 7th ed., continued in 8th Ed., by Sir John Leslie.

i46 A HISTORY OF MATHEMATICS

now began to blossom. Cardinal Richelieu, during the reign of Louis
XIII, pursued the broad policy of not favoring the opinions of any
sect, but of promoting the interests of the nation. His age was re-
markable for the progress of knowledge. It produced that great secu-
lar literature, the counterpart of which was found in England in the
sixteenth century. The seventeenth century was made illustrious
also by the great French mathematicians, Roberval, Descartes, Des-
argues, Fermat, and Pascal.

More gloomy is the picture in Germany. The great changes which
revolutionized the world in the sixteenth century, and which led Eng-
land to national greatness, led Germany to degradation. The first
effects of the Reformation there were salutary. At the close of the
fifteenth and during the sixteenth century, Germany had been con-
spicuous for her scientific pursuits. She had been a leader in as-
tronomy and trigonometry. Algebra also, excepting for the discoveries
in cubic equations, was, before the time of Vieta, in a more advanced
state there than elsewhere. But at the beginning of the seventeenth
century, when the sun of science began to rise in France, it set in Ger-
many. Theologic disputes and religious strife ensued. The Thirty
Years' War (1618-1648) proved ruinous. The German empire was
shattered, and became a mere lax confederation of petty despotisms.
Commerce was destroyed; national feeling died out. Art disappeared,
and in literature there was only a slavish imitation of French arti-
ficiality. Nor did Germany recover from this low state for 200 years;
for in 1756 began another struggle, the Seven Years' War, which
turned Prussia into a wasted land. Thus it followed that at the be-
ginning of the seventeenth century, the great Kepler was the only
German mathematician of eminence, and that in the interval of 200
years between Kepler and Gauss, there arose no great mathematician
in Germany excepting Leibniz.

Up to the seventeenth century, mathematics was cultivated but little
in Great Britain. During the sixteenth century, she brought forth
no mathematician comparable with Vieta, Stifel, or Tartaglia. But
with the time of Recorde, the English became conspicuous for numeri-
cal skill. The first important arithmetical work of English authorship
was published in Latin in 1522 by Cuthbert Tonstall (1474-1559). He
had studied at Oxford, Cambridge, and Padua, and drew freely from
the works of Pacioli and Regiomontanus. Reprints of his arithmetic
appeared in England and France. After Recorde the higher branches
of mathematics began to be studied. Later, Scotland brought forth
John Napier, the inventor of logarithms. The instantaneous appre-
ciation of their value is doubtless the result of superiority in calcula-
tion. In Italy, and especially in France, geometry, which for a long
time had been an almost stationary science, began to be studied with
success. Galileo, Torricelli, Roberval, Fermat, Desargues, Pascal,
Descartes, and the English Wallis are the great revolutioners of this

VIETA TO DESCARTES 147

science. Theoretical mechanics began to be studied. The foundations
were laid by Fermat and Pascal for the theory of numbers and the
theory of probability.

We shall first consider the improvements made in the art of calcu-
lating. The nations of antiquity experimented thousands of years
upon numeral notations before they happened to strike upon the so-
called "Arabic notation." In the simple expedient of the cipher,
which was permanently introduced by the Hindus, mathematics re-
ceived one of the most powerful impulses. It would seem that after
the "Arabic notation" was once thoroughly understood, decimal
fractions would occur at once as an obvious extension of it. But "it
is curious to think how much science had attempted in physical re-
search and how deeply numbers had been pondered, before it was per-
ceived that the all-powerful simplicity of the ' Arabic notation ' was as
valuable and as manageable in an infinitely descending as in an in-
finitely ascending progression." Simple as decimal fractions ap-
pear to us, the invention of them is not the result of one mind or even
of one age. They came into use by almost imperceptible degrees. The
first mathematicians identified with their history did not perceive
their true nature and importance, and failed to invent a suitable no-
tation. The idea of decimal fractions makes its first appearance in
methods for approximating to the square roots of numbers. ' Thus
John of Seville, presumably in imitation of Hindu rules, adds 2n ci-
phers to the number, then finds the square root, and takes this as the
numerator of a fraction whose denominator is i followed by n ciphers.
The same method was followed by Cardan, but it failed to be generally
adopted even by his Italian contemporaries; for otherwise it would
certainly have been at least mentioned by Pietro Calaldi (died 1626)
in a work devoted exclusively to the extraction of roots. Cataldi,
and before him Bombelli in 1572, find the square root by means of
continued fractions — a method ingenious and novel, but for practical
purposes inferior to Cardan's. Oronce Fine (1494-1555) in France
(called also Orontius Finaeus), and William Buckley (died about
1550) in England extracted the square root in the same way as
Cardan and John of Seville. The invention of decimals has been
frequently attributed to Regiomontanus, on the ground that in-
stead of placing the sinus totus, in trigonometry, equal to a multiple
of 60, like the Greeks, he put it= 100,000. But here the trigonomet-
rical lines were expressed in integers, and not in fractions. Though
he adopted a decimal division of the radius, he and his successors
did not apply the idea outside of trigonometry and, indeed, had no
notion whatever of decimal fractions. To Simon Stevin (1548-
1620) of Bruges in Belgium, a man who did a great deal of work in
most diverse fields of science, we owe the first systematic treatment of
decimal fractions. In his La Disme (1585) he describes in very express
1 Mark Napier, Memoirs of John Napier of Mcrchislon. Edinburgh, 1834.

148 A HISTORY OF MATHEMATICS

terms the advantages, not only of decimal fractions, but also of the
decimal division in systems of weights and measures. Stevin applied
the new fractions "to all the operations of ordinary arithmetic." l
What he lacked was a suitable notation. In place of our decimal point,
he used a cipher; to each place in the fraction was attached the cor-
responding index. Thus, in his notation, the number 5.912 would be
0123

5912 or 5®9©i©20. These indices, though cumbrous in practice, are
of interest, because they embody the notion of powers of numbers.
Stevin considered also fractional powers. He says that "f " placed
within a circle would mean x2/3, but he does not actually use his nota-
tion. This notion had been advanced much earlier by Oresme, but
it had remained unnoticed. Stevin found the greatest common di-
visor of x3+x2 and x^+jx+6 by the process of continual division,
thereby applying to polynomials Euclid's mode of finding the greatest
common divisor of numbers, as explained in Book VII of his Elements.
Stevin was enthusiastic not only over decimal fractions, but also over
the decimal division of weights and measures. He considered it the
duty of governments to establish the latter. He advocated the deci-
mal subdivision of the degree. No improvement was made in the
notation of decimals till the beginning of the seventeenth century.
After Stevin, decimals were used by Joost Bilrgi (1552-1632), a Swiss
by birth, who prepared a manuscript on arithmetic soon after 1592, and
by Johann Hartmann Beyer, who assumes the invention as his own.
In 1603, he published at Frankfurt on the Main a Logistica Decimalis.
Historians of mathematics do not yet agree to whom the first intro-
duction of the decimal point or comma should be ascribed. Among
the candidates for the honor are Pellos (1492), Bu'rgi (1592), Pitiscus
(1608, 1612), Kepler (1616), Napier (1616, 1617). This divergence
of opinion is due mainly to different standards of judgment. If the
requirement made of candidates is not only that the decimal point or
comma was actually used by them, but that they must give evidence
that the numbers used were actually decimal fractions, that the point
or comma was with them not merely a general symbol to indicate
a separation, that they must actually use the decimal point in opera-
tions including multiplication or division of decimal fractions, then
it would seem that the honor falls to John Napier, who exhibits such
use in his Rabdologia, 1617. Perhaps Napier received the suggestion
for this notation from Pitiscus who, according to G. Enestrom,2 uses
the point in his Trigonometria of 1608 and 1612, not as a regular deci-
mal point, but as a more general sign of separation. Napier's decimal
point did not meet with immediate adoption. W. Oughtred in 1631
designates the fraction .56 thus, 0)56. Albert Girard, a pupil of Stevin,
in 1629 uses the point on one occasion. John Walk's in 1657 writes

1 A. Quetelet, op. cit., p. 158.

2 Bibliotheca mathematica, 3. S., Vol. 6, 1905, p. 109.

VIETA TO DESCARTES 149

12(345, but afterwards in his algebra adopts the usual point. A. De
Morgan says that "to the first quarter of the eighteenth century we
must refer not only the complete and final victory of the decimal point,
but also that of the now universal method of performing the operations
of division and extraction of the square root." l We have dwelt at
some length on the progress of the decimal notation, because "the
history of language ... is of the highest order of interest, as well as
utility: its suggestions are the best lesson for the future which a reflect-
ing mind can have."

The miraculous powers of modern calculation are due to three in-
ventions: the Arabic Notation, Decimal Fractions, and Logarithms.
The invention of logarithms in the first quarter of the seventeenth
century was admirably timed, for Kepler was then examining plane-
tary orbits, and Galileo had just turned the telescope to the stars.
During the Renaissance German mathematicians had constructed
trigonometrical tables of great accuracy, but its greater precision
enormously increased the work of the calculator. It is no exaggera-
tion to say that the invention of logarithms "by shortening the labors
doubled the life of the astronomer." Logarithms were invented by
John Napier (1550-1617), Baron of Merchiston, in Scotland. It is
one of the greatest curiosities of the history of science that Napier
constructed logarithms before exponents were used. To be sure,
Stifel and Stevin made some attempts to denote powers by indices,
but this notation was not generally known, — not even to T. Harriot,
whose algebra appeared long after Napier's death. That logarithms
flow naturally from the exponential symbol was not observed until
much later. What, then, was Napier's line of thought?

Let AB be a definite line, DE a line extending from D indefinitely.
Imagine two points starting at the same moment; the one moving

AC B

l i

D F E

from A toward B, the other from D toward E. Let the velocity during
the first moment be the same for both: let that of the point on line DE
be uniform; but the velocity of the point on AB decreasing in such
a way that when it arrives at any point C, its velocity is proportional
to the remaining distance EC. While the first point moves over a dis-
tance AC, the second one moves over a distance DF. Napier calls
DF the logarithm of EC.

He first sought the logarithms only of sines; the line AB was the
sine of 90° and was taken =io7; EC was the sine of the arc, and

1 A. De Morgan, Arithmetical Books from the Invention of Printing to the Present
Time, London, 1847, p. xxvii.

i5o A HISTORY OF MATHEMATICS

DF its logarithm. We notice that as the motion proceeds, BC
decreases in geometrical progression, while DF increases in arith-
metical progression. Let AB=a=io7, let x=DF, y = BC, then

AC=a— y. The velocity of the point C is — jT^-y/ this gives

— nat. log y=t+c. When /=o, then y=a and c= -nat. log a. Again,

let -jr=a be the velocity of the point F, then x=at. Substituting for
at

t and c their values and remembering that 0=io7 and that by defini-
tion #=Nap. log y, we get

Nap. log y=io7 nat. log — .

It is evident from this formula that Napier's logarithms are not the
same as the natural logarithms. Napier's logarithms increase as the
number itself decreases. He took the logarithm of sin 9o°=o; i. e.
the logarithm of io7=o. The logarithm of sin a increased from zero
as a decreased from 90°. Napier's genesis of logarithms from the con-
ception of two flowing points reminds us of Newton's doctrine of
fluxions. The relation between geometric and arithmetical progres-
sions so skilfully utilized by Napier, had been observed by Archi-
medes, Stifel, and others. What was the base of Napier's system of
logarithms? To this we reply that not only did the notion of a
"base" never suggest itself to him, but it is inapplicable to his
system. This notion demands that zero be the logarithm of i; in
Napier's system, zero is the logarithm of io7. Napier's great in-
vention was given to the world in 1614 in a work entitled Mirifici
logarithmorum canonis descriptio. In it he explained the nature of
his logarithms, and gave a logarithmic table of the natural sines of
a quadrant from minute to minute. In 1619 appeared Napier's
Mirifici logarithmorum canonis conslruclio, as a posthumous work, in
which his method of calculating logarithms is explained. An English
translation of the Conslructio, by W. R. Macdonald, appeared in
Edinburgh, in 1889.

Henry Briggs (1556-1631), in Napier's time professor of geometry
at Gresham College, London, and afterwards professor at Oxford,
was so struck with admiration of Napier's book, that he left his studies
in London to do homage to the Scottish philosopher. Briggs was de-
layed in his journey, and Napier complained to a common friend, "Ah,
John, Mr. Briggs will not come." At that very moment knocks were
heard at the gate, and Briggs was brought into the lord's chamber.
Almost one-quarter of an hour wa,s spent, each beholding the other
without speaking a word. At last Briggs began: "My lord, I have
undertaken this long journey purposely to see your person, and to
know by what engine of wit or ingenuity you came first to think of

VIETA TO DESCARTES 151

this most excellent help in astronomy, viz. the logarithms; but, my
lord, being by you found out, I wonder nobody found it out before,
when now known it is so easy.." Briggs suggested to Napier the ad-
vantage that would result from retaining zero for the logarithm of the
whole sine, but choosing 10,000,000,000 for the logarithm of the loth
part of that same sine, i. e. of 5° 44' 22". Napier said that he had al-
ready thought of the change, and he pointed out a slight improvement
on Briggs' idea; viz. that zero should be the logarithm of i, and
10,000,000,000 that of the whole sine, thereby making the character-
istic of numbers greater than unity positive and not negative, as sug-
gested by Briggs. Briggs admitted this to be more convenient. The
invention of "Briggian logarithms" occurred, therefore, to Briggs
and Napier independently. The great practical advantage of the new
system was that its fundamental progression was accommodated to
the base, 10, of our numerical scale. Briggs devoted all his energies
to the construction of tables upon the new plan. Napier died in 1617,
with the satisfaction of having found in Briggs an able friend to bring
to completion his unfinished plans. In 1624 Briggs published his
Arilhmetica logarithmica, containing the logarithms to 14 places of
numbers, from i to 20,000 and from 90,000 to 100,000. The gap from
20,000 to 90,000 was filled up by that illustrious successor of Napier
and Briggs, Adrian Vlacq (i6oo?-i667). He was born at Gouda in
Holland and lived ten years in London as a bookseller and publisher.
Being driven out by London bookdealers, he settled in Paris where he
met opposition again, for selling foreign books. He died at The Hague.
John Milton, in his Defensio secunda, published an abuse of him.
Vlacq published in 1628 a table of logarithms from i to 100,000, of
which 70,000 were calculated by himself. The first publication of
Briggian logarithms of trigonometric functions was made in 1620 by
Edmund Gunter (1581-1626) of London, a colleague of Briggs, who
found the logarithmic sines and tangents for every minute to seven
places. Gunter was the inventor of the words cosine and cotangent
(1620).

The word cosine was an abbreviation of complemental sine. The
invention of the words tangent and secant is due to the physician and
mathematician, Thomas Finck, a native of Flensburg, who used them
in his Geometria rotundi, Basel, 1583. Gunter is known to engineers
for his "Gunter's chain." It is told of him that "When he was a stu-
dent at Christ College, it fell to his lot to preach the Passion sermon,
which some old divines that I knew did hear, but they said that it
was said of him then in the University that our Savior never suffered
so much since his passion as in that sermon, it was such a lamented
one." Briggs devoted the last years of his life to calculating more
extensive Briggian logarithms of trigonometric functions, but he died
in 1631, leaving his work unfinished. It was carried on by Henry Gel-
1 Aubrey's Brief Lives, Edition A. Clark, 1898, Vol. I, p. 276.

152 A HISTORY OF MATHEMATICS

librand (1597-1637) of Gresham College in London, and then pub-
lished by Vlacq at his own expense. Briggs divided a degree into
100 parts, as was done also by N. Roe in 1633, W. Oughtred in 1657,
John Newton in 1658, but owing to the publication by Vlacq of trigo-
nometrical tables constructed on the old sexagesimal division, Briggs'
innovation did not prevail. Briggs and Vlacq published four funda-
mental works, the results of which have not been superseded by any
subsequent calculations until very recently.

The word "characteristic," as used in logarithms, first occurs in
Briggs' Arithmetica logarithmica, 1624; the word "mantissa" was in-
troduced by John Wallis in the Latin edition of his Algebra, 1693,
p. 41, and was used by L. Euler in his Introductio in analysin in 1748,
p. 85.

The only rival of John Napier in the invention of logarithms was the
Swiss Joost Biirgi (1552-1632). He published a table of logarithms,
Arithmetische und Geometrische Progresstabulen, Prague, 1620, but he
conceived the idea and constructed his table independently of Napier.
He neglected to have it published until Napier's logarithms were
known and admired throughout Europe.

Among the various inventions of Napier to assist the memory of
the student or calculator, is "Napier's rule of circular parts" for the
solution of spherical right triangles. It is, perhaps, "the happiest
example of artificial memory that is known." Napier gives in the
Descriptio a proof of his rule; proofs were given later by Johann
Heinrich Lambert (1765) and Leslie Ellis (1863). * Of the four for-
mulas for oblique spherical triangles which are sometimes called "Na-
pier's Analogies," only two are due to Napier himself; they are given
in his Construct™. The other two were added by Briggs in his an-
notations to the Constructio.

A modification of Napier's logarithms was made by John Speidell,
a teacher of mathematics in London, who published the New Loga-
rithmes, London, 1619, containing the logarithms of sines, tangents
and secants. Speidell did not advance a new theory. He simply
aimed to improve on Napier's tables by making all logarithms posi-
tive. To achieve this end he subtracted Napier's logarithmic numbers
from io8 and then discarded the last two digits. Napier gave log sin
3o'=474i3852. Subtracting this from io8 leaves 52586148. Speidell
wrote log sin 3o'=52586i. It has been said that Speidell's logarithms
of 1619 are logarithms to the natural base e. This is not quite true,
on account of complications arising from the fact that the logarithms
in Speidell's table appear as integral numbers and that the natural
trigonometric values (not printed in Speidell's tables) are likewise
written as integral numbers. If the last five figures in Speidell's log-
arithms are taken as decimals (mantissas), then the logarithms are
the natural logarithms (with io added to every negative character-
1 R. Mortiz, Am. Math. Manthly, Vol. 22, 1915, p. 221.

VIETA TO DESCARTES 153

istic) of the trigonometric values, provided the latter are expressed
decimally as ratios. For instance, Napier gives sin 30^87265, the
radius being io7. In reality, sin 30'=. 0087265. The natural log-
arithm of this fraction is approximately 5.25861. Adding io gives
5.25861. As seen above, Speidell writes log sin 30'= 525861. The
relation between the natural logarithms and the logarithms in Spei-
dell's trigonometric tables is shown by the formula, Sp. log x=io^

( io+loge — i } . For secants and the latter half of the tangents the

addition of io is omitted. In Speidell's table, log tan 89°=4048i2, the
natural logarithm of tan 89° being 4.04812. In the 1622 edition of his
New Logarithmes, Speidell included also a table of logarithms of the
numbers i-iooo. Except for the omission of the decimal point, the loga-
rithms in this table are genuinely natural logarithms. Thus, he gives log
10=2302584; in modern notation, /ogeio= 2.302584. J. W. L. Glaisher
has pointed out l that these are not the earliest natural logarithms. The
second (1618) edition of Edward Wright's translation of Napier's De-
scriptio contains an anonymous Appendix, very probably written by
William Oughtred, describing a process of interpolation with the aid
of a small table containing the logarithms of 72 sines. The latter
are natural logarithms with the decimal point omitted. Thus, log
10=2302584, log 50=3911021. This Appendix is noteworthy also as
containing the earliest account of the radix method of computing log-
arithms. After the time of Speidell no tables of natural logarithms
were published until 1770, when J. H. Lambert inserted a seven place
table of natural logarithms of the numbers i-ioo in his Zusatze :« den
Logarithmischen und Trigonometrischen Tabellen. Most of the early
methods of computing logarithms originated in England. Napier
begins the computations of his logarithms of 1614 by forming a geo-
metric progression of 101 terms, the first term being io7 and the corn-
ratio ( i 7 land the last term 9,999,900.0004950. This progres-

V io /

sion constitutes the "First Table" given in his Construct™. Omitting
the decimal part of the last term, he takes 9,999,900 as the second
term of a new progression of 51 terms whose first term is io7, the corn-
ratio being (i A and the last term 9,995,001.222927 (should

be 9,995001.224804). A third geometric progression of 21 terms has
io7 as its first term, 9,995000 for its second term, the common ratio

i ) and 9,900,473.57808 as its last term. This progression of

2OOO/

21 terms constitutes the first of 69 columns of numbers in Napier's
1 Quarterly Jour, of Pure &• Appl. Math., Vol. 46, 1915, p. 145.

mon

i54 A HISTORY OF MATHEMATICS

"Third Table." Each column is a geometric progression of 21 terms

with ( i ) as the common ratio. The 60 first or top numbers in

\ 2000/

the 69 columns themselves constitute a geometric progression having

the ratio! i-- - ) , the first top number being io7, the second 9900000,
V loo/

and so on. The last number in the 6gth column is 4998609.4034.
Thus this ''Third Table" gives a series of numbers very nearly, but
not exactly in geometrical progression, and lying between io7 and
very nearly ^.io7. Says Hutton, these tables were "found in the
most simple manner, by little more than easy subtractions." The
numbers are taken as the sines of angles between 90° and 30°. Kine-
matical considerations yield him an upper and a lower limit for the
logarithm of a given sine. By these limits he obtains the logarithm
of each number in his "Third Table." To obtain the logarithms of
sines between o° and 30° Napier indicates two methods. By one of
them he computes log sin0, i5°< 0<3Q°, by the aid of his "Third
Table" and the formula sin 26=2 sin 6 sin (90°— 6). A repetition
of this process gives the logarithms of sines down to 6=7° 30', and
so on.

Biirgi's method of computation was more primitive than Napier's.
In his table the logarithms were printed in red and were called "red
numbers " ; the antilogarithms were in black. The expressions ra = ion,

bn=bu—I ( iH 4), where r0=o, b0= 100,000,000, and »=i, 2, 3, . . . ,

indicate the mode of computation. Any term bn of the geometric
series is obtained by adding to the preceding term 6*— i, the — ^th part

of that term. Proceeding thus Biirgi arrives at r= 230,270,022 and
6=1,000,000,000, this last pair of numbers being obtained by inter-
polation.

In the Appendix to the Construct™ there are described three meth-
ods of computing logarithms which are probably the result of the
joint labors of Napier and Briggs. The first method rests on the
successive extractions of fifth roots. The second calls for square
roots only. Taking log 1=0 and log io=io10, find the logarithm
of the mean proportion between i and io. There follows log Vi X io
=log 3.16227766017 = ! (io10); then log \/ 10x3. 162 27766017=^
5.62341325191 = ! (io10), and so on. Substantially this method was
used by Kepler in his book on logarithms of 1624 and by Vlacq. The
third method in the Appendix to the Constructio lets log 1 = 0, log 10=
io10, and takes 2 as a factor io10 times, yielding a number composed
of 301029996 figures; hence log 2=0,301029996.

VIETA TO DESCARTES 155

A famous method of computing logarithms is the so-called "radix
method." It requires the aid of a table of radices or numbers of the

f
form i± — -, with their logarithms. The logarithm of a number is

found by resolving the number into factors of the form i=*= — and

then adding the logarithms of the factors. The earliest appearance
of this method is in the anonymous " Appendix" (very probably due
to Oughtred) to Edward Wright's 1618 edition of Napier's Description
It is fully developed by Briggs who, in his Arithmetic*! logarithmica,
1624, gives a table of radices. The method has been frequently re-
dis"covered and given in various forms.2 A slight simplification of
Briggs' process was given as one of three methods by Robert Flower in
a tract, The Radix a new way of making Logarithms, London, 1771.
He divides a given number by a power of 10 and a single digit, so as
to reduce the first figure to .9, and then multiplies by a procession of
radices until all the digits become nines. The radix method was re-
discovered in 1786 by George Atwood (1746-1807), the inventor of
"Atwood's machine," in An essay on the Arithmetic of Factors, and
again by 7,ecchini Leonelli in 1802, by Thomas Manning (1772-1840),
scholar of Caius College, Cambridge, in 1806, by Thomas Weddle in
1845, Hearn in 1847 and Orchard in 1848. Extensions and variations
of the radix method have been published by Peter Gray (1807?- 1887),
a writer on life contingencies, Thoman, A. J. Ellis (1814-1890), and
others. The three distinct methods of its application are due to Briggs,
Flower and Weddle.

Another method of computing common logarithms is by the re-
peated formation of geometric means. If A = i, B=io, then C=
•\/~AB=$. 162278 has the logarithm .5, D=\/BC= 5.623413 has the
logarithm .75, etc. Perhaps suggested by Napier's remarks in the
Constructio, this method was developed by French writers, of whom
Jacques Ozanam (1640-1717) in 1670 was perhaps the first.3 Ozanam
is best known for his Recreations mathematiques et physiques, 1694.

Still different devices for the computation of logarithms were in-
vented by Brook Taylor (1717), John Long (1714), William Jones,
Roger Cotes (1722), Andrew Reid (1767), James Dodson (1742), Abel
Biirja (1786), and others.4

1 J. W. L. Glaisher, in Quarterly Jour, of Math's, Vol. 46, 1915, p. 125.

2 For the detailed history of this method consult also A. J. Ellis in Proceedings
of the Royal Society (London), Vol. 31, 1881, pp. 398-413; S. Lupton, Mathematical
Gazette, Vol. 7, 1913, pp. 147-150, 170-173; Ch. Hutton's Introduction to his
Mathematical Tables.

3 See J. W. L. Glaisher in Quarterly Journal of Pure and Appl. Math's, Vol. 47,
1916, pp. 249-301.

4 For details see Ch. Hutton's Introduction to his Mathematical Tables, also the
Encyclopedic dcs sciences malhematiqucs , lyoS; I, 23, "Tables dc logarithmcs."

156 A HISTORY OF MATHEMATICS

After the labor of computing logarithms was practically over, the
facile methods of computing by infinite series came to be discovered.
James Gregory, Lord William Brounker (1620-1684), Nicholas Mer-
cator (1620-1687), John Wallis and Edmund Halley are the pioneer
workers. Mercator in 1668 derived what amounts to the infinite
series for log (i+a). Transformations of this series yielded rapidly
converging results. Wallis in 1695 obtained \ log (i+2)/log (i— z) =
z+\ z3+5 25+ . . . . G. Vega in his Thesaurus of 1794 lets z= i/(2;y2— i).

The theoretic view point of the logarithm was broadened somewhat
during the seventeenth century by the graphic representation, both
in rectangular and polar coordinates, of a variable and its variable
logarithm. Thus were invented the logarithmic curve and the loga-
rithmic spiral. It has been thought that the earliest reference to the
logarithmic curve was made by the Italian Evangelista Torricelli in
a letter of the year 1644, but Paul Tannery made it practically certain
that Descartes knew the curve in 1639. * Descartes described the log-
arithmic spiral in 1638 in a letter to P. Mersenne, but does not give its
equation, nor connect it with logarithms. He describes it as the curve
which makes equal angles with all the radii drawn through the origin.
The name "logarithmic spiral" was coined by Pierre Varignon in a
paper presented to the Paris academy in 1704 and published in I722.2

The most brilliant conquest in algebra during the sixteenth century
had been the solution of cubic and biquadratic equations. All at-
tempts at solving algebraically equations of' higher degrees remaining
fruitless, a new line of inquiry — the properties of equations and their
roots— was gradually opened up. We have seen that Vieta had at-
tained a partial knowledge of the relations between roots and co-
efficients. Jacques Peletier (1517-1582), a French man of letters, poet
and mathematician, had observed as early as 1558, that the root of
an equation is a divisor of the last term. In passing he writes equa-
tions with all terms on one side, and equated to zero. This was done
also by Buteo and Harriot. One who extended the theory of equa-
tions somewhat further than Vieta, was Albert Girard (i59O?-i633?),
a mathematician of Lorraine. Like Vieta, this ingenious author ap-
plied algebra to geometry, and was the first who understood the use
of negative roots in the solution of geometric problems. He spoke of
imaginary quantities, inferred by induction that every equation has
as many roots as there are units in the number expressing its degree,
and first showed how to express the sums of their powers in terms of
the coefficients. Another algebraist of considerable power was the
English Thomas Harriot (1560-1621). He accompanied the first
colony sent out by Sir Walter Raleigh to Virginia. After having sur-

1 See G. Loria, Bibliotheca math., 3. S., Vol. i, 1900, p. 75; L' inter mldiaire des
mathematiciens, Vol. 7, 1900, p. 95.

2 For details and references, see F. Cajori, "History of the Exponential and
Logarithmic Concepts," Am. Math. Monthly, Vol. 20, 1913, pp. 10, u.

VIETA TO DESCARTES 157

veyed that country he returned to England. As a mathematician, he
was the boast of his country. He brought the theory of equations
under one comprehensive point of view by grasping that truth in its
full extent to which Vieta and Girard only approximated; viz. that
in an equation in its simplest form, the coefficient of the second term
with its sign changed is equal to the sum of the roots; the coefficient
of the third is equal to the sum of the products of every two of the
roots, etc. He was the first to decompose equations into their simple
factors; but, since he failed to recognize imaginary and even negative
roots, he failed also to prove that every equation could be thus de-
composed. Harriot made some changes in algebraic notation, adopt-
ing small letters of the alphabet in place of the capitals used by Vieta.
The symbols of inequality > and < were introduced by him. The
signs ^ and ^_ were first used about a century later by the Parisian
hydrographer, Pierre Bouguer.1 Harriot's work, A rtis A nalyti ca praxis,
was published in 1631, ten years after his death. William Oughtred
(1574-1660) contributed vastly to the propagation of mathematical
knowledge in England by his treatises, the Clams mathematics, 1631
(later Latin editions, 1648, 1652, 1667, 1693; English editions, 1647,
1694), Circles of Proportion, 1632, Trigonometric , 1657. 2 Oughtred
was an episcopal minister at Albury, near London, and gave private
lessons, free of charge, to pupils interested in mathematics. Among
his most noted pupils are the mathematician John Wallis and the
astronomer Seth Ward. Oughtred laid extraordinary emphasis upon
the use of mathematical symbols; altogether he used over 150 of them.
Only three have come down to modern times, namely X as the symbol
of multiplication, :: as that of proportion, and -^- as that for "differ-
ence." The symbol X occurs in the Clavis, but the letter X which
closely resembles it, occurs as a sign of multiplication in the anony-
mous "Appendix to the Logarithmes" in Edward Wright's transla-
tion of Napier's Descriptio, published in i6i8.3 This appendix was
most probably written by Oughtred. A proportion A:B = C:D he
wrote A- B :: C- D. Oughtred's notation for ratio and proportion was
widely used in England and on the Continent, but as early as 1651
the English astronomer Vincent Wing began to use (:) for ratio,4 a
notation which gained ground and freed the dot (.) for use as the sym-
bol of separation in decimal fractions. It is interesting to note the
attitude of Leibniz toward some of these symbols. On July 29, 1698,
he wrote in a letter to John Bernoulli: "I do not like X as a symbol
for multiplication, as it is easily confounded with x; . . . often I simply
relate two quantities by an interposed dot and indicate multiplication

1 P. H. Fuss, Corresp. math, phys., I, 1843, p. 304; Encyclopedic dcs sciences mathi-
matiques, T. I, Vol. I, 1904, p. 23.

"See F. Cajori, William Oughtred, Chicago and London, 1916.
3 F. Cajori, in Nature, Vol. XCIV, 1914, p. 363.
< Ibid., p. 477-

158 A HISTORY OF MATHEMATICS

by ZC-LM. Hence in designating ratio I use not one point but two
points, which I use, at the same time, for division; thus, for your
dy. x:\dt.a I write dy:x=dt:a; forffy is to x as dt is to a, is indeed the
same as, dy divided by x is equal to dt divided by a. From this equa-
tion follow then all the rules of proportion." This conception of
ratio and proportion was far in advance of that in contemporary
arithmetics. Through the aid of Christian Wolf the dot was generally
adopted in the eighteenth century as a symbol of multiplication.
Presumably Leibniz had no knowledge that Harriot in his Artis
analytics praxis, 1631, used a dot for multiplication, as in aaa— 3.
bba= + 2.ccc. Harriot's dot received no attention, not even from Wallis.
Oughtred and some of his English contemporaries, Richard Nor-
wood, John Speidell and others were prominent in introducing abbre-
viations for the trigonometric functions: s, si, or sin for sine; s co or
si co for "sine complement" or cosine; se for secant, etc. Oughtred
did not use parentheses. Terms to be aggregated were enclosed be-
tween double colons. He wrote ^(A+E) thus, ^q:A+E: The two
dots at the end were sometimes omitted. Thus, CiA+B—E meant
(A+B— E).3 Before Oughtred the use of parentheses had been sug-
gested by Clavius in 1608 and Girard in 1629. In fact, as early as

1556 Tartaglia wrote \/^2S — Vio thus R v. (R28 men Rio), where
R v. means "radix universalis," but he did not use parentheses in in-
dicating the product of two expressions.1 Parentheses were used by
I. Errard de Bar-le-Duc (1619), Jacobo de Billy (1643), Richard
Norwood (1631), Samuel Foster (1659); nevertheless parentheses did
not become popular in algebra before the time of Leibniz and the
Bernoullis.

It is noteworthy that Oughtred denotes 3^-and f-ff, the approxi-
mate ratios of the circumference to the diameter, by the symbol ^; it

occurs in the 1647 edition and in the later editions of his Clavis mathe-
matics. Oughtred's notation was adopted and used extensively by
Isaac Barrow. It was the forerunner of the notation ^=3.14159 . . . ,
first used by William Jones in 1706 in his Synopsis palmariorum ma-
theseos, London, 1706, p. 263. L. Euler first used ^=3.14159 ... in
1737. In his time, the symbol met with general adoption.

Oughtred stands out prominently as the inventor of the circular
and the rectilinear slide rules. The circular slide rule was described
in print in his book, the Circles of Proportion, 1632. His rectilinear
slide rule was described in 1633 in an Addition to the above work.
But Oughtred was not the first to describe the circular slide rule in
print; this was done by one of his pupils, Richard Delamain, in 1630,
in a booklet, entitled Grammelogia.2 A bitter controversy arose be-

1 G. Enestrom in Bibliotheca malhematica, 3. S., Vol. 7, p. 296.

2 See F. Cajori, William Oughtred, Chicago and London, 1916, p. 46.

VIETA TO DESCARTES 159

tween Delamain and Oughtred. Each accused the other of having
stolen the invention from him. Most probably each was an in-
dependent inventor. To the invention of the rectilinear slide rule
Oughtred has a clear title. He states that he designed his slide rules
as early as 1621. The slide rule was improved in England during the
seventeenth and eighteenth centuries and was used quite extensively.1

Some of the stories told about Oughtred are doubtless apocryphal,
as for instance, that his economical wife denied him the use of a candle
for study in the evening, and that he died of joy at the Restoration,
after drinking "a glass of sack" to his Majesty's health. De Morgan
humorously remarks, "It should be added, by way of excuse, that he
was eighty-six years old."

Algebra was now in a state of sufficient perfection to enable Des-
cartes and others to take that important step which forms one of the
grand epochs in the history of mathematics, — the application of alge-
braic analysis to define the nature and investigate the properties of
algebraic curves.

In geometry, the determination of the areas of curvilinear figures
was diligently studied at this period. Paul Guldin (1577-1643), a
Swiss mathematician of considerable note, rediscovered the following
theorem, published in his Centrobaryca, which has been named after
him, though first found in the Mathematical Collections of Pappus:
The volume of a solid of revolution is equal to the area of the generat-
ing figure, multiplied by the circumference described by the centre of
gravity. We shall see that this method excels that of Kepler and
Cavalieri in following a more exact and natural course; but it has the
disadvantage of necessitating the determination of the centre of grav-
ity, which in itself may be a more difficult problem than the original
one of finding the volume. Guldin made some attempts to prove his
theorem, but Cavalieri pointed out the weakness of his demonstration.

Johannes Kepler (1571-1630) was a native of Wurtemberg and im-
bibed Copernican principles while at the University of Tubingen. His
pursuit of science was repeatedly interrupted by war, religious perse-
cution, pecuniary embarrassments, frequent changes of residence,
and family troubles. In 1600 he became for one year assistant to the
Danish astronomer, Tycho Brahe, in the observatory near Prague.
The relation between the two great astronomers was not always of an
agreeable character. Kepler's publications are voluminous. His first at-
tempt to explain the solar system was made in 1596, when he thought
he had discovered a curious relation between the five regular solids
and the number and distance of the planets. The publication of this
pseudo-discovery brought him much fame. At one time he tried to
represent the orbit of Mars by the oval curve which we now write in
polar coordinates, p=2r cos3 6. Maturer reflection and intercourse
with Tycho Brahe and Galileo led him to investigations and results
1 See F. Cajori, History of the Logarithmic Slide Rule, New York, 1909.

160 A HISTORY OF MATHEMATICS

worthy of his genius — "Kepler's laws." He enriched pure mathe-
matics as well as astronomy. It is not strange that he was interested
in the mathematical science which had done him so much service; for
"if the Greeks had not cultivated conic sections, Kepler could not
have superseded Ptolemy." *• The Greeks never dreamed that these '
curves would ever be of practical use; Aristaeus and Apollonius \
studied them merely to satisfy 'their intellectual cravings after the i
ideal; yet the conic sections assisted Kepler in tracing the march of .*
the planets in their elliptic orbits. Kepler made also extended use of
logarithms and decimal fractions, and was enthusiastic in diffusing
a knowledge of them. At one tune, while purchasing wine, he was
struck by the inaccuracy of the ordinary modes of determining the
contents of kegs. This led him to the study of the volumes of solids
of revolution and to the publication of the Stereometria Doliorum in
1615. In it he deals first with the solids known to Archimedes and
then takes up others. Kepler made wide application of an old but |
neglected idea, that of infinitely great and infinitely small quantities, f
Greek mathematicians usually shunned this notion, but with it modern
mathematicians completely revolutionized the science. In comparing
rectilinear figures, the method of superposition was employed by the
ancients, but in comparing rectilinear and curvilinear figures with
each other, this method failed because no addition or subtraction of
rectilinear figures could ever produce curvilinear ones. To meet this
case, they devised the Method of ExhaustiQn,_which was long and
difficult; it was purely synthetical, and in general required that the
conclusion should be known at the^ufset. The new notion of infinity
led gradually to the invention of methods immeasurably more power-
ful. Kepler conceived the circle to be composed oi an irmmteli umber
of triangles having their commoli veilices at the^LeiitTe, and their
bases in the circumfemite, and the spht!le~To~coris^t of an infinite
nnmKre_nf pyramids. He" applied conceptions of this kind to the de-
termination of the areas and volumes of figures generated by curves
revolving about any line as axis, but succeeded in solving only a few
of the simplest out of the 84 problems which he proposed for investi-
gation in. his Stereometria.

Other points of mathematical interest in Kepler's works are (i) the
assertion that the circumference of an ellipse^whose axes are 2 a and
26, is nearly TT (a+b) ;~(2") a passage from whTch it has, been inferred
that Kepler knew the variation of a function near its maximum value
tojHsappeaTfts) the~assumption of the principle of continuity (which
differentiates modern from anciient_geoHief ry ) , when he shows that
a parabola has"a focus at" infinity, "that lines radiating from this " caecus
focus" arc parallel and have-nrrDTEerpoinrgt infinity.

The Stereometria led Cavalieri, an Italian Jesuit, to the consideration

1 William Whewell, History of the Inductive Sciences, 3rd Ed., New York, 1858,
Vol. I, p. 311.

VIETA TO DESCARTES 161

of infinitely small quantities. Bonaventura Cavalieri (1598-1647),
a pupil of Galileo and professor at Bologna, is celebrated for his Geo-
metria indivisibilibus continuorum nova quadam ratione promota, 1635.
This work expounds his method of Indivisibles, which occupies an
intermediate place between the method of exhaustion of the Greeks
and the methods of Newton and Leibniz. "TndTvisiblesIljvere dis-
cussed by Aristotle_and the scholastic philosophers. They commanded
the attention of Galileo. Cavalieri does not define the term. He
borrows the concept from the scholastic philosophy of Bradwardine
and Thomas Aquinas, in which a point is the indivisible of a line, a line
the indivisible of a surface, etc. Each indivisible is capable of gener-
ating thejiexthigher continuum by motion; a moving point generates
a line, etc The relative magnitude of two solids or surfaces could
tKenbe found simply by the summation of series of planes or lines.
For example, Cavalieri finds the sum of the squares oi all lines making
up a triangle equal to one-third the sum of the squares of all lines of
a parallelogram of equal base and altitude; for if in a triangle, the first
line at the apex be i, then the second is 2, the third is 3, and so on;
and the sum of their squares is

i2+22+32+ • . . +n2=n(n+i) (2n+i)-s-6.

In the parallelogram, each of the lines is n and their number is n; hence
the total sum of their squares is n3. The ratio between the two sums
is therefore

since n is infinite. From this he concludes that the pyramid or cone is
respectively ^ of a prism or cylinder of equal base and altitude, since
the polygons or circles composing the former decrease from the base
to the apex in the same way as the squares of the lines parallel to the
base in a triangle decrease from base to apex. By the Method of In-
divisibles, Cavalieri solved the majority of the problems proposed by
Kepler. Though expeditious and yielding correct results, Cavalieri's
method lacks a scientific foundation. If a line has absolutely no width,
then the addition of no number, however great, of lines can ever yield
an area; if a plane has no thickness whatever, then even an infinite
number of planes cannot form a solid. Though unphilosophical,
Cavalieri's method was used for fifty years as a sort of integral
calculus. It yielded solutions to some difficult problems. Guldin
made a severe attack on Cavalieri and his method. The latter
published in 1647, after the death of Guldin, a treatise entitled
Exercitationes geometries sex, in which he replied to the objections
of his opponent and attempted to give a clearer explanation of his
method. Guldin had never been able to demonstrate the theorem
named after him, except by metaphysical reasoning, but Cavalieri
proved it by the method of indivisibles. A revised edition of the
Geometria appeared in 1653.

162 A HISTORY OF MATHEMATICS

There is an important curve, not known to the ancients, which now
began to be studied with great zeal. Roberval gave it the name of
"trochoid," Pascal the name of "roulette," Galileo the name of "cy-
cloid." The invention of this curve seems to be due to Charles Bou-
velles who in a geometry published in Paris in 1501 refers to this curve
in connection with the problem of the squaring of the circle. Galileo
valued it for the graceful form it would give to arches in architecture.
He ascertained its area by weighing paper figures of the cycloid against
that of the generating circle, and found thereby the first area to be
nearly but not exactly thrice the latter. A mathematical determina-
tion was made by his pupil, Evangelista Torricelli (1608-1647), who
is more widely known as a physicist than as a mathematician.

By the Method of Indivisibles he demonstrated its area to be triple
that of the revolving circle, and published his solution. This same
quadrature had been effected a few years earlier (about 1636) by
Roberval in France, but his solution was not known to the Italians.
Roberval, being a man of irritable and violent disposition, unjustly
accused the mild and amiable Torricelli of stealing the proof. This
accusation of plagiarism created so much chagrin with Torricelli that
. it is considered to have been the cause of his early death. Vincenzo
Viviani (1622-1703), another prominent pupil of Galileo, determined
the tangent to the cycloid. This was accomplished in France by
Descartes and Fermat.

In France, where geometry began to be cultivated with greatest
success, Roberval, Fermat, Pascal, employed the Method of Indivis-
ibles and made new improvements in it. Giles Persone de Roberval
(1602-1675), for forty years professor of mathematics at the College
of France in Paris, claimed for himself the invention of the Method of
Indivisibles. Since his complete works were not published until after
his death, it is difficult to settle questions of priority. Montucla and
Chasles are of the opinion that he invented the method independently
of and earlier than the Italian geometer, though the work of the latter
was published much earlier than Roberval's. Marie finds it difficult
to believe that the Frenchman borrowed nothing whatever from the
Italian, for both could not have hit independently upon the word
Indivisibles, which is applicable to infinitely small quantities, as con-
ceived by Cavalieri, but not as conceived by Roberval. Roberval
and Pascal improved the rational basis of the Method of Indivisibles,
by considering an area as made up of an indefinite number of rectangles
instead of lines, and a solid as composed of indefinitely small solids
instead of surfaces. Roberval applied the method to the finding of
areas, volumes, and centres of gravity. He effected the quadrature
of a parabola of any degree ya=am~1x, and also of a parabola ym =
am~axn. We have already mentioned his quadrature of the cycloid.
Roberval is best known for his method of drawing tangents, which,
however, was invented at the same time, if not earlier, by Torricelli.

VIETA TO DESCARTES 16,3

Torricelli's appeared in 1644 under the title Opera geometrica. Rober-
val gives the fuller exposition of it. Some of his special applications
were published at Paris as early as 1644 in Mersenne's Cogitata physico-
mathematica. Roberval presented the full development of the sub-
ject to the French Academy of Sciences in 1668 which published it
in its Memoires. This academy had grown out of scientific meetings
held with Mersenne at Paris. It was founded by Minister Richelieu
in 1635 and reorganized by Minister Colbert in 1666. Marin Mersenne
(1588-1648) rendered great services to science. His polite and en-
gaging manners procured him many friends, including Descartes and
Fermat. He encouraged scientific research, carried on an extensive
correspondence, and thereby was the medium for the intercommunica-
tion of scientific intelligence.

Roberval's method of drawing tangents is allied to Newton's prin-
ciple of fluxions. Archimedes conceived his spiral to be generated by
a double motion. This idea Roberval extended to all curves. Plane
curves, as for instance the conic sections, may be generated by a point
acted upon by two forces, and are the resultant of two motions. If
at any point of the curve the resultant be resolved into its components,
then the diagonal of the parallelogram determined by them is the tan-
gent to the curve at that point. The greatest difficulty connected
with this ingenious method consisted in resolving the resultant into
components having the proper lengths and directions. Roberval did
not always succeed in doing this, yet his new idea was a great step in
advance. He broke off from the ancient definition of a tangent as
a straight line having only one point in common with a curve, — a defi-
nitioir which by the methods then available was not adapted to bring
out ffft properties of tangents to curves of higher degrees, nor even of
curves of the second degree and the parts they may be made to play
in the generation of the curves. The subject of tangents received
special attention also from Fermat, Descartes, and Barrow, and
reached its highest development after the invention of the differential
calculus. Fermat and Descartes defined tangents as secants whose
two points of intersection with the curve coincide; Barrow considered
a curve a polygon, and called one of its sides produced a tangent.

A profound scholar in all branches of learning and a mathematician
of exceptional powers was Pierre de Fermat (1601-1665). He studied
law at Toulouse, and in 1631 was made councillor for the parliament
of Toulouse. His leisure time was mostly devoted to mathematics,
which he studied with irresistible passion. Unlike Descartes and
Pascal, he led a quiet and unaggressive life. Fermat has left the im-
press of his genius upon all branches of mathematics then known.
A great contribution to geometry was his De maximis et minimis.
About twenty years earlier, Kepler had first observed that the incre-
ment of a variable, as, for instance, the ordinate of a curve, is evan-
escent for values very near a maximum or a minimum value of the

164 A HISTORY OF MATHEMATICS

variable. Developing this idea, Fermat obtained his rule for maxima
and minima. He substituted x+e for x in the given function of x and
then equated to each other the two consecutive values of the function
and divided the equation by e. If e be taken o, then the roots of this
equation are the values of x, making the function a maximum or a
minimum. Fermat was in possession of this rule in 1629. The main
difference between it and the rule of the differential calculus is that it
introduces the indefinite quantity e instead of the infinitely small d.\\
Fermat made it the basis for his method of drawing tangents, which
involved the determination of the length of the subtangent for a given
point of a curve.

Owing to a want of explicitness in statement, Fermat's method of
maxima and minima, and of tangents, was severely attacked by his
great contemporary, Descartes, who could never be brought to render
due justice to his merit. In the ensuing dispute, Fermat found two
zealous defenders in Roberval and Pascal, the father; while C. My-
dorge, G. Desargues, and Claude Hardy supported Descartes.

Since Fermat introduced the conception of infinitely small differ-
ences between consecutive values of a function and arrived at the
principle for finding the maxima and minima, it was maintained by
Lagrange, Laplace, and Fourier, that Fermat may be regarded as the
first inventor of the differential calculus. This point is not well taken,
as will be seen from the words of Poisson, himself a Frenchman, who
rightly says that the differenti^calculus "consists in a system of rules
proper for finding the differentials of all functions, rather than in the
use which may be made of these infinitely small variations in the so-
lution of one or two isolated problems.")

A contemporary mathematician, whose genius perhaps equalled that
of the great Fermat, was Blaise Pascal (1623-1662). He was born at
Clermont in Auvergne. In 1626 his father retired to Paris, where he
devoted himself to teaching his son, for he would not trust his educa-
tion to others. Blaise Pascal's genius for geometry showed itself when
he was but twelve years old. His father was well skilled in mathe-
matics, but did not wish his son to study it until he was perfectly
acquainted with Latin and Greek. All mathematical books were
hidden out of his sight. The boy once asked his father what mathe-
matics treated of, and was answered, in general, "that it was the
method of making figures with exactness, and of finding out what
proportions they relatively had to one another." He was at the same
time forbidden to talk any more about it, or ever to think of it. But
his genius could not submit to be confined within these bounds. Start-
ing with the bare fact that mathematics taught the means of making
figures infallibly exact, he employed his thoughts about it and with
a piece of charcoal drew figures upon the tiles of the pavement, trying
the methods of drawing, for example, an exact circle or equilateral
triangle. He gave names of his own to these figures and then formed

VIETA TO DESCARTES 165'

axioms, and, in short, came to make demonstrations. In this way he
is reported to have arrived unaided at the theorem that the sum of
the three angles of a triangle is equal to two right angles. His father
caught him in the act of studying this theorem, and was so astonished
at the sublimity and force of his genius as to weep for joy. Thefather
now gave him Euclid's Elements, which he, without assistance, mas-
tered easily. His regular studies being languages, the boy employed
only his hours of amusement on the study of geometry, yet he had so
ready and lively a penetration that, at the age of sixteen, he wrote
a treatise upon conies, which passed for such a surprising effort of
genius, that it was said nothing equal to it in strength had been pro-
duced since the time of Archimedes. Descartes refused to believe
that it was written by one so young as Pascal. This treatise was never
published, and is now lost. Leibniz saw it in Paris and reported on
a portion of its contents. The precocious youth made vast progress
in all the sciences, but the constant application at so tender an age
greatly impaired his health. Yet he continued working, and at nine-
teen invented his famous machine for performing arithmetical opera-
tions mechanically. This continued strain from overwork resulted in
a permanent indisposition, and he would sometimes say that from the
time he was eighteen, he never passed a day free from pain. At the
age of twenty-four he resolved to lay aside the study of the human
sciences and to consecrate his talents to religion. His Provincial
Letters against the Jesuits are celebrated. But at times he returned
to the favorite study of his youth. Being kept awake one night by
a toothache, some thoughts undesignedly came into his head concern-
ing the roulette or cycloid; one idea followed another; and he thus
discovered properties of this curve even to demonstration. A corre-
spondence between him and Fermat on certain problems was the
beginning of the theory of probability. Pascal's illness increased, and
he died at Paris at the early age of thirty-nine years. By him the
answer to the objection to Cavalieri's Method of Indivisibles was put
in clearer form. Like Roberval, he explained " the sum of right lines "
to mean "the sum of infinitely small rectangles." Pascal greatly ad-
vanced the knowledge of the cycloid. He determined the area of a
section produced by any line parallel to the base; the volume gener-
ated by it revolving around its base or around the axis; and, finally,
the centres of gravity of these volumes, and also of half these volumes
cut by planes of symmetry. Before publishing his results, he sent,
in 1658, to all mathematicians that famous challenge offering prizes
for the first two solutions of these problems. Only Wallis and A. La
Louvere competed for them. The latter was quite unequal to the task ;
the former, being pressed for time, made numerous mistakes: neither
got a prize. Pascal then published his own solutions, which produced
a great sensation among scientific men. Wallis, too, published his,
with the errors corrected. Though not competing for the prizes,

i66 A HISTORY OF MATHEMATICS

Huygens, Wren, and Fermat solved some of the questions. The chief
discoveries of Christopher Wren (1632-1723), the celebrated architect
of St. Paul's Cathedral in London, were the rectification of a cycloida'l
arc and the determination of its centre of gravity. Fermat found the
area generated by an arc of the cycloid. Huygens invented the cy-
cloidal pendulum.

The beginning of the seventeenth century witnessed also a revival of
synthetic geometry. One who treated conies still by ancient methods,
but who succeeded in greatly simplifying many prolix proofs of Apollo-
nius, was Claude Mydorge (1585-1647), in Paris, a friend of Descartes.
But it remained for Girard Desargues (1593-1662) of Lyons, and for
Pascal, to leave the beaten track and cut out fresh paths. They intro-
duced the important method of Perspective. All conies on a cone with
circular base appear circular to an eye at the apex. Hence Desargues
and Pascal conceived the treatment of the conic sections as projections
of circles. Two important and beautiful theorems were given by Des-
argues: The one is on the "involution of the six points," in which a
transversal meets a conic and an inscribed quadrangle; the other is
that, if the vertices of two triangles, situated either in space or in
a plane, lie on three lines meeting in a point, then their sides meet in
three points lying on a line; and conversely. This last theorem has
been employed in recent times by Brianchon, C. Sturm, Gergonne,
and Poncelet. Poncelet made it the basis of his beautiful theory of
homological figures. We owe to Desargues the theory of involution
and of transversals; also the beautiful conception that the two ex-
tremities of a straight line may be considered as meeting at infinity,
and that parallels differ from other pairs of lines only in having their
points of intersection at infinity. He re-invented the epicycloid and
showed its application to the construction of gear teeth, a subject
elaborated more fully later by La Hire. Pascal greatly admired
Desargues' results, saying (in his Essais pour les Coniques), "I wish to
acknowledge that I owe the little that I have discovered on this sub-
ject, to his writings." Pascal's and Desargues' writings contained
some of the fundamental ideas of modern synthetic geometry. In
Pascal's wonderful work on conies, written at the age of sixteen and
now lost, were, given the theorem on the anharmonic ratio, first found
in Pappus, and also that celebrated proposition on the mystic hexagon,
known as "Pascal's theorem," viz. that the opposite sides of a hexa-
gon inscribed in a conic intersect in three points which are collinear.
This theorem formed the keystone to his theory. He himself said
that from this alone he deduced over 400 corollaries, embracing the
conies of Apollonius and many other results. Less gifted than Des-
argues and Pascal was Philippe de la Hire (1640-1718). At first
active as a painter, he afterwards devoted himself to astronomy and
mathematics, and became professor of the College de France in Paris.
He wrote three works on conic sections, published in 1673, l679 arjd

VIETA TO DESCARTES 167

1685. The last of these, the Sectiones Conicae, was best known. La
Hire gave the polar properties of circles, and, by projection, transferred
his polar theory from the circle to the conic sections. In the construc-
tion of maps De la Hire used "globular" projection in which the eye
is not at the pole of the sphere, as in the Ptolemaic stereographic pro-
jection, but on the radius produced through the pole at a distance
r sin 45° outside the sphere. Globular projection has the advantage
that everywhere on the map there is approximately the same degree
of exaggeration of distances. This mode of projection was modified
by his countryman A. Parent. De la Hire wrote on roulettes, on
graphic methods, epicycloids, conchoids, and on magic squares. The
labors of De la Hire, the genius of Desargues and Pascal, uncovered
several of the rich treasures of modern synthetic geometry; but owing
to the absorbing interest taken in the analytical geometry of Descartes
and later in the differential calculus, the subject was almost entirely
neglected until the nineteenth century.

In the theory of numbers no new results of scientific value had been
reached for over 1000 years, extending from the times of Diophantus
and the Hindus until the beginning of the seventeenth century. But
the illustrious period we are now considering produced men who
rescued this science from the realm of mysticism and superstition,
in which it had been so long imprisoned; the properties of numbers
began again to be studied scientifically. Not being in possession of
the Hindu indeterminate analysis, many beautiful results of the
Brahmins had to be re-discovered by the Europeans. Thus a solution
in integers of linear indeterminate equations was re-discovered by the
Frenchman Bachet de Meziriac (1581-1638), who was the earliest
noteworthy European Diophantist. In 1612 he published Ptoblemes
plaisants et delectables qui se font par les nombres, and in 1621 a Greek
edition of Diophantus with notes. An interest in prime numbers is
disclosed in the so-called " Mersenne's numbers," of the form Mp=
2P — i, with p prime. Marin Mersenne asserted in the preface to his
Cogitata Physico-Mathematica, 1644, that the only values of p not
greater than 257 which make Mp a prime are i, 2, 3, 5, 7, 13, 17, 19,
31, 67, 127, and 257. Four mistakes have now been detected in
Mersenne's classification, viz., Mer is composite; Mei, Mas and Mm
are prime. Mm has been found to be composite. Mersenne gave in
1644 also the first eight perfect numbers 6, 28, 496, 8128, 23550336,
8589869056, 137438691328, 2305843008139952128. In Euclid's Ele-
ments, Bk. 9, Prop. 36, is given the formula for perfect numbers
2p— X(2P- 1) , where 21*"1 — i is prime. The above eight perfect numbers
are reproduced by taking p = 2, 3, 5, 7, 13, 17, 19, 31. A ninth perfect
number was found in 1885 by P. Seelhoff, for which />=6i, a tenth
in 1912 by R. E. Powers, for which £ = 89. The father of the modern
theory of numbers is Fermat. He was so uncommunicative in dis-
position, that he generally concealed his methods and made known

168 A HISTORY OF MATHEMATICS

his results only. In some cases later analysts have been greatly
puzzled in the attempt of supplying the proofs. Fermat owned a copy
of Bachet's Diophantus, in which he entered numerous marginal notes.
In 1670 these notes were incorporated in a new edition of Diophanlus,
brought out by his son. Other theorems on numbers, due to Fermat,
were published in his Opera varia (edited by his son) and in Wallis's
Commercium' epistolicum of 1658. Of the following theorems, the
first seven are found in the marginal notes: 1 —

(1) x"+yn=zn is impossible for integral values of x, y, and z, when
n>2.

This famous theorem was appended by Fermat to the problem of
Diophantus II, 8: " To divide a given square number into two squares."
Fermat's marginal note is as follows: "On the other hand it is im-
possible to separate a cube into two cubes, or a biquadrate into two
biquadrates, or generally any power except a square into two powers
with the same exponent. I have discovered a truly marvelous proof
of this, which however the margin is not large enough to contain."
That Fermat actually possessed a proof is doubtful. No general
proof has yet been published. Euler proved the theorem for n=$
and «=4; Dirichlet forw=5 and w=i4, G. Lame forw=7 and Kum-
mer for many other values. Repeatedly was the theorem made the
prize question of learned societies, by the Academy of Sciences in
Paris in 1823 and 1850, by the Academy of Brussels hi 1883. The
recent history of the theorem follows later.

(2) A prune of the form 4«+i is only once the hypothenuse of a
right triangle; its square is twice; its cube is three times, etc. Ex-
ample: 52=32+425 252=i52+202=72+242; i252=752+ioo2=352+
I202=442+ii72.

(3) A prime of the form 4«+i can be expressed once, and only
once, as the sum of two squares. Proved by Euler.

(4) A number composed of two cubes can be resolved into two
other cubes in an infinite multiplicity of ways.

(5) Every number is either a triangular number or the sum of two
or three triangular numbers; either a square or the sum of two, three,
or four squares; either a pentagonal number or the sum of two, three,
four, or five pentagonal numbers; similarly for polygonal numbers
in general. The proof of this and other theorems is promised by
Fermat in a future work which never appeared. This theorem is
also given, with others, in a letter of 1637 (?) addressed to Pater
'Mersenne.

(6) As many numbers as you please may be found, such that the
square of each remains a square on the addition to or subtraction from
it of the sum of all the numbers.

1 For a fuller historical account of Fermat's Diophantine theorems and prob-
lems, see T. L. Heath, Diophantus of Alexandria, 2. Ed., 1910, pp. 267-328. See
also Annals of Mathematics, 2. S., Vol. 18, 1917, pp. 161-187.

VIETA TO DESCARTES 169

(7) x4+y*=z* is impossible.

(8) In a letter of 1640 he gives the celebrated theorem generally
known as "Fermat's theorem," which we state in Gauss's notation:
If p is prime, and a is prime to p, then aP~l=i. (mod p). It was proved
by Leibniz and by Euler.

(9) Fermat died with the belief that he had found a long-sought-for
law of prime numbers in the formula 22"+i=a prime, but he admitted
that he was unable to prove it rigorously. The law is not true, as was
pointed out by Euler in the example 2^+1 =4,294,967, 297 = 6, 700,417
times 641. The American lightning calculator Zerah Colburn, when
a boy, readily found the factors, but was unable to explain the method
by which he made his marvellous mental computation.

(10) An odd prime number can be expressed as the difference of
two squares in one, and only one, way. This theorem, given in the
Relation, was used by Fermat for the decomposition of large numbers
into prime factors.

(n) If the integers a, b, c represent the sides of a right triangle,
then its area cannot be a square number. This was proved by La-
grange.

(12) Fermat's solution of axz+i=yz, where a is integral but not
a square, has come down in only the broadest outline, as given in the
Relation. He proposed the problem to the Frenchman, Bernhard
Frenicle de Bessy, and in 1657 to all living mathematicians. In Eng-
land, Wallis and Lord Brouncker conjointly found a laborious solution,
which was published in 1658, and also in 1668 in Thomas Brancker's
translation of Rahn's Algebra, "altered and augmented" by John
Pell (1610-1685). The first solution was given by the Hindus.
Though Pell had no other connection with the problem, it went by
the name of " Pell's problem." Pell held at one time the mathematical
chair at Amsterdam. In a controversy with Longomontanus who
claimed to have effected the quadrature of the circle, Pell first used
the now familiar trigonometric formula tan2yl = 2tarL4/(i — tan24).

We are not sure that Fermat subjected all his theorems to rigorous
proof. His methods of proof were entirely lost until 1879, when a
document was found buried among the manuscripts of Huygens in
the library of Leyden, entitled Relation des decouvertes en la science des
nombres. It appears from it that he used an inductive method, called
by him la descente infinie ou indefinie. He says that this was particu-
larly applicable in proving the impossibility of certain relations, as,
for instance, Theorem n, given above, but that he succeeded in using
the method also in proving affirmative statements. Thus he proved
Theorem 3 by showing that if we suppose there be a prime 4/1+1
which does not possess this property, then there will be a smaller
prime of the form 4«+i not possessing it; and a third one smaller
than the second, not possessing it; and so on. Thus descending in-
definitely, he arrives at the number 5, which is the smallest prime

i7o A HISTORY OF MATHEMATICS

factor of the form ^n+i. From the above supposition it would follow
that 5 is not the sum of two squares — a conclusion contrary to fact.
Hence the supposition is false, and the theorem is established. Fermat
applied this method of descent with success in a large number of
theorems. By this method L. Euler, A. M. Legendre, P. G. L. Dirich-
let, proved several of his enunciations and many other numerical
propositions.

Fermat was interested in magic squares. These squares, to which
the Chinese and Arabs were so partial, reached the Occident not later
than the fifteenth century. A magic square of 25 cells was found by
M. Curtze in a German manuscript of that time. The artist, Albrecht
Diirer, exhibits one of 16 cells in 1514 in his painting called "Melan-
cholic." The above-named Bernhard Frenicle de Bessy (about 1602-
1675) brought out the fact that the number of magic squares increased
enormously with the order by writing down 880 magic squares of
the order four. Fermat gave a general rule for finding the number of
magic squares of the order n, such that, for n=S, this number was
1,004,144,095,344; but he seems to have recognized the falsity of his
rule. Bachet de Meziriac, in his Problemes plaisants et deleciables,
Lyon, 1612, gave a rule "des terrasses" for writing down magic
squares of odd order. Frenicle de Bessy gave a process for those of
even order. In the seventeenth century magic squares were studied l
by Antoine Arnauld, Jean Prestet, J. Ozanam; in the eighteenth cen-
tury by Poignard, De la Hire, J. Sauveur, L. L. Pajot, J. J. Rallier
des Ourmes, L. Euler and Benjamin Franklin. In a letter B. Franklin
said of his magic square of i62 cells, "I make no question, but you
will readily allow the square of 16 to be the most magically magical
of any magic square ever made by any magician."

A correspondence between B, Pascal and P. Fermat relating to a
certain game of chance was the germ of the theory of probabilities,
of which some anticipations are found in Cardan, Tartaglia, J. Kepler
and Galileo. Chevalier de Mere proposed to B. Pascal the funda-
mental " Problem of Points," 2 to determine the probability which
each player has, at any given stage of the game, of winning the game.
Pascal and Fermat supposed that the players have equal chances of
winning a single point.

The former communicated this problem to Fermat, who studied
it with lively interest and solved it by the theory of combinations, a
theory which was diligently studied both by him and Pascal. The
calculus of probabilities engaged the attention also of C. Huygens.
The most important theorem reached by him was that> if A has p
chances of winning a sum a, and q chances of winning a sum b, then

1 Encyclopedic des sciences math's, T. I, Vol. 3, 1906, p. 66.

*0euvres completes de Blaise Pascal, T. I, Paris, 1866, pp. 220-237. See also I.
Todhunter, History of the Mathematical Theory of Probability, Cambridge and
London, 1865, Chapter II.

VIETA TO DESCARTES 171

he may expect to win the sum — — -. Huygens gave his results in

p+q

a treatise on probability (1657), which was the best account of the
subject until the appearance of Jakob Bernoulli's Ars conjectandi
which contained a reprint of Huygens' treatise. An absurd abuse of
mathematics in connection with the probability of testimony was
made by John Craig who in 1699 concluded that faith in the Gospel
so far as it depended on oral tradition expired about the year 800,
and so far as it depended on written tradition it would expire in the
year 3 i 50.

Connected with the theory of probability were the investigations
on mortality and insurance. The use of tables of mortality does not
seem to have been altogether unknown to the ancients, but the first
name usually mentioned in this connection is Captain John Graunt
who published at London in 1662 his Natural and Political Observa-
tions . . . made upon the bills of mortality, basing his deductions upon
records of deaths which began to be kept in London in 1592 and were
first intended to make known the progress of the plague. Graunt was
careful to publish the actual figures on which he based his conclusions,
comparing himself, when so doing, to a "silly schoolboy, coming to
say his lessons to the world (that peevish and tetchie master), who
brings a bundle of rods, wherewith to be whipped for every mistake
he has committed." : Nothing of marked importance was done after
Graunt until 1693 when Edmund Halley 1 published in the Philo-
sophical Transactions (London) his celebrated memoir on the Degrees
of Mortality of Mankind . . . with an Attempt to ascertain the Price of
Annuities upon Lives, To find the value of an annuity, multiply the
chance that the individual concerned will be alive after n years by
the present value of the annual payment due at the end of n years;
then sum the results thus obtained for all values of n from i to the
extreme possible age for the life of that individual. Halley considers
also annuities on joint lives.

Among the ancients, Archimedes was the only one who attained
clear and correct notions on theoretical statics. He had acquired
firm possession of the idea of pressure, which lies at the root of me-
chanical science. But his ideas slept nearly twenty centuries, until
the time of S. Stevin and Galileo Galilei (1564-1642). Stevin deter-
mined accurately the force necessary to sustain a body on a plane
inclined at any angle to the horizon. He was in possession of a com-
plete doctrine of equilibrium. While Stevin investigated statics,
Galileo pursued principally dynamics. Galileo was the first to abandon
the idea usually attributed to Aristotle that bodies descend more
quickly in proportion as they are heavier; he established the first law
of motion; determined the laws of falling bodies; and, having obtained

1 1. Todhunter, History of Hie Tltcory of Probability, pp. 38, 42.

172 A HISTORY OF MATHEMATICS

a clear notion of acceleration and of the independence of different
motions, was able to prove that projectiles move in parabolic curves.
Up to his time it was believed that a cannon-ball moved forward at
first in a straight line and then suddenly fell vertically to the ground.
Galileo had an understanding of centrifugal forces, and gave a correct
definition of momentum. Though he formulated the- fundamental
principle of statics, known as the parallelogram of forces, yet he did
not fully recognize its scope. The principle of virtual velocities was
partly conceived by Guido Ubaldo (died 1607), and afterwards more
fully by Galileo.

Galileo is the founder of the science of dynamics. Among his con-
temporaries it was chiefly the novelties he detected in the sky that
made him celebrated, but J. Lagrange claims that his astronomical
discoveries required only a telescope and perseverance, while it took
an extraordinary genius to discover laws from phenomena, which we
see constantly and of which the true explanation escaped all earlier
philosophers. Galileo's dialogues on mechanics, the Discorsi e demos-
irazioni matematiche, 1638, touch also the subject of infinite aggregates.
The author displays a keenness of vision and an originality which
was not equalled before the time of Dedekind and Georg Cantor.
Salviati, who in general represents Galileo's own ideas in these dia-
logues, says,1 "infinity and indivisibility are in their very nature in-
comprehensible to us." Simplicio, who is the spokesman of Aris-
totelian scholastic philosophy, remarks that "the infinity of points
in the long line is greater than the infinity of points in the short line."
Then come the remarkable words of Salviati: "This is one of the
difficulties which arise when we attempt, with our finite minds, to
discuss the infinite, assigning to it those properties which we give to
the finite and unlimited; but this I think is wrong, for we cannot
speak of infinite quantities as being the one greater or less than or
equal to another. . . . We can only infer that the totality of all
numbers is infinite, and that the number of squares is infinite, and
that the number of the roots is infinite; neither is the number of squares
less than the totality of all numbers, nor the latter greater than the
former; and finally the attributes 'equal,' 'greater,' and 'less' are
not applicable to infinite, but only to finite quantities. . . . One
line does not contain more or less or just as many points as another,
but . . . each line contains an infinite number." From the time of
Galileo and Descartes to Sir William Hamilton, there was held the
doctrine of the finitude of the human mind and its consequent in-
ability to conceive the infinite. A. De Morgan ridiculed this, saying,
the argument amounts to this, "who drives fat oxen should himself
be fat."

Infinite series, which sprang into prominence at the time of the

1 See Galileo's Dialogues concerning two new Sciences, translated by Henry Crew
and Alfonso de Salvio, New York, 1914, "First Day," pp. 30-32.

DESCARTES TO NEWTON 173

invention of the differential and integral calculus, were used by a few
writers before that time. Pietro Mengoli (1626-1686) of Bologna *
treats them in a book, Nova quadrature arithmetics, of 1650. He
proves the divergence of the harmonic series by dividing its terms
into an infinite number of groups, such that the sum of the terms in
each group is greater than i. The first proof of this was formerly
attributed to Jakob Bernoulli, 1689. Mengoli showed the conver-
gence of the reciprocals of the triangular numbers, a result formerly
supposed to have been first reached by C. Huygens, G. W. Leibniz,
or Jakob Bernoulli. Mengoli reached creditable results on the sum-
mation of infinite series.

Descartes to Newton

Among the earliest thinkers of the seventeenth and eighteenth
centuries, who employed their mental powers toward the destruction
of old ideas and the up-building of new ones, ranks Rene Descartes
(1596-1650). Though he professed orthodoxy in faith all his life,
yet in science he was a profound sceptic. He found that the world's
brightest thinkers had been long exercised in metaphysics, yet they
had discovered nothing certain; nay, had even flatly contradicted each
other. This led him to the gigantic resolution of taking nothing
whatever on authority, but of subjecting everything to scrutinous
examination, according to new methods of inquiry. The certainty of
the conclusions in geometry and arithmetic brought out in his mind
the contrast between the true and false ways of seeking the truth.
He thereupon attempted to apply mathematical reasoning to all
sciences. "Comparing the mysteries of nature with the laws of
mathematics, he dared to hope that the secrets of both could be un-
locked with the same key." Thus he built up a system of philosophy
called Cartesianism.

Great as was Descartes' celebrity as a metaphysician, it may be
fairly questioned whether his claim to be remembered by posterity
as a mathematician is not greater. His philosophy has long since
been superseded by other systems, but the analytical geometry of
Descartes will remain a valuable possession forever. At the age of
twenty-one, Descartes enlisted in the army of Prince Maurice of
Orange. His years of soldiering were years of leisure, in which he had
time to pursue his studies. At that time mathematics was his favorite
science. But in 1625 he ceased to devote himself to pure mathe-
matics. Sir William Hamilton 2 is in error when he states that

1 See G. Enestrom in Bibliolhcca malhemalica, 3. S., Vol. 12, 1911-12, pp. 135-148.

2 Sir William Hamilton, the metaphysician, made a famous attack upon the
study of mathematics as a training of the mind, which appeared in the Edinburgh
Review of 1836. It was shown by A. T. Bledsoe in the Southern Review for July,
1877, that Hamilton misrepresented the sentiments held by Descartes and other
scientists. See also J. S. Mill's Examination of Sir William Hamilton's Philosophy;

174 A HISTORY OF MATHEMATICS

Descartes considered mathematical studies absolutely pernicious as a
means of internal culture. In a letter to Mersenne, Descartes says:
"M. Desargues puts me under obligations on account of the pains
that it has pleased him to have in me, in that he shows that he is
sorry that I do not wish to study more in geometry, but I have re-
solved to quit only abstract geometry, that is to say, the consideration
of questions which serve only to exercise the mind, and this, in order to
study another kind of geometry, which has for its object the explana-
tion of the phenomena of nature. . . . You know that all my physics
is nothing else than geometry." The years between 1629 and 1649
were passed by him in Holland in the study, principally, of physics
and metaphysics. His residence in Holland was during the most
brilliant days of the Dutch state. In 1637 he published his Discours
de la Methode, containing among others an essay of 106 pages on
geometry. His Geometrie is not easy reading. An edition appeared
subsequently with notes by his friend De Beaune, which were intended
to remove the difficulties. The Geometrie of Descartes is of epoch-
making importance; nevertheless we cannot accept Michel Chasles'
statement that this work is proles sine matre creata — a child brought
into being without a mother. In part, Descartes' ideas are found in
Apollonius; the application of algebra to geometry is found in Vieta,
Ghetaldi, Oughtred, and even among the Arabs. Fermat, Descartes'
contemporary, advanced ideas on analytical geometry akin to his
own in a treatise entitled Ad locos pianos et solidos isagoge, which,
however, was not published until 1679 in Fermat's Varia opera. In
Descartes' Geometrie there is no systematic development of the
method of analytics. The method must be constructed from isolated
statements occurring in different parts of the treatise. In the 32
geometric drawings illustrating the text the axes of coordinates are
in no case explicitly set forth. The treatise consists of three "books."
The first deals with "problems which can be constructed by the aid
of the circle and straight line only." The second book is "on the
nature of curved lines." The third book treats of the "construction
of problems solid and more than solid." In the first book it is made
clear, that if a problem has a finite number of solutions, the final
equation obtained will have only one unknown, that if the final
equation has two or more unknowns, the problem "is not wholly
determined." * If the final equation has two unknowns "then since
there is always an infinity of different points which satisfy the de-
mand, it is therefore required to recognize and trace the line on which
all of them must be located" (p. 9). To accomplish this Descartes

C. J. Keyser, Mathematics, 1907, pp. 20-44; F. Cajori in Popular Science Monthly,
1912, pp. 360-372.

1 Descartes' Geometrie, ed. 1886, p. 4. We are here guided by G. Enestrom in
Bibliotheca mathematica, 3. S., Vol. n, pp. 240-243; Vol. 12, pp. 273, 274; Vol. 14,
p. 357, and by H. Wieleitner in Vol. 14, pp. 241-243, 329, 330.

DESCARTES TO NEWTON 175

selects a straight line which he sometimes calls a "diameter" (p. 31)
and associates each of its points with a point sought in such a way that
the latter can be constructed when the* former point is assumed as
known. Thus, on p. 18 he says, " Je choises une ligne droite comme
AB, pour rapporter a ses divers points tous ceux de cette ligne courbe
EC." Here Descartes follows Apollonius who related the points of ,
a conic to the points of a diameter, by distances (ordinates) which
make a constant angle with the diameter and are determined in length ,
by the position of the point on the diameter. This constant angle is I
with Descartes usually a right angle. The new feature introduced by
Descartes was the use of an equation "with more than one unknown, so
that (in case of two unknowns) for any value of one unknown (ab-
scissa), the length of the other (ordinate) could be computed. He
uses the letters x and y for the abscissa and ordinate. He makes it
plain that the x and y may be represented by other distances than the
ones selected by him (p. iq), that, for instance, the angle formed by
x and y need not be a right angle. It is noteworthy that Descartes
and Fermat, and their successors down to the middle of the eighteenth
century, used oblique coordinates more frequently than did later
analysts. It is also noteworthy that Descartes does not formally
introduce a second axis, our y-axis. Such formal introduction is found
in G. Cramer's Introduction d I' analyse des lignes courbcs algebriques,
1750; earlier publications by de Gua, L. Euler, W. Murdoch and others
contain only occasional references to a y-axis. The words "abscissa,"
"ordinate" were not used by Descartes. In the strictly technical
sense of analytics as one of the coordinates of a point, the word
"ordinate" was used by Leibniz in 1694, but in a less restricted sense
such expressions as "ordinatim applicatse" occur much earlier in
F. Commandinus and others. The technical use of "abscissa" is
observed in the eighteenth century by C. Wolf and others. In the
more general sense of a "distance" it was used earlier by B. Cavalieri
in his Indivisibles, by Stefano degli Angeli (1623-1697), a professor
of mathematics in Rome, and by others. Leibniz introduced the word
" coordinate " in 1692. To guard against certain current historical
errors we quote the following from P. Tannery: "One frequently
attributes wrongly to Descartes the introduction of the convention
of reckoning coordinates positively and negatively, in the sense in /
which we start them from the origin. The truth is that in this respect
the Geometric of 1637 contains only certain remarks touching the
interpretation of real or false (positive or negative) roots of equations.
"... If then we examine with care the rules given by Descartes in
his Gbometrie, as well as his application of them, we notice that he
adopts as a principle that an equation of a georrietric locus is not
valid except for the angle of the coordipates (quadrant) in which it
was established, and all his contemporaries do likewise. The extension
of an equation to other angles (quadrants) was freely made in particu-

176 A HISTORY OF MATHEMATICS

lar cases for the interpretation of the negative roots of equations;
but while it served particular conventions (for example for reckoning
distances as positive and negative), it was in reality quite long in
completely establishing itself, and one cannot attribute the honor for
it to any particular geometer."

Descartes' geometry was called "analytical geometry," partly
because, unlike the synthetic geometry of the ancients, it is actually
analytical, in the sense that the word is used in logic; and partly be-
cause the practice had then already arisen, of designating by the term
analysis the calculus with general quantities.

The first important example solved by Descartes in his geometry
is the "problem of Pappus"; viz. "Given several straight lines in a
plane, to find the locus of a point such that the perpendiculars, or more
generally, straight lines at given angles, drawn from the point to the
given lines, shall satisfy the condition that the product of certain of
them shall be in a given ratio to the product of the rest." Of this
celebrated problem, the Greeks solved only the special case when the
number of given lines is four, in which case the locus of the point
turns out to be a conic section. By Descartes it was solved com-
pletely, and it afforded an excellent example of the use which can be
made of his analytical method in the study of loci. Another solution
was given later by Newton in the Principia. Descartes illustrates
his analytical method also by the ovals, now named after him, "cer-
taines ovales que vous verrez etre tres-utiles pour la theorie de la
catoptrique." These curves were studied by Descartes, probably, as
early as 1629; they were intended by him to serve in the construction
of converging lenses, but yielded no results of practical value. In
the nineteenth century they received much attention.1

The power of Descartes' analytical method in geometry has been
vividly set forth recently by L. Boltzmann in the remark that the
formula appears at times cleverer than the man who invented it. Of
all the problems which he solved by his geometry, none gave him as
great pleasure as his mode of constructing tangents. It was published
earlier than the methods of Fermat and Roberval which were noticed
on a preceding page.

Descartes' method consisted in first finding the normal. Through
a given point x, y of the curve he drew a circle which had its centre
at the intersection of the normal and the .r-axis. Then he imposed
the condition that the circle cut the curve in two coincident points
x, y. In 1638 Descartes indicated in a letter that, in place of the
circle, a straight line may be used. This idea is elaborated by Flori-
mond de Beaune in his notes to the 1649 edition of Descartes' Geometric.
In finding the point of intersection of the normal and £-axis, Descartes
used the method of Indeterminate Coefficients, of which he bears the
honor of invention. Indeterminate coefficients were employed by
1 See G. Loria Ebene Curoen (F. Schiitte), I, 1910, p. 174.

DESCARTES TO NEWTON 177

him also in solving biquadratic equations. Descartes' method of
tangents is profound, but operose, and inferior to Fermat's method.
In the third book of his Geometric he points out that if a cubic equation
(with rational coefficients) has a rational root, then it can be factored
and the cubic can be solved geometrically by the use of ruler and
compasses only. He derives the cubic zs=^z—q as the equation upon
which the trisection of an angle depends. He effects a trisection by the
aid of a parabola and circle, but does not consider the reducibility of
the equation. Hence he left the question of the " insolvability " of
the problem untouched. Not till the nineteenth century were con-
clusive proofs advanced of the impossibility of trisecting any angle
and of duplicating a cube, culminating at last in the clear and simple
proofs given by F. Klein in 1895 in his Atisgewahlte Fragen der Elemen-
iar geometric, translated into English in 1897 by W. W. Beman and
D. E. Smith. Descartes proved that every geometric problem giving
rise to a cubic equation can be reduced either to the duplication of a
cube or to the trisection of an angle. This fact had been previously
recognized by Vieta.

The essays of Descartes on dioptrics and geometry were sharply
criticised by Fermat, who wrote objections to the former, and sent
his own treatise on "maxima and minima" to show that there were
omissions in the geometry. Descartes thereupon made an attack on
Fermat's method of tangents. Descartes was in the wrong in this
attack, yet he continued the controversy with obstinacy. In a letter
of 1638, addressed to Mersenne and to be transmitted to Fermat,
Descartes gives x3+y3=axy, now known as the "folium of Descartes,"
as representing a curve to which Fermat's method of tangents would
not apply.1 The curve is accompanied by a figure which shows that
Descartes did not then know the shape of the curve. At that time
the fundamental agreement about algebraic signs of coordinates had
not yet been hit upon; only finite values of variables were used. Hence
the infinite branches of the curve remained unnoticed; some investi-
gators thought there were four leaves instead of only one. C. Huygens
in 1692 gave the correct shape and the asymptote of the curve.

Parabolas of higher order, ytt=pn~1x, are mentioned by Descartes
in a letter of July 13, 1638, in which the centre of mass and the volume
obtained by revolution are considered. Cognate considerations are
due to G. P. Roberval, P. Fermat and B. Cavalieri. Apparently, the
shapes of these curves were not studied, and it remained for C. Mac-
laurin (1748) and G. F. A. 1'Hospital (1770) to remark that they
have wholly different shapes, according to whether n is a positive or
a negative integer.

Descartes had a controversy with G. P. Roberval on the cycloid.
This curve has been called the "Helen of geometers," on account

de Descartes (Tannery et Adam), 1897, I, 400; II, 316. See also G.
Loria, Ebene Curven (F. Schiitte), I, 1910, p. 54.

178 A HISTORY OF MATHEMATICS

of its beautiful properties and the controversies which their discovery
occasioned. Its quadrature by Roberval was generally considered a
brilliant achievement, but Descartes commented on it by saying that
any one moderately well versed in geometry might have done this.
He then sent a short demonstration of his own. On Roberval's in-
timating that he had been assisted by a knowledge of the solution,
Descartes constructed the tangent to the curve, and challenged
Roberval and Fermat to do the same. Fermat accomplished it, but
Roberval never succeeded in solving this problem, which had cost
the genius of Descartes but a moderate degree of attention.

The application of algebra to the doctrine of curved lines reacted
favorably upon algebra. As an abstract science, Descartes improved
it by the introduction of the modern exponential notation. In his
Geometric, 1637, he writes "aa ou a2 pour multiplier a par soimeme;
et as pour le multiplier encore une fois par a, et ainsi a 1'infini."
Thus, while F. Vieta represented A3 by "A cubus" and Stevin x3
by a figure 3 within a small circle, Descartes wrote a3. In his Geometne
he does not use negative and fractional exponents, nor literal ex-
ponents. His notation was the outgrowth and an improvement of
notations employed by writers before him. Nicolas Chuquet's manu-
script work, Le Tiiparty en la science des nombres,1 1484, gives I2X3
and i ox5, and their product I2OA;8, by the symbols I23, io5, I2O8,
respectively. Chuquet goes even further and writes i2x° and jx~]
thus 12°, 7lm; he represents the product of 8x3 and jx~ l by 56 2. J.
Biirgi, Reymer and J. Kepler use Roman numerals for the exponen-
tial symbol. J. Biirgi writes i6#2 thus jr. Thomas Harriot .simply

repeats the letters; he writes in his Artis analytics praxis (1631),
a4— io24a2+6254a, thus: aaaa— io24aa+6254a.

Descartes' exponential notation spread rapidly; about 1660 or
1670 the positive integral exponent had won an undisputed place in
algebraic notation. In 1656 J. Wallis speaks of negative and fractional
"indices," in his Arithmelica infinitorum, but he does not actually

write a~l for ~, or a2/3 for \/03- It was I. Newton who, in his famous

letter to H. Oldenburg, dated June 13, 1676, and containing his an-
nouncement of the binomial theorem, first uses negative and fractional
exponents.

With Descartes a letter represented always only a positive number.
It was Johann Hudde who in 1659 first let a letter stand for negative
as well as positive values.

Descartes also established some theorems on the theory of equa-
tions. Celebrated is his "rule of signs" for determining the number

1 Chuquet's "Le Triparty," Buttdtino Boncompagni, Vol. 13, 1880, p. 740.

DESCARTES TO NEWTON 179

of positive and negative roots. He gives the rule after pointing out
the roots 2, 3, 4, —5 and the corresponding binomial factors of the
equation x4—4x3—igx2+io()x— 120=0.- His exact words are as
follows:

"On connolt aussi de ceci combien il peut y avoir de vraies racines
et combien de fausses en chaque equation: a savoir 51 y en peut avoir
autant de vraies que les signes + et — s'y trouvent de fois etre
changes, et autant de fausses qu'il s'y trouve de fois deux signes -f
ou deux signes — qui s'entre-suivent. Comme en la derniere, a cause
qu'apres +x* il y a — 4.v3, qui est un changement du signe + en — ,
et apres — ig.r2 il y a +106:*:, et apres +io6.r il y a —120, qui sont
encore deux autres changements, ou connoit qu'il y a trois vraies
racines; et une fausse, a cause que les deux signes — de 4x3 et i$x2
s'entre-suivent."

This statement lacks completeness. For this reason he has been
frequently criticized. J. Wallis claimed that Descartes failed to
notice that the rule breaks down in case of imaginary roots, but
Descartes does not say that the equation always has, but that it may
have, so many roots. Did Descartes receive any suggestion of his
rule from earlier writers? He might have received a hint from H.
Cardan, whose remarks on this subject have been summarized by
G. Enestrom * as follows: If in an equation of the second, third or
fourth degree, (i) the last term is negative, then one variation of sign
signifies one and only one positive root, (2) the last term is positive,
then two variations indicate either several positive roots or none.
Cardan does not consider equations having more than two variations.
G. W. Leibniz was the first to erroneously attribute the rule of signs
to T. Harriot. Descartes was charged by J. Wallis with availing
himself, without acknowledgment, of Harriot's theory of equations,
particularly his mode of generating equations; but there seems to be
no good ground for the charge.

In mechanics, Descartes can hardly be said to have advanced be-
yond Galileo. The latter had overthrown the ideas of Aristotle on
this subject, and Descartes simply "threw himself upon the enemy"
that had already been "put to the rout." His statement of the first
and second laws of motion was an improvement in form, but his third
law is false in substance. The motions of bodies in their direct impact
was imperfectly understood by Galileo, erroneously given by Descartes,
and first correctly stated by C. Wren, J. Wallis, and C. Huygens.

One of the most devoted pupils of Descartes was the learned
Princess Elizabeth, daughter of Frederick V. She applied the new
analytical geometry to the solution of the "Apollonian problem."
His second royal follower was Queen Christina, the daughter of Gus-
tavus Adolphus. She urged upon Descartes to come to the Swedish
court. After much hesitation he accepted the invitation in 1649.
1 Bibliotheca mathcmatica, 3rd S., Vol. 7, 1906-7, p. 293.

i8o A HISTORY OF MATHEMATICS

He died at Stockholm one year later. His life had been one long war-
fare against the prejudices of men.

It is most remarkable that the mathematics and philosophy of
Descartes should at first have been appreciated less by his country-
men than by foreigners. The indiscreet temper of Descartes alienated
the great contemporary French mathematicians, Roberval, Fermat,
Pascal. They continued in investigations of their own, and on some
points strongly opposed Descartes. The universities of France were
under strict ecclesiastical control and did nothing to introduce his
mathematics and philosophy. It was in the youthful universities of
Holland that the effect of Cartesian teachings was most immediate
and strongest.

The only prominent Frenchman who immediately followed in the
footsteps of the great master was Florimond de Beaune (1601-1652).
He was one of the first to point out that the properties of a curve
can be deduced from the properties of its tangent. This mode of
inquiry has been called the inverse method of tangents. He contributed
to the theory of equations by considering for the first time the upper
and lower limits of the roots of numerical equations.

In the Netherlands a large number of distinguished mathematicians
were at once struck with admiration for the Cartesian geometry.
Foremost among these are van Schooten, John de Witt, van Heuraet,
Sluze, and Hudde. Franciscus van Schooten (died 1660), professor
of mathematics at Leyden, brought out an edition of Descartes'
geometry, together with the notes thereon by De Beaune. His chief
work is his Mxercitationes Mathematics, 1657, in which he applies the
analytical geometry to the solution of many interesting and difficult
problems. The noble-hearted Johann de Witt (1625-1672), grand-
pensioner of Holland, celebrated as a statesman and for his tragical
end, was an ardent geometrician. He conceived a new and ingenious
way of generating conies, which is essentially the same as that by
projective pencils of rays in modern synthetic geometry. He treated
the subject not synthetically, but with aid of the Cartesian analysis.
Rene Frangois de Sluse (1622-1685) and Johann Hudde (1633-
1704) made some improvements on Descartes' and Fermat's methods
of drawing tangents, and on the theory of maxima and minima. With
Hudde, we find the first use of three variables in analytical geometry.
He is the -author of an ingenious rule for finding equal roots. We
illustrate it by the equation x3—xz—Sx+i2=o. Taking an arith-
metical progression 3, 2, i, o, of which the highest term is equal to
the degree of the equation, and multiplying each term of the equation
respectively by the corresponding term of the progression, we get
3-v3— 2x2— 8.r=o, or $x2— 2x—8=o. This last equation is by one
degree lower than the original one. Find the G.C.D. of the two
equations. This is x— 2; hence 2 is one of the two equal roots. Had
there been no common divisor, then the original equation would not

DESCARTES TO NEWTON 181

have possessed equal roots. Hudde gave a demonstration for this
rule.1

Heinrich van Heuraet must be mentioned as one of the earliest
geometers who occupied themselves with success in the rectification
of curves. He observed in a general way that the two problems of
quadrature and of rectification are really identical, and that the one
can be reduced to the other. Thus he carried the rectification of the
hyperbola back to the quadrature of the hyperbola. The curve which
John Wallis named the "semi-cubical parabola," y3=ax2, was the
first curve to be rectified absolutely. This appears to have been
accomplished independently by P. Fermat in France, Van Heuraet
in Holland and by William Neil (1637-1670) in England. According
to J. Wallis the priority belongs to Neil. Soon after, the cycloid was
rectified by C. Wren and Fermat.

A mathematician of no mean ability was Gregory St. Vincent
(1584-1667), a Belgian, who studied under C. Clavius in Rome and
was two years professor at Prague, where, during war time, his manu-
script volume on geometry and statics was lost in a fire. Other papers
of his were saved but carried about for ten years before they came
again into his possession, at his home in Ghent. They became the
groundwork of his great book, the Opus geometricum quadratures
circuit et sectionum coni, Antwerp, 1647. It consists of 1225 folio
pages, divided into ten books. St. Vincent proposes four methods for
squaring the circle, but does not actually carry them out. The work
was attacked by R. Descartes, M. Mersenne and G. P. Roberval,
and defended by the Jesuit Alfons Anton de Sarasa. and others.
Though erroneous on the possibility of squaring the cinile, the Opus
contains solid achievements, which were the more remarkable, because
at that time only four of the seven books of the conies of Apollonius
of Perga were known in the Occident. St. Vincent deals with conies,
surfaces and solids from a new point of view, employing infinitesimals
in a way perhaps less objectionable than in B. Cavalieri's book. St.
Vincent was probably the first to use the word exhaurire in a geo-
metrical sense. From this word arose the name of "method of ex-
haustion," as applied to the method of Euclid and Archimedes. St.
Vincent used a method of transformation of one conic into another,
called per subtendas (by chords), which contains germs of analytic
g-jometry. He created another special method which he called Ductus
plani in planum and used in the study of solids.2 Unlike Archimedes
who kept on dividing distances, only until a certain degree of small-
ness was reached, St. Vincent permitted the subdivisions to continue

1 Heinrich Suter, Geschkhtc der M athcmalischen Wissenschaften Zurich, 2. Theil,
1875, p. 25.

2 See M. Marie, Hlstoire dcs sciences math., Vol. 3, 1884, pp. 186-193; Karl Bopp,
Kcgdschnitte des Gregorius a St. Vincento in Abhandl. z. Gcsch. d. math. Wissensch.,
XX Heft, 1907, pp. 83-314.

i82 A HISTORY OF MATHEMATICS

ad infinitum and obtained a geometric series that was infinite. How-
ever, infinite series had been obtained before him by Alvarus Thomas,
a native of Lisbon, in a work, Liber de triplici motu, iscx),1 and possibly
by others. But St. Vincent was the first to apply geometric series to
the "Achilles" and to look upon the paradox as a question in the
summation of an infinite series. Moreover, St. Vincent was the first
to state the exact time and place of overtaking the tortoise. He
spoke of the limit as an obstacle against further advance, similar to
a rigid wall. Apparently he was not troubled by the fact that in his
theory, the variable does not reach its limit. His exposition of the
"Achilles" was favorably received by G. W. Leibniz and by writers
over a century afterward. The fullest account and discussion of
Zeno's arguments on motion that was published before the nineteenth
century was given by the noted French skeptical philosopher, Pierre
Bayle, in an article "Zenon d'Elee" in his Dictionnaire historique el
critique, 1696.*

The prince of philosophers in Holland, and one of the greatest
scientists of the seventeenth century, was Christian Huygens (1629-
1695) , a native of The Hague. Eminent as a physicist and astronomer,
as well as mathematician, he was a worthy predecessor of Sir Isaac
Newton. He studied at Leyden under Frans Van Schooten. The
perusal of some of his earliest theorems led R. Descartes to predict
his future greatness. In 1651 Huygens wrote a treatise in which he
pointed out the fallacies of Gregory St. Vincent on the subject of
quadratures. He himself gave a remarkably close and convenient
approximation to the length of a circular arc. In 1660 and 1663 he
went to Paris and to London. In 1666 he was appointed by Louis
XIV member of the French Academy of Sciences. He was induced
to remain in Paris from that time until 1681, when he returned to his
native city, partly for consideration of his health and partly on ac-
count of the revocation of the Edict of Nantes.

The majority of his profound discoveries were made with aid of the
ancient geometry, though at times he used the geometry of R. Des-
cartes or of B. Cavalieri and P. Fermat. Thus, like his illustrious
friend, Sir Isaac Newton, he always showed partiality for the Greek
geometry. Newton and Huygens were kindred minds, and had the
greatest admiration for each other. Newton always speaks of him
as the "Summus Hugenius."

To the two curves (cubical parabola and cycloid) previously recti-
fied he added a third, — the cissoid. A French physician, Claudius
Perrault, proposed the question, to determine the path in a fixed plane
of a heavy point attached to one end of a taut string whose other end
moves along a straight line in that plane. Huygens and G. W. Leibniz
studied this problem in 1693, generalized it, and thus worked out the.

1 H. Wieleitner, in Bibliotheca mathematica, 3. F., Bd. 1914, 14, p. 152.

2 See F. Cajori in Am. Math. Monthly, Vol. 22, 1915, pp. 109-112.

DESCARTES TO NEWTON 183

geometry of the "tractrix." * Huygens solved the problem of the
catenary, determined the surface of the parabolic and hyperbolic
conoid, and discovered the properties of the logarithmic curve and
the solids generated by it. Huygens' De horologio oscillatorio (Paris,
1673) is a work that ranks second only to the Principia of Newton
and constitutes historically a necessary introduction to it. The book
opens with a description of pendulum clocks, of which Huygens is the
inventor. Then follows a treatment of accelerated motion of bodies
falling free, or sliding on inclined planes, or on given curves, — cul-
minating in the brilliant discovery that the cycloid is the tautochronous
curve. To the theory of curves he added the important theory of
"evolutes." After explaining that the tangent of the evolute is
normal to the involute, he applied the theory to the cycloid, and
showed by simple reasoning that the evolute of this curve is an equal
cycloid. Then comes the complete general discussion of the centre
of oscillation. This subject had been proposed for investigation by
M. Mersenne and discussed by R. Descartes and G. P. RobervaL
In Huygens' assumption that the common centre of gravity of a
group of bodies, oscillating about a horizontal axis, rises to its original
height, but no higher, is expressed for the first time one of the most
beautiful principles of dynamics, afterwards called the principle of
the conservati6n of vis viva. The thirteen theorems at the close of
the work relate to the theory of centrifugal force in circular motion.
This theory aided Newton in discovering the law of gravitation.2

Huygens wrote the first formal treatise on probability. He pro-
posed the wave-theory of light and with great skill applied geometry
to its development. This theory was long neglected, but was revived
and elaborated by Thomas Young and A. J. Fresnel a century later.
Huygens and his brother improved the telescope by devising a better
way of grinding and polishing lenses. With more efficient instru-
ments he determined the nature of Saturn's appendage and solved
other astronomical questions. Huygens' Opuscula posthuma appeared
in 1703.

The theory of combinations, the primitive notions of which go
back to ancient Greece, received the attention of William Buckley
of King's College, Cambridge (died 1550), and especially of Blaise
Pascal who treats of it in his Arithmetical Triangle. Before Pascal,
this Triangle had been constructed by N. Tartaglia and M. Stifel.
Fermat applied combinations to the study of probability. The earliest
mathematical work of Leibniz was his De arte combinatoria. The
subject was treated by John Wallis in his Algebra.

John, Wallis (1616-1703) was one of the most original mathemati-
cians of his day. He was educated for the Church at Cambridge and en-

1 G. Loria, Ebene Curven (F. Schiitte) It, IQII, p. 188.

2 K. Diilirintf, Kritische Gescliichle der Allgemeinen Principien der Mechanik.

1887, p. 135.

184 A HISTORY OF MATHEMATICS

tered Holy Orders. But his genius was employed chiefly in the study of
mathematics. In 1649 he was appointed Savilian professor of geometry
at Oxford. He was one of the original members of the Royal Society,
which was founded in 1663. He ranks as one of the world's greatest de-
cipherers of cryptic writing.1 Wallis thoroughly grasped the mathemat-
ical methods both of B. Cavalieri and R. Descartes. His Conic Sections
is the earliest work in which these curves are no longer considered as
sections of a cone, but as curves of the second degree, and are treated
analytically by the Cartesian method of co-ordinates. In this work
Wallis speaks of Descartes in the highest terms, but in his Algebra
(1685. Latin edition 1693), he, without good reason, accuses Descartes
of plagiarizing from T. Harriot. It is interesting to observe that, in
his Algebra, Wallis discusses the possibility of a fourth dimension.
Whereas nature, says Wallis, "doth not admit of more than three
(local) dimensions ... it may justly seem very improper to talk of
a solid . . . drawn into a fourth, fifth, sixth, or further dimension. . . .
Nor can our fansie imagine how there should be a fourth local dimen-
sion beyond these three." : The first to busy himself with the number
of dimensions of space was Ptolemy. Wallis felt the need of a method
of representing imaginaries graphically, but he failed to discover a
general and consistent representation.3 He published Nasir-Eddin's
proof of the parallel postulate and, abandoning the idea of equi-
distance that had been employed without success by F. Commandino,
C. S. Clavio, P. A. Cataldi and G. A. Borelli, gave a proof of his own
based on the axiom that, to every figure there exists a similar figure
of arbitrary magnitude.4 The existence of similar triangles was as-
sumed 1000 years before Wallis by Aganis, who was probably a
teacher of Simplicius. We have already mentioned elsewhere Wallis's
solution of the prize questions on the cycloid, which were proposed by
Pascal.

The Arithmetica infinitorum, published in 1655, is his greatest work.
By the application of analysis to the Method of Indivisibles, he greatly
increased the power of this instrument for effecting quadratures. He
created the arithmetical conception of a limit by considering the
successive values of a fraction, formed in the study of certain ratios;
these fractional values steadily approach a limiting value, so that
the difference becomes less than any assignable one and vanishes
when the process is carried to infinity. He advanced beyond J. Kepler
by making more extended use of the "law of continuity" and placing

1 D. E. Smith in Bull. Am. Math. Soc., Vol. 24, 1917, p. 82.

2 G. Enestrom in Bibliotheca mathematica, 3. S., Vol. 12, 1911-12, p. 88.

3 See Wallis' Algebra, 1685, pp. 264-273; see also Enestrom in Bibliotheca mathe-
matica, 3. S., Vol. 7, pp. 263-269.

4 R. Bonola, op. tit., pp. 12-17. See also F. Engel u. P. Stackel, Theorie der
Parallellinien von Euclid bis auf Gauss, Leipzig, 1895, pp. 21-36. This treatise
gives translations into German of Saccheri, also the essays of Lambert and Taurinus,
and letters of Gauss.

DESCARTES TO NEWTON 185

full reliance in it. By this law he was led to regard the denominators
of fractions as powers with negative exponents. Thus, the descending
geometrical progression x3, x2, xl, x°, if continued, gives x~1, x~2, x~*,

etc. ; which is the same thing as -, —$, -3. The exponents of this geo-

OC 00 OC

metric series are in continued arithmetical progression, 3, 2, i, o,

— i, — 2, —3. However, Wallis does not actually use here the no-
tation x~l, x~2, etc.; he merely speaks of negative exponents. He
also used fractional exponents, which, like the negative, had been
invented long before, but had failed to be generally introduced. The
symbol oo for infinity is due to him. Wallis introduces the name,
" hypergeometric series" for a series different from a, ab, ab2, . . . ;
he did not look upon this new series as a power-series nor as a function
of x.

B. Cavalieri and the French geometers had ascertained the formula
for squaring the parabola of any degree, y=xm, m being a positive
integer. By the summation of the powers of the terms of infinite
arithmetical series, it was found that the curve y=xm is to the area
of the parallelogram having the same base and altitude as i is to
m+i. Aided by the law of continuity, Wallis arrived at the result
that this formula holds true not only when m is positive and integral,
but also when it is fractional or negative. Thus, in the parabola
y=\/~px, m=\; hence the area of the parabolic segment is to that
of the circumscribed rectangle as i : i^, or as 2:3. Again, suppose
that in y=xm, m=—\; then the curve is a kind of hyperbola referred
to its asymptotes, and the hyperbolic space between the curve and
its asymptotes is to the corresponding parallelogram as i : |. If m=

— i, as in the common equilateral hyperbola y=x~1 or xy=i, then
this ratio is i : — i+i, or i : o, showing that its asymptotic space
is infinite. But in the case when m is greater than unity and negative,
Wallis was unable to interpret correctly his results. For example,
if m=— 3, then the ratio becomes i : — 2, or as unity to a negative
number. What is the meaning of this? Wallis reasoned thus: If the
denominator is only zero, then the area is already infinite; but if it is
less than zero, then the area must be more than infinite. It was
pointed out later by P. Varignon, that this space, supposed to exceed
infinity, is really finite, but taken negatively; that is, measured in a
contrary direction.1 The method of Wallis was easily extended to

m p

cases such as y=ax^+bxq by performing the quadrature for each term
separately, and then adding the results.

The manner in which Wallis studied the quadrature of the circle
and arrived at his expression for the value of TT is extraordinary. He
found that the areas comprised between the axes, the ordinate cor-

1 J. F. Montucla, Hisloire des math6maliqitcs , Paris, Tome 2, An VII, p. 350.

1 86 A HISTORY OF MATHEMATICS

responding to x, and the curves represented by the equations y=
(i — x2)0, y = (i— a;2)1, y=(i— x2)2, y=(i — a;2)3, etc., are expressed in
functions of the circumscribed rectangles having x and y for their
sides, by the quantities forming the series
x,

,
x—$x3+%xb— %x7, etc.

When x=i, these values become respectively i, |, -£s, 14^-, etc. Now
since the ordinate of the circle is y=(i — rc2)^, the exponent of which is
\ or the mean value between o and i, the question of this quadrature
reduced itself to this: If o, i, 2, 3, etc., operated upon by a certain law,
give i, |, i85, ^5' what will \ give, when operated upon by the same
law? He attempted to solve this by interpolation, a method first
brought into prominence by him, and arrived by a highly complicated
and difficult analysis at the following very remarkable expression:

7T_2.2.4.4.6.6.8.8. . .

2~i-3-3-5-5-7-7-9- ••'

He did not succeed in making the interpolation itself, because he
did not employ literal or general exponents, and could not conceive a
series with more than one term and less than two, which it seemed
to him the interpolated series must have. The consideration of this
difficulty led I. Newton to the discovery of the Binomial Theorem.
This is the best place to speak of that discovery. Newton virtually
assumed that the same conditions which underlie the general ex-
pressions for the areas given above must also hold for the expression
to be interpolated. In the first place, he observed that in each ex-
pression the first term is x, that x increases in odd powers, that the
signs alternate + and — , and that the second terms §#3, f.v3, f.r3, fx3,
are in arithmetical progression. Hence the first two terms of the

1^3

interpolated series must be x— — . He next considered that the de-

nominators i, 3, 5, 7, etc., are in arithmetical progression, and that
the coefficients in the numerators in each expression are the digits
of some power of the number n; namely, for the first expression, 11°
or i; for the second, n1 or i, i; for the third, n2 or i, 2, i; for the
fourth, ii3 or i, 3, 3, i; etc. He then discovered that, having given
the second digit (call it m), the remaining digits can be found by con-

m— o m—i m — 2

tmual multiplication of the terms of the series - . - . - .

i 2 3

*»— 3 „, ., m—i . m — 2 .

- - . etc. Thus, if w=4, then 4. - gives 6; 6 . - gives 4;

DESCARTES TO NEWTON 187

tn — 3

4 . - - gives i. Applying this rule to the required series, since the
4

ir3

second term is -1— , we have m=%, and then get for the succeeding
o

coefficients in the numerators respectively — f, — jV, — Tfj, etc.;

1^3 1^5 1-7

hence the required area for the circular segment is x — —

O J /

etc. Thus he found the interpolated expression to be an infinite series,
instead of one having more than one term and less than two, as Wallis
believed it must be. This interpolation suggested to Newton a mode
of expanding (i— .v2)*, or, more generally, (i — x2)m, into a series. He
observed that he had only to omit from the expression just found the
denominators i, 3, 5, 7, etc., and to lower each power of x by unity,
and he had the desired expression. In a letter to H. Oldenburg
(June 13, 1676), Newton states the theorem as follows: The extraction
of roots is much shortened by the theorem

where A means the first term, /*», B the second term, C the third
term, etc. He verified it by actual multiplication, but gave no regular
proof of it. He gave it for any exponent whatever, but made no dis-
tinction between the case when the exponent is positive and integral,
and the others.

It should here be mentioned that very rude beginnings of the bi-
nomial theorem are found very early. The Hindus and Arabs used
the expansions of (a+b)2 and (a+b)s for extracting roots; Vieta knew
the expansion of (a+b)4; but these were the results of simple multi-
plication without the discovery of any law. The binomial coefficients
for positive whole exponents were known to some Arabic and Euro-
pean mathematicians. B. Pascal derived the coefficients from the
method of what is called the "arithmetical triangle." Lucas de
Burgo, M. Stifel, S. Stevinus, H. Briggs, and others, all possessed
something from which one would think the binomial theorem could
have been gotten with a little attention, "if we did not know that
such simple relations were difficult to discover."

Though Wallis had obtained an entirely new expression for TT, he
was not satisfied with it; for instead of a finite number of terms yield-
ing an absolute value, it contained an infinite number, approaching
nearer and nearer to that value. He therefore induced his friend, Lord
Brouncker, the first president of the Royal Society, to investigate
this subject. Of course Lord Brouncker did not find what they were
after, but he obtained the following beautiful equality : —

i88

2+ etc.

Continued fractions, both ascending and descending, appear to have
been known already to the Greeks and Hindus, though not in our
present notation. Brouncker's expression gave birth to the theory of
continued fractions.

Wallis' method of quadratures was diligently studied by his dis-
ciples. Lord Brouncker obtained the first infinite series for the area
of the equilateral hyperbola xy=i between one of its asymptotes and

the ordinates for x=i and x=2\ viz. the area h— H — 7+ • - • The

1.2 3.4 5-6

Logarithmotechnia (London, 1668) of Nicolaus M creator is often said

aa a3
to contain the series log (i+a) = a 1 ... In reality it con-

" O

tains the numerical values of the first few terms of that series, tak-
ing a=.i, also a =.2 1. He adhered to the mode of exposition which
favored the concrete special case to the general formula. Wallis was
the first to state Mercator's logarithmic series in general symbols.
Mercator deduced his results from the grand property of the hyper-
bola deduced by Gregory St. Vincent in Book VII of his Opus geo-
metricum, Antwerp, 1647: If parallels to one asymptote are drawn
between the hyperbola and the other asymptote, so that the successive
areas of the mixtilinear quadrilaterals thus formed are equal, then
the lengths of the parallels form a geometric progression. Apparently
the first writer to state this theorem in the language of logarithms
was the Belgian Jesuit Alfons Anton de Sarasa, who defended Gregory
St. Vincent against attacks made by Mersenne. Mercator showed
how the construction of logarithmic tables could be reduced to the
quadrature of hyperbolic spaces. Following up some suggestions of
Wallis, William Neil succeeded in rectifying the cubical parabola, and
C. Wren in rectifying any cycloidal arc. Gregory St. Vincent, in
Part X of his Opus describes the construction of certain quartic curves,
often called virtual parabolas of St. Vincent, one of which has a shape
much like a lemniscate and in Cartesian co-ordinates is d2(y2— x2) =y4.
Curves of this type are mentioned in the correspondence of C. Huy-
gens with R. de Sluse, and with G. W. Leibniz.

A prominent English mathematician and contemporary of Wallis
was Isaac Barrow (1630-1677). He was professor of mathematics
in London, and then in Cambridge, but in 1669 he resigned his chair

DESCARTES TO NEWTON

to his illustrious pupil, Isaac Newton, and renounced the study of
mathematics for that of divinity. As a mathematician, he is most
celebrated for his method of tangents. He simplified the method of
P. Fermat by introducing two infinitesimals instead of one, and ap-
proximated to the course of reasoning afterwards followed by Newton
in his doctrine on Ultimate Ratios. The following books are Barrow's:
Lectiones geometries (1670), Lectiones mathematics (1683-1685).

He considered the infinitesimal right triangle ABB' having for its
sides the difference between two successive ordinates, the distance
between them, and the portion of the curve intercepted by them.
This triangle is similar to BPT, formed by the ordinate, the tangent,
and the sub-tangent. Hence, if we
know the ratio of B'A to BA, then
we know the ratio of the ordinate
and the sub-tangent, and the tangent
can be constructed at once. For any
curve, say yz=px, the ratio of B'A to
BA is determined from its equation
as follows: If x receives an infinitesi-
mal increment PP' = e, then yfeceives
an increment B'A'=a, and the equation for the ordinate B'P' becomes
y2+2ay+a2=px+pe. Since y2=px, we get 2ay+a2=pe; neglecting
higher powers of the infinitesimals, we have 2'ay=pe, which gives

a:e=p: 2y=p
But a: e=the ordinate: the sub-tangent; hence

p: 2\^px= \/px\ sub-tangent,

giving 2X for the value of the sub-tangent. This method differs from
that of the differential calculus chiefly in notation. In fact, a recent
investigator asserts, "Isaac Barrow was the first inventor of the in-
finitesimal calculus." 1

Of the integrations that were performed before the Integral Calculus
was invented, one of the most difficult grew out of a practical problem
of navigation in connection with Gerardus M creator's map. In 1599
Edward Wright published a table of latitudes giving numbers express-
ing the length of an arc of the nautical meridian. The table was com-
puted by the continued addition of the secants of i", 2", 3", etc. In

/e
sec 6 d 0 = r log tan (90° — 0)/2.
0

It was Henry Bond who noticed by inspection about 1645 tnat
Wright's table was a table of logarithmic tangents. Actual demon-
strations of this, thereby really establishing the above definite integral,
were given by James Gregory in 1668, Isaac Barrow in 1670, John

1 J. M. Child, The Geometrical Lectures of Isaac Barrow, Chicago and London,
1916, preface,

1 90 A HISTORY OF MATHEMATICS

Wallis in 1685, and Edmund Halley in 1698. l James Gregory and

/e
tan 6 d 0 = log sec 6; B. Cavalieri in
0

/a
xn dx. Similar results were obtained
o

by E. Torricelli, Gregory St. Vincent, P. Fermat, G. P. Roberval and
B. Pascal.2

Newton to Euler

It has been seen that in France prodigious scientific progress was
made during the beginning and middle of the seventeenth century.
The toleration which marked the reign of Henry IV and Louis XIII
was accompanied by intense intellectual activity. Extraordinary con-
fidence came to be placed in the power of the human mind. The bold
intellectual conquests of R. Descartes, P. Fermat, and B. Pascal en-
riched mathematics with imperishable treasures. During the early
part of the reign of Louis XIV we behold the sunset splendor of this
glorious period. Then followed a night of mental effeminacy. This
lack of great scientific thinkers during the reign of Louis XIV may be
due to the simple fact that no great minds were born; but, according
to Buckle, it was due to the paternalism, to the spirit of dependence
and subordination, and to the lack of toleration, which marked the
policy of Louis XIV.

In the absence of great French thinkers, Louis XIV surrounded
himself by eminent foreigners. O. Romer from Denmark, C. Huygens
from Holland, Dominic Cassini from Italy, were the mathematicians
and astronomers adorning his court. They were in possession of a
brilliant reputation before going to Paris. Simply because they per-
formed scientific work in Paris, that work belongs no more to France
than the discoveries of R. Descartes belong to Holland, or those of
J. Lagrange to Germany, or those of L. Euler and J. V. Poncelet to
Russia. We must look to other countries than France for the great
scientific men of the latter part of the seventeenth century.

About the time when Louis XIV assumed the direction of the
French government Charles II became king of England. At this
time England was extending her commerce and navigation, and ad-
vancing considerably in material prosperity. A strong intellectual
movement took place, which was unwittingly supported by the king.
The age of poetry was soon followed by an age of science and philos-
ophy. ' In two successive centuries England produced Shakespeare
and I. Newton!

1 See F. Cajori in Bibliotheca malhematica, 3. S., Vol. 14, 1915, pp. 312-319.
2H. G. Zeuthen, Geschichte der Math, (deutsch v. R. Meyer), Leipzig, 1903,
pp. 256 ff.

NEWTON TO EULER 191

Germany still continued in a state of national degradation. The
Thirty Years' War had dismembered the empire and brutalized the
people. Yet this darkest period of Germany's history produced G. W.
Leibniz, one of the greatest geniuses of modern times.

There are certain focal points in history toward which the lines of
past progress converge, and from which radiate the advances of the
future. Such was the age of Newton and Leibniz in the history of
mathematics. During fifty years preceding this era several of the
brightest and acutest mathematicians bent the force of their genius
in a direction which finally led to the discovery of the infinitesimal
calculus by Newton and Leibniz. B. Cavalieri, G. P. Roberval, P.
Fermat, R. Descartes, J. Wallis, and others had each contributed to
the new geometry. So great was the advance made, and so near was
their approach toward the invention of the infinitesimal analysis, that
both J. Lagrange and P. S. Laplace pronounced their countryman,
P. Fermat, to be the first inventor of it. The differential calculus,
therefore, was not so much an individual discovery as the grand result
of a succession of discoveries by different minds. Indeed, no great
discovery ever flashed upon the mind at once, and though those of
Newton will influence mankind to the end of the world, yet it must be
admitted that Pope's lines are only a "poetic fancy": —

"Nature and Nature's laws lay hid in night;
God said, 'Let Newton be,' and all was light."

Isaac Newton (1642-1727) was born at Woolsthorpe, in Lincoln-
shire, the same year in which Galileo died. At his birth he was so
small and weak that his life was despaired of. His mother sent him
at an early age to a village school, and in his twelfth year to the public
school at Grantham. At first he seems to have been very inattentive
to his studies and very low in the school; but when, one day, the little
Isaac received a severe kick upon his stomach from a boy who was
above him, he labored hard till he ranked higher in school than his
antagonist. From that time he continued to rise until he was the
head boy.1 At Grantham, Isaac showed a decided taste for mechan-
ical inventions. He constructed a water-clock, a wind-mill, a carriage
moved by the person who sat in it, and other toys. When he had at-
tained his fifteenth year his mother took him home to assist her in
the management of the farm, but his great dislike for farmwork and
his irresistible passion for study, induced her to send him back to
Grantham, where he remained till his eighteenth year, when he en-
tered Trinity College, Cambridge (1660). Cambridge was the real
birthplace of Newton's genius. Some idea of his strong intuitive
powers may be drawn from the fact that he regarded the theorems of
ancient geometry as self-evident truths, and that, without any pre-
liminary study, he made himself master of Descartes' Geometry. He
1 D. Brewster, The Memoirs of Newton, Edinburgh, Vol. I, 1855, p. 8.

192 A HISTORY OF MATHEMATICS

afterwards regarded this neglect of elementary geometry a mistake
in his mathematical studies, and he expressed to Dr. H. Pemberton
his regret that "he had applied himself to the works of Descartes and
other algebraic writers before he had considered the Elements of Euclid
with that attention which so excellent a writer deserves." Besides R.
Descartes' Geometry, he studied W. Oughtred's Clams, J. Kepler's
Optics, the works of F. Vieta, van Schooten's Miscellanies, I. Barrow's
Lectures, and the works of J. Wallis. He was particularly delighted
with Wallis' Arithmetic of Infinites, a treatise fraught with rich and
varied suggestions. Newton had the good fortune of having for a
teacher and fast friend the celebrated Dr. Barrow, who had been
elected professor of Greek in 1660, and was made Lucasian professor
of mathematics in 1663. The mathematics of Barrow and of Wallis
were the starting-points from which Newton, with a higher power
than his masters', moved onward into wider fields. \Vallis had ef-
fected the quadrature of curves whose ordinates are expressed by any
integral and positive power of (i— x~). We have seen how Wallis
attempted but failed to interpolate between the areas thus calculated,
the areas of other curves, such as that of the circle; how Newton at-
tacked the problem, effected the interpolation, and discovered the
Binomial Theorem, which afforded a much easier and direct access to
the quadrature of curves than did the method of interpolation; for
even though the binomial expression for the ordinate be raised to a
fractional or negative power, the binomial could at once be expanded
into a series, and the quadrature of each separate term of that series
could be effected by the method of Wallis. Newton introduced the
system of literal indices.

Newton's study of quadratures soon led him to another and most
profound invention. He himself says that in 1665 and 1666 he con-
ceived the method of fluxions and applied them to. the quadrature of
curves. Newton did not communicate the invention to any of his
Trt£hds till 1669, when he placed in the hands of Barrow a tract, en-
titled De A nalysi per ALquationes Numero Terminorum Infinitas, which
was sent by Barrow to John Collins, who greatly admired it. In
this treatise the principle of fluxions, .though distinctly pointed out,
is only partially developed and explained. Supposing the abscissa to
increase uniformly in proportion to the time, he looked upon the area
of a curve as a nascent quantity increasing by continued fluxion in
the proportion of the length of the ordinate. The expression which
was obtained for the fluxion he expanded into a finite or infinite series
of monomial terms, to which Wallis' rule was applicable. Barrow
urged Newton to publish this treatise; "but the modesty of the author,
of which the excess, if not culpable, was certainly in the present in-
stance very unfortunate, prevented his compliance." [ Had this tract

'•John Playfair, "Progress of the Mathematical and Physical Sciences" in En-
i •:chp(cdia Britannica, 7th Edition.

NEWTON TO EULER 193

been published then, instead of forty-two years later, there probably
would have been no occasion for that long and deplorable controversy
between Newton and Leibniz.

For a long time Newton's method remained unknown, except to his
friends and their correspondents. In a letter to Collins, dated De-
cember loth, 1672, Newton states the fact of his invention with one
example, and then says: "This is one particular, or rather corollary,
of a general method, which extends itself, without any troublesome
calculation, not only to the drawing of tangents to any curve lines,
whether geometrical or mechanical, or anyhow respecting right lines
or other curves, but also to the resolving other abstruser kinds of
problems about the crookedness, areas, lengths, centres of gravity of
curves, etc. ; nor is it (as Hudde's method of Maximis and Minimis)
limited to equations which are free from surd quantities. This method
I have interwoven with that other of working in equations, by reducing
them to infinite series."

These last words relate to a treatise he composed in the year 1671,
entitled Method of Fluxions, in which he aimed to represent his method
as an independent calculus and as a complete system. This tract was
intended as an introduction to an edition of Kinckhuysen's Algebra,
which he had undertaken to publish. "But the fear of being involved
in disputes about this new discovery, or perhaps the wish to render
it more complete, or to have the sole advantage of employing it in his
physical researches, induced him to abandon this design."

Excepting two papers on optics, all of his works appear to have
been published only after the most pressing solicitations of his friends
and against his own wishes. His researches on light were severely
criticised, and he wrote in 1675: "I was so persecuted with discussions
arising out of my theory of light that I blamed my own imprudence
for parting with so substantial a blessing as my quiet to run after a
shadow."

The Method of Fluxions, translated by J. Colson from Newton's
Latin, was first published in 1736, or sixty-five years after it was
written. In it he explains first the expansion into series of fractional
and irrational quantities, — a subject which, in his first years of study,
received the most careful attention. He then proceeds to the solution
of the two following mechanical problems, which constitute the pillars,
so to speak, of the abstract calculus: —

"I. The length ofthe space described being rpntiiyially^fi'. e. at
all times) given; to "find the velocity of the motion at any time pro-
posed.

"II. The velocity of the motion being continually given; to find
the length of the space described at any time proposed."

Preparatory to the solution, Newton says: "Thus, in the equation
y=x2, if y represents the length of the space at any time described,
1 1). lirewster, op. cit., Vol. 2, 1855, p. 15.

194 A HISTORY OF MATHEMATICS

which (time) another space x, by increasing with an uniform celerity
x, measures and exhibits as described: then zxx will represent the
celerity by which the space y, at the same moment of time, proceeds
to be described; and contrarywise."

"But whereas we need not consider the time here, any farther than
it is expounded and measured by an equable local motion; and be-
sides, whereas only quantities of the same kind can be compared to-
gether, and also their velocities of increase and decrease; therefore, in
what follows I shall have no regard to time formally considered, but
I shall suppose some one of the quantities proposed, being of the same
kind, to be increased by an equable fluxion, to which the rest may be
referred, as it were to time; and, therefore, by way of analogy, it
may not improperly receive the name of time." In this statement of
Newton there is contained his answer to the objection which has been
raised against his method, that it introduces into analysis the foreign
idea of motion. A quantity thus increasing by uniform fluxion, is
what we now call an independent variable.

Newton continues: "Now those quantities which I consider as
gradually and indefinitely increasing, I shall hereafter call fluents, or
flowing quantities, and shall represent them by the final letters of the
alphabet, v, x, ;y,_and z; . . . and the velocities by which every fluent
is increased by its generating motion (which I may__£all fluxions, or
simply velocities, or celerities), I shall represent bythe same letters
pointed, thus^, x, y, z. That is, for the celerity of the quantity v
I shall put v, and so for the celerities of the other quantities x, y, and
z, I shall put x, y, and z, respectively." It must here be observed that
Newton does not take the fluxions themselves infinitely small. The
"moments of fluxions." a term introduced further on, are infinitely
small quajiiifTes. These "moments," as defined and used in the
Method of Fluxions, are substantially the differentials of Leibniz. De
Morgan points out that no small amount of confusion has arisen from
the use of the word fluxion and the notation x by all the English writers
previous to 1704, excepting Newton and George Cheyne, in the sense
of an infinitely small increment.1 Strange to say, even in the Com-
mercium epistolicum the words moment and fluxion appear to be used
as synonymous.

After showing by examples how to solve the first problem, Newton
proceeds to the demonstration of his solution : —

"The moments of flowing quantities (that is, their indefinitely
small parts^ by the 3rrpggl'rm of which^in infinitely small portions of
time, they- are continually increased) are as the velocities of. their
flowing_or_increasing.

"Wherefore, if the moment of any one (as x} be represented by the
product of its celerity x into an infinitely small quantity o (i. e. by

'A. De Morgan, "On the Early History of Infinitesimals," in Philosophical
Magazine, November, 1852.

NEWTON TO EULER 195

xo), the moments of the others, v, y, z, will be represented by vo, yo,
zo; because vo, xo, yo, and 00 are to each other as v, x, y, and z.

"Now since the moments, as xo and yo, are the indefinitely little
accessions of the flowing quantities x and y, by which those quantities
are increased through the several indefinitely little intervals of time,
it follows that those quantities, x and y, after any indefinitely small
interval of time, become x+xo and y+yo, and therefore the equation,
which at all times' indifferently expresses the relation of the flowing
quantities, will as well express the relation between x+xo and y+yo,
as between x and y; so that x+xo and y+yo may be substituted in
the same equation for those quantities, instead of x and y. Thus let
any equation x3 — ax2+axy-ys=o be given, and substitute x+xo for
x, and y+yo for y, and there will arise

=o.

"Now, by supposition, x3 -ax^+axy — y3=o, which therefore, being
expunged and the remaining terms being divided by o, there will
remain

— ax — 2axxo — axoxo
+axy+ayxo +axoyo

+axyo
—y3 —^y^yo

$x~x — 2axx+ayx+axy — $y y+^xxxo — axxo+axyo - $yyyo
+x3oo—y3oo=o.

But whereas zero is supposed to be infinitely little, that it may repre-
sent the moments of quantities, the terms that are multiplied by it
will be nothing in respect of the rest (termini in earn ducti pro nihilo
possunt haberi cum aliis cc^o^Tplherefore I reject them, and there
remains

T,X^X — 2dxx+dyx+dxy—^y2y=o,

as above in Example I." Newton here uses infinitesimals.

Much greater than in the first problem were the difficulties en-
countered in the solution of the second problem, involving, as it does,
inverse operations which have been taxing the skill of the best ana-
"lystsjincehis time. Newton gives first a special solution to the second
problem in which he resorts to a rule for which he has given no proof.

In the general solution of his second problem, Newton assumed
homogeneity with respect to the fluxions and then considered three
cases: (i) when the equation contains 4wo-fl.ujnnns of quantities and
but one of the_fluents; (2) when the equation involves both the fluents
as welt^Tbothlhe fluxions; (3) when the equation contains the flu-
ents and the fluxions of three or more quantities. The first case is the

easiest since it requires simply the integration of -j- =/(*), to which

I96 A HISTORY OF MATHEMATICS

his "special solution" is applicable. The second case demanded
nothing less than the general solution of a differential equation of the
first order. Those who know what efforts were afterwards needed
for the complete exploration of this field in analysis, will not depre-
ciate Newton's work even though he resorted to solutions in form of
infinite series. Newton's third case comes now under the solution of
partial differential equations. He took the equation 2x— z+xy=o
and succeeded in finding a particular integral of it.

The rest of the treatise is devoted to the determination of maxima
and minima, the radius of curvature of curves, and other geometrical
applications of his fluxionary calculus. All this was done previous
to the year 1672.

It must be observed that in the Method of Fluxions (as well as in
his De Analyst and all earlier papers) the method employed by New-
ton is strictly infinitesimal, and in substance like that of Leibniz.
Thus, the original conception of the calculus in England, as well as
on the Continent, was based on infinitesimals.^ The fundamental
principles of the fluxionary calculus were first given to the world in
the Principia; but its peculiar notation did not appear until published
in the second volume of Wallii:£!2Ig£/>ra in 1693. The exposition
given in the Algebra was a contribution of Newton; it rests on in-
finitesimals. In the first edition of the Principia (1687) the descrip-
tion of fluxions is likewise founded on infinitesimals, but in the second
(1713) the foundation is somewhat altered. In Book II, Lemma II,
of the first edition we read: "Cave tamen intellexeris particulas
finitas. Momenta quam primum finite sunt magnitudinis, desinunt
esse momenta. Finiri enim repugnat aliquatenus perpetuo eorum
incremento vel decremento. Intelligenda sunt principia jamjam nas-
centia finitarum magnitudinum." In the second edition the two
sentences which we print in italics are replaced by the following:
"Particular finitae non sunt momenta sed quantitates ipsae ex mo-
men tis genitae." Through the difficulty of the phrases in both ex-
tracts, this much distinctly appears, that in the first, moments are
infinitely small quantities. What else they are in the second is not
clear.1 In the Quadrature of Curves of_^7o4) the infinitely small
quantity is completely abandoned. It hasoSeTTshown that in the
Method of Fluxions JNewton rejected terms involving the quantity o,
because they are infinitely small compared with other terms. This
reasoning is unsatisfactory; for as long as o is a quantity, though
ever so small, this rejection cannot be made without affecting the
result. Newton seems to have felt this, for in the Quadrature of Curves
he remarked that "in mathematics the minutest errors are 'not to be
neglected" (errores quam minimi in rebus mathematicis non sunt
contemnendi).

The early distinction between the system of Newton and Leibniz
1 A. De Morgan, loc. ciL, 1852.

NEWTON TO EULER

197

lies in this, that Newton, holding to the conception of velocity or
fluxion, used the infinitely small increment as a mean's ot determining
it, white^With .Leibniz the relation ot tne infinitely small increments
is itself the object of determination. The difference between the two
rests mainly upon a difference" in the mode of generating quantities.

We give Newton's statement of the method of fluxions or rates, as
given in the introduction to his Quadrature of Curves. "I consider
mathematical quantities in this place not as consisting of very small
parts, but as described by a continued motion. Lines are described,
and thereby generated, not by the apposition of parts, but by the
continued motion .oLpoints; superficies by the molion_oUines; solids
by the motion of superficies; angks^by the rotation of the sides;
portions' uf Liiiiti by (-totinjial flux: and so on in other quantities.
These geneses really take place in the nature of things, and are daily
seen in the motion of bodies. . . .

"Fluxions are, as near as we please (auam proxime], as the incre-
mentajjf fluents generated in times, eoua! and as small as possible,
and tospeaTt accurately, they are in the prime ratio of nascent in-
creinerTts"fyet they can be expressed by any lines whatever, which are
proportional to them."

Newton exempiifies this last assertion by the problem of tangency:
Let AB be the abscissa, BC the ordinate, VCH the tangent, EC the
increment of the ordinate, which produced meets VH at T, and Cc
the increment of the curve. The right line Cc being produced to K,
there are formed three small triangles, the rectilinear CEc, the mix-
tilinear CEc, and the rectilinear CET. Of these, the first is evidently
the smallest, and the last the greatest. Now suppose the ordinate be
to move into the place BC, so that the point c exactly coincides with
the point C; CK, and
therefore the curve Cc,
is coincident with the tan-
gent CH, EC is absolutely
equal to ET, and the
mixtilinear evanescent tri-
angle CEc is, in the last
form, similar to the tri-
angle CET; and its eva-
nescent sides CE, EC, Cc,
will be proportional to
CE, ET, and CT, the
sides of the triangle CET.

Hence it follows that the fluxions of the lines AB, BC, AC, being in
the last ratio of their evanescent increments, are proportional to the
sides of the triangle CET, or, which is all one, of the triangle VBC
similar thereunto. As long as the points C and c are distant from
each other by an interval, however small, the line CK will stand

1 98 A HISTORY OF MATHEMATICS

apart by a small angle from the tangent CH. But when CK co-
incides with CH, and the lines CE, EC, cC reach their ultimate
ratios, then the points C and c accurately coincide and are one
and the same. Newton then adds that "in mathematics the
minutest errors are not to be neglected." This is plainly a re-
jection of the postulates of Leibniz. The doctrine of infinitely
small quantities is here renounced in a manner which would lead
one to suppose that Newton had never held it himself. Thus it
appears that Newton's doctrine was different in different periods.
Though, in the above reasoning, the Charybdis of infinitesimals is
safely avoided, the dangers of a Scylla stare us in the face. We are
required to believe that a point may be considered a triangle, or that
a triangle can be inscribed in a point; nay, that three dissimilar tri-
angles become similar and equal when they have reached their ulti-
mate form in one and the same point.

In the introduction to the Quadrature of Curves the fluxion of xn
is determined as follows: —

"In the same time that x, by flowing, becomes x+o, the power
xn becomes (x+ o)n, i. e. by the method of infinite series

— ft

V*~2+etc.,

and the increments

ft ~~~~^t

o and noxn~l-\ o2x"~2+etc.,

are to one another as

n2— n
i to nxn~l-\ cxtn~2+etc.

"Let now the increments vanish, and their last proportion will be
i to nx" ~ J : hence the fluxion of the quantity x is to the fluxion of the
quantity xn as i: nxn~I.

"The fluxion of lines, straight or curved, in all cases whatever, as
also the fluxions of superficies, angles, and other quantities, can be
obtained in the same manner by the method of prime and ultimate ^
ratios. But to establish in this way the analysis of infinite quantities,
and to investigate prime and ultimate ratios of finite quantities, nas-
cent or evanescent, is in harmony with the geometry of the ancients;
and I have endeavored to show that, in the method of fluxions, it is
not necessary to introduce into geometry infinitely small quantities."
This mode of differentiating does not remove all the difficulties con-
nected with the subject. When o becomes nothing, then we get the

ratio -=nx"~l, which needs further elucidation. Indeed, the method
o

of Newton, as delivered by himself, is encumbered with difficulties

NEWTON TO EULER 199

and objections. Later we shall state Bishop Berkeley's objection to
this reasoning. Even among the ablest admirers of Newton, there
have been obstinate disputes respecting his explanation of his method
of "prime and ultimate ratios."

The so-called "method of limits" is frequently attributed to New-
ton, but the pure method of limits was never adopted by him as his
method of constructing the calculus. All he did was to establish in
his Principia certain principles which are appUcablej,aJjiat-methc)d,
but which he used for a differest-purpose. The "rTrst lemma of the
first book has been made the foundation of the method of limits: —

"Quantities and the ratios of quantities, which in any finite time
converge continually to equality, and before the end of that time ap-
proach nearer the one to the other than by any given difference, be-
come ultimately equal. '^

In this, as well as in the lemmas following this, there are obscurities
and difficulties. Newton appears to teach that a variable quantity
and its limit will ultimately coincide and be equal.

The full title of Newton's Principia is Philosophic Naturalis Prin-
cipia Iflathematica. It was printe""d Inj^Sy under Che direction, and
at the expense, of Edmund Halley. A second edition was brought
out in 1713 with many alterations and improvements, and accom-
panied by a preface from Roger Cotes. It was sold out in a few
months, but a pirated edition published in Amsterdam sup'plied the
demand. The third and last edition which appeared in England during
Newton's lifetime was published in 1726 by Henry Pemberton. The
Principia consists of three books, of which the first two, constituting
the great bulk of the work, treat of the mathematical principles of
natural philosophy, namely, the laws and conditions of motions and
forces" In the Lllhd book Is1 diawn up the constitution of the universe
as deduced from the foregoing principles! The great principle under-
lying this memo rabTeT work is that of universal gravitation. The first
book was completed on April 28, 1686. After the remarkably short
period of three months, the second book was finished. The third book
is the result of the next nine or ten months' labors. It is only a sketch
of a much more extended elaboration of the subject which he had
planned, but which was never brought to completion.

The law of gravitation is enunciated in the first book. Its discovery
envelops the name of Newton in a halo of perpetual glory. The cur-
rent version of the discovery is as follows: it was conjectured by
Robert Hooke (1635-1703), C. Huygens, E. Halley, C. Wren, I. New-
ton, and others, that, if J. Kepler's third law was true (its absolute
accuracy was doubted at that time), then the attraction between the
earth and other members of the solar system varied inversely as the
square of the distance. But the proof of the truth or falsity of the
guess was wanting. In 1666 Newton reasoned, in substance, that if
g represent the acceleration of gravity on the surface of the earth, r

200 A HISTORY OF MATHEMATICS

be the earth's radius, R the distance of the moon from the earth, T
the time of lunar revolution, and a a degree at the equator, then, if
the law is true,

gF2=47r r2' or g=Y2\r) ' '

The data at Newton's command gave R = 6o.4r, T= 2,360,628 seconds,
but a only 60 instead of 69! English miles. This wrong value of a
rendered the calculated value of g smaller than its true value, as
known from actual measurement. It looked as though the law of
inverse squares were not the true law, and Newton laid the calculation
aside. In 1684 he casually ascertained at a meeting of the Royal
Society that Jean Picard had measured an arc of the meridian, and
obtained a more accurate value for the earth's radius. Taking the
corrected value for a, he found a figure for g which corresponded to
the known value. Thus the law of inverse squares was verified. In a
scholium in the Principia, Newton acknowledged his indebtedness to
Huygens for the laws on centrifugal force employed in his calculation.

The perusal by the astronomer Adams of a great mass of unpub-
lished letters and manuscripts of Newton forming the Portsmouth
collection (which remained private property until 1872, when its
owner placed it in the hands of the University of Cambridge) seems to
indicate 'that the difficulties encountered by Newton in the above
calculation were of a different nature. According to Adams, Newton's
numerical verification was fairly complete in 1666, but Newton had
not been able to determine what the attraction of a spherical shell
upon an external point would be. His letters to E. Halley show
that he did not suppose the earth to attract as though all its mass
were concentrated into a point at the centre. He could not have
asserted, therefore, that the assumed law of gravity was verified by
the figures, though for long distances he might have claimed that it
yielded close approximations. When Halley visited Newton in 1684,
he requested Newton to determine what the orbit of a planet would
be if the law of attraction were that of inverse squares. Newton had
solved a similar problem for R. Hooke in 1679, and replied at once
that it was an ellipse. After Halley's visit, Newton, with Picard's
new value for the earth's radius, reviewed his early calculation, and
was able to show that if the distances between the bodies in the solar
system were so great that the bodies might be considered as points,
then their motions were in accordance with the assumed law of gravi-
tation. In 1685 he completed his discovery by showing that a sphere
whose density at any point depends only on the distance from the
centre attracts an external point as though its whole mass were con-
centrated at the centre.

Newton's unpublished manuscripts in the Portsmouth collection
show that he had worked out, by means of fluxions and fluents, his

NEWTON TO EULER 201

lunar calculations to a higher degree of approximation than that given
in the Principia, but that he was unable to interpret his results geo-
metrically. The papers in that collection throw light upon the mode
by which Newton arrived at some of the results in the Principia, as,
for instance, the famous, solution in Book II, Prop. 35, Scholium, of
the problem of the solid of revolution which moves through a resisting
medium with the least resistance. The solution is unproved in the
Principia, but is demonstrated by Newton in the draft of a letter to
David Gregory of Oxford, found in the Portsmouth collection.1

It is chiefly upon the Principia that the fame of Newton rests.
David Brewster calls it "the brightest page in the records of human
reason." Let us listen, for a moment, to the comments of P. S. La-
place, the foremost among those followers of Newton who grappled
with the subtle problems of the motions of planets under the influence
of gravitation: "Newton has well established the existence of the
principle which he had the merit of discovering, but the development
of its consequences and advantages has been the work of the successors
of this great mathematician. The imperfection of the infinitesimal
calculus, when first discovered, did not allow him completely to re-
solve the difficult problems which the theory of the universe offers;
and he was oftentimes forced to give mere hints, which were always
uncertain till confirmed by rigorous analysis. Notwithstanding these
unavoidable defects, the importance and the generality of his dis-
coveries respecting the system of the universe, and the most interesting
points of natural philosophy, the great number of profound and orig-
inal views, which have been the origin of the most brilliant discoveries
of the mathematicians of the last century, which were all presented
with much elegance, will insure to the Principia a lasting pre-eminence
over all other productions of the human mind."

Newton's Arithmetica universalis, consisting of algebraical lectures
delivered by him during the first nine years he was professor at Cam-
bridge, were published in 1707, or more than thirty years after they
were written. This work was published by William Whiston (1667-
1752). We are not accurately informed how Whiston came in pos-
session of it, but according to some authorities its publication was a
breach of confidence on his part. He succeeded Newton in the
Lucasian professorship at Cambridge.

The Arithiv^" "wv^fiU* contains new and important results on
the theory of equations. Newton states Descartes' rule of signs in
accurate form and gives formulae expressing thp <^rp nLthe powers
of the roots urj_to the sixth power and by an Kand so on" makes it
TTTdeTltTHaTthey can be extended to any higher power. Newton's
formulae take the implicit form, while similar formula:; given earlier

1O. Bolza, in Bibliotheca tnathematica, 3. S., Vol. 13, 1913, pp. 1467149- For
a bibliography of this "problem of Newton" on the surface of least resistance, see
L'Inlcrmidiairc d^s matltimaticiens, Vol. 23, 1916, pp. 81-84.

202 A HISTORY OF MATHEMATICS

by Albert Girard take the explicit form, as do also the general formulae
derived later by E. Waring. Newton uses his formulae for fixing an
i^s upper limit of real roots; (the sum of any even power of all the roots
must exceed the same even power of any one of the roots? He estab-
lished also another limit ? A number is an upper limit, if, when sub-
stituted for x, it gives to f(x) and to all its derivatives the same sign.v
In 1748 Colin Maclaurin proved that an "upper limit is obtained by
adding unity to the absolute value of the largest negative coefficient
of the equation. Newton showed that in equations with real co-
efficients, imaginary roots always occur in pairs. *His inventive genius
is grandly displayed in his rule for determining the inferior limit of the
number of imaginary roots, and the superior limits for the number
of positive and negative roots. ^ Though less expeditious than Des-
cartes', Newton's rule always gives as close, and generally closer,
limits to the number of positive and negative roots. Newton did
not prove his rule.

Some light was thrown upon it by George Campbell and Colin
Maclaurin, in the Philosophical Transactions, of the years 1728 and
1729. But no complete demonstration was found for a century and a
half, until, at last, Sylvester established a remarkable general theorem
which includes Newton's rule as a special case. Not without interest
is Newton's suggestion that the conchoid be admitted as a curve to
be used in geometric constructions, along with the straight line and
circle, since the conchoid can be used for the duplication of a cube and
trisection of an angle — to one or the other of which every problem
involving curves of the third or fourth degree can be reduced.

The treatise on Method of Fluxions contains Newton's method of
approximating to the roots of numerical equations. Substantially
the same explanation is given in his De analysi per c&quationes numero
terminorum infinitas. He explains it by working one example, namely
the now famous cubic 1.^v3— 2V — _g_=o. The earliest printed account
appeared in Walfis' Algebra, 1685, chapter 94. Newton assumes that
an approximate value is already known, which differs from the true
value by less than one-tenth of that value. He takes y=2 and sub-
stitutes y^2+p in the equation, which becomes~^3+6/>2+io/>— 1=0.
Neglecting the higher powers of p, he gets io/>— 1=0. Taking
p=.i+q, he gets <73+6.3<72+n. 239+ .061=0. From ii.23<7+.o6i=o
he gets q= — .0054+^, and by the same process, r= —.00004853.
Finally y=2+.i — .0054 — .00004853 = 2.09455147. Newton arranges
his work in a paradigm. He seems quite aware that his method may
fail. If there is doubt, he says, whether p=.i is sufficiently close to
the truth, find p from 6p2+iop — 1 = 0. He does not show that even
this latter method will always answer. By the same mode of pro-

1 For quotations from Newton, see F. Cajori, "Historical Note on the Newton-
Raphson Method of Approximation," Amer. Math. Monthly, Vol. 18, 1911, pp. 29-
33-

NEWTON TO EULER 203

cedure, Newton finds, by a rapidly converging series, the value of y
in terms of a and x, in the equation y3+axy+aay — xs — 2a3=o.

In f&go, Joseph Raphson (1648-1715), a iellow ot the Royal Society
of London, published a tract, Analysis aquationum universalis. His
method closely resembles that of Newton. The only difference is
this, that Newton derives each successive step, p, q, r, of approach to
the root, from a new equation, while Raphson finds it each time by
substitution in the original equation. In Newton's cubic, Raphson
would not find the second correction by the use of x3+6xz+iox — 1=0,
but would substitute 2.i+q in the original equation, finding q =
- .0054. He would then substitute 2.og46+r in the original equation,
finding r= —.00004853, and so on. Raphson does not mention
Newton; he evidently considered the difference sufficient for his
method to be classed independently. To be emphasized is the fact
that the process which in modern texts goes by the name of "New-
ton's method of approximation," is really not Newton's method, but

Raphson's modification of it. The/om now so familiar, a— 4rr4 was

/(«)

not used by Newton, but was used by Raphson. To be sure, Raphson
does not use this notation; he writes /(a) and f'(a) out in full as poly-
nomials. It is doubtful, whether this method should be named after
Newton alone. Though not identical with Vieta's process, it re-
sembles Vieta's. The chief difference lies in the divisor used. The
divisor f'(a) is much simpler, and easier to compute than Vieta's
divisor. Raphson's version of the process represents what J. Lagrange
recognized as an advance on the scheme of Newton. The method is
"plus simple que celle de Newton." * Perhaps the name "Newton-
Raphson method" would be a designation more nearly representing
the facts of history. We may add that the solution of numerical
equations was considered geometrically by Thomas Baker in 1684
and Edmund Halley in 1687, but in 1694 Halley "had a very great
desire of doing the same in numbers." The only difference between
Halley's and Newton's own method is that Halley solves a quadratic
equation at each step, Newton a linear equation. Halley modified
also certain algebraic expressions yielding approximate cube and
fifth roots, given in 1692 by the Frenchman, Thomas Fantet de Lagny
(1660-1734). In 1705 and 1706 Lagny outlines a method of differences;
such a method, less systematically developed, had been previously
explained in England by John Collins. By this method, if a, b, c, . . .
are in arithmetical progression, then a root may be found approxi-
mately from the first, second, and higher differences of /(-a),

, • • •

Newton's Method of Fluxions contains also "Newton's parallelo-
gram," which enabled him, in an equation, /(.v, y)=o, to find a series

'Lagrange, Resolution des equal, num., 1798, Note V, p. 138.

204 A HISTORY OF MATHEMATICS

in powers of x equal to the variable y. The great utility of this rule
lay in its determining the form of the series; for, as soon as the law was
known by which the exponents in the series vary, then the expansion
could be effected by the method of indeterminate coefficients. The
rule is still used in determining the infinite branches to curves, or their
figure at multiple points. Newton gave no proof for it, nor any clue
as to how he discovered it. The proof was supplied half a century
later, by A. G. Kastner and G. Cramer, independently.1

In 1704 was published, as an appendix to the Opticks, the Enu-
meratio Unearum tertii ordinis, which contains theorems on the theory
of curves. Newton divides cubics into seventy-two species, arranged
in larger groups, for which his commentators have supplied the names
"genera" and "classes," recognizing fourteen of the former and seven
(or four) of the latter. He overlooked six species demanded by his
principles of classification, and afterwards added by J. Stirling, Wil-
liam Murdoch (1754-1839), and G. Cramer. He enunciates the re-
markable theorem that the five species which he names "divergent
parabolas" give by their projection every cubic curve whatever. As
a rule, the tract contains no proofs. It has been the subject of frequent
conjecture how Newton deduced his results. Recently we have gotten
at the facts, since much of the analysis used by Newton and a few
additional theorems have been-sdiscovered among the Portsmouth
papers. An account of the four holograph manuscripts on this sub-
ject has been published by W. W. Rouse Ball, in the Transactions of
the London Mathematical Society (vol. xx, pp. 104-143). It is inter-
esting to observe how Newton begins his research on the classification
of cubic curves by the algebraic method, but, finding it laborious,
attacks the problem geometrically, and afterwards returns again to
analysis.

Space does not permit us to do more than merely mention Newton's
prolonged researches in other departments of science. He conducted
a long series of experiments in optics and is the author of the corpus-
cular theory of light. The last of a number of papers on optics,
which he contributed to the Royal Society, 1687, elaborates the theory
of "fits." He explained the decomposition of light and the theory
of the rainbow. By him were invented the reflecting telescope and
the sextant (afterwards re-invented by Thomas Godfrey of Phila-
delphia 2 and by John Hadley). He deduced a theoretical expression
for the velocity of sound in air, engaged in experiments on chemistry,
elasticity, magnetism, and the law of cooling, and entered upon geo-
logical speculations.

During the two years following the close of 1692, Newton suffered

1 S. Giinther, Vermischle Unlersuchimgen zur Geschichle d. math. Wiss,, Leipzig'
1876, pp. 136-187.

2 F. Cajori, Teaching and History of Mathematics in Hie U. S., Washington, 1890,
p. 42.

NEWTON TO EULER 205

from insomnia and nervous irritability. Some thought that he la-
bored under temporary mental aberration. Though he recovered his
tranquillity and strength of mind, the time of great discoveries was
over; he would study out questions propounded to him, but no longer
did he by his own accord enter upon new fields of research. The
most noted investigation after his sickness was the testing of his lunar
theory by the observations of Flamsteed, the astronomer royal. In
1695 ne was appointed warden, and in 1699 master of the mint, which
office he held until his death. His body was interred in Westminster
Abbey, where in 1731 a magnificent monument was erected, bearing
an inscription ending with, " Sibi gratulentur mortales tale tantumque
exstitisse humani generis decus." It is not true that the Binomial
Theorem is also engraved on it.

We pass to Leibniz, the second and independent inventor of the
calculus. Gottfried Wilhelm Leibniz (1646-1716) was born in Leip-
zig. No period in the history of any civilized nation could have been
less favorable for literary and scientific pursuits than the middle of
the seventeenth century in Germany. Yet circumstances seem to
have happily combined to bestow on the youthful genius an education
hardly otherwise obtainable during this darkest period of German
history. He was brought early in contact with the best of the culture
then existing. In his fifteenth year he entered the University of
Leipzig. Though law was his principal study, he applied himself
with great diligence to every branch of knowledge. Instruction in
German universities was then very low. The higher mathematics
was not taught at all. We are told that a certain John Kuhn lectured
on Euclid's Elements, but that his lectures were so obscure that none
except Leibniz could understand them. Later on, Leibniz attended,
for a half-year, at Jena, the lectures of Erhard Weigel, a philosopher
and mathematician of local reputation. In 1666 Leibniz published
a treatise, De Arte Combinatoria, in which he does not pass beyond
the rudiments of mathematics, but which contains remarkable plans
for a theory of mathematical logic, a symbolic method with formal
rules obviating the necessity of thinking. Vaguely such plans had
been previously suggested by R. Descartes and Pierre Herigone. In
manuscripts which Leibniz left unpublished he enunciated the princi-
pal properties of what is now called logical multiplication, addition,
negation, identity, class-induction and the null-class.1 Other theses
written by him at this time were metaphysical and juristical in char-
acter. A fortunate circumstance led Leibniz abroad. In 1672 he was
sent by Baron Boineburg on a political mission to Paris. He there
formed the acquaintance of the most distinguished men of the age.
Among these was C. Huygens, who presented a copy of his work on
the oscillation of the pendulum to Leibniz, and first led the gifted
young German to the study of higher mathematics. In 1673 Leibniz
1 See Philip E. B. Jourdain in Quarterly Jour, of Math., Vol. 41, 1910, p. 329.

206 A HISTORY OF MATHEMATICS

went to London, and remained there from January till March. He
there became incidentally acquainted with the mathematician John
Pell, to whom he explained a method he had found on the summation
of series of numbers by their differences. Pell told him that a similar
formula had been published by Gabriel Mouton (1618-1694) as
early as 1670, and then called his attention to N. Mercator's work
on the rectification of the parabola. While in London, Leibniz ex-
hibited to the Royal Society his arithmetical machine, which was
similar to B. Pascal's, but more efficient and perfect. After his re-
turn to Paris, he had the leisure to study mathematics more system-
atically. With indomitable energy he set about removing his igno-
rance of higher mathematics. C. Huygens was his principal master.
He studied the geometric works of R. Descartes, Honorarius Fabri,
Gregory St. Vincent, and B. Pascal. A careful study of infinite
series led him to the discovery of the following expression for the
ratio of the circumference to the diameter of the circle, previously
discovered by James Gregory: —

7 = i-^+i-y+!-etc.
4

This elegant series was found in the same way as N. Mercator's on
the hyperbola. C. Huygens was highly pleased with it and urged
him on to new investigations. In 1673 Leibniz derived the series

arc ta,nx=x-lx3+^x5— . . . ,

from which most of the practical methods of computing TT have been
obtained. This series had been previously discovered by James
Gregory, and was used by Abraham Sharp (1651-1742) under in-
structions from E. Halley for calculating TT to 72 places. In 1706
John Machin (1680-1751), professor of astronomy at Gresham Col-
lege in London, obtained 100 places by using an expression that is
obtained from the relation

— =4 arc tan £— arc tan -*^-§,
4

by substituting Gregory's infinite series for
arc tan and arc tan

Machin's formula was used in 1874 by William Shanks (1812-1882)
for computing TT to 707 places.

Leibniz entered into a detailed study of the quadrature of curves
and thereby became intimately acquainted with the higher math-
ematics. Among the papers of Leibniz is still found a manuscript
on quadratures, written before he left Paris in 1676, but which was
never printed by him. The more important parts of it were embodied
in articles published later in the Acta eruditorum.

In the study of Cartesian geometry the attention of Leibniz was

NEWTON TO EULER 207

drawn early to the direct and inverse problems of tangents. The
direct problem had been solved by Descartes for the simplest curves
only; while the inverse had completely transcended the power of his
analysis. Leibniz investigated both problems for any curve; he
constructed what he called the triangulum characteristicum — an
infinitely small triangle between the infinitely small part of the curve
coinciding with the tangent, and the differences of the ordinates and
abscissas. A curve is here considered to be a polygon. The triangulum
characteristicum is similar to the triangle formed by the tangent, the
ordinate of the point of contact, and the sub-tangent, as well as to
that between the ordinate, normal, and sub-normal. It was employed
by I. Barrow in England, but Leibniz states that he obtained it from
Pascal. From it Leibniz observed the connection existing between the
direct and inverse problems of tangents. He saw also that the latter
could be carried back to the quadrature of curves. All these results
are contained in a manuscript of Leibniz, written in 1673. One mode
used by him in effecting quadratures was as follows: The rectangle
formed by a sub-normal p and an element a (i. e. infinitely small part
of the abscissa) is equal to the rectangle formed by the ordinate y
and the element I of that ordinate; or in symbols, pa=yl. But the
summation of these rectangles from zero on gives a right triangle
equal to half the square of the ordinal Thus, using Cavalieri's no-
tation, he gets

omn. pa=omn. yl=^- (own. meaning omnia, all).
But y=omn. I; hence

— ,1 omn. I"

omn. omn. l-= .

a 20.

This equation is especially interesting, since it is here that Leibniz
first introduces a new notation. He says: "It will be useful to write

I for omn., as I / for omn. I, that is, the sum of the £'s"; he then
writes the equation thus: —

From this he deduced tiie simplest integrals, such as

Since the symbol of summation I raises the dimensions, he con-
cluded that the opposite calculus, or that of differences d, would

208 A HISTORY OF MATHEMATICS

lower them. Thus, if I l=ya, then J=^T. The symbol d was at

first placed by Leibniz in the denominator, because the lowering of
the power of a term was brought about in ordinary calculation by
division. The manuscript giving the above is dated October 2gth,
1675. 1 This, then, was the memorable day on which the notation
of the new calculus came to be, — a notation which contributed enor-
mously to the rapid growth and perfect development of the calculus.
Leibniz proceeded to apply his new calculus to the solution of
certain problems then grouped together under the name of the In-
verse Problems of Tangents. He found the cubical parabola to be
the solution to the following: To find the curve in which the sub-
normal is reciprocally proportional to the ordinate. The correctness
of his solution was tested by him by applying to the result the method
of tangents of Baron Rene Francois de Sluse (1622-1685) and reason-
ing backwards to the original supposition. In the solution of the

third problem he changes his notation from -r to the now usual nota-

tion dx. It is worthy of remark that in these investigations, Leibniz
nowhere explains the significance of dx and dy, except at one place

oc
in a marginal note: "Idem est dx et -:, id est, differentia inter duas

x proximas." Nor does he use the term differential, but always differ-
ence. Not till ten years later, in the Acta eriiditorum, did he give
further explanations of these symbols. What he aimed at principally
was to determine the change an expression undergoes when the sym-
bol I or d is placed before it. It may be a consolation to students

wrestling with the elements of the differential calculus to know that
it required Leibniz considerable thought and attention 2 to determine

doc J£

whether dx dy is the same as d(xy). and -:- the same as &-. After

dy y

considering these questions at the close of one of his manuscripts, he
concluded that the expressions were not the same, though he could
not give the true value for each. Ten days later, in a manuscript
dated November 21, 1675, he found the equation ydx =dxy—xdy,
giving an expression for d(xy), which he observed to be true for all
curves. He succeeded also in eliminating dx from a differential
equation, so that it contained only dy, and thereby led to the solution
of the problem under consideration. "Behold, a most elegant way

1 C. J. Gerhardt, Entdeckung der hoheren Analysis. Halle, 1855, p. 125.

2 C. J. Gerhardt, Entdeckung der Differenzialrechnung durch Leibniz, Halle, 1848,
pp. 25, 41.

NEWTON TO EULER 209

by which the problems of the inverse method of tangents are solved,
or at least are reduced to quadratures!" Thus he saw clearly that
the inverse problems of tangents could be solved by quadratures, or,
in other words, by the integral calculus. In course of a half-year he
discovered that the direct problem of tangents, too, yielded to the
power of his new calculus, and that thereby a more general solution
than that of R. Descartes could be obtained. He succeeded in solving
all the special problems of this kind, which had been left unsolved
by Descartes. Of these we mention only the celebrated problem
proposed to Descartes by F. de Beaune, viz. to find the curve whose
ordinate is to its sub-tangent as a given line is to that part of the
ordinate which lies between the curve and a line drawn from the
vertex of the curve at a given inclination to the axis.

Such was, in brief, the progress in the evolution of the new calculus
made by Leibniz during his stay in Paris. Before his departure, in
October, 1676, he found himself in possession of the most elementary
rules and formulae of the infinitesimal calculus.

From Paris, Leibniz returned to Hanover by way of London and
Amsterdam. In London he met John Collins, who showed him a
part of his scientific correspondence. Of this we shall speak later.
In Amsterdam he discussed mathematics with R. F. de Sluse, and
became satisfied that his own method of constructing tangents not
only accomplished all that Sluse's did, but even more, since it could
be extended to three variables, by which tangent planes to surfaces
could be found; and especially, since neither irrationals nor fractions
prevented the immediate application of his method.

In a paper of July n, 1677, Leibniz gave correct rules for the dif-
ferentiation of sums, products, quotients, powers, and roots. He had
given the differentials of a few negative and fractional powers, as
early as November, 1676, but had made some mistakes. For d\/x

he had given the erroneous value — -—, and in another place the value

\/x

T ^

— \x~^\ for d-$ occurs in one place the wrong value, -2, while a few

X X

lines lower is given — -3, its correct value.

In 1682 was founded in Berlin the Ada emditorum, a journal
sometimes known by the name of Leipzig Acts. It was a partial imi-
tation of the French Journal des Savans (founded in 1665), and the
literary and scientific review published in Germany. Leibniz was a
frequent contributor. E. W. Tschirnhausen, who had studied mathe-
matics in Paris with Leibniz, and who was familiar with the new
analysis of Leibniz, published in the Acta eruditorum a paper on quad-
ratures, which consists principally of subject-matter communicated

2io A HISTORY OF MATHEMATICS

by Leibniz to Tschirnhausen during a controversy which they had
had on this subject. Fearing that Tschirnhausen might claim as his
own and publish the notation and rules of the differential calculus,
Leibniz decided, at last, to make public the fruits of his inventions.
In 1684, or nine years after the new calculus first dawned upon the
mind of Leibniz, and nineteen years after Newton first worked at
fluxions, and three years before the publication of Newton's Principia,
Leibniz published, in the Ada eniditorum, his first paper on the differ-
ential calculus. He was unwilling to give to the world all his treasures,
but chose those parts of his work which were most abstruse and least
perspicuous. This epoch-making paper of only six pages bears the
title: "Nova methodus pro maximis et minimis, itemque tangentibus,
quae nee fractas nee irrationales quantitates moratur, et singulare
pro illis calculi genus." The rules of calculation are briefly stated
without proof, and the meaning of dx and dy is not made clear.
Printer's errors increased the difficulty of comprehending the subject.
It has been inferred from this that Leibniz himself had no definite
and settled ideas on this subject. Are dy and dx finite or infinitesimal
quantities? At first they appear, indeed, to have been taken as finite,
when he says: "We now call any line selected at random dx, then
we designate the line which is to dx as y is to the sub-tangent, by dy,
which is the difference of y." Leibniz then ascertains, by his calculus,
in what way a ray of light passing through two differently refracting
media, can travel easiest from one point to another; and then closes
his article by giving his solution, in a few words, of F. de Beaune's
problem. Two years later (1686) Leibniz published in the Ada
eruditorum a paper containing the rudiments of the integral calculus.
The quantities dx and dy are there treated as infinitely small. He
showed that by the use of his notation, the properties of curves could
be fully expressed by equations. Thus the equation

/ . C dx

y=-V2x—x*+ I ~7= =f

J \/2X—X2

characterizes the cycloid.1

The great invention of Leibniz, now made public by his articles in
the Ada eruditorum, made little impression upon the mass of mathe-
maticians. In Germany no one comprehended the new calculus
except Tschirnhausen, who remained indifferent to it. The author's
statements were too short and succinct to make the calculus generally
understood. The first to take up the study of it were two foreigners, — •
the Scotchman John Craig, and the Swiss Jakob (James) Bernoulli.
The latter wrote Leibniz a letter in 1687, wishing to be initiated into
the mysteries of the new analysis. Leibniz was then travelling abroad,
so that this letter remained unanswered till 1690. James Bernoulli

1 C. J. Gerhardt, Geschichle der Mathematik in Dcutschland, Miinchen, 1877,
P- 159-

NEWTON TO EULER 211

succeeded, meanwhile, by close application, in uncovering the secrets
of the differential calculus without assistance. He and his brother
John proved to be mathematicians of exceptional power. They applied
themselves to the new science with a success and to an extent which
made Leibniz declare that it was as much theirs as his. Leibniz
carried on an extensive correspondence with them, as well as with other
mathematicians. In a letter to John Bernoulli he suggests, among
other things, that the integral calculus be improved by reducing in-
tegrals back to certain fundamental irreducible forms. The integra-
tion of logarithmic expressions was then studied. The writings of
Leibniz contain many innovations, and anticipations of since prom-
nent methods. Thus he made use of variable parameters, laid the
foundation of analysis in situ, introduced in a manuscript of 1678 the
notion of determinants (previously used by the Japanese), in his
effort to simplify the expression arising in the elimination of the un-
known quantities from a set of linear equations. He resorted to the
device of breaking up certain fractions into the sum of other fractions
for the purpose of easier integration; he explicitly assumed the prin-
ciple of continuity; he gave the first instance of a "singular solution,"
and laid the foundation to the theory of envelopes in two papers, one
of which contains for the first time the terms co-ordinate and axes of
co-ordinates. He wrote on osculating curves, but his paper contained
the error (pointed out by John Bernoulli, but not admitted by Leibniz)
that an osculating circle will necessarily cut a curve in four consecutive
points. Well known is his theorem on the nih differential coefficient
of the product of two functions of a variable. Of his many papers on
mechanics, some are valuable, while others contain grave errors.
Leibniz introduced in 1694 the use of the word function, but not in
the modern sense. Later in that year Jakob Bernoulli used the word
in the Leibnizian sense. In the appendix to a letter to Leibniz, dated
July 5, 1698, John Bernoulli uses the word in a more nearly modern
sense: "earum [applicatarum] quascunque functiones per alias appli-
catas PZ expresses. " In 1718 John Bernoulli arrives at the definition
of function as a "quantity composed in any manner of a variable and
any constants." (On appelle ici fonction d'une grandeur variable,
une quantite composee de quelque maniere que ce soit de cette gran-
deur variable et de constantes.) 1

Leibniz made important contributions to the notation of mathe-
matics. Not only is our notation, of the differential and integral
calculus due to him, but he used the sign of equality in writing pro-
portions, thus a:b=c:d. In Leibnizian manuscripts occurs ~ for
"similar" and ~ for "equal and similar" or "congruent." Says

1 See M. Cantor, op. cit., Vol. Ill, 2 Ed., 1901, pp. 215, 216, 456, 457; Encyclo-
pedic dcs sciences malMmatiqiics , Tome II, Vol. I, pp. 3-5.

2 Leibniz, Werke Ed. Gerhardt, 3. Folge, Bd. V, p. 153. See also J. Tropfke,
op. cit., Vol. IT, 1903, p. 12.

212 A HISTORY OF MATHEMATICS

P. E. B. Jourdain,1 "Leibniz himself attributed all of his mathe-
matical discoveries to his improvements in notation."

Before tracing the further development of the calculus we shall
sketch the history of that long and bitter controversy between English
and Continental mathematicians on the invention of the calculus.
The question was, did Leibniz invent it independently of Newton, or
was he a plagiarist?

We must begin with the early correspondence between the parties
appearing in this dispute. Newton had begun using his notation of
fluxions in 1665. 2 In 1669 I. Barrow sent John Collins Newton's
tract, De Analyst per equaliones, etc.

The first visit of Leibniz to London extended from the nth of Jan-
uary until March, 1673. He was in the habit of committing to writing
important scientific communications received from others. In 1890
C. J. Gerhardt discovered in the royal library at Hanover a sheet of
manuscript with notes taken by Leibniz during this journey.3 They
are headed " Observata Philosophica in itinere Anglicano sub initium
anni 1673." The sheet is divided by horizontal lines into sections.
The sections given to Chymica, Mechanica, Magnetica, Botanica,
Anatomica, Medica, Miscellanea, contain extensive memoranda, while
those devoted to mathematics have very few notes. Under Geo-
metrica he says only this: " Tangentes omnium figurarum. Figurarum
geometricarum explicatio per motum puncti in moto lati." We sus-
pect from this that Leibniz had read Isaac Barrow's lectures. Newton
is referred to only under Optica. Evidently Leibniz did not obtain a
knowledge of fluxions during this visit to London, nor is it claimed
that he did by his opponents.

Various letters of I. Newton, J. Collins, and others, up to the be-
ginning of 1676, state that Newton invented a method by which tan-
gents could be drawn without the necessity of freeing their equations
from irrational terms. Leibniz announced in 1674 to H. Oldenburg,
then secretary of the Royal Society, that he possessed very general
analytical methods, by which he had found theorems of great im-
portance on the quadrature of the circle by means of series. In answer,
Oldenburg stated Newton and James Gregory had also discovered
methods of quadratures, which extended to the circle. Leibniz de-
sired to have these methods communicated to him; and Newton, at
the request of Oldenburg and Collins, wrote to the former the cele-
brated letters of June 13 and October 24, 1676. The first contained
the Binomial Theorem and a variety of other matters relating to
infinite series and quadratures; but nothing directly on the method of

1 P. E. B. Jourdain, The Nature of Mathematics, London, p. 71.

2 J. Edleston, Correspondence of Sir Isaac Newton and Professor Cotes, London,
1850, p. xxi; A. De Morgan, "Fluxions" and "Commercium Epistoh'cum " in
the Penny Cyclopedia.

3 C. J. Gerhardt, "Leibniz in London" in Sitzungsberichte der K. Preussischcn
Academic d. Wissensch. zu Berlin, Feb., 1891.

NEWTON TO EULER 213

fluxions. Leibniz in reply speaks in the highest terms of what Newton
had done, and requests further explanation. Newton in his second
letter just mentioned explains the way in which he found the Binomial
Theorem, and also communicates his method of fluxions and fluents
in form of an anagram in which all the letters in the sentence com-
municated were placed in alphabetical order. Thus Newton says
that his method of drawing tangents was

6a cedes lyff "ji $1 gn 40 $qrr 45 gt I2VX.

The sentence was, "Data aequatione quotcunque fluentes quantitates
involvente fluxiones invenire, et vice versa." ("Having any given
equation involving never so many flowing quantities, to find the
fluxions, and vice versa.") Surely this anagram afforded no hint.
Leibniz wrote a reply to John Collins, in which, without any desire
of concealment, he explained the principle, notation, and the use of
the differential calculus.

The death of Oldenburg brought this correspondence to a close.
Nothing material, happened till 1684, when Leibniz published his
first paper on the differential calculus in the Ada eruditorum, so that
while Newton's claim to the priority of invention must be admitted
by all, it must also be granted that Leibniz was the first to give the
full benefit of the calculus to the world. Thus, while Newton's in-
vention remained a secret, communicated only to a few friends, the
calculus of Leibniz was spreading over the Continent. No rivalry or
hostility existed, as yet, between the illustrious scientists. Newton
expressed a very favorable opinion of Leibniz's inventions, known to
him through the above correspondence with Oldenburg, in the follow-
ing celebrated scholium (Principia, first edition, 1687, Book II,
Prop. 7, scholium): —

"In letters which went between me and that most excellent geom-
eter, G. G. Leibniz, ten years ago, when I signified that I was in the
knowledge of a method of determining maxima and minima, of draw-
ing tangents, and the like, and when I concealed it in transposed letters
involving this sentence (Data aequatione, etc., above cited), that most
distinguished man wrote back that he had also fallen upon a method
of the same kind, and communicated his method, which hardly dif-
fered from mine, except in his forms of words and symbols."

As regards this passage, we shall see that Newton was afterwards
weak enough, as De Morgan says: "First, to deny the plain and ob-
vious meaning, and secondly, to omit it entirely from the third edition
of the Principia" On the Continent, great progress was made in
the calculus by Leibniz and his coadjutors, the brothers James and
John Bernoulli, and Marquis de 1'Hospital. In 1695 John Wallis in-
formed Newton by letter that "he had heard that his notions of
fluxions passed in Holland with great applause by the name of ' Leib-
niz's Calculus Differentialis.' " Accordingly Wallis stated in the pref-

214 A HISTORY OF MATHEMATICS

ace to a volume of his works that the calculus differentialis was New-
ton's method of fluxions which had been communicated to Leibniz
in the Oldenburg letters. A review of Wallis' works, in the Ada
eruditorum for 1696, reminded the reader of Newton's own admission
in the scholium above cited.

For fifteen years Leibniz had enjoyed unchallenged the honor of
being the inventor of his calculus. But in 1699 Fatio de Duillier
(1664-1753), a Swiss, who had settled in England, stated in a mathe-
matical paper, presented to the Royal Society, his conviction that
I. Newton was the first inventor; adding that, whether Leibniz, the
second inventor, had borrowed anything from the other, he would
leave to the judgment of those who had seen the letters and manu-
scripts of Newton. This was the first distinct insinuation of plagiar-
ism. It would seem that the English mathematicians had for some
time been cherishing suspicions unfavorable to Leibniz. A feeling
had doubtless long prevailed that Leibniz, during his second visit to
London in 1676, had or might have seen among the papers of John
Collins, Newton's Analysis per czquationes, etc., which contained ap-
plications of the fluxionary method, but no systematic development
or explanation of it. Leibniz certainly did see at least part of this
tract. During the week spent in London, he took note of whatever
interested him among the letters and papers of Collins. His memo-
randa discovered by C. J. Gerhardt in 1849 in the Hanover library
fill two sheets.1 The one bearing on our question is headed " Excerpta
ex tractatu Newtoni Msc. de Analysi per aequationes numero ter-
minorum infinitas." The notes are very brief, excepting those De
resolutione aquationum affectarum, of which there is an almost com-
plete copy. This part was evidently new to him. If he examined
Newton's entire tract, the other parts did not particularly impress
him. From it he seems to have gained nothing pertaining to the in-
finitesimal calculus. By the previous introduction of his own al-
gorithm he had made greater progress than by what came to his
knowledge in London. Nothing mathematical that he had received
engaged his thoughts in the immediate future, for on his way back
to Holland he composed a lengthy dialogue on mechanical subjects.

Fatio de Duillier's insinuations lighted up a flame of discord which
a whole century was hardly sufficient to extinguish. Leibniz, who
had never contested the priority of Newton's discovery, and who
appeared to be quite satisfied with Newton's admission in his scholium,
now appears for the first time in the controversy. He made an ani-
mated reply in the A eta eruditorum and complained to the Royal
Society of the injustice done him.

Here the affair rested for some time. In the Quadrature of Curves,
published 1704, for the first time, a formal exposition of the method
and notation of fluxions was made public. In 1705 appeared an un-
1 C. J. Gerhardt, "Leibniz in London," loc. oil.

NEWTON TO EULER 215

favorable review of this in the Ada eruditorum, stating that Newton
uses and always has used fluxions for the differences of Leibniz. This
was considered by Newton's friends an imputation of plagiarism on
the part of their chief, but this interpretation was always strenuously
resisted by Leibniz. John Keill (1671-1721), professor of astronomy
at Oxford, undertook with more zeal than judgment the defence of
Newton. In a paper inserted in the Philosophical Transactions of
1708, he claimed that Newton was the first inventor of fluxions and
"that the same calculus was afterward published by Leibniz, the
name and the mode of notation being changed." Leibniz complained
to the secretary of the Royal Society of bad treatment and requested
the interference of that body to induce Keill to disavow the intention
of imputing fraud. John Keill was not made to retract his accusation ;
on the contrary, was authorized by Newton and the Royal Society
to explain and defend his statement. This he did in a long letter.
Leibniz thereupon complained that the charge was now more open
than before, and appealed for justice to the Royal Society and to
Newton himself. The Royal Society, thus appealed to as a judge,
appointed a committee which collected and reported upon a large
mass of documents — mostly letters from and to Newton, Leibniz,
Wallis, Collins, etc. This report, called the Commercium epistolicum,
appeared in the year 1712 and again in 1722 and 1725, with a Recensio
prefixed, and additional notes by Keill. The final conclusion in the
Commercium epistolicum was that Newton was "the first inventor."
But this was not to the point. The question was not whether Newton
was the first inventor, but whether Leibniz had stolen the method.
The committee had not formally ventured to assert their belief that
Leibniz was a plagiarist. In the following sentence they insinuated
that Leibniz did take or might have taken, his method from that of
Newton: "And we find no mention of his (Leibniz's) having any other
Differential Method than Mouton's before his Letter of 2ist of June,
1677, which was a year after a Copy of Mr. Newton's Letter, of the
loth of December, 1672, had been sent to Paris to be communicated
to him; and about four years after Mr. Collins began to communicate
that Letter to his Correspondents; in which Letter the Method of
Fluxions was sufficiently describ'd to any intelligent Person."

About 1850 it was shown that what H. Oldenburg sent to Leibniz
was not Newton's letter of Dec. 10, 1672, but only excerpts from it
which omitted Newton's method of drawing tangents and could not
possibly convey an idea of fluxions. Oldenburg's letter was found
among the Leibniz manuscripts in the Royal Library at Hanover, and
was published by C. J. Gerhardt in 1846, 1848, 1849 and 1855, * and
again later.

1 See Essays on the Life and Work of Newton by Augustus De Morgan, edited, with
notes and appendices, by Philip E. B. Jourdain, Chicago and London, 1914. Jour-
dain gives on p. 102 the bibliography of the publications of Newton and Leibniz.

2i6 A HISTORY OF MATHEMATICS

Moreover, when J. Edleston in 1850 published the Correspondence
of Sir Isaac Newton and Professor Cotes, it became known that the
Royal Society in 1712 had not one, but two, parcels of Collins. One
parcel contained letters of James Gregory, and Isaac Newton's letter
of Dec. 10, 1672, in full; the other parcel, which was marked "To
Leibnitz, the i4th of June, 1676 About Mr. Gregories remains,"
contained an abridgment of a part of the contents of the first parcel,
with nothing but an allusion to Newton's method described in his
letter of Dec. 10, 1672. In the Commercium ejristolicum Newton's
letter was printed in full and no mention was made of the existence
of the second parcel that was marked "To Leibnitz. ..." Thus the
Commercium epistolicum conveyed the impression that Newton's un-
curtailed letter of Dec. 10, 1672, had reached Leibniz in which fluxions
"was sufficiently described to any intelligent person," while as a
matter of fact the method is not described at all in the letter which
Leibniz received.

Leibniz protested only in private letters against the proceeding of
the Royal Society, declaring that he would not answer an argument
so weak. John Bernoulli, in a letter to Leibniz, which was published
later in an anonymous tract, is as decidedly unfair towards Newton
as the friends of the latter had been towards Leibniz. John Keill
replied, and then Newton and Leibniz appear as mutual accusers in
several letters addressed to third parties. In a letter dated April 9,
1716, and sent to Antonio Schinella Conti (1677-1749), an Italian
priest then residing in London, Leibniz again reminded Newton of
the admission he had made in the scholium, which he was now desirous
of disavowing; Leibniz also states that he always believed Newton,
but that, seeing him connive at accusations which he must have
known to be false, it was natural that he (Leibniz) should begin to
doubt. Newton did not reply to this letter, but circulated some re-
marks among his friends which he published immediately after hearing
of the death of Leibniz, November 14, 1716. This paper of Newton
gives the following explanation pertaining to the scholium in question:
"He [Leibniz] pretends that in my book of principles I allowed him
the invention of the calculus diff erentialis, independently of my own ;
and that to attribute this invention to myself is contrary to my
knowledge there avowed. But in the paragraph there referred unto
I do not find one word to this purpose." In the third edition of the
Principia, 1726, Newton omitted the scholium and substituted in its
place another, in which the name of Leibniz does not appear.

National pride and party feeling long prevented the adoption of
impartial opinions in England, but now it is generally admitted by

We recommend J. B. Biot and F. Lefort's edition of the Commercium efnstolictim>
Paris, 1856, which exhibits all the alterations made in the different reprints of this
publication and reproduces also H. Oldenburg's letter to Leibniz of July 26, 1676,
and other important documents bearing on the controversy.

NEWTON TO EULER 217

nearly all familiar with the matter, that Leibniz really was an inde-
pendent inventor. Perhaps the most telling evidence to show that
Leibniz was an independent inventor is found in the study of his
mathematical papers (collected and edited by C. J. Gerhardt, in seven
volumes, Berlin, 1849-1863), which point out a gradual and natural
evolution of the rules of the calculus in his own mind. "There was
throughout the whole dispute," says De Morgan, "a confusion be-
tween the knowledge of fluxions or differentials and that of a calculus
of fluxions or differentials; that is, a digested method with general
rules."

This controversy is to be regretted on account of the long and bitter
alienation which it produced between English and Continental
mathematicians. It stopped almost completely all interchange of
ideas on scientific subjects. The English adhered closely to Newton's
methods and, until about 1820, remained, in most cases, ignorant of
the brilliant mathematical discoveries that were being made on the
Continent. The loss in point of scientific advantage was almost
entirely on the side of Britain. The only way in which this dispute
may be said, in a small measure, to have furthered the progress of
mathematics, is through the challenge problems by which each side
attempted to annoy its adversaries.

The recurring practice of issuing challenge problems was inaugurated
at this time by Leibniz. They were, at first, not intended as defiances,
but merely as exercises in the new calculus. Such was the problem
of the isochronous curve (to find the curve along which a body falls
with uniform velocity), proposed by him to the Cartesians in 1687, and
solved by Jakob Bernoulli, himself, and Johann Bernoulli. Jakob Ber-
noulli proposed in the Acta eruditorum of 1690 the question to find the
curve (the catenary) formed by a chain of uniform weight suspended
freely from its ends. It was resolved by C. Huygens, G. W. Leibniz,
Johann Bernoulli, and Jakob Bernoulli himself; the properties of the
catenary were worked out methodically by David Gregory l of Oxford
and himself. In 1696 Johann Bernoulli challenged the best mathemati-
cians in Europe to solve the difficult problem, to find the curve (the
cycloid) along which a body falls from one point to another in the
shortest possible time. Leibniz solved it the day he received it.
Newton, de 1'Hospital, and the two Bernoullis gave solutions. New-
ton's appeared anonymously in the Philosophical Transactions, but
Johann Bernoulli recognized in it his powerful mind, "tanquam," he
Ws, "ex ungue leonem." The problem of orthogonal trajectories (a
system of curves described by a known law being given, to describe
a curve which shall cut them all at right angles) was proposed by
Johann Bernoulli in a letter to G. W. Leibniz in 1694. Later it was
Ipng printed in the Acta eruditorum, but failed at first to receive much

1 Phil. Trans., London, 1697.

2i8 A HISTORY OF MATHEMATICS

attention. It was again proposed in 1716 by Leibniz, to feel the pulse
of the English mathematicians.

This may be considered as the first defiance problem professedly
aimed at the English. Newton solved it the same evening on which
it was delivered to him, although he was much fatigued by the day's
work at the mint. His solution, as published, was a general plan of
an investigation rather than an actual solution, and was, on that
account, criticised by Johann Bernoulli as being of no value. Brook
Taylor undertook the defence of it, but ended by using very repre-
hensible language. Johann Bernoulli was not to be outdone in in-
civility, and made a bitter reply. Not long afterwards Taylor sent
an open defiance to Continental mathematicians of a problem on the
integration of a fluxion of complicated form which was known to
very few geometers in England and supposed to be beyond the power
of their adversaries. The selection was injudicious, for Johann
Bernoulli had long before explained the method of this and similar
integrations. It served only to display the skill and augment the
triumph of the followers of Leibniz. The last and most unskilful
challenge was by John Keill. The problem was to find the path of a
projectile in a medium which resists proportionally to the square of
the velocity. Without first making sure that he himself could solve
it, Keill boldly challenged Johann Bernoulli to produce a solution.
The latter resolved the question in very short time, not only for a
resistance proportional to the square, but to any power of the velocity.
Suspecting the weakness of the adversary, he repeatedly offered to
send his solution to a confidential person in London, provided Keill
would do the same. Keill never made a reply, and Johann Bernoulli
abused him and cruelly exulted over him.1

The explanations of the fundamental principles of the calculus, as
given by Newton and Leibniz, lacked clearness and rigor. For that
reason it met with opposition from several quarters. In 1694 Bernhard
Nieuwentijt (1654-1718) of Holland denied the existence of differentials
of higher orders and objected to the practice of neglecting infinitely
small quantities. These objections Leibniz was not able to meet

satisfactorily. In his reply he said the value of ~ in geometry could

be expressed as the ratio of finite quantities. In the interpretation
of dx and dy Leibniz vacillated.2 At one time they appear in his
writings as finite lines; then they are called infinitely small quantities,
and again, quantitates inassignabiles, which spring from quantitates
assignabiles by the law of continuity. In this last presentation Leibniz
approached nearest to Newton.

1 John Playfair, "Progress of the Mathematical and Physical Sciences" in
Encyclopedia Brilannica, 7th Ed., continued in the 8th Ed. by Sir John Leslie.

2 Consult G. Vivanti, II concetto d'lnfinitesimo. Saggio storico. Nuova edizione.
Napoli, 1901.

NEWTON TO EULER 219

In England the principles of fluxions were boldly attacked by
Bishop George Berkeley (1685-1753), the eminent metaphysician, in
a publication called the Analyst (1734). He argued with great acute-
ness, contending, among other things, that the fundamental idea of
supposing a finite ratio to exist between terms absolutely evanescent —
"the ghosts of departed quantities," as he called them — was absurd
and unintelligible. Berkeley claimed that the second and third
fluxions were even more mysterious than the first fluxion. His con-
tention that no geometrical quantity can be exhausted by division is
in consonance with the claim made by Zeno in his "dichotomy,"
and the claim that the actual infinite cannot be realized. Most modern
readers recognize these contentions as untenable. Berkeley declared
as axiomatic a lemma involving the shifting of the hypothesis: If .v
receives ah increment i, where i is expressly supposed to be some
quantity, then the increment of xn, divided by i, is found to be nxtl~~ '+
n(n—i)/-2 xn~~2i+ ... If now you take i=o, the hypothesis is shifted,
and there is a manifest sophism in retaining any result that was ob-
tained on the supposition that i is not zero. Berkeley's lemma found
no favor among English mathematicians until 1803 when Robert
Woodhouse openly accepted it. The fact that correct results are
obtained in the differential calculus by incorrect reasoning is explained
by Berkeley on the theory of " a compensation of errors." This theory
was later advanced also by Lagrange and L. N. M. Carnot. The
publication of Berkeley's Analyst was the most spectacular mathe-
matical event of the eighteenth century in England. Practically all
British discussions of fluxional concepts of that time involve issues
raised by Berkeley. Berkeley's object in writing the Analyst was to
show that the principles of fluxions are no clearer than those of Chris-
tianity. He referred to an " infidel m ithematician " (Edmund Halley) ,
of whom the story is told 1 that, when he jested concerning theological
questions, he was repulsed by Newton with the remark, "I have
studied these things; you have not." A friend of Berkeley, when on a
bed of sickness, refused spiritual consolation, because the great
mathematician Halley had convinced him of the inconceivability of
the doctrines of Christianity. This induced Berkeley to write the
Analyst.

Replies to the Analyst were published by James Jurin (1684-1750)
of Trinity College, Cambridge under the pseudonym of "Philalethes
Cantabrigiensis " and by John Walton of Dublin. There followed
several rejoinders. Jurin's defence of Newton's fluxions did not meet
the approval of the mathematician, Benjamin Robins (1707-1751).
In a Journal, called the Republick of Letters (London) and later in
the Works of the Learned, a long and acrimonious controversy was
carried on between Jurin and Robins, and later between Jurin and
Henry Pemberton (1694-1771), the editor of the third edition of
1 Mach Mechanics, 1907, pp. 448-449.

220 A HISTORY OF MATHEMATICS

Newton's Principia. The question at issue was the precise meaning
of certain passages in the writings of Newton: Did Newton hold that
there are variables which reach their limits? Jurin answered "Yes";
Robins and Pemberton answered "No." The debate between Jurin
and Robins is important in the history of the theory of limits. Though
holding a narrow view of the concept of a limit Robins deserves credit
for rejecting all infinitely small quantities and giving a logically quite,
coherent presentation of fluxions in a pamphlet, called A Discourse
concerning the Nature and Certainty of Sir Isaac Newton's Methods of
Fluxions, 1735. This and Maclaurin's Fluxions, 1742, mark the top-
notch of mathematical rigor, reached during the eighteenth century
in the exposition of the calculus. Both before and after the period
of eight years, 1834-1842, there existed during the eighteenth century
in Great Britain a mixture of Continental and British conceptions of
the new calculus, a superposition of British symbols and phraseology
upon the older Continental concepts. Newton's notation was poor and
Leibniz's philosophy of the calculus was poor. The mixture repre-
sented the temporary survival of the least fit of both systems. The
subsequent course of events was a superposition of the Leibnizian
notation and phraseology upon the limit-concept as developed by
Newton, Jurin, Robins, Maclaurin, D'Alembert and later writers.

In France Michel Rolle for a time rejected the differential calculus
and had a controversy with P. Varignon on the subject. Perhaps the
most powerful argument in favor of the new calculus was the con-
sistency of the results to which it led. Famous is D'Alemhert's advice
to young students: "Allez en avant, et la foi vous viendra."

Among the most vigorous promoters of the calculus on the Conti-
nent were the Bernoullis. They and Euler made Basel in Switzerland
famous as the cradle of great mathematicians. The family of Ber-
noullis furnished in course of a century eight members who distin-
guished themselves in mathematics. We subjoin the following genea-
logical table: —

Nicolaus Bernoulli, the Father
Jakob, 1654-1705 Nicolaus Johann, 1667-1748

Nicolaus, 1687-1759 Nicolaus, 1695-1726
Daniel, 1700-1782
Johann, 1710-1790

Daniel Johann, 1744-1807 Jakob, 1759-1789

Most celebrated were the two brothers Jakob (James) and Johann
(John), and Daniel, the son of John. Jakob and Johann were staunch
friends of G. W. Leibniz and worked hand in hand with him. Jakob
(James) Bernoulli (1654-1705) was born in Basel. Becoming inter-

NEWTON TO EULER 221

ested in the calculus, he mastered it without aid from a teacher.
From 1687 until his death he occupied the mathematical chair at the
University of Basel. He was the first to give a solution to Leibniz's
problem of the isochronous curve. In his solution, published in the
Acta eruditorwn, 1690, we meet for the first time with the word
integral. Leibniz had called the integral calculus calculus summatorius,
but in 1696 the term calculus integralis was agreed upon between
Leibniz and Johann Bernoulli. Jakob Bernoulli gave in 1694 in the
Acta eruditorum the formula for the radius of curvature in rectangular
co-ordinates. At the same time he gave the formula also in polar co-
ordinates. He was one of the first to use polar co-ordinates in a gen-
eral manner and not simply for spiral shaped curves.1 Jakob proposed
the problem of the catenary, then proved the correctness of Leibniz's
construction of this curve, and solved the more complicated problems,
supposing the string to be (i) of variably density, (2) extensible,
(3) acted upon at each point by a force directed to a fixed centre. Of
these problems he published answers without explanations, while his
brother Johann gave in addition their theory. He determined the
shape of the "elastic curve" formed by an elastic plate or rod fixed
at one end and bent by a weight applied to the other end; of the
"lintearia," a flexible rectangular plate with two sides fixed hori-
zontally at the same height, filled with a liquid; of the "velaria," a
rectangular sail filled with wind. In the Acta erudilorum of 1694 he
makes reference to the lemniscate, a curve which "formam refert
jacentis notoe octonarii oo, seu complicity in nodum fasciae, sive
lemnisci." That this curve is a special case of Cassini's oval remained
long unnoticed and was first pointed out by Pietro Ferroni in 1782
and G. Saladini in 1806. Jakob Bernoulli studied the loxodromic and
logarithmic spirals, in the last of which he took, particular delight
from its remarkable property of reproducing itself under a variety of
conditions. Following the example of Archimedes, he willed that the
curve be engraved upon his tombstone with the inscription "eadent
mutata resurgo." In 1696 he proposed the famous problem of isoper-
imetrical figures, and in 1701 published his own solution. He wrote
a work on Ars Conjectandi, published in 1713, eight years after his
death. It consists of four parts. The first contains Huygens' treatise
on probability, with a valuable commentary. The second part is on
permutations and combinations, which he uses in a proof of the bi-
nomial theorem for the case of positive integral exponents; it contains
a formula for the sum of the rth powers of the first n integers, which in-
volves the so-called "numbers of Bernoulli." He could boast that
by means of it he calculated intra semi-quadrantem horae the sum of
the loth powers of the first thousand integers. The third part con-
tains solutions of problems on probability. The fourth part is the
most important, even though left incomplete. It contains "Ber-
1 G. Enestrom in Bibliolheca mathematica, 3. S., Vol. 13, 1912, p. 76.

222 A HISTORY OF MATHEMATICS

noulli's theorem": If (r+s)nt, where the letters are integers and
l=r+s, is expanded by the binomial theorem, then by taking n large
enough the ratio of u (denoting the sum of the greatest term and the
n preceding terms and the n following terms) to the sum of the re-
maining terms may be made as great as we please. Letting r and s
be proportional to the probability of the happening and failing of an
event in a single trial, then u corresponds to the probability that in nt
trials the number of times the event happens will lie between n(r— i)
and n(r+i), both inclusive. Bernoulli's theorem "will ensure him a
permanent place in the history of the theory of probability." l Prom-
inent contemporary workers on probability were Montmort in France
and De Moivre in England. In December, 1913, the Academy of
Sciences of Petrograd celebrated the bicentenary of the "law of large
numbers," Jakob Bernoulli's Ars conjectandi having been published
at Basel in 1713. Of his collected works, in three volumes, one was
printed in 1713, the other two in 1744.

Johann (John) Bernoulli (1667-1748) was initiated into mathe-
matics by his brother. He afterwards visited France, where he met
Nicolas Malebranche, Giovanni Domenico Cassini, P. de Lahire, P.
Varignon, and G. F. de 1'Hospital. For ten years he occupied the
mathematical chair at Grdningen and then succeeded his brother at
Basel. He was one of the most enthusiastic teachers and most suc-
cessful original investigators of his time. He was a member of almost
every learned society in Europe. His controversies were almost as
numerous as his discoveries. He was ardent in his friendships, but
unfair, mean, and violent toward all who incurred his dislike-^even
his own brother and son. He had a bitter dispute with Jakob on the
isoperimetrical problem. Jakob convicted him of several paralogisms.
After his brother's death he attempted to substitute a disguised solu-
tion of the former for an incorrect one of his own. Johann admired
the merits of G. W. Leibniz and L. Euler, but was blind to those of
I. Newton. He immensely enriched the integral calculus by his labors.
Among his discoveries are the exponential calculus, the line of swiftest
descent, and its beautiful relation to the path described by a ray
passing through strata of variable density. In 1694 he explained in
a letter to 1'Hospital the method of evaluating the indeterminate

form -. He treated trigonometry by the analytical method, studied

caustic curves and trajectories. Several times he was given prizes
by the Academy of Science in Paris.

Of his sons, Nicolaus and Daniel were appointed professors of
mathematics at the same time in the Academy of St. Petersburg. The
former soon died in the prime of life; the latter returned to Basel in
1733, where he assumed the chair of experimental philosophy. His

1 1. Todhunter, History of Theor. of Prob., p. 77.

NEWTON TO EULER 223

first mathematical publication was the solution of a differential equa-
tion proposed by J. F. Riccati. He wrote a work on hydrodynamics.
He was the first to use a suitable notation for inverse trigonometric
functions; in 1729 he used AS. to represent arcsine; L. Euler in 1736
used At for arctangent.1 Daniel Bernoulli's investigations on prob-
ability are remarkable for their boldness and originality. He pro-
posed the theory of moral expectation, which he thought would give
results more in accordance with our ordinary notions than the theory
of mathematical probability. He applies his moral expectation to the
so-called "Petersburg problem": A throws a coin in the air; if head
appears at the first throw he is to receive a shilling from B, if head
does not appear until the second throw he is to receive 2 shillings, if
head does not appear until the third throw he is to receive 4 shillings,
and so on: required the expectation of A. By the mathematical
theory, A's expectation is infinite, a paradoxical result. A given sum
of money not being of equal importance to every man, account should
be taken of relative values. Suppose A starts with a sum a, then the
moral expectation in the Petersburg problem is finite, according to
Daniel Bernoulli, when a is finite; it is 2 when a=o, about 3 when
a= 10, about 6 when a- iooo.2 The Petersburg problem was discussed
by P. S. Laplace, S. D. Poisson and G. Cramer. Daniel Bernoulli's
"moral expectation" has become classic, but no one ever makes use
of it. He applies the theory of probability to insurance; to determine
the mortality caused by small-pox at various stages of life; to deter-
mine the number of survivors at a given age from a given number of
births; to determine how much inoculation lengthens the average
duration of life. He showed how the differential calculus could be
used in the theory of probability. He and L. Euler enjoyed the honor
of having gained or shared no less than ten prizes from the Academy
of Sciences in Paris. Once, while travelling with a learned stranger
who asked his name, he said, "I am Daniel Bernoulli." The stranger
could not believe that his companion actually was that great celebrity,
and replied "I am Isaac Newton."

Johann Bernoulli (born 1710) succeeded his father in the professor-
ship of mathematics at Basel. He captured three prizes (on the cap-
stan, the propagation of light, and the magnet) from the Academy of
Sciences at Paris. Nicolaus Bernoulli (born 1687) held for a time the
mathematical chair at Padua which Galileo had once filled. He proved

92A
in 1742 that -- = -- . Johann Bernoulli (born 1744) at the age

QlQU

of nineteen was appointed astronomer royal at Berlin, and after-
wards director of the mathematical department of the Academy. His
brother Jakob took upon himself the duties of the chair of experi-

1 G. Enestrom in Bibliotheca mathematica, Vol. 6, pp. 319-321; Vol. 14, p. 78.

2 1. Todhunter, Hist, of the Theor. of Prob., p. 220.

224 A HISTORY OF MATHEMATICS

mental physics at Basel, previously performed by his uncle Jakob,
and later was appointed mathematical professor in the Academy at
St. Petersburg.

Brief mention will now be made of some other mathematicians
belonging to the period of Newton, Leibniz, and the elder Bernoullis.

Guillaume Francois Antoine PHospital (1661-1704), a pupil of
Johann Bernoulli, has already been mentioned as taking part in the
challenges issued by Leibniz and the Bernoullis. He helped power-
fully in making the calculus of Leibniz better known to the mass of
mathematicians by the publication of a treatise thereon, the Analyse
des infiniment petits, Paris, 1696. This contains the method of finding
the limiting value of a fraction whose two terms tend toward zero
at the same time, due to Johann Bernoulli.

Another zealous French advocate of the calculus was Pierre Varig-
non (1654-1722). In Mem. de Paris, Annee MDCCIV, Paris, 1722, he
follows Ja. Bernoulli in the use of polar co-ordinates, p and co. Letting
x= p and y=lu>, the equations thus changed represent wholly different
curves. For instance, the parabolas xm=am~ly become Fermatian
spirals. Joseph Saurin (1659-1737) solved the delicate problem of
how to determine the tangents at the multiple points of algebraic
curves. Francois Nicole (1683-1758) in 1717 issued an elementary
treatise on finite differences, in which- he finds the sums of a consider-
able number of interesting series. He wrote also on roulettes, particu-
larly spherical epicycloids, and their rectification. Also interested in
finite differences was Pierre Raymond de Montmort (1678-1719).
His chief writings, on the theory of probability, served to stimulate
his more distinguished successor, De Moivre. Montmort gave the
first general solution of the Problem of Points. Jean Paul de Gua
(1713-1785) gave the demonstration of Descartes' rule of signs, now
given in books. This skilful geometer wrote in 1740 a work on analyt-
ical geometry, the object of which was to show that most investiga-
tions on curves could be carried on with the analysis of Descartes quite
as easily as with the calculus. He shows how to find the tangents,
asymptotes, and various singular points of curves of all degrees, and
proves by perspective that several of these points can be at infinity.
Michel Rolle (1652-1719) is the author of a theorem named after him.
That theorem is not found in his Traite d'algebre of 1690, but occurs
in his Methode pour rcsoudre les egalitez, Paris, 1691. l The name
"Rolle's theorem" was first used by M. W. Drobisch (1802-1896) of
Leipzig in 1834 and by Giusto Bella vitis in 1846. His Algebre contains
his "method of cascades." In an equation in v which he has trans-
formed so that its signs become alternately plus and minus, he puts
v=x+z and arranges the result according to the descending powers
of x. The coefficients of xn, x"'1, . . ., when equated to zero, are

1 See F. Cajori on the history of Rolle's Theorem in Bibliotheca mathemalica,
3rd S., Vol. II, 1911, pp. 300-313.

NEWTON TO EULER 225

called "cascades." They are the successive derivatives of the original
equation in v, each put equal to zero. Now comes a theorem which
in modern version is: Between two successive real roots of f(v) = o
there cannot be more than one real root of f(v)=o. To ascertain the
root-limits of a given equation, Rolle begins with the cascade of lowest
degree and ascends, solving each as he proceeds. This process is very
laborious.

Of Italian mathematicians, Riccati and Fagnano must not remain
unmentioned. Jacopo Francesco, Count Riccati (1676-1754) is best
known in connection with his problem, called Riccati's equation,
published in the Acta eruditorum in 1724. He succeeded in integrating
this differential equation for some special cases. Long before this
Jakob Bernoulli had made attempts to solve this equation, but with-
out success. A geometrician of remarkable power was Giulio Carlo,
Count de Fagnano (1682-1766). He discovered the following for-

' T _ /t

mula, ir=2i log - ., in which he anticipated L. Euler in the use of

imaginary exponents and logarithms. His studies on the rectification
of the ellipse and hyperbola are the starting-points of the theory of
elliptic functions. He showed, for instance, that two arcs of an ellipse
can be found in an indefinite number of ways, whose difference is
expressible by a right line. In the rectification of the lemniscate he
reached results which connect with elliptic functions; he showed that
its arc can be divided geometrically in n equal parts, if n is 2 • 2m,
3 • 2m, or 5 • 2OT. He gave expert advice to Pope Benedict XIV re-
garding the safety of the cupola of St. Peter's at Rome. In return
the Pope promised to publish his mathematical productions. For
some reason the promise was not fulfilled and they were not published
until 1750. Fagnano's mathematical works were re-published in 1911
and 1912 by the Italian Society for the Advancement of Science.

In Germany the only noted contemporary of Leibniz is Ehrenfried
Walter Tschirnhausen (1651-1708), who discovered the caustic of
reflection, experimented on metallic reflectors and large burning-
glasses, and gave a method of transforming equations named after
him. He endeavored to solve equations of any degree by removing
all the terms except the first and last. This procedure had been tried
before him by the Frenchman Francois Dulaurens and by the Scotch-
man James Gregory.1 Gregory's Vera circuli et hyperboles quadrature,
(Patavii, 1667) is noteworthy as containing a novel attempt, namely,
to prove that the quadrature of the circle cannot be effected by the
aid of algebra. His ideas were not understood in his day, not even by
C. Huygens with whom he had a controversy on this subject. James
Gregory's proof could not now be considered binding. Believing that
the most simple methods (like those of the ancients) are the most

1 G. Enestrom in Bibliolhcca mathemalica, 3. S., Vol. 9, 1908-9, pp. 258, 259.

226 A HISTORY OF MATHEMATICS

correct, Tschirnhausen concluded that in the researches relating to
the properties of curves the calculus might as well be dispensed with.

After the death of Leibniz there was in Germany not a single mathe-
matician of note. Christian Wolf (1679-1754), professor at Halle,
was ambitious to figure as successor of Leibniz, but he "forced the
ingenious ideas of Leibniz into a pedantic scholasticism, and had the
unenviable reputation of having presented the elements of the arith-
metic, algebra, and analysis developed since the time of the Renais-
sance in the form of Euclid, — of course only in outward form, for into
the spirit of them he was quite unable to penetrate" (H. Hankel).

The contemporaries and immediate successors of Newton in Great
Britain were men of no mean merit. We have reference to R. Cotes,
B. Taylor, L. Maclaurin, and A. de Moivre. We are told that at the
death of Roger Cotes (1682-1716), Newton exclaimed, "If Cotes had
lived, we might have known something." It was at the request of
Dr. Bentley that R. Cotes undertook the publication of the second
edition of Newton's Principia. His mathematical papers were pub-
lished after his death by Robert Smith, his successor in the Plumbian
professorship at Trinity College. The title of the work, Harmonia
Mensurarum, was suggested by the following theorem contained in it:
If on each radius vector, through a fixed point O, there be taken a
point R, such that the reciprocal of OR be the arithmetic mean of the
reciprocals of OR i,OR 2, . . . ORn, then the locus of R will be a straight
line. In this work progress was made in the application of logarithms
and the properties of the circle to the calculus of fluents. To Cotes
we owe a theorem in trigonometry which depends on the forming of
factors of xn — i. In the Philosophical Transactions of London, pub-
lished 1714, he develops an important formula, reprinted in his Har-
monia Mensurarum, which in modern notation is i 0=log (cos<f>+i.
sintf).} Usually this formula is attributed to L. Euler. Cotes studied
the curve p20=a2, to which he gave the name "lituus." Chief among
the admirers of Newton were B. Taylor and C. Maclaurin. The quar-
rel between English and Continental mathematicians caused them to
work quite independently of their great contemporaries across the
Channel.

Brook Taylor (1685-1731) was interested in many branches of
learning, and in the latter part of his life engaged mainly in religious
and philosophic speculations. His principal work, Methodus incre-
mentorum directa et inversa, London, 1715-1717, added a new branch
to mathematics, now called "finite differences," of which he was the
inventor. He made many important applications of it, particularly
to the study of the form of movement of vibrating strings, the reduc-
tion of which to mechanical principles was first attempted by him.
This work contains also "Taylor's theorem," and, as a special case
of it, what is now called "Maclaurin's Theorem." Taylor discovered
his theorem at least three years before its appearance in print. He

NEWTON TO EULER 227

gave it in a letter to John Machin, dated July 26, 1712. Its importance
was not recognized by analysts for over fifty years, until J. Lagrange
pointed out its power. His proof of it does not consider the question
of convergency, and is quite worthless. The first more rigorous proof
was given a century later by A. L. Cauchy. Taylor gave a singular
solution of a differential equation and the method of finding that
solution by differentiation of the differential equation. Taylor's
work contains the first correct explanation of astronomical refraction.
He wrote also a work on linear perspective, a treatise which, like his
other writings, suffers for want of fulness and clearness of expression.
At the age of twenty-three he gave a remarkable solution of the prob-
lem of the centre of oscillation, published in 1714. His claim to
priority was unjustly disputed by Johann Bernoulli. In the Philo-
sophical Transactions, Vol. 30, 1717, Taylor applies "Taylor's series"
to the solution of numerical equations. He assumes that a rough
approximation, a, to a root of f(x)=o has been found. Let/(a)=&,
f'(a) = k', f"(a) = k", and x=a+s. He expands o=/(a+s) by his
theorem, discards all powers of s above the second, substitutes the
values of k, k', k", and then solves for s. By a repetition of this
process, close approximations are secured. He makes the important
observation that his method solves also equations involving radicals
and transcendental functions. The first application of the Newton-
Raphson process to the solution of transcendental equations, was
made by Thomas Simpson in his Essays . . . on Mathematicks,
London, 1740.

The earliest to suggest the method of recurring series for finding
roots was Daniel Bernoulli (1700-1782) who in 1728 brought the
quartic to the form i = ax+bx2+cx3+ex4, then selected arbitrarily
four numbers A, B,C, D, and a fifth, E, thus, E=aD+bC+cB+eA,
also a sixth by the same recursion formula F=aE+bD+cC+eB,
and so on. If the last two numbers thus found are M and N, then
x=M-T-N is an approximate root. Daniel Bernoulli gives no proof,
but is aware that there is not always convergence to the root. This
method was perfected by Leonhard Euler in his Introductio in analysin
infinitorum, 1748, Vol. I, Chap. 17, and by Joseph Lagrange in Note
VI of his Resolution des equations numeriques.

Brook Taylor in 1717 expressed a root of a quadratic equation in
the form of an infinite series; for the cubic Francois Nicole did simi-
larly in 1738 and Clairaut in 1746. A. C. Clairaut inserted the process
in his Elements d'algebre. Thomas Simpson determined roots by re-
version of series in 1743 and by infinite series in 1745. Marquis de
Courtivron (1715-1785) also expressed the roots in the form of in-
finite series, while L. Euler devoted several articles to this topic.1

At this time the matter of convergence of the series did not receive

1 For references see F. Cajori, in Colorado College Publication, General Series 51,
p. 212.

228 A HISTORY OF MATHEMATICS

proper attention, except in some rare instances. James Gregory of
Edinburgh, in his Vera circuit el hyperbolae, quadratura (1667), first
used the terms "convergent" and "divergent" series, while William
Brouncker gave an argument which amounted to a proof of the con-
vergence of his series, noted above.

Colin Maclaurin (1698-1746) was elected professor of mathematics
at Aberdeen at the age of nineteen by competitive examination, and
in 1725 succeeded James Gregory at the University of Edinburgh.
He enjoyed the friendship of Newton, and, inspired by Newton's
discoveries, he published in 1719 his Geometria Organica, containing
a new and remarkable mode of generating conies, known by his name,
and referring to the fact which became known later as "Cramer's
paradox," that a curve of the nth order is not always determined by
%n(n+$) points, that the number may be less. A second tract, De
Limarum geometricarum proprietatibus, 1720, is remarkable for the
elegance of its demonstrations. It is based upon two theorems: the
first is the theorem of Cotes; the second is Maclaurin's: If through
any point 0 a line be drawn meeting the curve in n points, and at
these points tangents be drawn, and if any other line through 0 cut
the curve in RI, RZ, etc., and the system of n tangents in r\, TZ, etc.,

j j

then Z:^T;=27r- This and Cotes' theorem are generalizations of
OR Or

theorems of Newton. Maclaurin uses these in his treatment of curves
of the second and third degree, culminating in the remarkable theorem
that if a quadrangle has its vertices and the two points of intersection
of its opposite sides upon a curve of the third degree, then the tangents
drawn at two opposite vertices cut each other on the curve. He de-
duced independently B. Pascal's theorem on the hexagram. Some
of his geometrical results were reached independently by William
Braikenridge (about 1700 — after 1759), a clergyman in Edinburgh.
The following is known as the " Braikenridge-Maclaurin theorem":
If the sides of a polygon are restricted to pass through fixed points
while all the vertices but one lie on fixed straight lines, the free vertex
describes a conic section or a straight line. Maclaurin's more general
statement (Phil. Trans., 1735) is thus: If a polygon move so that each
of its sides passes through a fixed point, and if all its summits except
one describe curves of the degrees m, n, p, etc., respectively, then the
free summit moves on a curve of the degree 2 mnp . . ., which reduces
to mnp . . . when the fixed points all lie on a straight line. Mac-
laurin was the first to write on "pedal curves," a name due to Olry
Terquem (1782-1862). Maclaurin is the author of an Algebra. The
object of his treatise on Fluxions was to found the doctrine of fluxions
on geometric demonstrations after the manner of the ancients, and
thus, by rigorous exposition, answer such attacks as Berkeley's that
the doctrine rested on false reasoning. The Fluxions contained for

NEWTON TO EULER 229

the first time the correct way of distinguishing between maxima and
minima, and explained their use in the theory of multiple points.
"Maclaurin's theorem" was previously given by B. Taylor and James
Stirling, and is but a particular case of " Taylor's theorem." Maclaurin
invented the trisectrix, x(x2+y2)=a(yz — 3#2), which is akin to the
Folium of Descartes. Appended to the treatise on Fluxions is the
solution of a number of beautiful geometric, mechanical, and as-
tronomical problems, in which he employs ancient methods with
such consummate skill as to induce A. C. Clairaut to abandon analytic
methods and to attack the problem of the figure of the earth by pure
geometry. His solutions commanded the liveliest admiration of J.
Lagrange. Maclaurin investigated the attraction of the ellipsoid of
revolution, and showed that a homogeneous liquid mass revolving
uniformly around an axis under the action of gravity must assume
the form of an ellipsoid of revolution. Newton had given this theorem
without proof. Notwithstanding the genius of Maclaurin, his in-
fluence on the progress of mathematics in Great Britain was unfortu-
nate; for, by his example, he induced his countrymen to neglect
analysis and to be indifferent to the wonderful progress in the higher
analysis made on the Continent.

James Stirling (1692-1770), whom we have mentioned in connec-
tion with C. Maclaurin's theorem and Newton's enumeration of 72
forms of cubic curves (to which Stirling added 4 forms), was educated at
Glasgow and Oxford. He was expelled from Oxford for corresponding
with Jacobites. For ten years he studied in Venice. He enjoyed the
friendship of Newton. His Methodus differentialis appeared in 1730.

It remains for us to speak of Abraham de Moivre (1667-1754)-,
who was of French descent, but was compelled to leave France at
the age of eighteen, on the Revocation of the Edict of Nantes. He
settled in London, where he gave lessons in mathematics. He ranked
high as a mathematician. Newton himself, in the later years of his
life, used to reply to inquirers respecting mathematics, even respecting
his Principia: "Go to Mr. De Moivre; he knows these things better
than I do." He lived to the advanced age of eighty-seven and sank
into a state of almost total lethargy. His subsistence was latterly
dependent on the solution of questions on games of chance and
problems on probabilities, which he was in the habit of giving at a
tavern in St. Martin's Lane. Shortly before his death he declared
that it was necessary for him to sleep ten or twenty minutes longer
every day. The day after he had reached the total of over twenty-
three hours, he slept exactly twenty-four hours and then passed away
in his sleep. De Moivre enjoyed the friendship of Newton and Halley.
His power as a mathematician lay in analytic rather than geometric
investigation. He revolutionized higher trigonometry by the dis-
covery of the theorem known by his name and by extending the
theorems on the multiplication and division of sectors from the circle

230 A HISTORY OF MATHEMATICS

to the hyperbola. His work on the theory of probability surpasses
anything done by any other mathematician except P. S. Laplace.
His principal contributions are his investigations respecting the
Duration of Play, his Theory of Recurring Series, and his extension
of the value of Daniel Bernoulli's theorem by the aid of Stirling's
theorem.1 His chief works are the Doctrine of Chances, 1716, the
Miscellanea Analytica, 1730, and his papers in the Philosophical
Transactions. Unfortunately he did not publish the proofs of his
results in the doctrine of chances, and J. Lagrange more than fifty
years later found a good exercise for his skill in supplying the proofs.
A generalization of a problem first stated by C. Huygens has re-
ceived the name of "De Moivre's Problem:" Given n dice, each
having / faces, determine the chances of throwing any given number
of points. It was solved by A. de Moivre, P. R. de Montmort, P. S.
Laplace and others. De Moivre also generalized the Problem on the
Duration of Play, so that it reads as follows: Suppose A has m counters,
and B has n counters; let their chances of winning in a single game be
as a to b; the loser in each game is to give a counter to his adversary:
required the probability that when or before a certain number of games
has been played, one of the players will have won all the counters of
his adversary. De Moivre's solution of this problem constitutes his
most substantial achievement in the theory of chances. He employed
in his researches the method of ordinary finite differences, or as he
called it, the method of recurrent series.

A famous theory involving the. notion of inverse probability was
advanced by Thomas Bayes. It was published in the London Philo-
sophical Transactions, Vols. 53 and 54 for the years 1763 and 1764,
after the death of Bayes, which occurred in 1761. These researches
originated the discussion of the probabilities of causes as inferred
from observed effects, a subject developed more fully by P. S. Laplace.
Using modern symbols, Bayes' fundamental theorem may be stated
thus: 2 If an event has happened p times and failed q times, the
probability that its chance at a single trial lies between a and b is

/•b pi

XP (i -*)« dx-r- I x* (i — x)« dx.
a Jo

A memoir of John Michell "On the probable Parallax, and Magni-
tude of the fixed Stars" in the London Philosophical Transactions,
Vol. 57 I, for the year 1767, contains the famous argument for the
existence of design drawn from the fact of the closeness of certain
stars, like the Pleiades. "We may take the six brightest of the
Pleiades, and, supposing the whole number of those stars, which are
equal in splendor to the faintest of these, to be about 1500, we shall

1 1. Todhunter, A History of the Mathematical Theory of Probability, Cambridge
and London, 1865, pp. 135-193.
2 I. Todhunter, op. cit., p. 295.

EULER, LAGRANGE AND LAPLACE 231

find the odds to be near 500,000 to i, that no six stars, out of that
number, scattered at random, in the whole heavens, would be within
so small a distance from each other, as the Pleiades are."

Euler, Lagrange, and Laplace

In the rapid development of mathematics during the eighteenth
century the leading part was taken, not by the universities, but by
the academies. Particularly prominent were the academies at Berlin
and Petrograd. This fact is the more singular, because at that time
Germany and Russia did not produce great mathematicians. The
academies received their adornment mainly from the Swiss and
French. It was after the French Revolution that schools gained their
ascendancy over academies.

During the period from 1730 to 1820 Switzerland had her L. Euler;
France, her J. Lagrange, P. S. Laplace, A. M. Legendre, and G. Monge.
The mediocrity of French mathematics which marked the time of
Louis XIV was now followed by one of the very brightest periods of
all history. England, on the other hand, which during the unpro-
ductive period in France had her Newton, could now boast of no great
mathematician. Except young Gauss, Germany had no great name.
France now waved the mathematical sceptre. Mathematical studies
among the English and German people had sunk to the lowest ebb.
Among them the direction of original research was ill chosen. The
former adhered with excessive partiality to ancient geometrical
methods; the latter produced the combinatorial school, which brought
forth nothing of great value.

The labors of L. Euler, J. Lagrange, and P. S. Laplace lay in higher
analysis, and this they developed to a wonderful degree. By them
analysis came to be completely severed from geometry. During the
preceding period the effort of mathematicians not only in England,
but, to some extent, even on the continent, had been directed toward
the solution of problems clothed in geometric garb, and the results of
calculation were usually reduced to geometric form. A change now
took place. Euler brought about an emancipation of the analytical
calculus from geometry and established it as an independent science.
Lagrange and Laplace scrupulously adhered to this separation.
Building on the broad foundation laid for higher analysis and me-
chanics by Newton and Leibniz, Euler, with matchless fertility of
mind, erected an elaborate structure. There are few great ideas pur-
sued by succeeding analysts which were not suggested by L. Euler,
or of which he did not share the honor of invention. With, perhaps,
less exuberance of invention, but with more comprehensive genius and
profounder reasoning, J. Lagrange developed the infinitesimal calculus
and put analytical mechanics into the form in which we now know it.
P. S. Laplace applied the calculus and mechanics to the elaboration

232 A HISTORY OF MATHEMATICS

of the theory of universal gravitation, and thus, largely extending and
supplementing the labors of Newton, gave a full analytical discussion
of the solar system. He also wrote an epoch-marking work on Prob-
ability. Among the analytical branches created during this period
are the calculus of Variations by Euler and Lagrange, Spherical Har-
monics by Legendre and Laplace, and Elliptic Integrals by Legend re.

Comparing the growth of analysis at this time with the growth
during the time of K. F. Gauss, A. L. Cauchy, and recent mathe-
maticians, we observe an important difference. During the former
period we witness mainly a development with reference to form. Plac-
ing almost implicit confidence in results of calculation, mathemati-
cians did not always pause to discover rigorous proofs, and were thus
led to general propositions, some of which have since been found to
be true in only special cases. The Combinatorial School in Germany
carried this tendency to the greatest extreme; they worshipped
formalism and paid no attention to the actual contents of formulae.
But in recent times there has been added to the dexterity in the formal
treatment of problems, a much-needed rigor of demonstration. A
good example of this increased rigor is seen in the present use of in-
finite series as compared to that of Euler, and of Lagrange in his earlier
works.

The ostracism of geometry, brought about by the master-minds of
this period, could not last permanently. Indeed, a new geometric
school sprang into existence in France before the close of this period.
J. Lagrange would not permit a single diagram to appear in his
Mecanique analytique, but thirteen years before his death, G. Monge
published his epoch-making Geometric descriptive.

Leonhard Euler (1707-1783) was born in Basel. His father, a
minister, gave him his first instruction in mathematics and then sent
him to the University of Basel, where he became a favorite pupil of
Johann Bernoulli. In his nineteenth year he composed a dissertation
on the masting of ships, which received the second prize from the
French Academy of Sciences. When Johann Bernoulli's two sons,
Daniel and Nicolaus, went to Russia, they induced Catharine I, in
1727, to invite their friend L. Euler to St. Petersburg, where Daniel,
in 1733, was assigned to the chair of mathematics. In 1735 the solving
of an astronomical problem, proposed by the Academy, for which
several eminent mathematicians had demanded some months' time,
was achieved in three days by Euler with aid of unproved methods of
his own. But the effort threw him into a fever and deprived him of
the use of his right eye. With still superior methods this same problem
was solved later by K. F. Gauss in one hour! 1 The despotism of
Anne I caused the gentle Euler to shrink from public affairs and to
devote all his time to science. After his call to Berlin by Frederick the

1 W. Sartorius Waltershausen, Gauss, zum Gedachtniss, Leipzig, 1856.

EULER, LAGRANGE AND LAPLACE 233

Great in 1741, the queen of Prussia, who received him kindly, won-
dered how so distinguished a scholar should be so timid and reticent.
Euler naively replied, "Madam, it is because I come from a country
where, when one speaks, one is hanged." It was on the recommenda-
tion of D'Alembert that Frederick the Great had invited Euler to
Berlin. Frederick was no admirer of mathematicians and, in a letter
to Voltaire, spoke of Euler derisively as " un gros cyclope de geometre."
In 1766 Euler with difficulty obtained permission to depart from Berlin
to accept a call by Catharine II to St. Petersburg. Soon after his
return to Russia he became blind, but this did not stop his wonderful
literary productiveness, which continued for seventeen years, until
the day of his death. He dictated to his servant his Anleitung zur
Algebra, 1770, which, though purely elementary, is meritorious as
one of the earliest attempts to put the fundamental processes on a
sound basis. -

The story goes that when the French philosopher Denis Diderot
paid a visit to the Russian Court, he conversed very freely and gave
the younger members of the Court circle a good deal of lively atheism.
Thereupon Diderot was informed that a learned mathematician was
in possession of an algebraical demonstration of the existence of God,
and would give it to him before all the Court, if he desired to hear it.
Diderot consented. Then Euler advanced toward Diderot, and said
gravely, and in a tone of perfect conviction: Monsieur, (a+bn)/n=x,
done Dieu existe; rgpondez! Diderot, to whom algebra was Hebrew,
was embarrassed and disconcerted, while peals of laughter rose on all
sides. He asked permission to return to France at once, which was
granted.1

Euler was such a prolific writer that only in the present century
have plans been brought to maturity for a complete edition of his
works. In 1909 the Swiss Natural Science Association voted to publish
Euler's works in their original language. The task is being carried on
with the financial assistance of German, French, American and other
mathematical organizations and of many individual donors. The
expense of publication will greatly exceed the original estimate of
400,000 francs, owing to a mass of new manuscripts recently found in
Petrograd.

The following are his chief works: 2 Introductio in analysin in-
finitorum, 1748, a work that caused a revolution in analytical mathe-
matics, a subject which had hitherto never been presented in so general
and systematic manner; Institutiones calculi dijferentialis , 1755, and
Institutiones calculi integralis, 1768-1770, which were the most com-
plete and accurate works on the calculus of that time, and contained
not only a full summary of everything then known on this subject,

1 From DC Morgan's Budget of Paradoxes, 2. Ed., Chicago, igis, Vol. II, p. 4.

2 See G. Enestrom, Verzeichniss der Schriften Leonhard Eiders, i. Lieferung, 1910,
2. Lieferung, 1913, Leipzig.

234 A HISTORY OF MATHEMATICS

but also the Beta and Gamma Functions and other original investi-
gations; Methodus inveniendi tineas curvas maximi minimive proprietate
gaudentes, 1744, which, displaying an amount of mathematical genius
seldom rivalled, contained his researches on the calculus of variations
to the invention of which Euler was led by the study of the researches
of Johann and Jakob Bernoulli. One of the earliest problems bearing
on this subject was Newton's solid of revolution, of least resistance,
reduced by him in 1686 to a differential equation. (Principia, Bk. II,
Sec. VII, Prop. XXXIV, Scholium.) Johann Bernoulli's problem of
the brachistochrone, solved by him in 1697, and by his brother Jakob
in the same year, stimulated Euler. The study of isoperimetrical
curves, the brachistochrone in a resisting medium and the theory of
geodesies, previously treated by the elder Bernoullis and others, led
to the creation of this new branch of mathematics, the Calculus of
Variations. His method was essentially geometrical, which makes
the solution of the simpler problems very clear. Euler's Theoria
motuum planetarum et cometarum, 1744, Theoria motus luna, 1753,
Theoria motuum luna, 1772, are his chief works on astronomy; Ses
lettres a une princesse d'Allemagne sur quelques sujets de Physique et de
Philosophic, 1770, was a work which enjoyed great popularity.

We proceed to mention the principal innovations and inventions
of Euler. In his Introductio (1748) every "analytical expression" hi
x, i. e. every expression made up of powers, logarithms, trigonometric
functions, etc., is called a "function" of x. Sometimes Euler used
another definition of "function," namely, the relation between y
and x expressed in the x-y plane by any curve drawn freehand, "libero
manus ductu." In modified form, these two rival definitions are
traceable in all later history. Thus Lagrange proceeded on the idea
involved in the first definition, Fourier on the idea involved in the
second.

Euler treated trigonometry as a branch of analysis and consistently
treated trigonometric values as ratios. The term "trigonometric
function" was introduced in 1770 by Georg Simon Klugel (1739-1812)
of Halle, the author of a mathematical dictionary.2 Euler developed
and systematized the mode of writing trigonometric formulas, taking,
for instance, the sinus totus equal to i. He simplified formulas by
the simple expedient of designating the angles of a triangle by A, B, C,
and the opposite sides by a, b, c, respectively. Only once before have
we encountered this simple device. It was used in a pamphlet pre-
pared by Ri. Rawlinson at Oxford sometime between 1655 and 1668. 3
This notation was re-introduced simultaneously with Euler by Thomas
Simpson in England. We may add here that in 1734 Euler used the
notation f(x) to indicate "function of x" that the use of e as the

1 F. Klein, Elementarmathematik v. hoh. Standpunkte aus., I, Leipzig, 1908, p. 438.

2 M. Cantor, op. cit., Vol. IV, 1908, p. 413.

3 See F. Cajori in Nature, Vol. 94, 1915, p. 642.

EULER, LAGRANGE AND LAPLACE 235

symbol for the natural base of logarithms was introduced by him in
1728,* that in 1750 he used 5* to denote the half-sum of the sides of a
triangle, that in 1755 he introduced 2 to signify "summation," that
in 1777 he used i for \/ -i, a notation used later by K. F. Gauss.

We pause to remark that in Euler's time Thomas Simpson (1710-
1761), an able and self-taught English mathematician, for many years
professor at the Royal Military Academy at Woolwich, and author of
several text-books, was active in perfecting trigonometry as a science.
His Trigonometry, London, 1748, contains elegant proofs of two
formulas for plane triangles, (a+b) : c=cos %(A - B) : sin\C and (a — b):
c—sin \(A — B): cos^C, which have been ascribed to the German as-
tronomer Karl Brandan Mollweide (1774-1825), who developed them
much later. The first formula was given in different notation by I.
Newton in his Universal Arithmetique; both formulas are given by
Friedrich Wilhelm Op pel in I746.2

Euler laid down the rules for the transformation of co-ordinates in
space, gave a methodic analytic treatment of plane curves and of
surfaces of the second order. He was the first to discuss the equation
of the second degree in three variables, and to classify the surfaces
represented by it. By criteria analogous to those used in the classi-
fication of conies he obtained five species. He devised a method of
solving- biquadratic equations by assuming x=\/p+-\/q+\/f, with
the hope that it would lead him to a general solution of algebraic
equations. The method of elimination by solving a series of linear
equations (invented independently by E. Bezout) and the method of
elimination by symmetric functions, are due to him. Far reaching"!
are Euler's researches on logarithms. Euler defined logarithms as
exponents,3 thus abandoning the old view of logarithms as terms of
an arithmetic series in one-to-one correspondence with terms of a
geometric series. This union between the exponential and logarithmic
concepts had taken place somewhat earlier. The possibility of de-
fining logarithms as exponents had been recognized by John Wallis
in 1685, by Johann Bernoulli in 1694, but not till 1742 do we find a
systematic exposition of logarithms, based on this idea. It is given
in the introduction to Gardiner's Tables of Logarithms, London, 1742.
This introduction is "collected wholly from the papers" of William
Jones. Euler's influence caused the ready adoption of the new defini-
tion. That this view of logarithms was in every way a step in advance
has been doubted by some writers. Certain it is that it involves in-
ternal difficulties of a serious nature. Euler threw a stream of light
upon the subtle subject of the logarithms of negative and imaginary
numbers. In 1712 and 1713 this subject had been discussed in a

1 G. Enestrom, Bibliotheca mathematica, Vol. 14, 1913-1914, p. 81.

2 A. v. Braunmuhl, op. cit., 2. Toil, 1903, p. 93; H. Wieleitner in Bibliotheca
mathcmatica, 3. S., Vol. 14, pp. 348, 349.

3 See. L. Euler, Inlroductio, 1748, Chap. VI, § 102.

236 A HISTORY OF MATHEMATICS

correspondence between G. W. Leibniz and Johann Bernoulli.1 Leib-
niz maintained that since a positive logarithm corresponds to a number
larger than unity, and a negative logarithm to a positive number less
than unity, the logarithm of - 1 was not really true, but imaginary;
hence the ratio -I-M, having no logarithm, is itself imaginary.
Moreover, if there really existed a logarithm of - 1, then half of it
would be the logarithm of \^~^i, a conclusion which he considered
absurd. The statements of Leibniz involve a double use of the term
imaginary: (i) in the sense of non-existent, (2) in the sense of a number
of the type \/^i. Johann Bernoulli maintained that — i has a
logarithm. Since dx:x=—dx: —x, there results by integration
log (x) = \og ( — x); the logarithmic curve y=\og x has therefore two
branches, symmetrical to the y = axis, as has the hyperbola. The corre-
spondence between Leibniz and Johann Bernoulli was first published
in 1745. In 1714 Roger Cotes developed in the Philosophical Trans-
actions an important theorem which was republished in his Harmonia
mensurarum (1722). In modern notation it is i<f>=log (cos (f>+i sin </>).
In the exponential form it was discovered again by Euler in 1748.
Cotes was aware of the periodicity of the trigonometric functions.
Had he applied this idea to his formula, he might have anticipated
Euler by many years in showing that the logarithm of a number has
an infinite number of different values. A second discussion of the
logarithms of negative numbers took place in a correspondence be-
tween young Euler and his revered teacher, Johann Bernoulli, in the
years 1727-1731. 2 Bernoulli argued, as before, that log x=\og (— x).
Euler uncovered the difficulties and inconsistencies of his own and
Bernoulli's views, without, at that time, being able to advance a
satisfactory theory. He showed that Johann Bernoulli's expression

for the area of a circular sector becomes for a quadrant . — ,

4 V-i

which is incompatible with Bernoulli's claim that log (— i)=o. Be- .
tween 1731 and 1747 Euler made steady progress in the mastery of
relations involving imaginaries. In a letter of Oct. 18, 1740, to
Johann Bernoulli, he stated that y=2 cos x and y=exV^~l+e~x'^~I,

were both integrals of the differential equation -r^,+y=o an'd were

equal to each other. Euler knew the corresponding expression for
sin x. Both expressions are given by him in the Miscellanea Berolinen-
sia, 1743, and again in his Introductio, 1748, Vol. I, 104. He gave the
value V--^"V^~I=0>2078795763 as early as 1746, in a letter to Chris-
tian Goldbach (1690-1764), but makes no reference here to the in-

1 See F. Cajori, "History of the Exponential and Logarithmic Concepts," Amer-
ican Math. Monthly, Vol. 20, 1913, pp. 39-42.

2 See F. Cajori in Am. Math. Monthly, Vol. 20, 1913, pp. 44-46.

EULER, LAGRANGE AND LAPLACE 237

finitely many values of this imaginary expression.1 The creative work
on this topic appears to have been done in 1747. During that year
and the year following Euler debated this subject with D'Alembert
in a correspondence of which only a few letters of Euler are extant.2
In a letter of April 15, 1747, Euler disproves the conclusion upheld by
D'Alembert, that log (— i)=o, and states his own results indicating
that now he had penetrated the subject; log n has an infinite number
of values which are all imaginary, except when n is a positive number,
in which case one logarithm out of this infinite number is real. On
Aug. 19, 1747, he said that he had sent an article to the Berlin Acad-
emy; this is no doubt the article published in 1862 under the title, Sur
les logarithmes des nombres negatifs el imaginaires. The reason why
Euler did not publish it at the time when it was written can only be
conjectured. Our guess is that Euler became dissatisfied with the
article. At any rate, he wrote a new one in 1749, De la controverse
entre Mrs. Leibnitz et Bernoulli sur les logarithmes negatifs et imaginaires.
In 1747 he based the proof that a number has an infinity of logarithms
on the relation i<p=log(cos<p+i siiKp); in 1749 on the assumption
/0g(i + a>) = co, a) being infinitely small. He developed the theory of
logarithms of complex numbers a third time in a paper of 1749 on
Recherches sur les racines imaginaires des equations. The two papers
of 1749 were published in 1751 in the Berlin Memoirs. The latter
primarily aims to prove that every equation has a root; it was dis-
cussed in 1799 by K. F. Gauss in his inaugural dissertation.

Euler's papers were not fully understood and did not carry convic-
tion. D'Alembert still felt that the question was not settled, and ad-
vanced arguments of metaphysical, analytical and geometrical nature
which shrouded the subject into denser haze and helped to prolong
the controversy to the end of the century. In 1759 Darnel de Foncenex
(1734-1799), a young friend of J. Lagrange, wrote on this subject.
In 1768 W. J. G. Karsten (1732-1787), professor at Biitzow, later at
Halle, wrote a long treatise which contains an interesting graphic
representation of imaginary logarithms.3 The debate on Euler's
results was carried on with much warmth by the Italian mathemati-
cians.

The subject of infinite series received new life from him. To his
researches on series we owe the creation of the theory of definite in-
tegrals by the development of the so-called Eulerian integrals. He
warns his readers occasionally against the use of divergent series, but
is nevertheless very careless himself. The rigid treatment to which
infinite series are subjected now was then undreamed of. No clear
notions existed as to what constitutes a convergent series. Neither

1 P. H. Fuss, Corresp. math, et phys. de quelques ctlebres glom&res du
siecle, I, 1843, P- 3&3-

2 See F. Cajori, Am. Math. Monthly, Vol. 20, 1913, pp. 76-79.

3 Am. Math. Monthly, Vol. 20, 1913, p. in.

238 A HISTORY OF MATHEMATICS

G. W. Leibniz nor Jakob and Johann Bernoulli had entertained any
serious doubt of the correctness of the expression 3=1 — 1+1 — 1+ . . .
Guido Grandi (1671-1742) of Pisa went so far as to conclude from
this that ^=0+0+0+ ... In the treatment of series Leibniz ad-
vanced a metaphysical method of proof which held sway over the
minds of the elder Bernoullis, and even of Euler.1 The tendency of
that reasoning was to justify results which seem highly absurd to
followers of Abel and Cauchy. The looseness of treatment can
best be seen from examples. The very paper in which Euler cautions
against divergent series contains the proof that

.-H — hi+w+w2+. . . = o as follows:
w2 n

i — n n n' n—i

these added give zero. Euler has no hesitation to write 1—3+5 — 7
+. . . = o, and no one objected to such results excepting Nicolaus
Bernoulli, the nephew of Johann and Jakob. Strange to say, Euler
finally succeeded in converting Nicolaus Bernoulli to his own erroneous
views. At the present time it is difficult to believe that Euler should
have confidently written sin <j)—2 sin 2</>+3 sin 3</> — 4 sin 40+. . .
=o, but such examples afford striking illustrations of the want of
scientific basis of certain parts of analysis at that time. Euler's proof
of the binomial formula for negative and fractional exponents, which
was widely reproduced in elementary text-books of the nineteenth
century, is faulty. A remarkable development, due to Euler, is what
he named the hypergeometric series, the summation of which he
observed to be dependent upon the integration of a linear differential
equation of the second order, but it remained for K. F. Gauss to point
out that for special values of its letters, this series represented nearly
all functions then known.

Euler gave in 1779 a series for arc tan x, different from the series of
James Gregory, which he applied to the formula 7T = 2o arc tan ^ +
8 arc tan ~3& used for computing IT. The series was published in 1798.
Euler reached remarkable results on the summation of the reciprocal
powers of the natural numbers. In 1736 he had found the sum of the
reciprocal squares to be ?T2/6, and of the reciprocal fourth powers to
be 7T4/9o. In an article of 1743 which until recently has been gen-
erally overlooked,2 Euler finds the sums of the reciprocal even powers
of the natural numbers up to and including the 26th power. Later
he showed the connection of coefficients occurring in these sums with
the "Bernoullian numbers" due to Jakob Bernoulli.

Euler developed the calculus of finite differences in the first chapters

1 R. Reiff, Geschichte der Uncndlichen Reihen, Tubingen, 1889, p. 68.

2 P. Stackel in Bibliotheca mathematica, 3. S., Vol. 8, 1907-8, pp. 37-60.

EULER, LAGRANGE AND LAPLACE 239

of his Institutiones calculi differentiates, and then deduced the differen-
tial calculus from it. He established a theorem on homogeneous func-
tions, known by his name, and contributed largely to the theory of
differential equations, a subject which had received the attention of
I. Newton, G. W. Leibniz, and the Bernoullis, but was still unde-
veloped. A. C. Clairaut, Alexis Fontaine des Bertins (1705-1771),
and L. Euler about the same time observed criteria of integrability,
but Euler in addition showed how to employ them to determine in-
tegrating factors. The principles on which the criteria rested involved
some degree of obscurity. Euler was the first to make a systematic
study of singular solutions of differential equations of the first order.
In 1736, 1756 and 1768 he considered the two paradoxes which had
puzzled A. C. Clairaut: The first, that a solution may be reached by
differentiation instead of integration; the second, that a singular
solution is not contained in the general solution. Euler tried to es-
tablish an a priori rule for determining whether a solution is contained
in the general solution or not. Stimulated by researches of Count
de Fagnano on elliptic integrals, Euler established the celebrated
addition-theorem for these integrals. He invented a new algorithm
for continued fractions, which he employed in the solution of the
indeterminate equation ax+by=c. We now know that substantially
the same solution of this equation was given 1000 years earlier, by
the Hindus. Euler gave 62 pairs of amicable numbers, of which 3
pairs were previously known: one pair had been discovered by the
Pythagoreans, another by Fermat and a third by Descartes.1 By
giving the factors of the number 22"+i when w=5, he pointed out
that this expression did not always represent primes, as was supposed
by P. Fermat. He first supplied the proof to "Fermat's theorem,"
and to a second theorem of Fermat, which states that every prime
of the form 4^+1 is expressible as the sum of two squares in one and
only one way. A third theorem, "Fermat's last theorem," that
xn-\-yn = zn, has no integral solution for values of n greater than 2,
was proved by Euler to be correct when «=4 and w=3- Euler dis-
covered four theorems which taken together make out the great law
of quadratic reciprocity, a law independently discovered by A. M.
Legendre.2

In 1737 Euler showed that the sum of the reciprocals of all prime
numbers is log- (loge=o ), thereby initiating a line of research on the
distribution of primes which is usually not carried back further than
to A. M. Legendre.3

In 1741 he wrote on partitions of numbers ("partitio numerorum").
In 1782 he published a discussion of the problem of 36 officers of six
different grades and from six different regiments, who are to be placed

1See Bibliolhcca mathemalica, 3. S., Vol. q, p. 263; Vol. 14, pp. 351-354.

2 Oswald Baumgart, Ueber das Quadratische Rcciprocitatsgesclz. Leipzig, 1885.

3 G. Enestrom in Bibliolhcca mathematica, 3. S., Vol. 13, 1912, p. 81.

24o A HISTORY OF MATHEMATICS

in a square in such a way that in each row and column there are six
officers, all of different grades as well as of different regiments. Euler
thinks that no solution is obtainable when the order of the square is
of the form 2 mod. 4. Arthur Cayley in 1890 reviewed what had been
written; P. A. MacMahon solved it in 1915. It is called the problem
of the "Latin squares," because Euler, in his notation, used "n lettres
latines." Euler enunciated and proved a well-known theorem, giving
the relation between the number of vertices, faces, and edges of cer-
tain polyhedra, which, however, was known to R. Descartes. The
powers of Euler were directed also towards the fascinating subject
of the theory of probability, in which he solved some difficult
problems.

Of no little importance are Euler's labors in analytical mechanics.
Says Whewell: "The person who did most to give to analysis the
generality and symmetry which are now its pride, was also the person
who made mechanics analytical; I mean Euler." * He worked out
the theory of the rotation of a body around a fixed point, established
the general equations of motion of a free body, and the general equation
of hydrodynamics. He solved an immense number and variety of
mechanical problems, which arose in his mind on all occasions. Thus,
on reading Virgil's lines, "The anchor drops, the rushing keel is staid,"
he could not help inquiring what would be the ship's motion in such
a case. About the same time as Daniel Bernoulli he published the
Principle of the Conservation of Areas and defended the principle of
"least action," advanced by P. Maupertius. He wrote also on tides
and on sound.

Astronomy owes to Euler the method of the variation of arbitrary
constants. By it he attacked the problem of perturbations, explain-
ing, in case of two planets, the secular variations of eccentricities,
nodes, etc. He was one of the first to take up with success the theory
of the moon's motion by giving approximate solutions to the "problem
of three bodies." He laid a sound basis for the calculation of tables
of the moon. These researches on the moon's motion, which captured
two prizes, were carried on while he was blind, with the assistance of
his sons and two of his pupils. His Mechanica sive motus scientia
analytice exposita, Vol. I, 1736, Vol. II, 1742, is, in the language of
Lagrange, "the first great work in which analysis is applied to the
science of movement."

Prophetic was his study of the movements of the earth's pole. He
showed that if the axis around which the earth rotates is not coincident
with the axis of figure, the axis of rotation will revolve about the axis
of figure in a predictable period. On the assumption that the earth
is perfectly rigid he showed that the period is 305 days. The earth
is now known to be elastic. From observations taken in 1884-5,

1 W. Whewell, History of the Inductive Sciences, 3rd Ed., Vol. 1, New York,
1858, p. 363.

EULER, LAGRANGE AND LAPLACE 241

S. C. Chandler of Harvard found the period to be 428 days.1 For
an earth of steel the time has been computed to be 441 days.

Euler in his Introductio in analysin (1748) had undertaken a classi-
fication of quartic curves, as had also a mathematician of Geneva,
Gabriel Cramer (1704-1752), in his Introduction a V analyse des lignes
courbes algebraiques, Geneva, 1750. Both based their classifications
on the behavior of the curves at infinity, obtaining thereby eight
classes which were divided into a considerable number of -species.
Another classification was made by E. Waring, in his Miscellanea
analytica, 1792, which yielded 12 main divisions and 84551 species.
These classifications rest upon ideas hardly in harmony with the
more recent projective methods, and have been abandoned. Cramer
studied the quartic y4 — x*+ay2+bx2=o which later received the at-
tention of F. Moigno (1840), Charles Briot and Jean Claude Bouquet,
and B. A. Nievenglowski (1895), and because of its peculiar form was
called by the French "courbe du diable." Cramer gave also a classi-
fication of quintic curves.

Most of Euler's memoirs are contained in the transactions of the
Academy of Sciences at St. Petersburg, and in those of the Academy
at Berlin. From 1728^.0—1783 a large portion of the Petropolitan
transactions were filled by his writings. He had engaged to furnish
the Petersburg Academ/with memoirs in sufficient number to enrich
its acts for twenty yedrs — a promise more than fulfilled, for down to
1818 the volumes u^ua,lly contained one or more papers of his, and
numerous papers afVstill unpublished. His mode of working was,
first to concentrate his powers upon a special problem, then to solve
separately all problems growing out of the first. No one excelled
him in dexterity of accommodating methods to special problems. It
is easy to see that mathematicians could not long continue in Euler's
habit of writing and publishing. The material would soon grow to
such enormous proportions as to be unmanageable. We are not sur-
prised to see almost the opposite in J. Lagrange, his great successor.
The great Frenchman delighted in the general and abstract, rather
than, like Euler, in the special and concrete. His writings are con-
densed and give in a nutshell what Euler narrates at great length.

Jean-le-Rond D'Alembert (1717-1783) was exposed, when an in-
fant, by his mother in a market by the church of St. Jean-le-Rond,
near the Notre-Dame in Paris, from whjch he derived his Christian
name. He was brought up by the wifeSlf a poor glazier. It is said
that when he began to show signs of great talent, his mother sent for
him, but received the reply, "You are only my step-mother; the
glazier's wife is my mother." His father provided him with a yearly
income. D'Alembert entered upon the study of law, but such was his
love for mathematics, that law was soon abandoned. At the age of
twenty-four his reputation as a mathematician secured for him ad-
1 For details see Nature, Vol. 97, 1916, p. 530.

242 A HISTORY OF MATHEMATICS

mission to the Academy of Sciences. In 1754 he was made permanent
secretary of the French Academy. During the last years of his life
he was mainly occupied with the great French encyclopaedia, which
was begun by Denis Diderot and himself. D'Alembert declined, in
1762, an invitation of Catharine II to undertake the education of her
son. Frederick the Great pressed him to go to Berlin. He made a
visit, but declined a permanent residence there. In 1743 appeared
his Traite de dynamique, founded upon the important general principle
bearing his name: The impressed forces are equivalent to the effective
forces. D'Alembert's principle seems to have been recognized before
him by A. Fontaine, and in some measure by Johann Bernoulli and
I. Newton. D'Alembert gave it a clear mathematical form and made
numerous applications of it. It enabled the laws of motion and the
reasonings depending on them to be represented in the most general
form, in analytical language. D'Alembert applied it in 1744 in a
treatise on the equilibrium and motion of fluids, in 1746 to a treatise
on the general causes of winds, which obtained a prize from the Berlin
Academy. In both these treatises, as also in one of 1747, discussing
the famous problem of vibrating chords, he was led to partial differ-
ential equations. He was a leader among the pioneers in the study of

2 ^.2

such equations. To the equation — % = a? -^, arising in the problem

of vibrating chords, he gave as the general solution,
y=f(x+a£)+4>(x— at},

and showed that there is only one arbitrary function, if y be supposed
to vanish for x=o and x=l. Daniel Bernoulli, starting with a par-
ticular integral given by Brook Taylor, showed that this differential
equation is satisfied by the trigonometric series

. TTX TTt 2TTX 27T/

y= asm -j- • cos -j-+ p sin — j- . cos -y- + . . .,
II It

and claimed this expression to be the most general solution. Thus
Daniel Bernoulli was the first to introduce ''Fourier's series" into
physics. He claimed that his solution, being compounded of an in-
finite number of tones and overtones of all possible intensities, was
a general solution of the problem. Euler denied its generality, on
the ground that, if true, the doubtful conclusion would follow that
the above series represents any arbitrary function of a variable.
These doubts were dispelled by J. Fourier. J. Lagrange proceeded to
find the sum of the above series, but D'Alembert objected to his
process, on the ground that it involved divergent series.1

A most beautiful result reached by D'Alembert, with aid of his
principle, was the complete solution of the problem of the precession
of the equinoxes, which had baffled the talents of the best minds.
1 R. Reiff, op. cit., II. Abschnitt.

EULER, LAGRANGE AND LAPLACE 243

He sent to the French Academy in 1747, on the same day with A. C.
Clairaut, a solution of the problem of three bodies. This had become
a question of universal interest to mathematicians, in which each
vied to outdo all others. The problem of two bodies, requiring the
determination of their motion when they attract each other with
forces inversely proportional to the square of the distance between
them, had been completely solved by I. Newton. The "problem of
three bodies" asks for the motion of three bodies attracting each
other according to the law of gravitation. Thus far, the complete
solution of this has transcended the power of analysis. The general
differential equations of motion were stated by P. S. Laplace, but
the difficulty arises in their integration/ The "solutions" given at
that time are merely convenient methods of approximation in special
cases when one body is the sun, disturbing the motion of the moon
around the earth, or where a planet moves under the influence of the
sun and another planet. The most important eighteenth century
researches on the problem of three bodies are due to J. Lagrange. In
1772 a prize was awarded him by the Paris Academy for his Essai
sur le probleme des trois corps. He shows that a complete solution of
the problem requires only that we know every moment the sides of
the triangle formed by the three bodies, the solution of the . triangle
depending upon two differential equations of the second order and
one differential equation of the third. He found particular solutions
when the triangles remain all similar.

In the discussion of the meaning of negative quantities, of the
fundamental processes of the calculus, of the logarithms of complex
numbers, and of the theory of probability, D'Alembert paid some
attention to the philosophy of mathematics. In the calculus he
favored the theory of limits. He looked upon infinity as nothing but
a limit which the finite approaches without ever reaching it. His
criticisms were not always happy. When students were halted by
the logical difficulties of the calculus, D'Alembert would say, "Allez
en avant, et la foi vous viendra." He argued that when the prob-
ability of an event is very small, it ought to be taken o. A coin is to
be tossed 100 times and if head appear at the last trial, and not before,
A shall pay B 2100 crowns. By the ordinary theory B should give A

1 crown at the start, which should not be; argues D'Alembert, be-
cause B will certainly lose. This view was taken also by Count de
Buffon. D'Alembert raised other objections to the principles of
probability.

The naturalist, Comte de Buffon (1707-1788), wrote an Essai
d'arithmetique morale, 1777. In the study of the Petersburg problem,
he let a child toss a coin 2084 times, which produced 10057 crowns;
there were 1061 games which produced i crown, 494 which produced

2 crowns and so on.1 He was one of the first to emphasize the desir-

1 For references, see I. Todhunter, History of Theory of Probability, p. 346.

244 A HISTORY OF MATHEMATICS

ability of verifying the theory by actual trial. He also introduced
what is called "local probability" by the consideration of problems
that require the aid of geometry. Some studies along this line had
been carried on earlier by John Arbuthnot (1658-1735) and Thomas
Simpson in England. Count de Buffon derived the probability that
a needle dropped upon a plane, ruled with equidistant, parallel lines,
will fall across one of the lines.

The probability of the correctness of judgments determined by a
majority of votes was examined mathematically by J ' ean-Antoine-
Nicolas Caritat de Condor eel (1743-1794). His general conclusions
are not of great importance; they are that voters must be enlightened
men in order to ensure our confidence in their decisions.1 He held
that capital punishment ought to be abolished, on the ground that,
however large the probability of the correctness of a single decision,
there will be a large probability that in the course of many decisions
some innocent person will be condemned.1

Alexis Claude Clairaut (1713-1765) was a youthful prodigy. He
read G. F. de 1'HospitaFs works on the infinitesimal calculus and on
conic sections at the age of ten. In 1731 was published his Recherches
sur les courbes a double courbure, which he had ready for the press
when he was sixteen. It was a work of remarkable elegance and se-
cured his admission to the Academy of Sciences when still under legal
age. In 1731 he gave a proof of the theorem enunciated by I. Newton,
that every cubic is a projection of one of five divergent parabolas.
Clairaut formed the acquaintance of Pierre Louis Moreau de Mauper-
tius (1698-1759), whom he accompanied on an expedition to Lapland
to measure the length of a degree of the meridian. At that time the
shape of the earth was a subject of serious disagreement. I. Newton
and C. Huygens had concluded from theory that the earth was flat-
tened at the poles. About 1712 Jean-Dominique Cassini (1625-1712)
and his son Jacques Cassini (1677-1756) measured an arc extending
from Dunkirk to Perpignan and arrived at the startling result that
the earth is elongated at the poles. To decide between the conflicting
opinions, measurements were renewed. Maupertius earned by his
work in Lapland the title of "earth flattener" by disproving the
Cassinian tenet that the earth was elongated at the poles, and showing
that Newton was right. On his return, in 1743, Clairaut published
a work, Theorie de la figure de la Terre, which was based on the results
of C. Maclaurin on homogeneous ellipsoids. It contains a remarkable
theorem, named after Clairaut, that the sum of the fractions, ex-
pressing the ellipticity and the increase of gravity at the pole is equal
to 2\ times the fraction expressing the centrifugal force at the equator,
the unit of force being represented by the force of gravity at the
equator. This theorem is independent of any hypothesis with respect
to the law of densities of the successive strata of the earth. It em-
1 1. Todhunter, History oj Theory of Prob., Chapter 17.

EULER, LAGRANGE AND LAPLACE 245

bodies most of Clairaut's researches. I. Todhunter says that "in
the figure of the earth no other person has accomplished so much as
Clairaut, and the subject remains at present substantially as he left
it, though the form is different. The splendid analysis which Laplace
supplied, adorned but did not really alter the theory which started
from the creative hands of Clairaut."

In 1752 he gained a prize of the St. Petersburg Academy for his
paper on Theorie de la Lune, in which for the first time modern analysis
is applied to lunar motion. This contained the explanation of the
motion of the lunar apsides. This motion, left unexplained by I.
Newton, seemed to him at first inexplicable by Newton's law, and
he was on the point of advancing a new hypothesis regarding gravi-
tation, when, taking the precaution to carry his calculation to a higher
degree of approximation, he reached results agreeing with observation.
The motion of the moon was studied about the same time by L. Euler
and D'Alembert. Clairaut predicted that "Halley's Comet," then
expected to return, would arrive at its nearest point to the sun on
April 13, 1759, a date which turned out to be one month too late.
He applied the process of differentiation to the differential equation
now known by his name and detected its singular solution. The same
process had been used earlier by Brook Taylor.

In their scientific labors there was between Clairaut and D'Alembert
great rivalry, often far from friendly. The growing ambition of
Clairaut to shine in society, where he was a great favorite, hindered
his scientific work in the latter part of his life.

The astronomer Jean-Dominique Cassini, whom we mentioned
above, is the inventor of a quartic curve which was published in his
son's Elements d' astronomic, 1749. The curve bears the name of
"Cassini's oval" or "general lemniscate." It grew out of the study
of a problem in astronomy.1 Its equation is (x*+yz)2— 2a?(x2— y~) +
a4-c4=o.

Johann Heinrich Lambert (1728-1777), born at Miihlhausen in
Alsace, was the son of a poor tailor. While working at his father's
trade, he acquired through his own unaided efforts a knowledge of
elementary mathematics. At the age of thirty he became tutor in a
Swiss family and secured leisure to continue his studies. In his
travels with his pupils through Europe he became acquainted with
the leading mathematicians. In 1764 he settled in Berlin, where he
became member of the Academy, and enjoyed the society of L. Euler
and J. Lagrange. He received a small pension, and later became
editor of the Berlin Ephemeris. His many-sided scholarship reminds
one of Leibniz. It cannot be said that he was overburdened with
modesty. When Frederick the Great asked him 'in their first inter-
view, which science he was most proficient in, he replied curtly, "All."

1 G. Loria, Ebene Curven (F. Schiitte), I, 1910, p. 208.

246 A HISTORY OF MATHEMATICS

To the emperor's further question, how he attained this mastery, he
said, "Like the celebrated Pascal, by my own self."

In his Cosmological Letters he made some remarkable prophecies
regarding the stellar system. He entered upon plans for a mathe-
matical symbolic logic of the nature once outlined by G. W. Leibniz.
In mathematics he made several discoveries which were extended
and overshadowed by his great contemporaries. His first research
on pure mathematics developed in an infinite series the root x of the
equation xm+px=q. Since each equation of the form axr+bxs=d
can be reduced to xm+px=q in two ways, one or the other of the two
resulting series was always found to be convergent, and to give a
value of x. Lambert's results stimulated L. Euler, who extended the
method to an equation of four terms, and particularly J. Lagrange,
who found that a function of a root of a— x+<f)(x) =o can be expressed
by the series bearing his name. In 1761 Lambert communicated to
the Berlin Academy a memoir (published 1768), in which he proves
rigorously that TT is irrational. It is given in simplified form in Note IV
of A. M. Legendre's Geometric, where the proof is extended to 7T2.
Lambert proved that if x is rational, but not zero, then neither e*
nor tan x can be a rational number; since tan 7r/4= i, it follows that

— or TT cannot be rational. Lambert's proofs rest on the expression

for e as a continued fraction given by L. Euler 1 who in 1737 had sub-
stantially shown the irrationality of e and e2. There were at this
time so many circle squarers that in 1775 the Paris Academy found it
necessary to pass a resolution that no more solutions on the quadrature
of the circle should be examined by its officials. This resolution ap-
plied also to solutions of the duplication of the cube and the trisection
of an angle. The conviction had been growing that the solution of
the squaring of the circle was impossible, but an irrefutable proof
was not discovered until over a century later. Lambert's Freye Per-
spective, 1759 and 1773, contains researches on descriptive geometry,
and entitle him to the honor of being the forerunner of Monge. In
his effort to simplify the calculation of cometary orbits, he was led
geometrically to some remarkable theorems on conies, for instance
this: "If in two ellipses having a common major axis we take two
such arcs that their chords are equal, and that also the sums of the
radii vectores, drawn respectively from the foci to the extremities of
these arcs, are equal to each other, then the sectors formed in each
ellipse by the arc and the two radii vectores are to each other as the
square roots of the parameters'of the ellipses." 2

Lambert elaborated the subject of hyperbolic functions which he
designated by sink x, cosh x, etc. He was, however, not the first to

1 R. C. Archibald in Am. Math. Monthly, Vol. 21, 1914, p. 253.

2 M. Chasles, Geschichte der Geometric, 1839, p. 183.

EULER, LAGRANGE AND LAPLACE 247

introduce them into trigonometry. That honor falls upon Vincenzo
Riccati (1707-1775), a son of Jacopo Riccati.1

In 1770 Lambert published a 7-place table of natural logarithms
for numbers i-ioo. In 1778 one of his pupils, Johann Karl Schulze,
published extensive tables which included the 48-place table of nat-
ural logarithms of primes and many other numbers up to 10,009,
which had been computed by the Dutch artillery officer, Wolfram.
A feat even more remarkable than Wolfram's, was the computation
of the common logarithms of numbers i-ioo and of all primes from
100 to noo, to 61 places, by Abraham Sharp of Yorkshire, who was
some time assistant to Flamsteed at the English Royal Observatory.
They were published in Sharp's Geometry Improv'd, 1717.

John Landen (1719-1790) was an English mathematician whose
writings served as the starting-point of investigations by L. Euler,
J. Lagrange, and A. M. Legendre. Landen's capital discovery, con-
tained in a memoir of 1755, was that every arc of the hyperbola is
immediately rectified by means of two arcs of an ellipse. In his
"residual analysis" he attempted to obviate the metaphysical diffi-
culties of fluxions by adopting a purely algebraic method. J. La-
grange's Calcul des Fonctions is based upon this idea. Landen snowed
how the algebraic expression for the roots of a cubic equation could
be derived by application of the differential and integral calculus.
Most of the time of this suggestive writer was spent in the pursuits
of active life.

Of influence in the teaching of mathematics in England was Charles
Hutton (1737-1823), for many years professor at the Royal Military
Academy of Woolwich. In 1785 he published his Mathematical Tables,
and in 1795 his Mathematical and Philosophical Dictionary, the best
work of its kind that has appeared in the English language. His
Elements of Conic Sections, 1789, is remarkable as being the first work
in which each equation is rendered conspicuous by being printed in
a separate line by itself.2

It is well known that the Newton-Raphson method of approxima-
tion to the roots of numerical equations, as it was handed down from
the seventeenth century, labored under the defect of insecurity in
the process, so that the successive corrections did not always yield
results converging to the true value of the root sought. The removal
of this defect is usually attributed to J. Fourier, but he was anticipated
half a century by /. Raym. Mourraille in his Traite de la resolution
des equations en general, Marseille et Paris, 1768. Mourraille was for
fourteen years secretary of the academy of sciences in Marseille; later
he became mayor of the city. Unlike" I. Newton and J. Lagrange,
Mourraille and J. Fourier introduced also geometrical considerations.
Mourraille concluded that security is insured if the first approximation

1 M. Cantor, op. ciL, Vol. IV, 1908, p. 411.

2 M. Cantor, op. ell., Vol. IV, 1908, p. 465.

248 A HISTORY OF MATHEMATICS

a is so selected that the curve is convex toward the axis of x for the
interval between a and the root. He shows that this condition is
sufficient, but not necessary.1

In the eighteenth century proofs were given of Descartes' Rule of
Signs which its discoverer had enunciated without demonstration.
G. W. Leibniz had pointed out a line of proof, but did not actually
give it. In 1675 Jean Prestet (1648-1690) published at Paris in his
Elemens des mathcmatiques a proof which he afterwards acknowledged
to be insufficient. In 1728 Johann Andreas Segner (1704-1777) pub-
lished at Jena a correct proof for equations having only real roots.
In 1756 he gave a general demonstration, based on the consideration
that multiplying a polynomial by (x—a) increases the number of
variations by at least one. Other proofs were given by Jean Paul de
Gua de Halves (1741), Isaac Milner (1778), Friedrich Wilhelm Stubner,
Abraham Gotthelf Kastner (1745), Edward Waring (1782), /. A.
Grunert (1827), K. F. Gauss (1828). Gauss showed that, if the num-
ber of positive roots falls short of the number of variations, it does so
by an even number. E. Laguerre later extended the rule to poly-
nomials with fractional and incommensurable exponents, and to in-
finite series.2 It was established by De Gua de Malves that the
absence of 2m successive terms indicates 2m imaginary roots, while
the absence of 2w+i successive terms indicates 2W+2 or 2m imagin-
ary roots, according as the two terms between which the deficiency
occurs have like or unlike signs.

Edward Waring (1734-1798) was born in Shrewsbury, studied at
Magdalene College, Cambridge, was senior wrangler in 1757, and
Lucasian professor of mathematics since 1760. He published Mis-
cellanea analytica in 1762, Meditationes algebraicee in 1770, Propridatis
algebraicarum curuarum in 1772, and Meditationes analytica in 1776.
These works contain many new results, but are difficult of compre-
hension on account of his brevity and obscurity of exposition. He is
said not to have lectured at Cambridge, his researches being thought
unsuited for presentation in the form of lectures. He admitted that
he never heard of any one in England, outside of Cambridge, who had
read and understood his researches.

In his Meditationes algebraica are some new theorems on number.
Foremost among these is a theorem discovered by his friend John
Wilson (1741-1793) and universally known as "Wilson's theorem."
Waring gives the theorem, known as " Waring' s theorem," that every
integer is either a cube or the sum of 2, 3, 4, 5, 6, 7, 8 or 9 cubes, either
a fourth power or the sum of 2, 3 . . or 19 fourth powers; this has
never yet been fully demonstrated. Also without proof is given the
theorem that every even integer is the sum of two primes and every

1 See F. Cajori in Bibliothcca mathematics, 3rd S., Vol. n, 1911, pp. 132-137.

2 For references to the publications of these writers, see F. Cajori in Colorado
College Publication, General Series No. 51, 1910, pp. 186, 187.

EULER, LAGRANGE AND LAPLACE 249

odd integer is a prime or the sum of three primes. The part relating
to even integers is generally known as "Goldbach's theorem," but
was first published by Waring. Christian Goldbach communicated
the theorem to L. Euler in a letter of June 30, 1742, but the letter
was not published until 1843 (Corr. math., P. H. Fuss).
Waring held advanced views on the convergence of series.1 He

taught that iH — -H — -H — -+. . . converges when n> i and diverges

3 •"

when n<i. He gave the well-known test for convergence and
divergence which is often ascribed to A. L. Cauchy, in which the
limit of the ratio of the (n+i)th to the nth term is considered. As
early as 1757 he had found the necessary and sufficient relations which
must exist between the coefficients of a quartic and quintic equation,
for two and for four imaginary roots. These criteria were obtained
by a new transformation, namely the one which yields an equation
whose roots are the squares of the differences of the roots of the given
equation. To solve the important problem of the separation of the
roots Waring transforms a numerical equation into one whose roots
are reciprocals of the differences of the roots of the given equation.
The reciprocal of tr;e largest of the roots of the transformed equation
is less than the smallest difference D, between any two roots of the
given equation. If M is an upper limit of the roots of the given equa-
tion, then the subtraction of D, 2D, -$D, etc., from M will give values
which separate all the real roots. In the Meditationes algebraicce of
1770, Waring gives for the first time a process for the approximation
to the values of imaginary roots. If x is approximately a+ib, sub-
stitute x=a+a'+(b+b'}i, expand and reject higher powers of a' and
b'. Equating real numbers to each other and imaginary numbers to
each other, two equations are obtained which yield values of a! and 6'.
Etienne Bezout (1730-1783) was a French writer of popular mathe-
matical school-books. In his Theorie generate des Equations Alge-
briques, 1779, he gave the method of elimination by linear equations
(invented also by L. Euler). This method was first published by him
in a memoir of 1764, in which he uses determinants, without, however,
entering upon their theory. A beautiful theorem, as to the degree of
the resultant goes by his name. He and L. Euler both gave the degree
as in general m . n, the product of the orders of the intersecting loci,
and both proved the theorem by reducing the problem to one of
elimination from an auxiliary set of linear equations. The determi-
nant resulting from Bezout's method is what J. J. Sylvester and later
writers call the Bezoutiant. Bezout fixed the degree of the eliminant
also for a large number of particular cases. "One may say that he
determined the number of finite intersections of algebraic loci, not
only when all the intersections are finite, but also when singular

1 M. Cantor, op. cit., Vol. IV, 1908, p. 275.

250 A HISTORY OF MATHEMATICS

points, or singular lines, planes, etc., at infinity occasion the with-
drawal to infinity of certain of the intersection points; and this at a
time when the nature of such singularities had not been developed." l
Louis Arbogaste (1759-1803) of Alsace was professor of mathe-
matics at Strasburg. His chief work, the Calcul des Derivations, 1800,
gives the method known by his name, by which the successive coeffi-
cients of a development are derived from one another when the ex-
pression is complicated. A. De Morgan has pointed out that the
true nature of derivation is differentiation accompanied by integration.
In this book for the first time are the symbols of operation separated

from those of quantity. The notation Dxy for -p is due to him.

Maria Gaetana Agnesi (1718-1799) of Milan, distinguished as a
linguist, mathematician, and philosopher, filled the mathematical
chair at the University of Bologna during her father's sickness.
Agnesi was a somnambulist. Several times it happened to her that
she went to her study, while in the somnambulist state, made a light,
and solved some problem she had left incomplete when awake. In
the. morning she was surprised to find the solution carefully worked
out on paper.2 In 1748 she published her Instituzioni Analitiche,
which was translated into English in 1801. The "witch of Agnesi"
or "Versiera" is a cubic curve xiy=ai(a — y) treated in Agnesi's In-
stituzioni, but given earlier by P. Fermat in the form (a2 — #2) y=a?.
The curve was discussed by Guido Grandi in his Quadrature, circuit
et hyperboles, Pisa, 1703 and i7io.3 In two letters from Grandi to
Leibniz, in 1713, curves resembling flowers are discussed; in 1728
Grandi published at Florence his Flores geometrici. He considered
curves in a plane, of the typep=r sinwco, and also curves on a sphere.
Recent studies along this line are due to Bodo Habenicht (1895),
E. W. Hyde (1875), H. Wieleitner (1906).

The leading eighteenth century historian of mathematics was Jean
Etienne Montucla (1725-1799) who published a Histoire des mathe-
matiques, in two volumes, Paris, 1758. A second edition of these two
volumes appeared in 1799. A third volume, written by Montucla,
was partly printed when he died; the rest of it was seen through the
press by the astronomer Joseph. Jerome le Francois de Lalande
(1732-1807), who prepared a fourth volume, mainly on the history of
astronomy.4

Joseph Louis Lagrange (1736-1813), one of the greatest mathe-
maticians of all times, was born at Turin and died at Paris. He was
of French extraction. His father, who had charge of the Sardinian

1 H. S. White in Bull. Am. Math. Soc., Vol. 15, 1909, p. 331.

2 Ulntermtdiaire des math&maticiens, Vol. 22, 1915, p. 241.

3 G. Loria, Ebene Cunen (F. Schiitte), I, 1910, p. 79.

4 For details on other mathematical historians, see S. Giinther's chapter in
Cantor, op. cit., Vol. IV, 1908, pp. 1-36.

EULER, LAGRANGE AND LAPLACE 251

military chest, was once wealthy, but lost all he had in speculation.
Lagrange considered this loss his good fortune, for otherwise he might
not have made mathematics the pursuit of his life. While at the
college in Turin his genius did not at once take its true bent. Cicero
and Virgil at first attracted him more than Archimedes and Newton.
He soon came to admire the geometry of the ancients, but the perusal
of a tract of E. Halley roused his enthusiasm for the analytical method,
in the development of which he was destined to reap undying glory.
He now applied himself to mathematics, and in his seventeenth year
he became professor of mathematics in the royal military academy at
Turin. Without assistance or guidance he entered upon a course of
study which in two years placed him on a level with the greatest of
his contemporaries. With aid of his pupils he established a society
which subsequently developed into the Turin Academy. In the first
five volumes of its transactions appear most of his earlier papers.
At the age of nineteen he communicated to L. Euler a general method
of dealing with " isoperimetrical problems," known now as the Cal-
culus of Variations. This commanded Euler's lively admiration, and
he courteously withheld for a time from publication some researches
of his own on this subject, so that the youthful Lagrange might com-
plete his investigations and claim the invention. Lagrange did quite
as much as Euler towards the creation of the Calculus of Variations.
As it came from Euler it lacked an analytic foundation, and this
Lagrange supplied. He separated the principles of this calculus from
geometric considerations by which his predecessor had derived them.
Euler had assumed as fixed the limits of the integral, i. e. the extrem-
ities of the curve to be determined, but Lagrange removed this re-
striction and allowed all co-ordinates of the curve to vary at the same
time. Euler introduced in 1766 the name "calculus of variations,"
and did much to improve this science along the lines marked out by
Lagrange. Lagrange's investigations on the calculus of variations
were published in 1762, 1771, 1788, 1797, 1806.

Another subject engaging the attention of Lagrange at Turin was
the propagation of sound. In his papers on this subject in the Mis-
cellanea Taurinensia, the young mathematician appears as the critic
of I. Newton, and the arbiter between Euler and D'Alembert. By
considering only the particles which are in a straight line, he reduced
the problem to the same partial differential equation that represents
the motions of vibrating strings.

Vibrating strings had been discussed by Brook Taylor, Johann
Bernoulli and his son Daniel, by D'Alembert and L. Euler. In solving
the partial differential equations, D'Alembert restricted himself to
functions which can be expanded by Taylor's series, while Euler
thought that no restriction was necessary, that they could be arbi-
trary, discontinuous. The problem was taken up with great skill by
Lagrange who introduced new points of view, but decided in favor of

2S2 A HISTORY OF MATHEMATICS

Euler. Later, de Condorcet and P. S. Laplace stood on the side of
D'Alembert since in their judgment some restriction upon the arbi-
trary functions was necessary. From the modern point of view,
neither D'Alembert nor Euler was wholly in the right: D'Alembert
insisted upon the needless restriction to functions with a limitless
number of derivatives, while Euler assumed that the differential and
integral calculus could be applied to any arbitrary function.1

It now appears that Daniel Bernoulli's claim that his solution was
a general one (a claim disputed by D'Alembert, J. Lagrange and L.
Euler) was fully justified. The problem of vibrating strings stimu-
lated the growth of the theory of expansions according to trigonometric
functions of multiples of the argument. H. Burkhardt has pointed
out that there was also another line of growth of this subject, namely
the growth in connection with the problem of perturbations, where
L. Euler started out with the development of the reciprocal distance
of two planets according to the cosine of multiples of the angle be-
tween their radii vectoris.

By constant application during nine years, Lagrange, at the age
of twenty-six, stood at the summit of European fame. But his intense
studies had seriously weakened a constitution never robust, and though
his physicians induced him to take rest and exercise, his nervous
system never fully recovered its tone, and he was thenceforth subject
to fits of melancholy.

In 1764 the French Academy proposed as the subject of a prize
the theory of the libration of the moon. It demanded an explanation,
on the principle of universal gravitation, why the moon always turns,
with but slight variations, the same phase to the earth. Lagrange
secured the prize. This success encouraged the Academy to propose
for a prize the theory of the four satellites of Jupiter, — a problem of
six bodies, more difficult than the one of three bodies previously
treated by A. C. Clairaut, D'Alembert, and L. Euler. Lagrange over-
came the difficulties by methods of approximation. Twenty-four
years afterwards this subject was carried further by P. S. Laplace.
Later astronomical investigations of Lagrange are on cometary per-
turbations (1778 and 1783), and on Kepler's problem. His researches
on the problem of three bodies has been referred to previously.

Being anxious to make the personal acquaintance of leading mathe-
maticians, Lagrange visited Paris, where he enjoyed the stimulating
delight of conversing with A. C. Clairaut, D'Alembert, de Condorcet,
the Abbe Marie, and others. He had planned a visit to London, but
he fell dangerously ill after a dinner in Paris, and was compelled to
return to Turin. In 1766 L. Euler left Berlin for St. Petersburg, and
he pointed out Lagrange as the only man capable of filling the place.

1 For details see H. Burkhardt's Entwicklungen nock oscillirenden Funktionen
und Integration der Dijferentidlglelchungcn d-:r mathcmatisclien Physik. Leipzig,
1908, p. 18. This is an exhaustive and valuable history of this topic.

EULER, LAGRANGE AND LAPLACE 253

1 D'Alembert recommended him at the same time. Frederick the Great
thereupon sent a message to Turin, expressing the wish of "the great-
est king of Europe" to have "the greatest mathematician" at his
court. Lagrange went to Berlin, and staid there twenty years. Find-
ing all his colleagues married, and being assured by their wives that
the marital state alone is happy, he married. The union was not a
happy one. His wife soon died. Frederick the Great held him in
high esteem, and frequently conversed with him on the advantages
of perfect regularity of life. This led Lagrange to cultivate regular
habits. He worked no longer each day than experience taught him
he could without breaking down. His papers were carefully thought
out before he began writing, and when he wrote he did so without a
single correction.

During the twenty years in Berlin he crowded the transactions of
the Berlin Academy with memoirs, and wrote also the epoch-making
work called the Mccanique Analytique. He enriched algebra by re^
searches on the solution of equations. There are two methods of;
solving directly algebraic equations, — that of substitution and that
of combination. The former method was developed by L. Ferrari,
F. Vieta, E. W. Tchirnhausen, L. Euler, E. Bezout, and Lagrange;
the latter by C. A. Vandermonde and Lagrange.1 In the method of
substitution the original forms are so transformed that the determina-
tion of the roots is made to depend upon simpler functions (resolvents).
In the method of combination auxiliary quantities are substituted
for certain simple combinations ("types") of the unknown roots of
the equation, and auxiliary equations (resolvents) are obtained for
these quantities with aid of the coefficients of the given equation. In
his Reflexions sur la resolution algebrique des equations, published in
Memoirs of the Berlin Academy for the years 1770 and 1771, Lagrange
traced all known algebraic solutions of equations to the uniform prin-
ciple consisting in the formation and solution of equations of lower
degree whose roots are linear functions of the required roots, and of
the roots of unity. He showed that the quintic cannot be reduced in
this way, its resolvent being of the sixth degree. In this connection
Lagrange had occasion to consider the number of values a rational
function can assume when its variables are permuted in every possible
way. In these studies we see the beginnings of the theory of groups.
.The theorem, that the order of a subgroup is a divisor of the order
of the group is practically established, and is known now as "La-
grange's theorem," although its complete proof was first given about
thirty years later by Pietro Abbati (1768-1842) of Modena in Italy.
Lagrange's researches on the theory of equations were continued after
he left Berlin. In the Resolution des equations numeriques (1798) he
gave among other things, a proof that every equation must have a
root, — a theorem which before this usually had been considered
1 L. Matthieaaen, op, oil., pp. 80-84.

254 A HISTORY OF MATHEMATICS

self-evident. Other proofs of this were given by J. R. Argand, K. F.
Gauss, and A. L. Cauchy. In a note to the above work Lagrange uses
Fermat's theorem and certain suggestions of Gauss in effecting a com-
plete algebraic solution of any binomial equation.

In the Berlin Memoires for the year 1767 Lagrange contributed a
paper, Sur la resolution des equations numeriques. He explains the
separation of the real roots by substituting for x the terms of the
progression, o, D, 2D, . . ., where D must be less than the least dif-
ference between the roots. Lagrange suggested three ways of com-
puting D: One way in 1767, another in 1795 and a third in 1798. The
first depends upon the equation of the squared differences of the roots
of the given equation. E. Waring before this had derived this im-
portant equation, but in 1767 Lagrange had not yet seen Waring's
writings. Lagrange finds equal roots by computing the highest com-
mon factor between f(x) and /'(#). He proceeds to develop a new
mode of approximation, that by continued fractions. P. A. Cataldi
had used these fractions in extracting square roots. Lagrange enters
upon greater details in his Additions to his paper of 1767. Unlike
the older methods of approximation, Lagrange's has no cases of
failure. "Cette methode ne laisse, ce me semble, rien a desirer," yet,
though theoretically perfect, it yields the root in the form of a con-
tinued fraction which is undesirable in practice.

While in Berlin Lagrange published several papers on the theory
of numbers. In 1769 he gave a solution in integers of indeterminate
equations of the second degree, which resembles the Hindu cyclic
method; he was the first to prove, in 1771, "Wilson's theorem," enun-
ciated by an Englishman, John Wilson, and first published by E.
Waring in his Meditationes Algebraic^; he investigated in 1775 under
what conditions ± 2 and =t 5 (— i and =fc= 3 having been discussed by
L. Euler) are quadratic residues, or non-residues of odd prime num-
bers, q; he proved in 1770 Bachet de Meziriac's theorem that every
integer is equal to the sum of four, or a less number, of squares. He
proved Fermat's theorem on xn+yn=2n, for the case n—4, also Fer-
mat's theorem that, if a2+62=c2, then ab is not a square.

In his memoir on Pyramids, 1773, Lagrange made considerable use
of determinants of the third order, and demonstrated that the square
of a determinant is itself a determinant. He never, however, dealt
explicitly and directly with determinants; he simply obtained acci-
dentally identities which are now recognized as relations between
determinants.

Lagrange wrote much on differential equations. Though the sub-
ject of contemplation by the greatest mathematicians (L. Euler,
D'Alembert, A. C. Clairaut, J. Lagrange, P. S. Laplace), yet more
than other branches of mathematics do they resist the systematic
application of fixed methods and principles. The subject of singular
solutions, which had been taken up by P. S. Laplace in 1771 and 1774,

EULER, LAGRANGE AND LAPLACE 255

was investigated by Lagrange who gave the derivation of a singular
solution from the general solution as well as from the differential
equation itself. Lagrange brought to view the relation of singular
solutions to envelopes. Nevertheless, he failed to remove all mystery
surrounding this subtle subject. An inconsistency in his theorems
caused about 1870 a complete reconsideration of the entire theory of
singular solutions. Lagrange's treatment is given in his Calctd des
Fonctions, Lessons 14-17. He generalized Euler's researches on total
differential equations of two variables, and of the ninth order; he
gave a solution of partial differential equations of the first order (Berlin
Memoirs, 1772 and 1774), and spoke of their singular solutions, ex-
tending their solution in Memoirs of 1779 and 1785 to equations of
any number of variables. The Memoirs of 1772 and 1774 were refined
in certain points by a young mathematician Paul Charpit (?-i784)
whose method of solution was first printed in Lacroix's Traite du
calcul, 2. Ed., Paris, i§ 14, T. II, p. 548. The discussion on partial
differential equations of the second order, carried on by D'Alembert,
Euler, and Lagrange, has already been referred to in our account of
D'Alembert.

While in Berlin, Lagrange wrote the "Mccanique Analytique" the(
greatest of his works (Paris, 1788). From the principle of virtual,
velocities he deduced, with aid of the calculus of variations, the whole
system of mechanics so elegantly and harmoniously that it may fitly
be called, in Sir William Rowan Hamilton's words, "a kind of scien-
tific poem." It is a most consummate example of analytic generality.
Geometrical figures are nowhere allowed. "On ne trouvera point de
figures dans cet ouvrage" (Preface). The two divisions of mechanics
— statics and dynamics — are in the first four sections of each carried
out analogously, and each is prefaced by a historic sketch of principles.
Lagrange formulated the principle of least action. In their original
form, the equations of motion involve the co-ordinates x, y, z, of the
different particles m or dm of the system. But x, y, z, are in general
not independent, and Lagrange introduced in place of them any
variables £, i/', </>, whatever, determining the position of the point at
the time. These "generalized co-ordinates " may be taken to be inde-
pendent. The equations of motion may now assume the form

dLdT__dT
dtd? dt* =°;

or when H, i/', <£,... are the partial differential coefficients with
respect to 4, ^, <f>, . . . of one and the same function V, then the form

dT dV

_
dt d% dt d£ "

The latter is par excellence the Lagrangian form of the equations of
motion. With Lagrange originated the remark that mechanics may

256 A HISTORY OF MATHEMATICS

be regarded as a geometry of four dimensions. To him falls the honor
of the introduction of the potential into dynamics. Lagrange was
anxious to have his Mecanique Analytique published in Paris. The
work was ready for print in 1786, but not till 1788 could he find a
publisher, and then only with the condition that after a few years
he would purchase all the unsold copies. The work was edited by
A. M. Legendre.

After the death of Frederick the Great, men of science were no
longer respected in Germany, and Lagrange accepted an invitation
of Louis XVI to migrate to Paris. The French queen treated him
with regard, and lodging was procured for him in the Louvre. But
he was seized with a long attack of melancholy which destroyed his
taste for mathematics. For two years his printed copy of the Me-
canique, fresh from the press, — the work of a quarter of a century, —
lay unopened on his desk. Through A. L. Lavoisier he became in-
terested in chemistry, which he found "as easy as algebra." The
disastrous crisis of the French Revolution aroused him again to ac-
tivity. About this time the young and accomplished daughter of the
astronomer P. C. Lemonnier took compassion on the sad, lonely
Lagrange, and insisted upon marrying him. Her devotion to him
constituted the one tie to life which at the approach of death he found
it hard to break.

He was made one of the commissioners to establish weights and
measures having units founded on nature. Lagrange strongly favored
the decimal subdivision. Such was the moderation of Lagrange's
character, and such the universal respect for him, that he Avas retained
as president of the commission on weights and measures even after
it had been purified by the Jacobins by striking out the names of A. L.
Lavoisier, P. S. Laplace, and others. Lagrange took alarm at the
fate of Lavoisier, and planned to return to Berlin, but at the estab-
lishment of the Ecole Normale in 1795 in Paris, he was induced to
accept a professorship. Scarcely had he time to elucidate the founda-
tions of arithmetic and algebra to young pupils, when the school was
closed. His additions to the algebra of L. Euler were prepared at
this time. In 1797 the Ecole Polytechnique was founded, with Lagrange
as one of the professors. The earliest triumph of this institution was
the restoration of Lagrange to analysis. His mathematical activity
burst out anew. He brought forth the Theorie desf auctions analytiques
(1797), Lemons sur le calcul des fonctions, a treatise on the same lines
as the preceding (1801), and the Resolution des equations numeriques
(1798), which includes papers published much earlier; his memoir,
Nowuelle methode pour resoudre les equations litterales par le nwyen des

series, published 1770, gives the notation </ for -7-, which occurs
however much earlier in a part of a memoir by Francois Daviet de

EULER, LAGRANGE AND LAPLACE 257

Foncenex in the Miscellanea Taurinensia for 1759, believed to have
been written for Foncenex by Lagrange himself.1 In 1810 he began
a thorough revision of his Mecanique analytique, but he died before
its completion.

The Theorie des f auctions, the germ of which is found in a memoir \
of his of 1772, aimed to place the principles of the calculus upon a \
sound foundation by relieving the mind of the difficult conception of ;
a limit. John Landen's residual calculus, professing a similar object, /
was unknown to him. In a letter to L. Euler of Nov. 24, 1759, La-
grange says that he believed he had developed the true metaphysics
of the calculus; at that time he seems to have been convinced that
the use of infinitesimals was rigorous. He "used both the infinitesimal
method and the method of derived functions side by side during his
whole life" (Jourdain). Lagrange attempted to prove Taylor's
theorem (the power of which he was the first to point out) by simple
algebra, and then tc develop the entire calculus from that theorem.
The principles of the calculus were in his day involved in philosophic
difficulties of a serious nature. The infinitesimals of G. W. Leibniz
had no satisfactory metaphysical basis. In the differential calculus
of L. Euler they were treated as absolute zeros. In I. Newton's limit-
ing ratio, the magnitudes of which it is the ratio cannot be found,
for at the moment when they should be caught and equated, there is
neither arc nor chord. The chord and arc were not taken by Newton
as equal before vanishing, nor after vanishing, but when they vanish.
"That method," said Lagrange, "has the great inconvenience of con-
sidering quantities in the state in which they cease, so to speak, to be
quantities; for though we can always well conceive the ratios of two
quantities, as long as they remain finite, that ratio offers to the mind
no clear and precise idea, as soon as its terms become both nothing
at the same time." D'Alembert's method of limits was much the
same as the method of prime and ultimate ratios. When Lagrange
endeavored to free the calculus of its metaphysical difficulties, by
resorting to common algebra, he avoided the whirlpool of Charybdis
only to suffer wreck against the rocks of Scylla. The algebra of his
day, as handed down to him by L. Euler, was founded on a false
view of infinity. No rigorous theory of infinite series had then been
established. Lagrange proposed to define the differential coefficient
of f(x) with respect to x as the coefficient of h in the expansion of
f(x+h) by Taylor's theorem, and thus to avoid all reference to limits.
But he used infinite series without ascertaining carefully that they
were convergent, and his proof that f(x+ti) can always be expanded
in a series of ascending powers of h, labors under serious defects.
Though Lagrange's method of developing the calculus was at first
greatly applauded, its defects were fatal, and to-day his "method of

1 Philip E. B. Jourdain in Proceed. ^Ih Intern. Congress, Cambridge, 1912, Cam-
bridge, 1913, Vol. II, p. 540.

258 A HISTORY OF MATHEMATICS

derivatives," as it was called, has been generally abandoned. He
introduced a notation of his own, but it was inconvenient, and was
abandoned by him in the second edition of his Mecanique, in which
he used infinitesimals. The primary object of the Theorie desfonctions
was not attained, but its secondary results were far-reaching. It
was a purely abstract mode of regarding functions, apart from geo-
metrical or mechanical considerations. In the further development
of higher analysis a function became the leading idea, and Lagrange's
work may be regarded as the starting-point of the theory of functions
as developed by A. L. Cauchy, G. F. B. Riemann, K. Weierstrass,
and others.

The first to doubt the rigor of Lagrange's exposition of the calculus
were Abel Burja (iy52-i8i6)of Berlin, the two Polish mathematicians
H. Wronski and /. B. Sniadecki (1756-1830), and the Bohemian
B. Bolzano, who were all men of limited acquaintance and influence.
It remained for A. L. Cauchy really to initiate the period of greater
rigor.

Instructive is C. E. Picard's characterization of the time of La-
grange: "In all this period, especially in the second half of the eight-
eenth century, what strikes us with admiration and is also somewhat
confusing, is the extreme importance of the applications realized,
while the pure theory appeared still so ill assured. One perceives it
when certain questions are raised like the degree of arbitrariness in
the integral of vibrating chords, which gives place to an interminable
and inconclusive discussion. Lagrange appreciated these insufficiencies
when he published his theory of analytic functions, where he strove
to give a precise foundation to analysis. One cannot too much
admire the marvellous presentiment he had of the role which the
functions, which with him we call analytic, were to play; but we may
confess that we stand astonished before the demonstration he be-
lieved to have given of the possibility of the development of a function
in Taylor's series." 1

In the treatment of infinite series Lagrange displayed in his earlier
writings that laxity common to all mathematicians of his tune, ex-
cepting Nicolaus Bernoulli II and D'Alembert. But his later articles
mark the beginning of a period of greater rigor. Thus, in the Calcul des
f auctions he gives his theorem on the limits of Taylor's theorem. La-
grange's mathematical researches extended to subjects which have
not been mentioned here — such as probabilities, finite differences,
ascending continued fractions, elliptic integrals. Everywhere his
wonderful powers of generalization and abstraction are made manifest.
In that respect he stood without a peer, but his great contemporary,
P. S. Laplace, surpassed him in practical sagacity. Lagrange was
content to leave the application of his general results to others, and
some of the most important researches of Laplace (particularly those
1 Congress of Arts and Science, St. Louis, 1904, Vol. 1, p. 503.

EULER, LAGRANGE AND LAPLACE 259

on the velocity of sound and on the secular acceleration of the moon)
are implicitly contained in Lagrange's works.

Lagrange was an extremely modest man, eager to avoid contro-
versy, and even timid in conversation. He spoke in tones of doubt,
and his first words generally were, " Je ne sais pas." He would never
allow his portrait to be taken, and the only ones that were secured
were sketched without his knowledge by persons attending the meet-
ings of the Institute.

Pierre Simon Laplace (1749-1827) was born at Beaumont-en-Auge
in Normandy. Very little is known of his early life. When at the
height of his fame he was loath to speak of his boyhood, spent in
poverty. His father was a small farmer. Some rich neighbors who
recognized the boy's talent assisted him in securing an education.
As an extern he attended the military school in Beaumont, where at
an early age he became teacher of mathematics. At eighteen he went
to Paris, armed with letters of recommendation to D'Alembert, who
was then at the height of his fame. The letters remained unnoticed,
but young Laplace, undaunted, wrote the great geometer a letter on
the principles of mechanics, which brought the following enthusiastic
response: "You needed no introduction; you have recommended your-
self; my support is your due." D'Alembert secured him a position
at the Ecole Mttitaire of Paris as professor of mathematics. His future
was now assured, and he entered upon those profound researches
which brought him the title of "the Newton of France." With
wonderful mastery of analysis, Laplace attacked the pending problems
in the application of the law of gravitation to celestial motions. Dur-
ing the succeeding fifteen years appeared most of his original contri-
butions to astronomy. His career was one of almost uninterrupted
prosperity. In 1784 he succeeded E. Bezout as examiner to the royal
artillery, and the following year he became member of the Academy
of Sciences. He was made president of the Bureau of Longitude; he
aided in the introduction of the decimal system, and taught, with
J. Lagrange, mathematics in the Ecole Normale. When, during the
Revolution, there arose a cry for the reform of everything, even of
the calendar, Laplace suggested the adoption of an era beginning with
the year 1250, when, according to his calculation, the major axis of
the earth's orbit had been perpendicular to the equinoctial line. The
year was to begin with the vernal equinox, and the zero meridian was
to be located east of Paris by 185.30 degrees of the centesimal division
of the quadrant, for by this meridian the beginning of his proposed
era fell at midnight. But the revolutionists rejected this scheme, and
made the start of the new era coincide with the beginning of the
glorious French Republic.1

Laplace was justly admired throughout Europe as a most sagacious

1 Rudolf Wolf, GeschicMe der Astronomic, Miinchen, 1877, p. 334.

26o A HISTORY OF MATHEMATICS

and profound scientist, but, unhappily for his reputation, he strove
not only after greatness in science, but also after political honors.
The political career of this eminent scientist was stained by servility
and suppleness. After the i8th of Brumaire, the day when Napoleon
was made emperor, Laplace's ardor for republican principles suddenly
gave way to a great devotion to the emperor. Napoleon rewarded
this devotion by giving him the post of minister of the interior, but
dismissed him after six months for incapacity. Said Napoleon, "La-
place ne saisissait aucune question sous son veritable point de vue; il
cherchait des subtilites partout, n'avait que des idees problematiques,
et portait enfin 1'esprit des infiniment petits j usque dans 1'administra-
tion." Desirous to retain his allegiance, Napoleon elevated him to
the Senate and bestowed various other honors upon him. Neverthe-
less, he cheerfully gave his voice in 1814 to the dethronement of his
patron and hastened to tender his services to the Bourbons, thereby
earning the title of marquis. This pettiness of his character is seen
in his writings. The first edition of the Systeme du monde was dedi-
cated to the Council of Five Hundred. To the third volume of the
Mecanique Celeste is prefixed a note that of all the truths contained
in the book, the one most precious to the author was the declaration he
thus made of gratitude and devotion to the peace-maker of Europe.
After this outburst of affection, we are surprised to find in the editions
of the Theorie analytique des probabilites, which appeared after the
Restoration, that the original dedication to the emperor is suppressed.

Though supple and servile in politics, it must be said that in religion
and science Laplace never misrepresented or concealed his own con-
victions however distasteful they might be to others. In mathematics
and astronomy his genius shines with a lustre excelled by few. Three
great works did he give to the scientific world, — the Mecanique Celeste,
the Exposition du systeme du monde, and the Theorie analytique des
probabilites. Besides these he contributed important memoirs to the
French Academy.

We first pass in brief review his astronomical researches. In 1773
he brought out a paper in which he proved that the mean motions
or mean distances of planets are invariable or merely subject to small
periodic changes. This was the first and most important step in his
attempt to establish the stability of the solar system.1 To I. Newton
and also to L. Euler it had seemed doubtful whether forces so numer-
ous, so variable in position, so different in intensity, as those in the
solar system, could be capable of maintaining permanently a condition
of equilibrium. Newton was of the opinion that a powerful hand
must intervene from time to time to repair the derangements occa-
sioned by the mutual action of the different bodies. This paper was
the beginning of a series of profound researches by J. Lagrange and

1 D. F. J. Arago, "Eulogy on Laplace," translated by B. Powell, Smithsonian
Report, 1874.

EULER, LAGRANGE AND LAPLACE 261

Laplace on the limits of variation of the various elements of planetary
orbits, in which the two great mathematicians alternately surpassed
and supplemented each other. Laplace's first paper really grew out
of researches on the theory of Jupiter and Saturn. The behavior of
these planets had been studied by L. Euler and J. Lagrange without
receiving satisfactory explanation. Observation revealed the ex-
istence of a steady acceleration of the mean motions of our moon and
of Jupiter and an equally strange diminution of the mean motion of
Saturn. It looked as though Saturn might eventually leave the
planetary system, while Jupiter would fall into the sun, and the moon
upon the earth. Laplace finally succeeded in showing, in a paper of
1784-1786, that these variations (called the "great inequality") be-
longed to the class of ordinary periodic perturbations, depending upon
the law of attraction. The cause of so influential a perturbation was
found in the commensurability of the mean motion of the two planets.

In the study of the Jovian system, Laplace was enabled to deter-
mine the masses of the moons. He also discovered certain very
remarkable, simple relations between the movements of those bodies,
known as "Laws of Laplace." His theory of these bodies was com-
pleted in papers of 1788 and 1789. These, as well as the other papers
here mentioned, were published in the Memoirs presenter par divers
swans. The year 1787 was made memorable by Laplace's announce-
ment that the lunar acceleration depended upon the secular changes
in the eccentricity of the earth's orbit. This removed all doubt then
existing as to the stability of the solar system. The universal validity
of the law of gravitation to explain all motion in the solar system
seemed established. That system, as then known, was at last found
to be a complete machine.

In 1796 Laplace published his Exposition du systeme du monde,
a non-mathematical popular treatise on astronomy, ending with a
sketch of the history of the science. In this work he enunciates for
the first time his celebrated nebular hypothesis. A similar theory
had been previously proposed by I. Kant in 1755, and by E. Sweden-
borg; but Laplace does not appear to have been aware of this.

Laplace conceived the idea of writing a work which should contain
a complete analytical solution of the mechanical problem presented by
the solar system, without deriving from observation any but indis-
pensable data. The result was the Mccanique Celeste, which is a
systematic presentation embracing all the discoveries of I. Newton,
A. C. Clairaut, D'Alembert, L. Euler, J. Lagrange, and of Laplace
himself, on celestial mechanics. The first and second volumes of this
work were published in 1799; the third appeared in 1802, the fourth
in 1805. Of the fifth volume, Books XI and XII were published in
1823; Books XIII, XIV, XV in 1824, and Book XVI in 1825. The
first two volumes contain the general theory of the motions and figure
of celestial bodies. The third and fourth volumes give special theories

262 A HISTORY OF MATHEMATICS

of celestial motions, — treating particularly of motions of comets, of
our moon, and of other satellites. The fifth volume opens with a
brief history of celestial mechanics, and then gives in appendices the
results of. the author's later researches. The Mecanique Celeste was
such a master-piece, and so complete, that Laplace's immediate suc-
cessors were able to add comparatively little. The general part of
the work was translated into German by Johann Karl Burkhardt
(1773-1825), and appeared in Berlin, 1800-1802. Nathaniel Bowditch
(1773-1838) brought out an edition in English, with an extensive com-
mentary, in Boston, 1829-1839. The Mecanique Celeste is not easy
reading. The difficulties lie, as a rule, not so much in the subject
itself as in the want of verbal explanation. A complicated chain of
reasoning receives often no explanation whatever. J. B. Biot, who
assisted Laplace in revising the work for the press, tells that he once
asked Laplace some explanation of a passage in the book which had
been written not long before, and that Laplace spent an hour endeavor-
ing to recover the reasoning which had been carelessly suppressed
with the remark, "II est facile de voir." Notwithstanding the impor-
tant researches in the work, which are due to Laplace himself, it
naturally contains a great deal that is drawn from his predecessors.
It is, in fact, the organized result of a century of patient toil. But
Laplace frequently neglects properly to acknowledge the source from
which he draws, and lets the reader infer that theorems and formulae
due to a predecessor are really his own.

We are told that when Laplace presented Napoleon with a copy
of the Mecanique Celeste, the latter made the remark, "M. Laplace,
they tell me you have written this large book on the system of the
universe, and have never even mentioned its Creator." Laplace is
said to have replied bluntly, "Je n'avais pas besoin de cette hy-
pothese-la." This assertion, taken literally, is impious, but may it
not have been intended to convey a meaning somewhat different
from its literal one? I. Newton was not able to explain by his law of
gravitation all questions arising in the mechanics of the heavens.
Thus, being unable to show that the solar system was stable, and
suspecting in fact that it was unstable, Newton expressed the opinion
that the special intervention, from time to time, of a powerful hand
was necessary to preserve order. Now Laplace thought that he had
proved by the law of gravitation that the solar system is stable, and
in that sense may be said to have felt no necessity for reference to the
Almighty.

We now proceed to researches which belong more properly to pure
mathematics. Of these the most conspicuous are on the theory of
probability. Laplace has done more towards advancing this subject
than any one other investigator. He published a series of papers,
the main results of which were collected in his Theorie analytique des
probabilites, 1812. The third edition (1820) consists of an introduction

EULER, LAGRANGE AND LAPLACE 263

and two books. The introduction was published separately under
the title, Essai philosophique sur les probabilities, and is an admirable
and masterly exposition without the aid of analytical formulae of the
principles and applications of the science. The first book contains
the theory of generating functions, which are applied, in the second
book, to the theory of probability. Laplace gives in his work on
probability his method of approximation to the values of definite
integrals. The solution of linear differential equations was reduced
by him to definite integrals. The use of partial difference equations
was introduced into the study of probability by him about the same
time as by J. Lagrange. One of the most important parts of the
work is the application of probability to the method of least squares,
which is shown to give the most probable as well as the most conven-
ient results.

Laplace's work on probability is very difficult reading, particularly
the part on the method of least squares. The analytical processes
are by no means clearly established or free from error. " No one was
more sure of giving the result of analytical processes correctly, and
no one ever took so little care to point out the various small con-
siderations on which correctness depends" (De Morgan). Laplace's
comprehensive work contains all of his own researches and much
derived from other writers. He gives masterly expositions of the
Problem of Points, of Jakob Bernoulli's theorem, of the problems taken
from Bayes and Count de Buff on. In this work as in his Mecanique
Celeste, Laplace is not in the habit of giving due credit to writers that
preceded him. A. De Morgan1 says of Laplace: "There is enough
originating from himself to make any reader wonder that one who
could so well afford to state what he had taken from others, should
have set an example so dangerous to his own claims."

Of Laplace's papers on the attraction of ellipsoids, the most im-
portant is the one published in 1785, and to a great extent reprinted
in the third volume of the Mecanique Celeste. It gives an exhaustive
treatment of the general problem of attraction of any ellipsoid upon
a particle situated outside or upon its surface. Spherical harmonics,
or the so-called "Laplace's coefficients," constitute a powerful analytic
engine in the theory of attraction, in electricity, and magnetism. The
theory of spherical harmonics for two dimensions had been previously
given by A. M. Legendre. Laplace failed to make due acknowledg-
ment of this, and there existed, in consequence, between the two
great men, "a feeling more than coldness." The potential function,
V, is much used by Laplace, and is shown by him to satisfy the partial

differential equation — 5- + — H — 5-=o. This is known as Laplace's
d* Syz 5z2

1 A. De Morgan, An Essay on Probabilities, London, 1838 (date of Preface)
p. II of Appendix I.

264 A HISTORY OF MATHEMATICS

equation, and was first given by him in the more complicated form
which it assumes in polar co-ordinates. The notion of potential was,
however, not introduced into analysis by Laplace. The honor of
that achievement belongs to J. Lagrange.

Regarding Laplace's equation, P. E. Picard said in 1904: "Few
equations have been the object of so many works as this celebrated
equation. The conditions at the limits may be of divers forms. The
simplest case is that of the calorific equilibrium of a body of which
we maintain the elements of the surface at given temperatures; from
the physical point of view, it may be regarded as evident that the
temperature, continuous within the interior since no source of heat
is there, is determined when it is given at the surface. A more general
case is that where . . . the temperature may be given on one portion,
while there is radiation on another portion. These questions . . .
have greatly contributed to the orientation of the theory of partial
differential equations. They have called attention to types of deter-
minations of integrals, which would not have presented themselves
in remaining at a point of view purely abstract." 1

Among the minor discoveries of Laplace are his method of solving
equations of the second, third, and fourth degrees, his memoir on
singular solutions of differential equations, his researches in finite
differences and in determinants, the establishment of the expansion
theorem in determinants which had been previously given by A. T.
Vandermonde for a special case, the determination of the complete
integral of the linear differential equation of the second order. In
the Mecanique Celeste he made a generalization of Lagrange's theorem
on the development of functions in series known as Laplace's theorem.

Laplace's investigations in physics were quite extensive. We men-
tion here his correction of Newton's formula on the velocity of sound
in gases by taking into account the changes of elasticity due to the
heat of compression and cold of rarefaction; his researches on the
theory of tides; his mathematical theory of capillarity; his explanation
of astronomical refraction; his formulae for measuring heights by the
barometer.

Laplace's writings stand out in bold contrast to those of J. Lagrange
in their lack of elegance and symmetry. Laplace looked upon mathe-
matics as the tool for the solution of physical problems. The true
result being once reached, he devoted little time to explaining the
various steps of his analysis, or in polishing his work. The last years
of his life were spent mostly at Arcueil in peaceful retirement on a
country-place, where he pursued his studies with his usual vigor
until his death. He was a great admirer of L. Euler, and would often
say, "Lisez Euler, lisez Euler, c'est notre maitre a tous."

The latter part of the eighteenth century brought forth researches
on the graphic representation of imaginaries, all of which remained
1 Congress of Arts and Science, St. Louis, 1904, Vol. I, p. 506.

EULER, LAGRANGE AND LAPLACE 265

quite unnoticed at that time. During the time of R. Descartes, I.
Newton and L. Euler, the negative and the imaginary came to be
accepted as numbers, but the latter was still regarded as an algebraic
fiction. A little over a hundred years after J. Wallis's unsuccessful
efforts along the line of graphic representation of imaginaries, "a
modest scientist," Henri Dominique Truel, pictured imaginaries upon
a line that was perpendicular to the line representing real numbers.
So far as known, Truel published nothing, nor are his manuscripts
extant. All we know about him is a brief reference to him made by
A. L. Cauchy,1 who says that Truel had his graphic scheme as early
as 1786, and about 1810 turned his "manuscripts over to Augustin
Normauf , a ship builder in Havre. W. J. G. Karsten's graphic scheme
of 1768 was confined to imaginary logarithms. The earliest printed
graphic representation of V — i and a+b^ — i was given in an "Essay
on the Analytic Representation of Direction, with Applications in
Particular to the Determination of Plane and Spherical Polygons"
presented in 1797 by Caspar Wessel (1745-1818) to the Royal Academy
of Sciences and Letters of Denmark and published in Vol. V of its
Memoirs in 1799. Wessel was born in Jonsrud, in Norway. For
many years he was in the employ of the Danish Academy of Sciences
as a surveyor. His paper lay buried in the Transactions of the Danish
Academy for nearly a century. In 1897 a French translation was
brought out by the Danish Academy.2 Another noteworthy publica-
tion which remained unknown for many years is an Essay 3 published
in 1806 by Jean Robert Argand (£768-1822) of Geneva, containing a
geometric representation of a+V — ib. Some parts of his paper are
less rigorous than the corresponding parts of Wessel. Argand gave
some remarkable applications to trigonometry, geometry and algebra.
The word "modulus," to represent the length of the vector a+ib,
is due to Argand. The writings of Wessel and Argand being little
noticed, it remained for K. F. Gauss to break down the last opposition
to the imaginary. Gauss seems to have been in possession of a graphic
scheme as early as 1799, but its fuller exposition was deferred until
1831.

During the French Revolution the metric system was introduced.
The general idea of decimal subdivision was obtained from a work of
Thomas Williams, London, 1788. On April 14, 1790, Mathurin
Jacques Brisson (1723-1806) proposed before the Paris Academy the
establishment of a system resting on a natural unit of length. A
scheme was elaborated which originally included the decimal sub-
division of the quadrant of a circle, as is shown by the report made to

1 Cauchy, Exercices d' Analyse et de phys. math., T. IV, 1847, p. 157.

z See also an address on Wessel by W. W. Beman in the Proceedings of the A m.
Ass'n Adv. of Science, Vol. 46, 1897.

3 Imaginary Quantities. Their Geometrical Interpretation. Translated from the
French of M. Argand by A. S. Hardy, New York, 1881.

266 A HISTORY OF MATHEMATICS

the Academy of Sciences on March 19, 1791, by a committee con-
sisting of J. C. Borda, J. Lagrange, P. S. Laplace, G. Monge, de Con-
dorcet. This subdivision is found in the Francois Collet (1744-1798)
logarithmic tables of 1795, and other tables published in France and
Germany. Nevertheless the decimal subdivision of the quadrant did
not then prevail.1 The commission composed of Borda, Lagrange,
Laplace, Monge and Condorcet decided upon the ten-millionth part
of the earth's quadrant as the primitive unit of length. The length
of the second's pendulum had been under consideration, but was
finally rejected, because it rested upon two dissimilar elements,
gravity and time. In 1799 the measurement of the earth's quadrant
was completed and the meter established as the natural unit of length.
Alexandre-Theophile Vandermonde (1735-1796) studied music
during his youth in Paris and advocated the theory that all art rested
upon one general law, through which any one could become a com-
poser with the aid of mathematics. He was the first to give a con-
nected and logical exposition of the theory of determinants, and may,
therefore, almost be regarded as the founder of that theory. He and J.
Lagrange originated the method of combinations in solving equations.
Adrien Marie Legendre (1752-1833) was educated at the College
Mazarin in Paris, where he began the study of mathematics under
Abbe Joseph Francois Marie (1738-1801). His mathematical genius
secured for him the position of professor of mathematics at the mili-
tary school of Paris. While there he prepared an essay on the curve
| described by projectiles thrown into resisting media (ballistic curve),
1 which captured a prize offered by the Royal Academy of Berlin. In
'1780 he resigned his position in order to reserve more tune for the
study of higher mathematics. He was then made member of several
public commissions. In 1795 he was elected professor at the Normal
School and later was appointed to some minor government positions.
Owing to his timidity and to Laplace's unfriendliness toward him, but
few important public offices commensurate with his ability were
tendered to him.

As an analyst, second only to P. S. Laplace and J. Lagrange, Legen-
dre enriched mathematics by important contributions, mainly on
elliptic integrals, theory of numbers, attraction of ellipsoids, and least
squares. The most important of Legendre's works is his Fonctions
elliptiques, issued in two volumes in 1825 and 1826. He took up the
subject where L. Euler, John Landen, and J. Lagrange had left it,
and for forty years was the only one to cultivate this new branch of
analysis, until at last C. G. J. Jacobi and N. H. Abel stepped in with
admirable new discoveries.2 Legendre imparted to the subject that

1 For details, see R. Mehmke in Jahresb. d. d. Math. Vereinigung, Leipzig, 1900,
pp. 138-163.

2 M. Elie de Beaumont, "Memoir of Legendre." Translated by C. A. Alexander,
Smithsonian Report, 1867.

EULER, LAGRANGE AND LAPLACE 267

connection and arrangement which belongs to an independent science.
Starting with an integral depending upon the square root of a poly-
nomial of the fourth degree in x, he showed that such integrals can be
brought back to three canonical forms, designated by F((j>), £(<£), and
n(0), the radical being expressed in the form A($) = \/i — k2 sin'2</>.
He also undertook the prodigious task of calculating tables of
arcs of the ellipse for different degrees of amplitude and eccentricity,
which supply the means of integrating a large number of differentials.

An earlier publication which contained part of his researches on
elliptic functions was his Calcul integral in three volumes (1811, 1816,
1817), in which he treats also at length of the two classes of definite
integrals named by him Eulerian. He tabulated the values of log
T(/>) for values of p between i and 2.

One of the earliest subjects of research was the attraction of sphe-
roids, which suggested to Legendre the function Pn, named after him.
His memoir was presented to the Academy of Sciences in 1783. The
researches of C. Maclaurin and J. Lagrange suppose the point at-
tracted by a spheroid to be at the surface or within the spheroid, but
Legendre showed that in order to determine the attraction of a
spheroid on any external point it suffices to cause the surface of another
spheroid described upon the same foci to pass through that point.
Other memoirs on ellipsoids appeared later.

In a paper of 1788 Legendre published criteria for distinguishing
between maxima and minima in the calculus of variations, which were
shown by J. Lagrange in 1797 to be insufficient; this matter was set
right by C. G. J. Jacobi in 1836.

The two household gods to which Legendre sacrificed with ever-
renewed pleasure in the silence" of his closet were the elliptic functions
and the theory of numbers. His researches on the latter subject,
together with the numerous scattered fragments on the theory of
numbers due to his predecessors in this line, were arranged as far
as possible into a systematic whole, and published in two large quarto
volumes, entitled Thcorie des nombres, 1830. Before the publication
of this work Legendre had issued at divers times preliminary articles.
Its crowning pinnacle is the theorem of quadratic reciprocity, pre-
viously indistinctly given by L. Euler without proof, but for the first
time clearly enunciated and partly proved by Legendre.1

While acting as one of the commissioners to connect Greenwich
and Paris geodetically, Legendre calculated the geodetic triangles in
France. This furnished the occasion of establishing formulae and
theorems on geodesies, on the treatment of the spherical triangle as
if it were a plane triangle, by applying certain corrections to the
angles, and on the method of least squares, published for the first
time by him without demonstration in 1806.

1 O. Baumgart, Ueber das Quadratische Reciprocitalsgcsdz, Leipzig, 1883.

268 A HISTORY OF MATHEMATICS

Legendre wrote an Elements de Geometric, 1794, which enjoyed
great popularity, being generally adopted on the Continent and in
the United States as a substitute for Euclid. This great modern rival
of Euclid passed through numerous editions; some containing the
elements of trigonometry and a proof of the irrationality of TT and 7r2.
With prophetic vision Legendre remarks: "II est meme probable que
le nombre TT n'est pas meme compris dans les irrationelles algebriques,
c'est-a-dire qu'il ne peut pas etre la racine 'dune equation algebrique
d'un nombre fini de termes dont les coefficients sont rationels."
Much attention was given by Legendre to the subject of parallel lines.
In the earlier editions of the Elements, he made direct appeal to the
senses for the correctness of the "parallel-axiom." He then attempted
to demonstrate that "axiom," but his proofs did not satisfy even
himself. In Vol. XII of the Memoirs of the Institute is a paper by
Legendre, containing his last attempt at a solution of the problem.
Assuming space to be infinite, he proved satisfactorily that it is im-
possible for the sum of the three angles of a triangle to exceed two
right angles; and that if there be any triangle the sum of whose angles
is two right angles, then the same must be true of all triangles. But
in the next step, to show that this sum cannot be less than two right
angles, his demonstration necessarily failed. If it could be granted
that the sum of the three angles is always equal to two right angles,
then the theory of parallels could be strictly deduced.

Another author who made contributions to elementary geometry
was the Italian Lorenzo Mascheroni (1750-1800). He published his
Geometria del compasso (Pavia, 1797, Palermo, 1903; * French editions
by A. M. Carette appeared in 1798 and 1825, a German edition by
J. P. Griison in 1825). All constructions are made with a pair of
compasses, but without restriction tc a fixed radius. He proved that
all constructions possible with ruler and compasses are possible with
compasses alone. It was J. V. Poncelet who proved in 1822 that all
such construction are possible with ruler alone, if we are given a fixed
circle with its centre in the plane of construction; A. Adler of Vienna
proved in 1890 that these constructions are possible with ruler alone
whose edges are parallel, or whose edges converge in a point. Masch-
eroni claimed that constructions with compasses are more accurate
than those with a ruler. Napoleon proposed to the French mathe-
maticians the problem, to divide the circumference of a circle into
four equal parts by the compasses only. Mascheroni does this by
applying the radius three times to the circumference; he obtains the
arcs A B, B C, C D; then A D is a diameter; the rest is obvious.
E. W. Hobson (Math. Gazette, March i, 1913) and others have shown
that all Euclidean constructions can be carried out by the use of
compasses alone.

1 A list of Mascheroni's writings is given in L'Iuterm£diaire des malhfynaticiens,
Vol. 19, 1912, p. 92.

EULER, LAGRANGE AND LAPLACE 269

In 1790 Mascheroni published annotations to Euler's Integral
Calculus. D'Alembert had argued "le calcul en defaut" by declaring
that the astroid x$+y$=i yielded o as the length of the arc from
x= — i to x=i, y being taken positive. To this Mascheroni added in
his annotations another paradox, by the contention that, for x>i,
the curve is imaginary, yet has a real length of arc.1 These paradoxes
found no adequate explanation at the time, due to an inadequate
fixing of the region of variability.

Joseph Fourier (1768-1830) was born at Auxerre, in central France.
He became an orphan in his eighth year. Through the influence of
friends he was admitted into the military school in his native place,
then conducted by the Benedictines of the Convent of St. Mark. He
there prosecuted his studies, particularly mathematics, with sur-
prising success. He wished to enter the artillery, but, being of low
birth (the son of a tailor), his application was answered thus: " Fourier,
not being noble, could not enter the artillery, although he were a
second Newton." He was soon appointed to the mathematical
chair in the military school. At the age of twenty-one he went to
Paris to read before the Academy of Sciences a memoir on the reso-
lution of numerical equations, which was an improvement on Newton's
method of approximation. This investigation of his early youth he
never lost sight of. He lectured upon it in the Polytechnic School;
he developed it on the banks of the Nile; it constituted a part of a
work entitled Analyse des equationes determines (1831), which was
in press when death overtook him. This work contained "Fourier's
theorem" on the number of real roots between two chosen limits.
The French physician F. D. Budan had published a theorem nearly
identical in principle, although different in statement, as early as
1807, but in 1807 Budan had not only not proved the theorem known
by his name, but had not yet satisfied himself that it was really true.
He gave a proof in 1811, which was printed in 1822. Fourier taught
his theorem to his pupils in the Polytechnic School in 1796, 1797 and
1803 ; he first printed the theorem and its proof in 1820. His priority
over Budan is firmly established.

Fourier was anticipated in two of his important results. His im-
provement on the Newton-Raphson method of approximation, render-
ing the process applicable without the possibility of failure, was given
earlier by Mourraille, as was also Fourier's method of settling the
question whether two roots near the border line of equality are really
equal, or perhaps slightly different, or perhaps imaginary. These
theorems were eclipsed by that of Sturm, published in 1835.

About this time new upper and lower limits of the real roots were
discovered. In 1815 Jean Jacques Bret (1781-?) professor in Grenoble,
printed three theorems, of which the following is best known: If frac-

1 M. Cantor, op. cit., Vol. IV, 1908, p. 485.

2 D. F. J. Arago, "Joseph Fourier," Smithsonian Report, 1871.

27o A HISTORY OF MATHEMATICS

tions are formed by giving each fraction a negative coefficient in an
equation for its numerator, taken positively, and for its denominator
the sum of the positive coefficients preceding it, if moreover unity is
added to each fraction thus formed, then the largest number thus
obtainable is larger than any root of the equation. In 1822 A. A.
Vene, a French officer of engineers, showed: If P is the largest negative
coefficient, and if S be the greatest coefficient among the positive
terms which precede the first negative term, then will P-r-S+i be a
superior limit.

Fourier took a prominent part at his home in promoting the Revo-
lution. Under the French Revolution the arts and sciences seemed for
a tune to flourish. The reformation of the weights and measures was
planned with grandeur of conception. The Normal School was
created in 1795, of which Fourier became at first pupil, then lecturer.
His brilliant success secured him a chair in the Polytechnic School,
the duties of which he afterwards quitted, along with G. Monge and
C. L. Berthollet, to accompany Napoleon on his campaign to Egypt.
Napoleon founded the Institute of Egypt, of which J. Fourier became
secretary. In Egypt he engaged not only in scientific work, but dis-
charged important political functions. After his return to France he
held for fourteen years the prefecture of Grenoble. During this
period he carried on his elaborate investigations on the propagation of
heat in solid bodies, published in 1822 in his work entitled La Theorie
Analytique de la Chaleur. This work marks an epoch in the history of
both pure and applied mathematics. It is the source of all modern
methods in mathematical physics involving the integration of partial
differential equations in problems where the boundary values are
fixed (" boundary- value problems"). Problems of this type involve
L. Euler's second definition of a "function" in which the relation is
not necessarily capable of being expressed analytically. This concept
of a function greatly influenced P. G. L. Dirichlet. The gem of
Fourier's great book is "Fourier's series." By this research a long
controversy was brought to a close, and the fact recognized that any
[arbitrary function (i. e. any graphically given function) of a real
Fivariable can be represented by a trigonometric series. The first
announcement of this great discovery was made by Fourier in 1807,

rt =00

before the French Academy. The trigonometric series 2 (<*» sin nx+

n=o

bn cos nx) represents the function <£(#). for every value of x, if the

I •/*"*

coefficients an=— I </>(*) sm nxdx, and bn be equal to a similar in-

irj -IT

tegral. The weak point in Fourier's analysis lies in his failure to
prove generally that the trigonometric series actually converges to

EULER, LAGRANGE AND LAPLACE 271

the value of the function. William Thomson (later Lord Kelvin)
says that on May i, 1840 (when he was only sixteen), "I took Fourier
out of the University Library; and in a fortnight I had mastered it —
gone right through it." Kelvin's whole career was influenced by
Fourier's work on heat, of which, he said, " it is difficult to say whether
their uniquely original quality, or their transcendant interest, or their
perennially important instructiveness for physical science, is most
to be praised." 1 Clerk Maxwell pronounced it a great mathematical
poem. In 1827 Fourier succeeded P. S. Laplace as president of the
council of the Polytechnic School.

About the time of Budan and Fourier, important devices were
invented in Italy and England for the solution of numerical equations.
The Italian scientific society in 1802 offered a gold medal for improve-
ments in the solution of such equations; it was awarded in 1804 to
Paolo Ruffini. With aid of the calculus he develops the theory of
transforming one equation into another whose roots are all diminished
by a certain constant.2 Then follows the mechanism for the practical
computer, and here Ruffini has a device which is simpler than Homer's
scheme of 1819 and practically identical with what is now known as
Horner's procedure. Horner had no knowledge of Ruffini's memoir.
Nor did either Horner or Ruffini know that their method had been
given by the Chinese as early as the thirteenth century. Horner's
first paper was read before the Royal Society, July i, 1819, and pub-
lished in the Philosophical Transactions for 1819. Horner uses L. F. A.
Arbogast's derivatives. The modern reader is surprised to find that
Horner's exposition involves very intricate reasoning which is in
marked contrast with the simple and elementary explanations found
in modern texts. Perhaps this was fortunate; a simpler treatment
might have prevented publication in the Philosophical Transactions.
As it was, much demur was made to the insertion of the paper. "The
elementary character of the subject," said T. S. Davies, "was the
professed objection; his recondite mode of treating it was the professed
passport for its admission." A second article of Horner on his method
was refused publication in the Philosophical Transactions, and ap-
peared in 1765 in the Mathematician, after the death of Horner; a
third article was printed in 1830. Both Horner and Ruffini explained
their methods at first by higher analysis and later by elementary
algebra; both offered their methods as substitutes for the old process
of root-extraction of numbers. Ruffini's paper was neglected and
forgotten. Horner was fortunate in finding two influential champions
of his method — John Radford Young (1799-1885) of Belfast and A.
De Morgan. The Ruffini-Horner process has been used widely in
England and the United States, less widely in Germany, Austria and

1 S. P. Thompson Life of William Thomson, London, 1910, pp. 14, 689.

2 See F. Cajori, "Horner's method of approximation anticipated by Ruffini," Bull.
Am. Math. Soc. ad S., Vol. 17, 1911, pp. 409-414.

272 A HISTORY OF MATHEMATICS

Italy, and not at all in France. In France the Newton-Raphson
method has held almost undisputed sway.1

Before proceeding to the origin of modern geometry we shall speak
briefly of the introduction of higher analysis into Great Britain. This
took place during the first quarter of the last century. The British
began to deplore the very small progress that science was making in
England as compared with its racing progress on the Continent. The
first Englishman to urge the study of continental writers was Robert
Woodhouse (1773-1827) of Caius College, Cambridge. In 1813 the
"Analytical Society" was formed at Cambridge. This was a small
club established by George Peacock, John Herschel, Charles Babbage,
and a few other Cambridge students, to promote, as it was humorously
expressed by Babbage, the principles of pure "Z)-ism," that is, the
Leibnizian notation in the calculus against those of "dot-age," or
of the Newtonian notation. This struggle ended in the introduction

flV

into Cambridge of the notation -j-, to the exclusion of the fluxional

CL3C

notation y. This was a great step in advance, not on account of any
great superiority of the Leibnizian over the Newtonian notation, but
because the adoption of the former opened up to English students
the vast storehouses of continental discoveries. Sir William Thom-
son, P. G. Tait, and some other modern writers find it frequently con-
venient to use both notations. Herschel, Peacock, and Babbage
translated, in 1816, from the French, S. F. Lacroix's briefer treatise
on the differential and integral calculus, and added in 1820 two
volumes of examples. Lacroix's larger work, the Traitc du calad
differentiel et integral, first contained the term "differential coefficient"
and definitions of "definite" and "indefinite" integrals. It was one
of the best and most extensive works on the calculus of that time.
Of the three founders of the "Analytical Society," Peacock afterwards
did most work in pure mathematics. Babbage became famous for
his invention of a calculating engine superior to Pascal's. It was
never finished, owing to a misunderstanding with the government,
and a consequent failure to secure funds. John Herschel, the eminent
astronomer, displayed his mastery over higher analysis in memoirs
communicated to the Royal Society on new applications of mathe-
matical analysis, and in articles contributed to cyclopaedias on light,
on meteorology, and on the history of mathematics. In the Philo-
sophical Transactions of 1813 he introduced the notation sin~1x,
tan~1x, . . . for arcsin x, arctan x, . . . He wrote also Iog2x, cos^x, . . .
for log (log x), cos (cos x), . . ., but in this notation he was anticipated
by Heinrich Burmann (?-i8i7) of Mannheim, a partisan of the com-
binatory analysis of C. F. Hindenburg in Germany.

1 For references and further detail, see Colorado College Publication, General Series
52, 1910.

EULER, LAGRANGE AND LAPLACE 273

George Peacock (1791-1858) was educated at Trinity College,
Cambridge, became Lowndean professor there, and later, dean of
Ely. His chief publications are his Algebra, 1830 and 1842, and his
Report on Recent Progress in Analysis, which was the first of several
valuable summaries of scientific progress printed in the volumes of
the British Association. He was one of the first to study seriously
the fundamental principles of algebra, and to recognize fully its purely
symbolic character. He advances, though somewhat imperfectly,
the "principle of the permanence of equivalent forms." It assumes
that the rules applying to the symbols of arithmetical algebra apply
also in symbolical algebra. About this time Duncan Farquharson
Gregory (1813-1844), fellow of Trinity College, Cambridge, wrote
a paper "on the real nature of symbolical algebra," which brought
out clearly the commutative and distributive laws. These laws had
been noticed years before by the inventors of symbolic methods in
the calculus. It was F. Servois who introduced the names commutative
and distributive in Gergonne's Annales, Vol. 5, 1814-15, p. 93. The
term associative seems to be due to W. R. Hamilton. Peacock's
investigations on the foundation of algebra were considerably ad-
vanced by A. De Morgan and H. Hankel.

James Ivory (1765-1842) was a Scotch mathematician who for
twelve years, beginning in 1804, held the mathematical chair in the
Royal Military College at Marlow (now at Sandhurst). He was
essentially a self-trained mathematician, and almost the only one in
Great Britain previous to the organization of the Analytical Society
who was well versed in continental mathematics. Of importance is
his memoir (Phil. Trans., 1809) in which the problem of the attraction
of a homogeneous ellipsoid upon an external point is reduced to the
simpler problem of the attraction of a related ellipsoid upon a corre-
sponding point interior to it. This is known as "Ivory's theorem."
He criticised with undue severity Laplace's solution of the method
of least squares, and gave three proofs of the principle without re-
course to probability; but they are far from being satisfactory.

About this time began the aggressive investigation of "curves of
pursuit." The Italian painter Leonardo da Vinci seems to be the
first to have directed attention to such curves. They were first
investigated by Pierre Bouguer of Paris in 1732, then by the French
collector of customs, Dubois-Ayme (Corresp. sur I'ecole polyt. II, 1811,
p. 275) who stimulated researches carried on by Thomas de St. Laurent,
Ch. Sturm, Jean Joseph Querret and Tedenat (Ann. de Mathem., Vol. 13,
1822-1823).

By the researches of R. Descartes and the invention of the calculus,
the analytical treatment of geometry was brought into great prom-
inence for over a century. Notwithstanding the efforts to revive
synthetic methods made by G. Desargues, B. Pascal, De Lahire,
I: Newton, and C. Maclaurin, the analytical method retained almost

274 A HISTORY OF MATHEMATICS

undisputed supremacy. It was reserved for the genius of G. Monge
to bring synthetic geometry in the foreground, and to open up new
avenues of progress. His Geometric descriptive marks the beginning
of a wonderful development of modern geometry.

Of the two leading problems of descriptive geometry, the one — to
represent by drawings geometrical magnitudes — was brought to a
high degree of perfection before the time of Monge; the other — to
solve problems on figures in space by constructions in a plane — had
received considerable attention before his time. His most noteworthy
predecessor in descriptive geometry was the Frenchman Amedee
Francois Frezier (1682-1773). But it remained for Monge to create
descriptive geometry as a distinct branch of science by imparting to
it geometric generality and elegance. All problems previously treated
in a special and uncertain manner were referred back to a few general
principles. He introduced the line of intersection of the horizontal
and the vertical plane as the axis of projection. By revolving one
plane into the other around this axis or ground-line, many advantages
were gained.1

Gaspard Monge (1746-1818) was born at Beaune. The construc-
tion of a plan of his native town brought the boy under the notice of
a colonel of engineers, who procured for him an appointment in the
college of engineers at Mezieres. Being of low birth, he could not
receive a commission in the army, but he was permitted to enter the
annex of the school, where surveying and drawing were taught. Ob-
serving that all the operations connected with the construction of
plans of fortification were conducted by long arithmetical processes,
he substituted a geometrical method, which the commandant at first
refused even to look at; so short was the time in which it could be
practised that, when once examined, it was received with avidity.
Monge developed these methods further and thus created his descrip-
tive geometry. Owing to the rivalry between the French military
schools of that time, he was not permitted to divulge his new methods
to any one outside of this institution. In 1 768 he was made professor of
mathematics at Mezieres. In 1780, when conversing with two of his
pupils, S. F. Lacroix and S. F. Gay de Vernon in Paris, he was obliged
to say, "All that I have here done by calculation, I could have done
with the ruler and compasses, but I am not allowed to reveal these
secrets to you." But Lacroix set himself to examine what the secret
could be, discovered the processes, and published them in 1795. The
method was published by Monge himself in the same year, first hi
the form in which the shorthand writers took down his lessons given
at the Normal School, where he had been elected professor, and then
again, in revised form, in the Journal des ecoles twrmales. The next
edition occurred in 1798-1799. After an ephemeral existence of only
four months the Normal School was closed in 1795. In the same year

1 Christian Wiener, Lehrbuch der Darstelleiiden Geometric, Leipzig, 1884, p. 26.

EULER, LAGRANGE AND LAPLACE 275

the Polytechnic School was opened, in the establishing of which
Monge took active part. He taught there descriptive geometry until
his departure from France to accompany Napoleon on the Egyptian
campaign. He was the first president of the Institute of Egypt.
Monge was a zealous partisan of Napoleon and was, for that reason,
deprived of all his honors by Louis XVIII. This and the destruction
of the Polytechnic School preyed heavily upon his mind. He did not
long survive this insult.

Monge's numerous papers were by no means confined to descriptive
geometry. His analytical discoveries are hardly less remarkable. He
introduced into analytic geometry the methodic use of the equation
of a line. He made important contributions to surfaces of the second
degree (previously studied by C. Wren and L. Euler) and discovered
between the theory of surfaces and the integration of partial differ-
ential equations, a hidden relation which threw new light upon both
subjects. He gave the differential of curves of curvature, established
a general theory of curvature, and applied it to the ellipsoid. He
found that the validity of solutions was not impaired when imaginaries
are involved among subsidiary quantities. Usually attributed to
Monge are the centres of similitude of circles and certain theorems,
which were, however, probabty known to Apollonius of Perga.1 Monge
published the following books: Statics, 1786; Applications de Valgebre
a la geometric, 1805; Application de I' analyse d la geometrie. The last
two contain most of his miscellaneous papers.

Monge was an inspiring teacher, and he gathered around him a
large circle of pupils, among which were C. Dupin, F. Servois, C. J.
Brianchon, Hachette, J. B. Biot, and J. V. Poncelet. Jean Baptiste
Biot (1774-1862), professor at the College de France in Paris, came in
contact as a young man with Laplace, Lagrange, and Monge. In
1804 he ascended with Gay-Lussac in a balloon. They proved that
the earth's magnetism is not appreciably reduced in intensity in
regions above the earth's surface. Biot wrote a popular book on
analytical geometry and was active in mathematical physics and
geodesy. He had a controversy with Arago who championed A. J.
Fresnel's wave theory of light. Biot was a man of strong individuality
and great influence.

Charles Dupin (1784-1873), for many years professor of mechanics
in the Conservatoire des Arts et Metiers in Paris, published in 1813
an important work on Developpements de geontettie, in which is intro-
duced the conception of conjugate tangents of a point of a surface,
and of the indicatrix.2 It contains also the theorem known as "Du-
pin's theorem." Surfaces of the second degree and descriptive geom-

1 R. C. Archibald in Am. Math. Monthly, Vol. 22, 1915, pp. 6-12; Vol. 23, pp. 159-
161.

z Gino Loria, Die Haupts'dchlislen Theorien der Geometric (F. Schiitte), Leipzig,
1888, p. 49.

276 A HISTORY OF MATHEMATICS

etry were successfully studied by Jean Nicolas Pierre Hachette (1769-
1834), who became professor of descriptive geometry at the Poly-
technic School after the departure of Monge for Rome and Egypt.
In 1822 he published his Traite de geometric descriptive.

Descriptive geometry, which arose, as we have seen, in technical
schools in France, was transferred to Germany at the foundation of
technical schools there. G. Schreiber (1799-1871), professor in Karls-
ruhe, was the first to spread Monge's geometry in Germany by the
publication of a work thereon in I828-I829.1 In the United States
descriptive geometry was introduced in 1816 at the Military Academy
in West Point by Claude Crozet, once a pupil at the Polytechnic
School in Paris. Crozet wrote the first English work on the subject.2

Lazare Nicholas Marguerite Carnot (1753-1823) was born at
Nolay in Burgundy, and educated in his native province. He entered
the army, but continued his mathematical studies, and wrote in 1784
a work on machines, containing the earliest proof that kinetic energy
is lost in collisions of bodies. With the advent of the Revolution he
threw himself into politics, and when coalesced Europe, in 1793,
launched against France a million soldiers, the gigantic task of or-
ganizing fourteen armies to meet the enemy was achieved by him.
He was banished in 1796 for opposing Napoleon's coup d'etat. The
refugee went to Geneva, where he issued, in 1797, a work still fre-
quently quoted, entitled, Reflexions sur la Metaphysique du Calad
Infinitesimal. He declared himself as an "irreconcilable enemy of
kings." After the Russian campaign he offered to fight for France,
though not for the empire. On the restoration he was exiled. He
died in Magdeburg. His Geometric de position, 1803, and his Essay on
Transversals, 1806, are important contributions to modern geometry.
While G. Monge revelled mainly in three-dimensional geometry,
Carnot confined himself to that of two. By his effort to explain the
meaning of the negative sign in geometry he established a "geometry
of position," which, however, is different from the "Geometric der
Lage" of to-day. He invented a class of general theorems on pro-
jective properties of figures, which have since been pushed to great
extent by J. V. Poncelet, Michel Chasles, and others.

Thanks to Carnot's researches, says J. G. Darboux,3 "the con-
ceptions of the inventors of analytic geometry, Descartes and Fermat,.
retook alongside the infinitesimal calculus of Leibniz and Newton
the place they had lost, yet should never have ceased to occupy. With
his geometry, said Lagrange, speaking of Monge, this demon of a man
will make himself immortal."

While in France the school of G. Monge was creating modern

1 C. Wiener, op. tit. p. 36.

2 F. Cajori, Teaching and History of Mathematics in U. S., Washington, 1890,
pp. 114, 117.

* Congress of Arts and Science, St. Louis, 1904, Vol. i, p. 535.

EULER, LAGRANGE AND LAPLACE 277

geometry, efforts were made in England to revive Greek geometry by
Robert Simson (1687-1768) and Matthew Stewart (1717-1785).
Stewart was a pupil of Simson and C. Maclaurin, and succeeded the
latter in the chair at Edinburgh. During the eighteenth century he
and Maclaurin were the only prominent mathematicians in Great
Britain. His genius was ill-directed by the fashion then prevalent in
England to ignore higher analysis. In his Four Tracts, Physical and
Mathematical, 1761, he applied geometry to the solution of difficult
astronomical problems, which on the Continent were approached
analytically with greater success. He published, in 1746, General
Theorems, and in 1763, his Propositiones geometries more veterum de-
monstrate. The former work contains sixty-nine theorems, of which
only five are accompanied by demonstrations. It gives many inter-
esting new results on the circle and the straight line. Stewart ex-
tended some theorems on transversals due to Giovanni Ceva (1647-
1734), an Italian, who published in 1678 at Mediolani a work, De
lineis rectis se inwcem secantibus, containing the theorem now known
by his name.

THE NINETEENTH AND TWENTIETH CENTURIES

Introduction

NEVER more zealously and successfully has mathematics been
cultivated than during the nineteenth and the present centuries. Nor
has progress, as in previous periods, been confined to one or two
countries. While the French and Swiss, who during the preceding
epoch carried the torch of progress, have continued to develop mathe-
matics with great success, from other countries whole armies of en-
thusiastic workers have wheeled into the front rank. Germany awoke
from her lethargy by bringing forward K. F. Gauss, C. G. J. Jacobi,
P. G. L. Dirichlet, and hosts of more recent men; Great Britain
produced her A. De Morgan, G. Boole, W. R. Hamilton, A. Cayley,
J. J. Sylvester, besides champions who are still living; Russia entered
the arena with her N. I. Lobachevski; Norway with N. H. Abel;
Italy with L. Cremona; Hungary with her two Bolyais; the United
States with Benjamin Peirce and J. Willard Gibbs.

H. S. White of Vassar College estimated the annual rate of increase
in mathematical publication from 1870 to 1909, and ascertained the
periods between these years when different subjects of research re-
ceived the greatest emphasis.1 Taking the Jahrbuch iiber die Fort-
schritte der Mathematik, published since 1871 (founded by Carl Ohrt-
mann (1839-1885) of the Konigliche Realschule in Berlin and since
1885 under the chief editorship of Emil Lampe of the technische
Hochschule in Berlin), and also the Revue Semestrielle, published since
1893 (under the auspices of the Mathematical Society of Amsterdam),
he counted the number of titles, and in some cases also the number of
pages filled by the reviews of books and articles devoted to a certain
subject of research, and reached the following approximate results:

(1) The total annual publication doubled during the forty years;

(2) During these forty years, 30% of the publication was on applied
mathematics, 25% on geometry, 20% on analysis, 18% on algebra,
7% on history and philosophy; (3) Geometry, dominated by "Pliicker,
his brilliant pupil Klein, Clifford, and Cayley," doubled its rate of
production from 1870 to 1890, then fell off a third, to regain most of
its loss after 1899; Synthetic geometry reached its maximum in 1887
and then declined during the following twenty years; the amount of
analytic geometry always exceeded that of synthetic geometry, the

1 H. S. White, "Forty Years' Fluctuations in Mathematical Research," Science,
N. S., Vol. 42, 1915, pp. 105-113.

278

INTRODUCTION 279

excess being most pronounced since 1887; (4) Analysis, "which takes
its rise equally from calculus, from the algebra of imaginaries, from
the intuitions and the critically refined developments of geometry, and
from abstract logic: the common servant and chief ruler of the other
branches of mathematics," shows a trebling in forty years, reaching
its first maximum in 1890, "probably the culmination of waves set
in motion by Weierstrass and Fuchs in Berlin, by Riemann in Got-
tingen, by Hermite in Paris, Mittag-Leffler in Stockholm, Dini and
Brioschi in Italy;" before 1887 much of the growth of analysis is due
to the theory of functions which reaches a maximum about 1887,
with a sweep of the curve upward again after 1900, due to the theory
of integral equations and the influence of Hilbert; (5) Algebra, in-
cluding series and groups, experienced during the forty years a steady
gain to 2\ times its original output; the part of algebra relating to
algebraic forms, invariants, etc., reached its acme before 1890 and
then declined most surprisingly; (6) Differential equations increased
in amount slowly but steadily from 1870, "under the combined in-
fluence of Weierstrass, Darboux and Lie," showing a slight decline
in 1886, but "followed by a marked recovery and advance during the
publication of lectures by Forsyth, Picard, Goursat and Painleve;"
(7) The mathematical theory of electricity and magnetism remained
less than one-fourth of the whole applied mathematics, but rose after
1873 steadily toward one-fourth, by the labors of Clerk Maxwell,
W. Thomson (Lord Kelvin) and P. G. Tait; (8) The constant shifting
of mathematical investigation is due partly to fashion.

The progress of mathematics has been greatly accelerated by the
organization of mathematical societies issuing regular periodicals.
The leading societies are as follows: London Mathematical Society
organized in 1865, La sociele mathematique de France organized in
1872, Edinburgh Mathematical Society organized 1883, Circolo mate-
matico di Palermo organized in 1884, American Mathematical Society
organized in 1888 under the name of New York Mathematical Society
and changed to its present name in I894,1 Deutsche Mathematiker-
Vereinigung organized in 1890, Indian Mathematical Society organized
in 1907, Sociedad Metematica Espanola organized in 1911, Mathematical
Association of America organized in 1915.

The- number of mathematical periodicals has enormously increased
during the passed century. According to Felix Miiller z there were,
up to 1700, only 17 periodicals containing mathematical articles;
there were, in the eighteenth century, 210 such periodicals, in the
nineteenth century 950 of them.

1 Consult Thomas S. Fiske's address in Bull. Am. Math. Soc., Vol. n, 1905, p. 238.
Dr. Fiske himself was a leader in the organization of the Society.

2 Jahresb. d. deutsch. Malhem. Vereinigung, Vol. 12, 1903, p. 439. See also G. A.
Miller in Historical Introduction to Mathematical Literature, New York, 1916,
Chaps. I, II.

28o A HISTORY OF MATHEMATICS

A great stimulus toward mathematical progress have been the
international congresses of mathematicians. In 1889 there was held
in Paris a Congres international de bibliographic des sciences mathe-
matiques. In 1893, during the Columbian Exposition, there was held
in Chicago an International Mathematical Congress. But, by com-
mon agreement, the gathering held in 1897 at Zurich, Switzerland, is
called the "first international mathematical congress." The second
was held in 1900 at Paris, the third in 1904 at Heidelberg, the fourth
in 1908 at Rome, the fifth in 1912 at Cambridge in England. The
object of these congresses has been to promote friendly relations, to
give reviews of the progress and present state of different branches of
mathematics, and to discuss matters of terminology and bibliography.

One of the great co-operative enterprises intended to bring the
results of modern research in digested form before the technical reader
is the Encyklo p'ddie der Mathematischen Wissenschaften, the publica-
tion of which was begun in 1898 under the editorship of Wilhelm Franz
Meyer of Konigsberg. Prominent as joint editor was Heinrich Burk-
hardl (1861-1914) of Zurich, later of Munich. In 1904 was begun the
publication of the French revised and enlarged edition under the
editorship of Jules Molk (1857-1914) of the University of Nancy.

As regards the productiveness of modern writers, Arthur Cayley
said in 1883: l "It is difficult to give an idea of the vast extent of
modern mathematics. This word ' extent' is not the right one: I mean
extent crowded with, beautiful detail, — not an extent of mere uni-
formity such as an objectless plain, but of a tract of beautiful country
seen at first in the distance, but which will bear to be rambled through
and studied in every detail of hillside and valley, stream, rock, wood,
and flower." It is pleasant to the mathematician to think that in his,
as in no other science, the achievements of every age remain posses-
sions forever; new discoveries seldom disprove older tenets; seldom
is anything lost or wasted.

If it be asked wherein the utility of some modern extensions of
mathematics lies, it must be acknowledged that it is at present difficult
to see how some of them are ever to become applicable to questions
of common life or physical science. But our inability to do this should
not be urged as an argument against the pursuit of such studies. In
the first place, we know neither the day nor the hour when these
abstract developments will find application in the mechanic arts, in
physical science, or in other branches of mathematics. For example,
the whole subject of graphical statics, so useful to the practical en-
gineer, was made to rest upon von Staudt's Geometrie der Lage; W. R.
Hamilton's "principle of varying action" has its use in astronomy;
complex quantities, general integrals, and general theorems in inte-
gration offer advantages in the study of electricity and magnetism.

1 Arthur Cayley, Inaugural Address before the British Association, 1883, Re-
port, p. 25.

INTRODUCTION 281

"The utility of such researches," said Spottiswoode in 1878,* "can
in no case be discounted, or even imagined beforehand. Who, for
instance, would have supposed that the calculus of forms or the theory
of substitutions would have thrown much light upon ordinary equa-
tions; or that Abelian functions and hyperelliptic transcendents would
have told us anything about the properties of curves; or that the
calculus of operations would have helped us in any way towards the
figure of the earth?"

As a matter of fact in the nineteenth century, as in all centuries,
practical questions have been controlling factors in the growth of
mathematics. Says C. E. Picard: "The influence of physical theories
has been exercised not only on the general nature of the problems to
be solved, but even in the details of the analytic transformations.
Thus is currently designated in recent memoirs on partial differential
equations under the name of Green's formula, a formula inspired by
the primitive formula of the English physicist. The theory of dynamic
electricity and that of magnetism, with Ampere and Gauss, have been
the origin of important progress; the study of curvilinear integrals
and that of the integrals of surfaces have taken thence all their de-
velopments, and formulas, such as that of Stokes which might also
be called Ampere's formula, have appeared for the first time in mem-
oirs on physics. The equations on the propagation of electricity, to
which are attached the names of Ohm and Kirchhoff , while presenting
a great analogy with those of heat, offer often conditions at the limits
a little different; we know all that telegraphy by cables owes to the
profound discussion of a Fourier's equation carried over into elec-
tricity. The equations long ago written of hydrodynamics, the
equations of the theory of electricity, those of Maxwell and of Hertz
in electromagnetism, have offered problems analogous to those re-
called above, but under conditions still more varied." 2

Along similar lines are the remarks of A. R. Forsyth. In 1905 he
said: 3 "The last feature of the century that will be mentioned has
been the increase in the number of subjects, apparently dissimilar
from one another, which are now being made to use mathematics to
some extent. Perhaps the most surprising is the application of mathe-
matics to the domain of pure thought; this was effected by George
Boole in his treatise 'Laws of Thought,' published in 1854; and though
the developments have passed considerably beyond Boole's researches,
his work is one of those classics that mark a new departure. Political
economy, on the initiative of Cournot and Jevons, has begun to employ
symbols and to develop the graphical methods; but there the present
use seems to be one of suggestive record and expression, rather than

1 William SpottiswoQde, Inaugural Address before British Association, 1878,
Report, p. 25.

2 Congress of Arts and Science, St. Louis, 1904, Vol. I, pp. 507-508.

3 Report of the British Ass'n (South Africa), 1905, London, 1906, p. 317.

282 A HISTORY OF MATHEMATICS

of positive construction. Chemistry, in a modern spirit, is stretching
out into mathematical theories; Willard Gibbs, in his memoir on the
equilibrium of chemical systems, has led the way; and, though his
way is a path which chemists find strewn with the thorns of analysis,
his work has rendered, incidentally, a real service in co-ordinating
experimental results belonging to physics and to chemistry. A new
and generalized theory of statistics is being constructed ; and a school
has grown up which is applying them to biological phenomena. Its
activity, however, has not yet met with the sympathetic goodwill of
all the pure biologists; and those who remember the quality of the
discussion that took place last year at Cambridge between the biome-
tricians and some of the biologists will agree that, if the new school
should languish, it will not be for want of the tonic of criticism."

The great characteristic of modern mathematics is its generalizing
tendency. Nowadays little weight is given to isolated theorems,
says J. J. Sylvester, "except as affording hints of an unsuspected new
sphere of thought, like meteorites detached from some undiscovered
planetary orb of speculation." In mathematics, as in all true sciences,
no subject is considered in itself alone, but always as related to, or
an outgrowth of, other things. The development of the notion of
continuity plays a leading part in modern research. In geometry
the principle of continuity, the idea of correspondence, and the theory
of projection constitute the fundamental modern notions. Continuity
asserts itself in a most striking way in relation to the circular points
at infinity in a plane. In algebra the modern idea finds expression
in the theory of linear transformations and invariants, and in the
recognition of the value of homogeneity and symmetry.

H. F. Baker l said in 1913 that, with the aid of groups "a complete
theory of equations which are soluble algebraically can be given. . . .
But the theory of groups has other applications. . . . The group of
interchanges among four quantities which leave unaltered the product
of their six differences is exactly similar to the group of rotations of a
regular tetrahedron whose centre is fixed, when its corners are inter-
changed among themselves. Then I mention the historical fact that
the problem of ascertaining when that well-known differential equa-
tion called the hypergeometric equation has all its solutions expressible
in finite terms as algebraic functions, was first solved in connection
with a group of similar kind. For any linear differential equation it is
of primary importance to consider the group of interchanges of its
solutions when the independent variable, starting from an arbitrary
point, makes all possible excursions, returning tfl its initial value. . . .
There is, however, a theory of groups different from those so far
referred to, in which the variables can change continuously; this alone
is most extensive, as may be judged from one of its lesser applications,
the familiar theory of the invariants of quantics. Moreover, perhaps
1 Report British Ass'n (Birmingham), 1913, London, 1914, p. 371.

INTRODUCTION 283

the most masterly of the analytical discussions of the theory of
geometry has been carried through as a particular application of the
theory of such groups."

"If the theory of groups illustrates how a unifying plan works in
mathematics beneath the bewildering detail, the next matter I refer
to well shows what a wealth, what a grandeur, of thought may spring
from what seem slight beginnings. Our ordinary integral calculus is
well-nigh powerless when the result of integration is not expressible
by algebraic or logarithmic functions. The attempt to extend the
possibilities of integration to the case when the function to be inte-
grated involves the square root of a polynomial of the fourth order,
led first, after many efforts, ... to the theory of doubly-periodic
functions. To-day this is much simpler than ordinary trigonometry,
and, even apart from its applications, it is quite incredible that it
should ever again pass from being among the treasures of civilized
man. Then, at first in uncouth form, but now clothed with delicate
beauty, came the theory of general algebraical integrals, of which the
influence is spread far and wide; and with it all that is systematic
in the theory of plane curves, and all that is associated with the con-
ception of a Riemann surface. After this came the theory of multiply-
periodic functions of any number of variables, which, though still
very far indeed from being complete, has, I have always felt, a majesty
of conception which is unique. Quite recently the ideas evolved in
the previous history have prompted a vast general theory of the
classification of algebraical surfaces according to their essential prop-:
erties, which is opening endless new vistas of thought."

The nineteenth century and the beginning of the twentieth century
constitute a period when the very foundations of mathematics have
been re-examined and when fundamental principles have been worked
out anew. Says H. F. Baker: [ "It is a constantly recurring need of
science to reconsidpr the exact implication of the terms employed;
and as numbers and functions are inevitable in all measurement, the
precise meaning of number, of continuity, of infinity, of limit, and
so on, are fundamental questions. . . . These notions have many
pitfalls I may cite. . . . the construction of a function which is
continuous at all points of a range, yet possesses no definite differential
coefficient at any point. Are we sure that human nature is the only
continuous variable in the concrete world, assuming it be continuous,
which can possess such a vacillating character? . . . We could take
out of our life all the moments at which we can say that our age is a
certain number of years, and days, and fractions of day, and still
have appreciably as long to live; this would be true, however often,
to whatever exactness, we named our age, provided we were quick
enough in naming it. ... These inquiries . . . have been associated
also with the theory of those series which Fourier used so boldly, and
1 H. F. Baker, loc. cit., p. 369.

284 A HISTORY OF MATHEMATICS

so wickedly, for the conduction of heat. Like all discoverers, he took
much for granted. Precisely how much is the problem. This problem
has led to the precision of what is meant by a function of real variables,
to the question of the uniform convergence of an infinite series, as
you may see in early papers of Stokes, to new formulation of the
conditions of integration and of the properties of multiple integrals,
and so on. And it remains still incompletely solved.

"Another case in which the suggestions of physics have caused
grave disquiet to the mathematicians is the problem of the variation
of a definite integral. No one is likely to underrate the grandeur of
the aim of those who would deduce the whole physical history of the
world from the single principle of least action. Everyone must be
interested in the theorem that a potential function, with a given
value at the boundary of a volume, is such as to render a certain in-
tegral, representing, say, the energy, a minimum. But in that pro-
portion one desires to be sure that the logical processes employed are
free from objection. And, alas! to deal only with one of the earliest
problems of the subject, though the finally .sufficient conditions for
a minimum of a simple integral seemed settled long ago, and could
be applied, for example, to Newton's celebrated problem of the
solid of least resistance, it has since been shown to be a general fact
that such a problem cannot have any definite solution at all. And,
although the principle of Thomson and Dirichlet, which relates to
the potential problem referred to, was expounded by Gauss, and
accepted by Riemann, and remains to-day in our standard treatise
on Natural Philosophy, there can be no doubt that, in the form in
which it was originally stated, it proves just nothing. Thus a new
investigation has been necessary into the foundations of the principle.
There is another problem, closely connected with this subject, to
which I would allude: the stability of the solar system. For those
who can make pronouncements in regard to this I have a feeling of
envy; for their methods, as yet, I have a quite other feeling. The
interest of this problem alone is sufficient to justify the craving of
the Pure Mathematician for powerful methods and unexceptionable
rigour."

There are others who view this struggle for absolute rigor from a
different angle. Horace Lamb in 1904 spoke as follows: 1 "a traveller
who refuses to pass over a bridge until he has personally tested the
soundness of every part of it is not likely to go very far; something
must be risked, even in Mathematics. It is notorious that even in
this realm of ' exact ' thought, discovery has often been in advance of
strict logic, as in the theory of imaginaries, for example, and in the
whole province of analysis of which Fourier's theorem is a type."

Says Maxime Bocher: 2 "There is what may perhaps be called the

1 Address before Section A, British Ass'n, in Cambridge, 1904.

1 Maxime Bficher in Congress of Arts and Science, St. Louis, 1904, Vol. I, p. 472.

INTRODUCTION 285

method of optimism, which leads us either willfully or instinctively
to shut our eyes to the possibility of evil. Thus the optimist who
treats a problem in algebra or analytic geometry will say, if he stops
to reflect on what he is doing: 'I know that I have no right to divide
by zero; but there are so many other values which the expression by
which I am dividing might have that I will assume that the Evil
One has not thrown a zero in my denominator this time.' This
method . . . has been of great service in the rapid development of
many branches of mathematics."

Definitions of Mathematics

One of the phases of the quest for rigor has been the're-defining of
mathematics. "Mathematics, the science of quantity" is an old idea
which goes back to Aristotle. A modified form of this old definition
is due to Auguste Comte (1798-1857), the French philosopher and
mathematician, the founder of positivism. Since the most striking
measurements are not direct, b.ut are indirect, as the determination
of distances and sizes of the planets, or of the atoms, he defined mathe-
matics "the science of indirect measurement." These definitions
have been abandoned for the reason that several modern branches of
mathematics, such as the theory of groups, analysis situs, projective
geometry, theory of numbers and the algebra of logic, have no relation
to quantity and measurement. "For one thing," says C. J. Keyser,1
"the notion of the continuum — the 'Grand Continuum' as Sylvester
called it — that central supporting pillar of modern Analysis, has been
constructed by K. Weierstrass, R. Dedekind, Georg Cantor and
others, without any reference whatever to quantity, so that number
and magnitude are not only independent, they are essentially dis-
parate." Or, if we prefer to go back a few centuries and refer to a
single theorem, we may quote G. Desargues as saying that if the
vertices of two triangles lie in three lines meeting in a point, then their
sides meet in three points lying on a line. This beautiful theorem
has nothing to do with measurement.

In 1870 Benjamin Peirce wrote in his Linear Associative Algebra
that "mathematics is the science which draws necessary conclusions."
This definition has been regarded as including too much and also as
in need of elucidation as to what constitutes a "necessary" conclusion.
Reasoning which seemed absolutely conclusive to one generation no
longer satisfies the next. According to present standards no reasoning
which claims to be exact can make any use of intuition, but must
proceed from definitely and completely stated premises according to
certain principles of formal logic.2 Mathematical logicians from
George Boole to C. S. Peirce, E. Schroder, and G. Peano have pre-
pared the field so well that of late years Peano and his followers, and

1 C. J. Keyser, The Human Worth of Rigorous Thinking, New York, 1916, p. 277.

2 Maxime Bdcher, loc. oil., Vol. I, p. 459.

286 A HISTORY OF MATHEMATICS.

independently G. Frege, "have been able to make a rather short
list of logical conceptions and principles upon which it would seem
that all exact reasoning depends." But the validity of logical prin-
ciples must stand the test of use, and on this point we may never be
sure. Frege and Bertrand Russell independently built up a theory of
arithmetic, each starting with apparently self-evident logical prin-
ciples. Then Russell discovers that his principles, applied to a very
general kind of logical class, lead to an absurdity. There is evident
need of reconstruction somewhere. After all, are we merely making
successive approximations to absolute rigor?

A. B. Kempe's definition is as follows: l "Mathematics is the
science by which we investigate those characteristics of any subject-
matter of thought which are due to the conception that it consists
of a number of differing and non-differing individuals and pluralities."
Ten years later Maxime Bocher modified Kempe's definition thus: 2
"If we have a certain class of objects and a certain class of relations,
and if the only questions which we investigate are whether ordered
groups of those objects do or do not satisfy the relations, the results
of the investigation are called mathematics." Bocher remarks that
if we restrict ourselves to exact or deductive mathematics, then
Kempe's definition becomes coextensive with B. Peirce's.

Bertrand Russell, in his Principles of Mathematics, Cambridge,
1903, regards pure mathematics as consisting exclusively of deduc-
tions "by logical principles from logical principles." Another def-
inition given by Russell sounds paradoxical, but really expresses the
extreme generality and extreme subtleness of certain parts of modern
mathematics: "Mathematics is the subject in which we never know
what we are talking about nor whether what we are saying is true." 3
Other definitions along similar lines are due to E. Papperitz (1892),
G. Itelson (1904), and L. Couturat (1908).

Synthetic Geometry

The conflict between synthetic and analytic methods in geometry
which arose near the close of the eighteenth century and the beginning
of the nineteenth has now come to an end. Neither side has come
out victorious. The greatest strength is found to lie, not in the sup-
pression of either, but in the friendly rivalry between the two, and in
the stimulating influence of the one upon the other. Lagrange prided
himself that in his Mecanique Analytique he had succeeded in avoiding
all figures; but since his time mechanics has received much help from
geometry.

Modern synthetic geometry was created by several investigators
about the same time. It seemed to be the outgrowth of a desire for

1 Proceed. London Math. Soc., Vol. 26, i8q4, p. 15.

2 M. Bdcher, op. cit., p. 466.

3 B. Russell in International Monthly, Vol. 4, 1901, p. 84.

SYNTHETIC GEOMETRY 287

general methods which should serve as threads of Ariadne to guide
the student through the labyrinth of theorems, corollaries, porisms,
and problems. Synthetic geometry was first cultivated by G. Monge,
L. N. M. Carnot, and J. V. Poncelet in France; it then bore rich fruits
at the hands of A. F. Mobius and Jakob Steiner in Germany and
Switzerland, and was finally developed to still higher perfection by
M. Chasles in France, von Staudt in Germany, and L. Cremona in
Italy.

Jean Victor Poncelet (1788-1867), a native of Metz, took part in
the Russian campaign, was abandoned as dead on the bloody field
of Krasnoi, and taken prisoner to Saratoff. Deprived there of all
books, and reduced to the remembrance of what he had learned at
the Lyceum at Metz and the Polytechnic School, where he had studied
with predilection the works of G. Monge, L. N. M. Carnot, and C. J.
Brianchon, he began to study mathematics from its elements. He
entered upon original researches which afterwards made him illus-
trious. While in prison he did for mathematics what Bunyan did for
literature, — produced a much-read work, which has remained of great
value down to the present time. He returned to France in 1814, and
in 1822 published the work in question, entitled, Traite des Proprietes
projectiles des figures. In it he investigated the properties of figures
which remain unaltered by projection of the figures. The projection
is not effected here by parallel rays of prescribed direction, as with
G. Monge, but by central projection. Thus perspective projection,
used before him by G. Desargues, B. Pascal, I. Newton, and J. H.
Lambert, was elevated by him into a fruitful geometric method.
Poncelet formulated the so-called principle of continuity, which asserts
that properties of a figure which hold when the figure varies according
to definite laws will hold also when the figure assumes some limiting
position.

"Poncelet," says J. G. Darboux,1 "could not content himself with
the insufficient resources furnished by the method of projections; to
attain imaginaries he created that famous principle of continuity
which gave birth to such long discussions between him and A. L.
Cauchy. Suitably enunciated, this principle is excellent and can
render great service. Poncelet was wrong in refusing to present it
as a simple consequence of analysis; and Cauchy, on the other hand,
was not willing to recognize that his own objections, applicable with-
out doubt to certain transcendent figures, were without force in the
applications made by the author of the Traite des proprietes projec-
tives." J. D. Gergonne characterized the principle as a valuable
instrument for the discovery of new truths, which nevertheless did
not make stringent proofs superfluous.2 By this principle of geometric

1 Congress of Arts and Science, St. Louis, 1904, Vol. I, p. 539.
1 E. Kotter, Die Enlwickelung der synlhetischen Geometrie von Monge bis aitf
Slavdt, Leipzig, 1901, p. 123.

288 A HISTORY OF MATHEMATICS

continuity Poncelet was led to the consideration of points and lines
which vanish at infinity or become imaginary. The inclusion of such
ideal points and lines was a gift which pure geometry received from
analysis, where imaginary quantities behave much in the same way
as real ones. Poncelet elaborated some ideas of De Lahire, F. Servois,
and J. D. Gergonne into a regular method — the method of "recipro-
cal polars." To him we owe the Principle of Duality as a consequence
of reciprocal polars. As an independent principle it is due to Gergonne.
Darboux says that the significance of the principle of duality which
was "a little vague at first, was sufficiently cleared up by the dis-
cussions which took place on this subject between J. D. Gergonne,
J. V. Poncelet and J. Pliicker." It had the advantage of making
correspond to a proposition another proposition of wholly different
aspect. "This was a fact essentially new. To put it in evidence,
Gergonne invented the system, which since has had so much success,
of memoirs printed in double columns with correlative propositions
in juxtaposition" (Darboux).

Joseph Diaz Gergonne (1771-1859) was an officer of artillery, then
professor of mathematics at the lyceum in Nimes and later professor
at Montpellier. He solved the Apollonian Problem and claimed
superiority of analytic methods over the synthetic. Thereupon
Poncelet published a purely geometric solution. Gergonne and Ponce-
let carried on an intense controversy on the priority of discovering
the principle of duality. No doubt, Poncelet entered this field earlier,
while Gergonne had a deeper grasp of the principle. Some geometers,
particularly C. J. Brianchon, entertained doubts on the general valid-
ity of the principle. The controversy led to one new result, namely,
Gergonne's considerations of the class of a curve or surface, as well
as its order.1 Poncelet wrote much on applied mechanics. In 1838
the Faculty of Sciences was enlarged by his election to the chair of
mechanics.

J. G. Darboux says that, "presented in opposition to analytic
geometry, the methods of Poncelet were not favorably received by
the French analysts. But such were their importance and their
novelty, that without delay they aroused, from divers sides, the
most profound researches." Many of these appeared in the Annales
de malhemaliqties, published by J. D. Gergonne at Nimes from 1810
to 1831. During over fifteen years this was the only journal in the
world devoted exclusively to mathematical researches. Gergonne
" collaborated, often against their will, with the authors of the memoirs
sent him, rewrote them, and sometimes made them say more or less
than they would have wished. . . . Gergonne, having become rector
of the Academy of Montpellier, was forced to suspend in 1831 the
publication of his journal. But the success it had obtained, the taste
for research it had contributed to develop, had commenced to bear
1 E. Kotter, op. dt., pp. 160-164.

SYNTHETIC GEOMETRY 289

their fruit. L. A. J. Quetelet had established in Belgium the Corre-
spondance mathematique et physique. A. L. Crelle, from 1826, brought
out at Berlin the first sheets of his celebrated journal, where he pub-
lished the memoirs of N. H. Abel, of C. G. J. Jacobi, of J. Steiner"
(Darboux).

Contemporaneous with J. V. Poncelet was the German geometer,
Augustus Ferdinand Mobius (1790-1868), a native of Schulpforta in
Prussia. He studied at Gottingen under K. F. Gauss, also at Leipzig
and Halle. In Leipzig he became, in 1815, privat-docent, the next
year extraordinary professor of astronomy, and in 1844 ordinary
professor. This position he held till his death. The most important
of his researches are on geometry. They appeared in Crelle's Journal,
and in his celebrated work entitled Der Barycentrische Calcul, Leipzig,
1827, "a work truly original, remarkable for the profundity of its
conceptions, the elegance and the rigor of its exposition" (Darboux).
As the name indicates, this calculus is based upon properties of the
centre of gravity.1 Thus, that the point S is the centre of gravity of
weights a, b, c, d placed at the points A, B, C, D respectively, is ex-
pressed by the equation

(a+b+c+d}S=aA +bB+cC+dD.

His calculus is the beginning of a quadruple algebra, and contains
the germs of Grassmann's marvellous system. In designating seg-
ments of lines we find throughout this work for the first time con-
sistency in the distinction of positive and negative by the order of
letters AB, BA. Similarly for triangles and tetrahedra. The remark
that it is always possible to give three points A, B, C such weights
a, (3, 7 that any fourth point M in their plane will become a centre of
mass, led Mobius to a new system of co-ordinates in which the position
of a point was indicated by an equation, and that of a line by co-
ordinates. By this algorithm he found by algebra many geometric
theorems expressing mainly invariantal properties, — for example, the
theorems on the anharmonic relation. Mobius wrote also on statics
and astronomy. He generalized spherical trigonometry by letting
the sides or angles^f triangles exceed i8oj.

Not only Mobius but also H. G. Grassmann discarded the usual
co-ordinate systems, and used algebraic analysis. Later in the nine-
teenth century and at the opening of the twentieth century, these
ideas were made use of, notably by Cyparissos Stephanos (1857-1917)
of the National University of Athens, H. Wiener, C. Segre, G. Peano,
F. Aschieri, E. Study, C. Burali-Forti and Hermann Grassmann
(1859- ), a son of H. G. Grassmann. Their researches, covering
the fields of binary and ternary linear transformations, were brought
together by the younger Grassmann into a treatise, Projektive Geome-
tric der Ebene unter Benutzung der Punktrechnung dargesteilt, 1909.

1 J. W. Gibbs, "Multiple Algebra," Proceedings Am. Ass'n for the Advanc. oj
Science, 1886.

2oo A HISTORY OF MATHEMATICS

Jakob Steiner (1796-1863), "the greatest geometrician since the
time of Euclid," was born in Utzendorf in the Canton of Bern. He
did not learn to write till he was fourteen. At eighteen he became a
pupil of Pestalozzi. Later he studied at Heidelberg and Berlin.
When A. L. Crelle started, in 1826, the celebrated mathematical
journal bearing his name, Steiner and Abel became leading con-
tributors. Through the influence of C. G. J. Jacobi and others, the
chair of geometry was founded for him at Berlin in 1834. This posi-
tion he occupied until his death, which occurred after years of bad
health.

In 1832 Steiner published his Systematische Entwickelung der Ab-
hangigkeit geometrischer Gestalten wn einander, "in which is uncovered
the organism by which the most diverse phenomena (Erscheinungeri)
in the world of space are united to each other." Here for the first
time, is the principle of duality introduced at the outset. This book
and von Staudt's lay the foundation on which synthetic geometry in
its later form rested. The researches of French mathematicians, cul-
minating in the remarkable creations of G. Monge, J. V. Poncelet
and J. D. Gergonne, suggested a unification of geometric processes.
This work of "uncovering the organism by which the most different
forms in the world of space are connected with each other," this ex-
posing of "a small number of very simple fundamental relations in
which the scheme reveals itself, by which the whole body of theorems
can be logically and easily developed" was the task which Steiner
assumed. Says H. Hankel: J "In the beautiful theorem that a conic
section can be generated by the intersection of two projective pencils
(and the dually correlated theorem referring to projective ranges),
J. Steiner recognized the fundamental principle out of which the
innumerable properties of these remarkable curves follow, as it were,
automatically with playful ease." Not only did he fairly complete
the theory of curves and surfaces of the second degree, but he made
great advances in the theory of those of higher degrees.

In the Systematische Entwickelungen (1832) Steiner directed atten-
tion to the complete figure obtained by joining in every possible way
six points on a conic and showed that in this hexagrammum mysticum
the 60 "Pascal lines" pass three by three through 20 points ("Steiner
points ") which lie four by four upon 15 straight lines (" Pliicker lines ").
J. Pliicker had sharply criticized Steiner for an error that had crept
into an earlier statement (1828) of the last theorem. Now, Steiner
gave the correct statement, but without acknowledgment to Pliicker.
Further properties of the hexagrammum mysticum are due to T. P.
Kirkman, A. Cayley and G. Salmon. The Pascal lines of three hexa-
gons concur in a new point ("Kirkman point"). There are 60 Kirk-
man points. Corresponding to three Pascal lines which concur in a
Steiner point, there are three Kirkman points which lie upon a straight
1 H. Hankel, Elemente der Projectivisclten Geometric, 1875, p. 26.

SYNTHETIC GEOMETRY '291

line ("Cayley line"). There are 20 Cayley lines which pass four by
four through 15 " Salmon points." Other new properties of the mystic
hexagon were obtained in 1877 by G. Veronese and L. Cremona.1

In Steiner's hands synthetic geometry made prodigious progress.
New discoveries followed each other so rapidly that he often did not
take time to record their demonstrations. In an article in Crelle's
Journal on Allgemeine Eigenschaften Algebraischer Curven he gives
without proof theorems which were declared by L. O. Hesse to be
"like Fermat's theorems, riddles to the present and future genera-
tions." Analytical proofs of some of them have been given since by
others, but L. Cremona finally proved them all by a synthetic method.
Steiner discovered synthetically the two prominent properties of a
surface of the third order; viz. that it contains twenty-seven straight
lines and a pentahedron which has the double points for its vertices
and the lines of the Hessian of the given surface for its edges. This
subject will be discussed more fully later. Steiner made investigations
by synthetic methods on maxima and minima, and arrived at the
solution of problems which at that time surpassed the analytic power
of the calculus of variations. It will appear later that his reasoning
on this topic is not always free from criticism.

Steiner generalized Malfatti's problem.2 Giovanni Francesco Mal-
fatti (1731-1807) of the university of Ferrara, in 1803, proposed the
problem, to cut three cylindrical holes out of a three-sided prism in
such a way that the cylinders and the prism have the same altitude
and that the volume of the cylinders be a maximum. This problem
was reduced to another, now generally known as Malfatti's problem:
to inscribe three circles in a triangle so that each circle will be tangent to
two sides of the triangle and to the other two circles. Malfatti gave an
analytical solution, but Steiner gave without proof a construction,
remarked that there were thirty-two solutions, generalized the problem
by replacing the three lines by three circles, and solved the analogous
problem for three dimensions. This general problem was solved
analytically by C. H. Schellbach (1809-1892) and A. Cayley and by
R. F. A. Clebsch with the aid of the addition theorem of elliptic
functions.3 A simple proof of Steiner's construction was given by
A. S. Hart of Trinity College, Dublin, in 1856.

Of interest is Steiner's paper, Ueber die geometrischen Consiructionen,
ausgefuhrt mittels der geraden Linie und eines festen Kreises (1833),
in which he shows that all quadratic constructions can be effected
with the aid of only a ruler, provided that a fixed circle is drawn once
for all. It was generally known that all linear constructions could be
effected by the ruler, without other aids of any kind. The case of

1 G. Salmon, Conic Sections, 6th Ed., 1879, Notes, p. 382.

2 Karl Fink, A Brief History of Mathematics, transl. by W. W. Beman and I). K.
Smith, Chicago, 1900, p. 256.

3 A. Wittstein, zur Geschichle des Malfatti'schen Problems, Nordlingen, 1878.

292 A HISTORY OF MATHEMATICS

cubic constructions, calling for the determination of three unknown
elements (points) was worked out in 1868 by Ludwig Hermann Kor-
tum (1836-1904) of Bonn, and Stephen Smith of Oxford in two re-
searches which received the Steiner prize of the Berlin Academy; it
was shown that if a conic (not a circle) is given to start with, then all
such constructions can be done with a ruler and compasses. Franz
London (1863-1917) of Breslau demonstrated in 1895 that these cubic
constructions can be effected with a ruler only, as soon as a fixed
cubic curve is once drawn.1

F. Biitzberger 2 has recently pointed out that in an unpublished
manuscript, Steiner disclosed a knowledge of the principle of inver-
sion as early as 1824. In 1847 Liouville called it the transformation
by reciprocal radii. After Steiner this transformation was found
independently by J. Bellavitis in 1836, J. W. Stubbs and J. R. Ingram
in 1842 and 1843, and by William Thomson (Lord Kelvin) in 1845.

Steiner's researches are confined to synthetic geometry. He hated
analysis as thoroughly as J. Lagrange disliked geometry. Steiner's
Gesammdte Werke were published in Berlin in 1881 and 1882.

Michel Chasles (1793-1880) was born at Epernon, entered the Poly-
technic School of Paris in 1812, engaged afterwards in business, which
he later gave up that he might devote all his time to scientific pursuits.
In 1841 he became professor of geodesy and mechanics at the Ecole
poly technique; later, "Professeur de Geometric superieure a la Faculte
des Sciences de Paris." He was a voluminous writer on geometrical
subjects. In 1837 he published his admirable Aperqu historique sur
I'origine et le develop pement des methodes en geometric, containing a
history of geometry and, as an appendix, a treatise "sur deux principes
generaux de la Science." The Aperqu historique is still a standard
historical work; the appendix contains the general theory of Homog-
raphy (Collineation) and of duality (Reciprocit)7). The name duality
is due to J. D. Gergonne. Chasles introduced the term anharmonic
ratio, corresponding to the German Doppeherh'dltniss and to Clifford's
cross-ratio. Chasles and J. Steiner elaborated independently the
modern synthetic or projective geometry. Numerous original memoirs
of Chasles were published later in the Journal de V Ecole Poly technique.
He gave a reduction of cubics, different from Newton's in this, that
the five curves from which all others can be projected are symmetrical
with respect to a centre. In 1864 he began the publication, in the
Comptes rendus, of articles in which he solves by his "method of
characteristics" and the "principle of correspondence" an immense
number of problems. He determined, for instance, the number of
intersections of two curves in a plane. The method of characteristics
contains the basis of enumerative geometry.

As regards Chasles' use of imaginaries, J. G. Darboux says: "Here,

1 Jahrcsb. d. d. Math. Vrreinigung, Vol. 4, p. 163.
* Bull. Am. Math. Soc., Vol. 20, 1914, p. 414.

SYNTHETIC GEOMETRY 293

his method was really new. . . . But Chasles introduced imaginaries
only by their symmetric functions, and consequently would not have
been able to define the cross-ratio of four elements when these ceased
to be real in whole or in part. If Chasles had been able to establish
the notion of the cross-ratio of imaginary elements, a formula he
gives in the Geometric superieure (p. 118 of the new edition) would
have immediately furnished him that beautiful definition of angle as
logarithm of a cross-ratio which enabled E. Laguerre, our regretted
confrere, to give the complete solution, sought so long, of the problem
of the transformation of relations which contain at the same time angles
and segments in homography and correlation." The application of
the principle of correspondence was extended by A. Cayley, A. Brill,
H. G. Zeuthen, H. A. Schwarz, G. H. Halphen, and others. The full
value of these principles of Chasles was not brought out until the
appearance, in 1879, of the Kalkul der Abzahlenden Geometrie by
Hermann Schubert (1848-1911) of Hamburg. This work contains
a masterly discussion of the problem of enumerative geometry, viz.
to determine the number of points, lines, curves, etc., of a system
which fulfil certain conditions. Schubert extended his enumerative
geometry to w-dimensional space.1 The fundamental principle of
enumerative geometry is the law of the "preservation of the number,"
which, as stated by Schubert, was found by E. Study and by G. Kohn
in 1903 to be not always valid. The particular problem examined
by Study and later also by F. Severi, considers the number of pro-
jectivities of a line which transform into itself a given group of four
points. If the cross-ratio of the group is not a cube root of — i, the
number of projectivities is 4, otherwise there are more. A recent
book on this subject is H. G. Zeuthen's Abzdhlende Methoden der
Geometrie, 1914.

To Chasles we owe the introduction into projective geometry of
non-projective properties of figures by means of the infinitely distant
imaginary sphere-circle.2 Remarkable is his complete solution, in
1846, by synthetic geometry, of the difficult question of the attrac-
tion of an ellipsoid on an external point. This celebrated problem
was treated alternately by synthetic and by analytic methods. Colin
Maclaurin's results, obtained synthetically, had created a sensation.
Nevertheless, both A. M. Legendre and S. D. Poisson expressed the
opinion that the resources of the synthetic method were easily ex-
hausted. Poisson solved it analytically in 1835. Then Chasles sur-
prised every one by his synthetic investigations, based on the con-
sideration of confocal surfaces. Poinsot reported on the memoir and
remarked on the analytic and synthetic methods: "It is certain that
one cannot afford to neglect either."

1 Gino Loria, Die Ilauplsdchlichslcn Thcoricn der Geometrie, 1888, p. 124.

2 F. Klein, Vergleichende Belrachlungen iibcr neiiere gcomdrische Forschnngcr,
Erlangen, 1872, p. 12.

294 A HISTORY OF MATHEMATICS

The labors of Chasles and Steiner raised synthetic geometry to an
honored and respected position by the side of analysis.

Karl Georg Christian von Staudt (1798-1867) was born in Rothen-
burg on the Tauber, and at his death, was professor in Erlangen. His
great works are the Geometric der Lage, Niirnberg, 1847, ano^ n^s
Beitrdge zur Geometric der Lage, 1856-1860. The author cut loose
from algebraic formulae and from metrical relations, particularly the
anharmonic ratio of J. Steiner and M. Chasles, and then created a
geometry of position, which is a complete science in itself, independent
of all measurements. He shows that projective properties of figures
have no dependence whatever on measurements, and can be estab-
lished without any mention of them. In his theory of "throws" or
"Wiirfe," he even gives a geometrical definition of a number in its
relation to geometry as determining the position of a point. Gustav
Kohn of the University of Vienna about 1894 introduced the throw
as a fundamental concept underlying the projective properties of a
geometric configuration, such that, according to a principle of duality
of this geometry, throws of figures appear in pairs of reciprocal throws;
figures of reciprocal throws form a complete analogy to figures of
equal throws. Referring to Von Staudt's numerical co-ordinates,
defined without introducing distance as a fundamental idea, A. N.
Whitehead said in 1906: "The establishment of this result is one of
the triumphs of modern mathematical thought."

The Beitrdge contains the first complete and general theory of
imaginary points, lines, and planes in projective geometry. Repre-
sentation of an imaginary point is sought in the combination of an
involution with a determinate direction, both on the real line through
the point. While purely projective, von Staudt's method is inti-
mately related to the problem of representing by actual points and
lines the imaginaries of analytical geometry. Says Kotter: 1 Staudt
was the first who succeeded " in subjecting the imaginary elements to
the fundamental theorem of projective geometry, thus returning to
analytical geometry the present which, in the hands of geometri-
cians, had led to the most beautiful results." Von Staudt's geometry
of position was for a long time disregarded, mainly, no doubt, because
his book is extremely condensed. An impulse to the study of this
subject was given by Culmann, who rests his graphical statics upon
the work of von Staudt. An interpreter of von Staudt was at last
found in Theodor Reye of Strassburg, who wrote a Geometrie der
Lage in 1868.

The graphic representation of the imaginaries of analytical geom-
etry was systematically undertaken by C. F. Maximilien Marie (1819-
1891), who worked, however, on entirely different lines from those
of von Staudt. Another independent attempt was made in 1893 by
F. H. Loud of Colorado College.

1 E. Kotter, op. cit., p. 123.

SYNTHETIC GEOMETRY 295

Synthetic geometry was studied with much success by Luigi Cre-
mona (1830-1903), who was born in Pa via and became in 1860 pro-
fessor of higher geometry in Bologna, in 1866 professor of geometry and
graphical statics in Milan, in 1873 professor of higher mathematics
and director of the engineering school at Rome. He was influenced
by the writings of M. Chasles, later he recognized von Staudt as the
true founder of pure geometry. A memoir of 1866 on cubic surfaces
secured half of the Steiner prize from Berlin, the other half being
awarded to Rudolf Sturm, then of Bromberg. Cremona used the
method of enumeration with great effect. He wrote on plane curves,
on surfaces, on birational transformations of plane and solid space.
The birational transformations, the simplest class of which is now
called the "Cremona transformation," proved of importance, not
only in geometry, but in the analytical theory of algebraic functions
and integrals. It was developed more fully by M. Nother and othersf
H. S. White comments on this subject as follows: l "Beyond the
linear or projective transformations of the plane there were known
the quadric inversions of Ludwig Immanuel Magnus (1790-1861) of
Berlin, changing lines into conies through three fundamental points
and those exceptional points into singular lines, to be discarded.
Cremona described at once the highest generalization of these trans-
formations, one-to-one for all points of the plane except a finite set
of fundamental points. He found that it must be mediated by a
net of rational curves; any two intersecting in one variable point,
and in fixed points, ordinary or multiple, which are the fundamental
points and which are themselves tranformed into singular rational
curves of the same orders as the indices of multiplicity of the
points. When the fundamental points are enumerated by classes
according to their several indices, the set of class numbers for the in-
verse transformation is found to be the same as for the direct, but
usually related to different indices. Tables of such rational nets of
low orders were made out by L. Cremona and A. Cayley, and a
wide new vista seemed opening (such indeed it was and is) when si-
multaneously three investigators announced that the most general
Cremona transformation is equivalent to a succession of quadric
transformations of Magnus's type. This seemed a climax, and a
set-back to certain expectations." Cremona's theory of the trans-
formation of curves and of the correspondence of points on curves
was extended by him to three dimensions. There he showed how a
great variety of particular transformations can be constructed, "but
anything like a general theory is still in the future." Ruled surfaces,
surfaces of the second order, space-curves of the third order, and
the general theory of surfaces received much attention at his hands.
He was interested in map-drawing, which had engaged the attention
of R. Hooke, G. Mercator, J. Lagrange, K. F. Gauss and others. For
1 Bull. Am. Math. $oc., Vol. 24, 1918, p. 242.

296 A HISTORY OF MATHEMATICS

a one-one correspondence the surface must be unicursal, and this is
sufficient. L. Cremona is associated with A. Cayley, R. F. A. Clebsch,
M. Nother and others in the development of this theory.1 Cremona's
writings were translated into German by Maximilian Curtze (1837-
1903), professor at the gymnasium in Thorn. The Opera matematiche
di Luigi Cremona were brought at Milan in 1914 and 1915.

One of the pupils of Cremona was Giovanni Battista Guccia (1855-
1914). He was born in Palermo and studied at Rome under Cremona.
In 1889 he became extraordinary professor at the University of Pal-
ermo, in 1894 ordinary professor. He gave much attention to the
study of curves and surfaces. He is best known as the founder in
1884 of the Circolo matematico di Palermo, and director of its Rendi-
conli. The society has become international and has been a powerful
stimulus for mathematical research in Italy.

Karl Culmann (1821-1881), professor at the Polytechnicum in Zu-
rich, published an epoch-making work on Die graphische Statik, Zurich,
1864, which has rendered graphical statics a great rival of analytical
statics. Before Culmann, Barthelemy-Edouard Cousinery (1790-1851)
a civil engineer at Paris, had turned his attention to the graphical
calculus, but he made use of perspective, and not of modern geometry.2
Culmann is the first to undertake to present the graphical calculus
as a symmetrical whole, holding the same relation to the new geom-
etry that analytical mechanics does to higher analysis. He makes
use of the polar theory of reciprocal figures as expressing the relation
between the force and the funicular polygons. He deduces this rela-
tion without leaving the plane of the two figures. But if the polygons
be regarded as projections of lines in space, these lines may be treated
as reciprocal elements of a "Nullsystem." This was done by Clerk
Maxwell in 1864, and elaborated further by L. Cremona. The graphi-
cal calculus has been applied by O. Mohr of Dresden to the elastic
line for continuous spans. Henry T. Eddy (1844- ), then of the
Rose Polytechnic Institute, now of the University of Minnesota, gives
graphical solutions of problems on the maximum stresses in bridges
under concentrated loads, with aid of what he calls "reaction poly-
gons." A standard work, La Statique graphique, 1874, was issued by
Maurice Levy of Paris.

Descriptive geometry [reduced to a science by G. Monge in France,
and elaborated further by his successors, /. N. P. Hachette, C. Dupin,
Theodore Olivier (1793-1853) of Paris, Jules de la Gournerie of Paris]
was soon studied also in other countries. The French directed their
attention mainly to the theory of surfaces and their curvature; the
Germans and Swiss, through Guido Schreiber (1799-1871) of Karls-

1 Proceedings of the Roy. Soc. of London, Vol. 75, London, 1905, pp. 277-
279.

s A. Jay du Bois, Graphical Statics, New York, 1875, P- xxxii; M. d'Ocagne, Trailf
de N omographie, Paris, 1899, p. 5.

SYNTHETIC GEOMETRY 297

ruhe, Karl Pohlke (1810-1877) of Berlin,1 Josef Schlesinger (1831-
1901) of Vienna, and particularly W. Fiedler, interwove projective
and descriptive geometry.

Wilhelm Fiedler (1832-1912), the son of a shoe-maker in Chemnitz,
Saxony, taught mathematics and mechanics in a technical school of
Chemnitz, 1853 to 1864, and studied meanwhile the works of M.
Chasles, G. Lame, B. de St.-Venant, J. V. Poncelet, J. Steiner, J.
Pliicker, von Staudt, G. Salmon, A. Cayley, J. J. Sylvester. He was
self-taught. On the recommendation of A. F. Mobius he was awarded
in 1859 the degree of doctor of philosophy by the University of Leipsic
for a dissertation on central projection. At this time Fiedler made
arrangements with Salmon for a German elaborated edition of Sal-
mon's Conic Sections; it appeared in 1860. In the same way were
brought out by Friedler Salmon's Higher Algebra in 1863, Salmon's
Geometry of Three Dimensions in 1862, Salmon's Higher Plane Curves
in 1873. In 1864 Fiedler became professor at the technical high school
in Prag, and in 1867 at the Polytechnic School in Zurich, where he
was active until his retirement in 1907. The emphasis of Fiedler's
activity was placed upon descriptive geometry. His Darslellende
Geometric, 1871, was brought, in the third edition, in organic connec-
tion with v. Staudt's geometry of position. Especially after the death
of Culmann in 1881, Fiedler was criticised on pedagogic grounds for
excessive emphasis upon geometric construction. A harmonizing
effort was the text on descriptive geometry by Christian Wiener
(1826-1896) of the Polytechnic School in Karlsruhe. Of interest is
Fiedler's recognition in 1870 of homogeneous co-ordinates as cross-
ratios, invariant in all linear transformations; this idea had been ad-
vanced in 1827 by A. F. Mobius, but had remained unnoticed.2 Fied-
ler's Zyklographie, 1882, contained constructions of problems on circles
and spheres.

The interweaving of projective and descriptive geometry was
carried on in Italy by G. Bellavitis. The theory of shades and shadows
was first investigated by the French writers quoted above, and in Ger-
many treated most exhaustively by Ludwig Burmester of Munich.

Elementary Geometry of the Triangle and Circle

It is truly astonishing that during the nineteenth century new
theorems should have been found relating to such simple figures as
the triangle and circle, which had been subjected to such close exam-
ination by the Greeks and the long line of geometers which followed.
It was L. Euler who proved in 1765 that the orthocenter, circumcenter

1 F. J. Obenrauch, Geschichle der darslcllenden und projectiven Geometric, Briinn,
1897, pp. 350, 352.

2 A. Voss, "Wilhelm Fiedler," Jahresb. d. d. Math. Vercinigung, Vol. 22, 1913,
p. 107.

298 A HISTORY OF MATHEMATICS

and centroid of a triangle are collinear, lying on the "Euler line."
H. C. Gossard of the University of Oklahoma showed in 1916 that
the three Euler lines of the triangles formed by the Euler line and the
sides, taken by twos, of a given triangle, form a triangle triply per-
spective with the given triangle and having the same Euler line. Con-
spicuous among the new developments is the "nine-point circle," the
discovery of which has been erroneously ascribed to Euler. Among
the several independent discoverers is the Englishman, Benjamin
Bevan (?-i838) who proposed in Leybourn's Mathematical Repository,
I, 18, 1804, a theorem for proof which practically gives us the nine-
point circle. The proof was supplied to the Repository, I, Part i,
p. 143, by John Butterworth, who also proposed a problem, solved by
himself and John Whitley, from the general tenor of which it appears
that they knew the circle in question to pass through all nine points.
These nine points are explicitly mentioned by C. J. Brianchon and
J. V. Poncelet in Gergonne's Annales of 1821. In 1822, Karl Wilhelm
Feuerbach (1800-1834), professor at the gymnasium in Erlangen,
published a pamphlet in which he arrives at the nine-point circle,
and proves the theorem known by his name, that this circle touches
the incircle and the three excircles. The Germans call it " Feuerbach's
Circle." The last independent discoverer of this remarkable circle, so
far as known, is T. S. Dames, in an article of 1827 in the Philosophical
Magazine, II, 29-31. Feuerbach's theorem was extended by Andrew
Searle Hart (1811-1890), fellow of Trinity College, Dublin, who
showed that the circles which touch three given circles can be dis-
tributed into sets of four all touched by the same circle.

In 1816 August Leopold Crelle published in Berlin a paper dealing
with certain properties of plane triangles. He showed how to deter-
mine a point ft inside a triangle, so that the
angles (taken in the same order) formed by
the lines joining it to the vertices are equal.
In the adjoining figure the three marked angles
are equal. If the construction is made so that
angle ft'AC=Q'CB=O'BA, then a second point
O' is obtained. The study of these new
angles and new points led Crelle to exclaim:
"It is indeed wonderful that so simple a figure as the triangle is
so inexhaustible in properties. How many as yet unknown proper-
ties of other figures may there not be!" Investigations were made
also by Karl Friedrich Andreas Jacobi (1795-1855) of Pforta and
some of his pupils, but after his death, in 1855, the whole
matter was forgotten. In 1875 the subject was again brought before
the mathematical public by Henri Brocard (1845- ) whose re-
searches were followed up by a large number of investigators in France,
England and Germany. Unfortunately, the names of geometricians
which have been attached to certain remarkable points, lines and

SYNTHETIC GEOMETRY

299

circles are not always the names of the men who first studied their
properties. Thus, we speak of "Brocard points" and "Brocard
angles," but historical research brought out the fact, in 1884 and 1886,
that these were the points and lines which had been studied by A/ L.
Crelle and K. F. A. Jacobi. The "Brocard Circle" is Brocard's own
creation. In the triangle ABC, let O and O' be the first and second
"Brocard point." Let A' be the intersection of BO and CO'; B' of

AO' and CO; C' of BO' and AO. The circle passing through A',
B', C' is the "Brocard circle." A'B'C' is "Brocard's first triangle."
Another like triangle, A"B"C" is called "Brocard's second triangle."
The points A", B", C", together with O, Q', and two other points,
lie in the circumference of the "Brocard circle."

In 1873 fimile Lemoine (1840-1912), the editor of I' Intermediate
des mathematiciens , called attention to a particular point within a plane
triangle which has been variously called the "Lemoine point," " sym-
median point," and "Grebe point," named after Ernst Wilhelm Grebe
(1804-1874) of Kassel. If CD is so drawn
as to make angles a and b equal, then one
of the two lines AB and CD is the anti-
parallel of the other, with reference to
the angle O. Now OE, the bisector of
AB, is the median and OF, the bisector of
the anti-parallel of AB, is called the sym-
median (abbreviated from symetrique de la
mcdiane). The point of concurrence of the
three symmedians in a triangle is called,
after Robert Tucker (1832-1905) of University College School in Lon-
don, the "symmedian point." John Sturgeon Mackay (1843-1914) of
Edinburgh has pointed out that some of the properties of this point,
brought to light since 1873, were first discovered previously to that

300

date. The anti-parallels of a triangle which pass through its sym-
median point, meet its sides in six points which lie on a circle, called
the "second Lemoine circle." The "first Lemoine circle" is a special
case of a "Tucker circle" and concentric with the "Brocard circle."
The "Tucker circles" may be thus defined. Let DF' = FE'=ED';
let, moreover, the following pairs of lines be anti-parallels to each
other: AB and ED', BC and FE', CA and DF'; then the six points

D, D', E, E', F, F', lie on a "Tucker
circle." Vary the length of the equal
anti-parallels, and a family of "Tucker
circles" is obtained. Allied to these
are the "Taylor circles," due to H. M.
Taylor of Trinity College, Cambridge.
Still different types are the "Mackay
circles," and the "Neuberg circles"
due to Joseph Neuberg (1840- ) of
Luxemburg. A systematic treatise on
this topic, Die Brocardschen Gebilde, was

written by Albrecht Emmerich, Berlin, 1891. Of the almost in-
numerable mass of new theorems on the triangle and circle, a great
number is given in the Treatise on the Circle and the Sphere, Oxford,
1916, written by J. L. Coolidge of Harvard University.

Since 1888 E. Lemoine of Paris developed a system, called geomet-
rographics, for the purpose of numerically comparing geometric con-
structions with respect to their simplicity. Coolidge calls these
"the best known and least undesirable tests for the simplicity
of a geometrical construction"; A. Emch declares that "they are
hardly of any practical value, in so far as they do not indicate how to
simplify a construction or how to make it more accurate."

A new theorem upon the circumscribed tetraedron was propounded
in 1897 by A. S. Bang and proved by Joh. Gehrke. The theorem is:
Opposite edges of a circumscribed tetraedron subtend equal angles at
the points of contact of the faces which contain them. It has been
the starting-point for extended developments by Franz Meyer, J.
Neuberg and H. S. White.1

Link-motion

The generation of rectilinear motion first arose as a practical prob-
lem in the design of steam engines. A close approximation to such
motion is the "parallel motion" designed by James Watt in 1784:
In a freely jointed quadrilateral ABCD, with the side AD fixed, a
point M on the side BC moves in nearly a straight line. The equa-
tion of the curve traced by M, sometimes called "Watt's curve,"
was first derived by the French engineer, Francois Marie de Prony

1 Bull. Am. Math. Soc., Vol. 14, 1908, p. 220.

SYNTHETIC GEOMETRY

301

(1755-1839); the curve is of the sixth order and was studied by Dar-
boux in 1879. A generalization of this curve is the " three-bar curve"
studied by Samuel Roberts in 1876 and Reinhold Mutter in 1902. :

A beautiful discovery in link-motion which came to attract a great
deal of attention was made by A. Peaucellier, Capitaine du genie a
Nice; in 1864 he proposed in the Nouvelles annales the problem of
devising compound compasses for the generation of a straight line
and also a conic. It is evident from his remarks that he himself had a
solution. In 1873 he published his solution in the same journal.
When Peaucellier 's cell came to be appreciated, he was awarded the
great mechanical prize of the Institute of France, the "Prix Montyou."
The generation of exact rectilinear motion had long been believed im-
possible. Only recently has it been pointed out that straight-line
motion had been invented before Peaucellier by another Frenchman,
P. F. Sarrus of Strasbourg. He presented an article and a model to
the Paris Academy of Sciences; the article, without any figure, was
published in 1853 in the Comptes Rendus 2 of the academy, and reported
on by Poncelet. The paper was entirely forgotten until attention was
called to it in 1905 by G. T. Bennett of Emmanuel College, Cambridge.3
The pieces ARSB and ATUB are
each hinged by three parallel hori-
zontal hinges, the two sets of hinges
having different directions. Con-
nected thus with B, A has a recti-
linear movement, up and down. In
one respect Sarrus's solution of
the problem of rectilinear motion is
more complete than that of Peau-
cellier; for, while the Peaucellier cell
gives rectilinear motion only to a
single point, Sarrus's apparatus gives rectilinear motion to the whole
piece A. It was re-invented in 1880 by H. M. Brunei and in 1891
by Archibald Barr. Yet to this day Sarrus's device appears to re-
main practically unknown.

While Sarrus's device is three-dimensional, that of Peaucellier is
two-dimensional. An independent solution of straight-line motion
was given in 1871 by the Russian Lipkin, a pupil of P. Chebichev of
the University of Petrograd. In 1874 J. J. Sylvester became interested
in link-motion, and lectured on it at the Royal Institution. Dur-
ing the next few years several mathematicians worked on linkages.
H. Hart of Woolwich reduced Peaucellier's seven links to four links.
A new device by Sylvester has been called "Sylvester's linkage."

1 G. Loria, Ebene Curven (F. Schiitte), Vol. I, 1910, pp. 274-279.
* Comptes Rendus, Vol. 36, pp. 1036, 1125. The author's name is here spelled
"Sarrut," but R. C. Archibald has pointed out that this is a misprint.
3 See Philosoph. Magaz., 6. S., Vol. 9, 1905, p. 803.

302 A HISTORY OF MATHEMATICS

The barrister at law, Alfred Bray Kempe of London showed in 1876
that a link-motion can be found to describe any given algebraic curve;
he is the author of a popular booklet, How to draw a Straight Line,
London, 1877. Other articles of note on this subject were prepared
by Samuel Roberts, Arthur Cayley, W. Woolsey Johnson, V. Liguine
of the University of Odessa, and G. P. X. Koenigs of the ficole Poly-
technique in Paris. The determination of the linkage with minimum
number of pieces by which a given curve can be described is still an
unsolved problem.

Parallel Lines, Non-Euclidean Geometry and Geometry of n Dimensions

During the nineteenth century very remarkable generalizations were
made, which reach to the very root of two of the oldest branches
of mathematics, — elementary algebra and geometry. In geometry
the axioms have been searched to the bottom, and the con-
clusion has been reached that the space denned by Euclid's axioms
is not the only possible non-contradictory space. Euclid proved
(I, 27) that "if a straight line falling on two other straight lines make
the alternate angels equal to one another, the two straight lines shall
be parallel to one another." Being unable to prove that in every
other case the two lines are not parallel, he assumed this to be true in
what is now generally called the $th "axiom," by some the nth or
the 1 2th "axiom."

Simpler and more obvious axioms have been advanced as sub-
stitutes. As early as 1663, John Wallis of Oxford recommended: "To
any triangle another triangle, as large as you please, can be drawn,
which is similar to the given triangle." G. Saccheri assumed the ex-
istence of two similar, unequal triangles. Postulates similar to Wallis'
have been proposed also by J. H. Lambert, L. Carnot, P. S. Laplace,
J. Delboeuf. A. C. Clairaut assumes the existence of a rectangle;
W. Bolyai postulated that a circle can be passed through any three
points not in the same straight line, A. M. Legendre that there existed a
finite triangle whose angle-sum is two right angles, J. F. Lorenz and
Legendre that through every point within an angle a line can be
drawn interescting both sides, C. L. Dodgson that in any circle the
inscribed equilateral quadrangle is greater than any one of the seg-
ments which lie outside it. But probably the simplest is the assump-
tion made by Joseph Fenn in his edition of Euclid's Elements, Dub-
lin, 1769, and again sixteen years later by William Ludlam (1718-1788),
vicar of Norton, and adopted by John Playfair: "Two straight lines
which cut one another can not both be parallel to the same straight
line." It is noteworthy l that this axiom is distinctly stated in
Proclus's note to Euclid, I, 31.

But the most numerous efforts to remove the supposed defect in

1 T. L. Heath, The Thirteen Books of Euclid's Elements, Vol. I, p. 220.

SYNTHETIC GEOMETRY 303

Euclid were attempts to prove the parallel postulate. After centuries
of desperate but fruitless endeavor, the bold idea dawned upon the
minds of several mathematicians that a geometry might be built up
without assuming the parallel-axiom. While A. M. Legendre still
endeavored to establish the axiom by rigid proof, Lobachevski brought
out a publication which assumed the contradictory of that axiom,
and which was the first of a series of articles destined to clear up ob-
scurities in the fundamental concepts, and greatly to extend the field
of geometry.

Nicholaus Ivanovich Lobachevski (1793-1856) was born at Ma-
karief, in Nizhni-Novgorod, Russia, studied at Kasan, and from 1827
to 1846 was professor and rector of the University of Kasan. His views
on the foundation of geometry were first set forth in a paper laid before
the physico-mathematical department of the University of Kasan in
February, 1826. This paper was never printed and was lost. His
earliest publication was in the Kasan Messenger for 1829, and then in
the Gelehrte Schriften der Universitat Kasan, 1836-1838, under the
title, "New Elements of Geometry, with a complete theory of Par-
allels." Being in the Russian language, the work remained unknown to
foreigners, but even at home it attracted no notice. In 1840 he pub-
lished a brief statement of his researches in Berlin, under the title,
Geometrische Untersuchungen zur Theorle der Parallellinien. Loba-
chevski constructed an " imaginary geometry," as he called it, which
has been described by W. K. Clifford as "quite simple, merely Eu«lid
without the vicious assumption." A remarkable part of this geometry
is this, that through a point an indefinite number of lines can be drawn
in a plane, none of which cut a given line in the same plane. A similar
system of geometry was deduced independently by the Bolyais in
Hungary, who called it "absolute geometry."

Wolfgang Bolyai de Bolya (1775-1856) was born in Szekler-Land,
Transylvania. After studying at Jena, he went to Gottingen, where
he became intimate with K. F. Gauss, then nineteen years' old. Gauss
used to say that Bolyai was the only man who fully understood his
views on the metaphysics of mathematics. Bolyai became professor
at the Reformed College of Maros-Vasarhely, where for forty-seven
years he had for his pupils most of the later professors of Transyl-
vania. The first publications of this remarkable genius were dramas
and poetry. Clad in old-time planter's garb, he was truly original in
his .private life as well as in his mode of thinking. He was extremely
modest. No monument, said he, should stand over his grave, only an
apple-tree, in memory of the three apples; the two of Eve and Paris,
which made hell out of earth, and that of I. Newton, which elevated
the earth again into the circle of heavenly bodies.1 His son, Johann

1 F. Schmidt, "Aus dem Leben zweier ungarischer Mathematiker Johann und
Wolfgang Bolyai von Bolya," Grunert's Archiv, 48:2, 1868. Franz Schmidt (1827-
1901) was an architect in Budapest.

304 A HISTORY OF MATHEMATICS

Bolyai (1802-1860), was educated for the army, and distinguished
himself as a profound mathematician, an impassioned violin-player,
and an expert fencer. He once accepted the challenge of thirteen
officers on condition that after each duel he might play a piece on his
violin, and he vanquished them all.

The chief mathematical work of Wolfgang Bolyai appeared in
two volumes, 1832-1833, entitled Tentamen juventutem studiosam in
dementa matheseos puree . . . introducendi. It is followed by an ap-
pendix composed by his son Johann. Its twenty-six pages make the
name of Johann Bolyai immortal. He published nothing else, but he
left behind one thousand pages of manuscript.

While Lobachevski enjoys priority of publication, it may be that
Bolyai developed his system somewhat earlier. Bolyai satisfied him-
self of the non-contradictory character of his new geometry on or be-
fore 1825; there is some doubt whether Lobachevski had reached this
point in 1826. Johann Bolyai's father seems to have been the only
person in Hungary who really appreciated the merits of his son's
work. For thirty-five years this appendix, as also Lobachevski's
researches, remained in almost entire oblivion. Finally Richard
Baltzer of the University of Giessen, in 1867, called attention to the
wonderful researches.

In 1866 J. Hoiiel translated Lobachevski's Geometrische Unter-
suchungen into French. In 1867 appeared a French translation of
Johann Bolyai's Appendix. In 1891 George Bruce Halsted, then of
the University of Texas, rendered these treatises easily accessible to
American readers by translations brought out under the titles of
J. Bolyai's The Science Absolute of Space and N. Lobachevski's Geo-
metrical Researches on the Theory of Parallels of 1840.

The Russian and Hungarian mathematicians were not the only
ones to whom pangeometry suggested itself. A copy of the Tentamen
reached K. F. Gauss, the elder Bolyai's former roommate at Gottingen,
and this Nestor of German mathematicians was surprised to discover
in it worked out what he himself had begun long before, only to leave
it after him in his papers. As early as 1 792 he had started on researches
of that character. His letters show that in 1 799 he was trying to prove
a priori the reality of Euclid's system; but some time within the next
thirty years he arrived at the conclusion reached by Lobachevski
and Bolyai. In 1829 he wrote to F. W. Bessel,. stating that his "con-
viction that we cannot found geometry completely a priori has be-
come, if possible, still firmer," and that "if number is merely a product
of our mind, space has also a reality beyond our mind of which we
cannot fully foreordain the laws a priori" The term non-Euclidean
geometry is due to Gauss. It is surprising that the first glimpses of
non-Euclidean geometry were had in the eighteenth century. Geron-
nimo Saccheri (1667-1733), a Jesuit father of Milan, in 1733 wrote

SYNTHETIC GEOMETRY 305

Euclides ab omni naevo vindicatus l (Euclid vindicated from every
flaw). Starting with two equal lines AC and BD, drawn perpendicu-
lar to a line AB and on the same side of it, and joining C and D, he
proves that the angles at C and D are equal. These angles must be
either right, or obtuse or acute. The hypothesis of an obtuse angle
is demolished by showing that it leads to results in conflict with
Euclid I, 17: Any two angles of a triangle are together less than two
right angles. The hypothesis of the acute angle leads to a long pro-
cession of theorems, of which the one declaring that two lines which
meet in a point at infinity can be perpendicular at that point to the
same straight line, is considered contrary to the nature of the straight
line; hence the hypothesis of the acute angle is destroyed. Though
not altogether satisfied with his proof, he declared Euclid "vindi-
cated." Another early writer was J. H. Lambert who in 1766 wrote
a paper "Zur Theorie der Parallellinien," published in the Leipziger
Magazinfiir reine und angewandte Mathematik, 1786, in which: (i) The
failure of the parallel-axiom in 'surf ace-spherics gives a geometry with
angle-sum >2 right angles; (2) In order to make intuitive a geometry
with angle-sum < 2 right angles we need the aid of an "imaginary
sphere" (pseudo-sphere); (3) In a space with the angle-sum differing
from 2 right angles, there is an absolute measure (Bolyai's natural
unit for length). Lambert arrived at no definite conclusion on the
validity of the hypotheses of the obtuse and acute angles.

Among the contemporaries and pupils of K. F. Gauss, three deserve
mention as writers on the theory of parallels, Ferdinand Karl ScJrwei-
kart (1780-1859), professor of law in Marburg, Franz Adolf Taurinus
(1794-1874), a nephew of Schweikart, and Friedrich Ludwig Wackier
(179 2-1 8 1 7), a pupil of Gauss in 1809 and professor at Dan tzig. Schwei-
kart sent Gauss in 1818 a manuscript on "Astral Geometry" which
he never published, in which the angle-sum of a triangle is less than
two right angles and there is an absolute unit of length. He induced
Taurinus to study this subject. Taurinus published in 1825 his Theorie
der Parallellinien, in which he took the position of Saccheri and Lam-
bert, and in 1826 his Geometries prima elementa, in an appendix of
which he gives important trigonometrical formulae for non-Euclidean
geometry by using the formulae of spherical geometry with an imag-
inary radius. His Elementa attracted no attention. In disgust he
burned the remainder of his edition. Wachter's results are contained
in a letter of 1816 to Gauss and in his Demonstratio axiomatis geo-
metrici in Euclideis undecimi, 1817. He showed that the geometry
on a sphere becomes identical with the geometry of Euclid when the
radius is infinitely increased, though it is distinctly shown that the
limiting surface is not a plane.2 Elsewhere we have mentioned the

1 See English translation by G. B. Halsted in Am. Math. Monthly, Vols. 1-5, 1894-
1898.
1 D. M. Y. Sommerville, Elem. of Non-Euclidean Geometry, London, 1914, p. 15.

306 A HISTORY OF MATHEMATICS

contemporary researches on parallel lines due to A. M. Legendre in
France.

The researches of K. F. Gauss, N. I. Lobachevski and J. Bolyai
have been considered by F. Klein as constituting the first period in
the history of non-Euclidean geometry. It is a period in which the
synthetic methods of elementary geometry were in vogue. The
second period embraces the researches of G. F. B. Riemann, H. Helm-
holtz, S. Lie and E. Beltrami, and employs the methods of differential
geometry. It was in 1854 that Gauss heard from his pupil, Riemann,
a marvellous dissertation which considered the foundations of geome-
try from a new point of view. Riemann was not familiar with Lo-
bachevski and Bolyai. He developed the notion of w-ply extended
magnitude, and the measure-relations of which a manifoldness of n
dimensions is capable, on the assumption that every line may be
measured by every other. Riemann applied his ideas to space. He
taught us to distinguish between " unboundedness " and "infinite
extent." According to him we have in our mind a more general notion
of space, i. e. a notion of non-Euclidean space; but we learn by expe-
rience that our physical space is, if not exactly, at least to a high degree
of approximation, Euclidean space. Riemann's profound dissertation
was not published until 1867, when it appeared in the Gottingen Ab-
handlungen. Before this the idea of n dimensions had suggested itself
under various aspects to Ptolemy, J. Wallis, D'Alembert, J. Lagrange,
J. Pliicker, and H. G. Grassmann. The idea of time as a fourth di-
mension had occurred to D'Alembert and Lagrange. About the same
tune with Riemann's paper, others were published from the pens of
H. Helmholtz and E. Beltrami. This period marks the beginning
of lively discussions upon this subject. Some writers — J. Bellavitis,
for example — were able to see in non-Euclidean geometry and n-
dimensional space nothing but huge caricatures, or diseased out-
growths of mathematics. H. Helmholtz's article was entitled That-
sachen, welche der Geometric zu Grunde liegen, 1868, and contained
many of the ideas of Riemann. Helmholtz popularized the subject in
lectures, and in articles for various magazines. Starting with the
idea of congruence, and assuming the free mobility of a rigid body and
the return unchanged to its original position after rotation about an
axis, he proves that the square of the line-element is a homogeneous
function of the second degree in the differentials.1 Helmholtz's
investigations were carefully examined by S. Lie who reduced the
Riemann-Helmholtz problem to the following form: To determine
all the continuous groups in space which, in a bounded region, have
the property of displacements. There arose three types of groups,

Sommerville is the author of a Bibliography of non-Euclidean geometry including
the theory of parallels, the foundations of geometry, and space of n dimensions, London,
1911.
1 D. M. Y. Sommerville, op. cit., p. 195.

SYNTHETIC GEOMETRY 307

which characterize the three geometries of Euclid, of N. I. Lobachev-
ski and J. Bolyai, and of F. G. B. Riemann.1

Eugenio Beltrami (1835-1900), born at Cremona, was a pupil of
F. Brioschi. He was professor at Bologna as a colleague of L. Cremona,
at Pisa as an associate of E. Betti, at Pavia as a co-worker with F.
Casorati, and since 1891 at Rome where he spent the last years of
his career, "uno degli illustri maestri dell' analisi in Italia." Beltrami
wrote in 1868 a classical paper, Saggio di inter pretazione delta geometria
non-euclidea (Giorn. di Matem., 6), which is analytical (and, like
several other papers, should be mentioned elsewhere were we to adhere
to a strict separation between synthesis and analysis). He reached
the brilliant and surprising conclusion that in part the theorems of
non-Euclidean geometry find their realization upon surfaces of con-
stant negative curvature. He studied, also, surfaces of constant
positive curvature, and ended with the interesting theorem that
the space of constant positive curvature is contained in the space
of constant negative curvature. These researches of Beltrami, H.
Helmholtz, and G. F. B. Riemann culminated in the conclusion that
on surfaces of constant curvature we may have three geometries, —
the non-Euclidean on a surface of constant negative curvature,
the spherical on a surface of constant positive curvature, and the
Euclidean geometry on a surface of zero curvature. The three ge-
ometries do not contradict each other, but are members of a system, —
a geometrical trinity. The ideas of hyper-space were brilliantly ex-
pounded and popularised in England by Clifford.

William Kingdon Clifford (1845-1879) was born at Exeter, edu-
cated at Trinity College, Cambridge, and from 1871 until his death
professor of applied mathematics in University College, London. His
premature death left incomplete several brilliant researches which
he had entered upon. Among these are his paper On Classification
of Loci and his Theory of Graphs. He wrote articles On the Canonical
Form and Dissection of a Riemann's Surface, on Biquaternions, and
an incomplete work on the Elements of Dynamic. He gave exact
meaning in dynamics to such familiar words as "spin," "twist,"
"squirt," "whirl." The theory of polars of curves and surfaces was
generalized by him and by Reye. His classification of loci, 1878,
being a general study of curves, was an introduction to the study
of tt-dimensional space in a direction mainly projective. This study
has been continued since chiefly by G. Veronese of Padua, C. Segre
of Turin, E. Bertini, F. Aschieri, P. Del Pezzo of Naples.

Beltrami's researches on non-Euclidean geometry were followed,
in 1871, by important investigations of Felix Klein, resting upon
Cayley's Sixth Memoir on Quantics, 1859. The development of geom-
etry in the first half of the nineteenth century had led to the separation

'Lie, Theorie der Transformalionsgruppen, Bd. Ill, Leipzig, 1893, pp. 437-543;
Bonola, op. cit., p. 154.

3o8 A HISTORY OF MATHEMATICS

of this science into two parts: the geometry of position or descriptive
geometry which dealt with properties that are unaffected by projec-
tion, and the geometry of measurement in which the fundamental
notions of distance, angle, etc., are changed by projection. Cayley's
Sixth Memoir brought these strictly segregated parts together again
by his definition of distance between two points. The question
whether it is not possible so to express the metrical properties of
figures that they will not vary by projection (or linear transformation)
had been solved for special projections by M. Chasles, J. V. Poncelet,
and E. Laguerre, but it remained for A. Cayley to give a general
solution by defining the distance between two points as an arbitrary
constant multiplied by the logarithm of the anharmonic ratio in which
the line joining the two points is divided by the fundamental quadric.
These researches, applying the .principles of pure projective geometry,
mark the third period in the development of non-Euclidean geometry.
Enlarging upon this notion, F. Klein showed the independence of
projective geometry from the parallel-axiom, and by properly choosing
the law of the measurement of distance deduced from projective
geometry the spherical, Euclidean, and pseudospherical geometries,
named by him respectively the elliptic, parabolic, and hyperbolic
geometries. This suggestive investigation was followed up by numer-
ous writers, particularly by G. Battaglini of Naples, E. d'Ovidio of
Turin, R. de Paolis of Pisa, F. Aschieri, A. Cayley, F. Lindemann of
Munich, E. Schering of Gottingen, W. Story of Clark University,
H. Stahl of Tubingen, A. Voss of Munich, Homersham Cox, A. Buch-
heim.1 The notion of parallelism applicable to hyperbolic space was
the only extension of Euclid's notion of parallelism until Clifford dis-
covered in elliptic space straight lines which possess most of the prop-
erties of Euclidean parallels, but differ from them in being skew. Two
lines are right (or left) parallel, if they cut the same right (or left)
generators of the absolute. Later F. Klein and R. S. Ball made
extensive contributions to the knowledge of these lines. More re-
cently E. Study of Bonn, J. L. Coolidge of Harvard University, W.
Vogt of Heidelberg and others have been studying this subject. The
methods employed have been those of analytic and synthetic geometry
as well as those of differential geometry and vectorial analysis.2 The
geometry of n dimensions was studied along a line mainly metrical
by a host of writers, among whom may be mentioned Simon Newcomb
of the Johns Hopkins University, L. Schlafii of Bern, W. I. Stringham
(1847-1909) of the University of California, W. Killing of Miinster,
T. Craig of the Johns Hopkins, Rudolf Lipschitz (1832-1903) of Bonn.
R. S. Heath of Birmingham and W. Killing investigated the kinematics
and mechanics of such a space. Regular solids in w-dimensipnal space
were studied by Stringham, Ellery W. Davis (1857-1918) of the

1 G. Loria, Die hauptsachlichsten Theorien der Geometric, 1888, p. 102.

2 Bull. Am. Math. Soc., Vol. 17, 1911, p. 315.

ANALYTIC GEOMETRY 309

University of Nebraska, Reinhold Hoppe (1816-1900) of Berlin, and
others. Stringham gave pictures of projections upon our space of
regular solids in four dimensions, and V. Schlegel at Hagen constructed
models of such projections. These are among the most curious of a
series of models published by L. Brill in Darmstadt. It has been
pointed out that if a fourth dimension existed, certain motions could
take place which we hold to be impossible. Thus S. Newcomb showed
the possibility of turning a closed material shell inside out by simple
flexure without either stretching or tearing; F. Klein pointed out that
knots could not be tied; G. Veronese showed that a body could be
removed from a closed room without breaking the walls; C. S. Peirce
proved that a body in four-fold space either rotates about two axes
at once, or cannot rotate without losing one of its dimensions.

A fourth period in the history of non-Euclidean geometry, intro-
duced by the researches of Moritz Pasch, Giuseppe Peano, Mario
Pieri, David Hilbert, Oswald Veblen, concerns itself with the logical
grounding of geometry (including non-Euclidean forms) upon sets
of axioms.

The geometry of hyperspace was exploited by spiritualists and
mediums, of whom Henry Slade was the most notorious. He con-
verted to spiritualism the German scientist F. Zollner and his coterie,
to whom he gave a spiritual demonstration of the existence of a fourth
dimension of space. These events contributed to the severity with
which the philosopher R. H. Lotze, in his Metaphysik, 1879, criticised
the mathematical theories of hyperspace and non-Euclidean geometry.

Analytic Geometry

In the preceding chapter we endeavored to give a flashlight view
of the rapid advance of synthetic geometry. In some cases we also
mentioned analytical treatises. Modern synthetic and modern
analytical geometry have much in common, and may be grouped to-
gether under the common name "projective geometry." Each has
advantages over the other. The continual direct viewing of figures
as existing in space adds exceptional charm to the study of the former,
but the latter has the advantage in this, that a well-established routine
in a certain degree may outrun thought itself, and thereby aid original
research. While in Germany J. Steiner and von Staudt developed
synthetic geometry, Pliicker laid the foundation of modern analytic
geometry.

Julius Pliicker (1801-1868) was born at Elberfeld, in Prussia. After
studying at Bonn, Berlin, and Heidelberg, he spent a short time in
Paris attending lectures of G. Monge and his pupils. Between 1826
and 1836 he held positions successively at Bonn, Berlin, and Halle.
He then became professor of physics at Bonn. Until 1846 his original
researches were on geometry. In 1828 and in 1831 he published his

3io A HISTORY OF MATHEMATICS

Analytisch-Geometrische Entwicklungen in two volumes. Therein he
adopted the abbreviated notation [used before him in a more restricted
way by Etienne Bobillier (1797-1832), professor of mechanics at
Chalons-sur-Marne], and avoided the tedious process of algebraic
elimination by a geometric consideration. In the second volume the
principle of duality is formulated analytically. With him duality and
homogeneity found expression already in his system of co-ordinates.
The homogenous or trilinear system used by him is much the same as
the co-ordinates of A. F. Mobius. In the identity of analytical opera-
tion and geometric construction Pliicker looked for the source of his
proofs. The System der Analytischen Geometrie, 1835, contains a
complete classification of plane curves of the third order, based on the
nature of the points at infinity. The Theorie der Algebraischen Curven,
1839, contains, besides an enumeration of curves of the fourth order,
the analytic relations between the ordinary singularities of plane
curves known as "Pliicker's equations," by which he was able to
explain "Poncelet's paradox." The discovery of these relations is,
says A. Cayley, "the most important one beyond all comparison in
the entire subject of modern geometry." The four Pliicker equa-
tions have been expressed in different forms. Cayley studied higher
singularities of plane curves. M. W. Haskell of the University of
California, in 1914, showed from the Pliicker equations that the
maximum number of cusps possible for a curve of order m is the
greatest integer in m(m—2)/3 (except when m is 4 and 6, in which
case the maximum number is 3 and 9), and that there is always a
self-dual curve with this maximum number of cusps.

Certain interrelations of the various geometrical researches of the
first half and middle of the nineteenth century are brought out by
J. G. Darboux in the following passage:1 "While M. Chasles, J.
Steiner, and, later, . . . von Staudt, were intent on constituting a
rival doctrine to analysis and set in some sort altar against altar,
J. D. Gergonne, E. Bobillier, C. Sturm, and above all J. Pliicker, per-
fected the geometry of R. Descartes and constituted an analytic sys-
tem in a manner adequate to the discoveries of the geometers. It is
to E. Bobillier and to J. Pliicker that we owe the method called
abridged notation. Bobillier consecrated to it some pages truly new
in the last volumes of the Annales of Gergonne. Pliicker commenced
to develop it in his first work, soon followed by a series of works where
are established in a fully conscious manner the foundations of the
modern analytic geometry. It is to him that we owe tangential co-
ordinates, trilinear co-ordinates, employed with homogeneous equa-
tions, and finally the employment of canonical forms whose validity
was recognized by the method, so deceptive sometimes, but so fruit-
ful, called the enumeration of constants"

In Germany J. Pliicker's researches met with no favor. His method
1 Congress of Arts and Science, St. Louis, 1904, Vol. 1, pp. 541, 542.

ANALYTIC GEOMETRY 311

was declared to be unproductive as compared with the synthetic
method of J. Steiner and J. V. Poncelet! His relations with C. G. J.
Jacobi were not altogether friendly. Steiner once declared that he
would stop writing for Crelle's Journal if Plucker continued to con-
tribute to it.1 The result was that many of Pliicker's researches were
published in foreign journals, and that his work came to be better
known in France and England than in his native country. The charge
was also brought against Plucker that, though occupying the chair
of physics, he was no physicist. This induced him to relinquish
mathematics, and for nearly twenty years to devote his energies to
physics. Important discoveries on Fresnel's wave-surface, magnetism,
spectrum-analysis were made by him. But towards the close of his
life he returned to his first love, — mathematics, — and enriched it with
new discoveries. By considering space as made up of lines he created
a "new geometry of space." Regarding a right line as a curve in-
volving four arbitrary parameters, one has the whole system of lines
in space. By connecting them by a single relation, he got a " complex "
of lines; by connecting them with a twofold relation, he got a "con-
gruency" of lines. His first researches on this subject were laid before
the Royal Society in 1865. His further investigations thereon ap-
peared in 1868 in a posthumous work entitled Neue Geometric des
Raumes gegrundet auf die Betrachtung der geraden Linie als Raumele-
ment, edited by Felix Klein. Pliicker's analysis lacks the elegance
found in J. Lagrange, C. G. J. Jacobi, L. O.. Hesse, and R. F. A.
Clebsch. For many years he had not kept up with the progress of
geometry, so that many investigations in his last work had already
received more general treatment on the part of others. The work
contained, nevertheless, much that was fresh and original. The theory
of complexes of the second degree, left unfinished by Plucker, was
continued by Felix Klein, who greatly extended and supplemented
the ideas of his master.

Ludwig Otto Hesse (1811-1874) was bora at Konigsberg, and
studied at the university of his native place under F. W. Bessel, C. G. J.
Jacobi, F. J. Richelot, and F. Neumann. Having taken the doctor's
degree in 1840, he became decent at Konigsberg, and in 1845 extraor-
dinary professor there. Among his pupils at that time were Heinrich
Durege (1821-1893) of Prague, Carl Neumann, R. F. A. Clebsch,
G. R. Kirchhoff. The Konigsberg period was one of great activity
for Hesse. Every new discovery increased his zeal for still greater
achievement. His earliest researches were on surfaces of the second
order, and were partly synthetic. He solved the problem to construct
any tenth point of such a surface when nine points are given. The
analogous problem- for a conic had been solved by Pascal by means
of the hexagram. A difficult problem confronting mathematicians
of this time was that of elimination. J. Plucker had seen that the
1 Ad. Dronkc, Julius Plttckcr, Bonn, 1871.

312 A HISTORY OF MATHEMATICS

main advantage of his special method in analytic geometry lay in
the avoidance of algebraic elimination. Hesse, however, showed how
by determinants to make algebraic elimination easy. In his earlier
results he was anticipated by J. J. Sylvester, who published his dialytic
method of elimination in 1840. These advances in algebra Hesse
applied to the analytic study of curves of the third order. By linear
substitutions, he reduced a form of the third degree in three variables
to one of only four terms, and was led to an important determinant
involving the second differential coefficient of a form of the third
degree, called the "Hessian." The "Hessian" plays a leading part
in the theory of invariants, a subject first studied by A. Cay ley.
Hesse showed that his determinant gives for every curve another
curve, such that the double points of the first are points on the second,
or "Hessian." Similarly for surfaces (Crelle, 1844). Many of the
most important theorems on curves of the third order are due to
Hesse. He determined the curve of the i4th order, which passes
through the 56 points of contact of the 28 bi-tangents of a curve of
the fourth order. His great memoir on this subject (Crelle, 1855)
was published at the same tune as was a paper by J. Steiner treating
of the same subject.

Hesse's income at Konigsberg had not kept pace with his growing
reputation. Hardly was he able to support himself and family. In
1855 he accepted a more lucrative position at Halle, and in 1856 one
at Heidelberg. Here he remained until 1868, when he accepted a
position at a technic school in Munich.1 At Heidelberg he revised
and enlarged upon his previous researches, and published in 1861 his
Vorlestmgen uber die Analytische Geometric des Ranmes, insbesoitdere
iiber Fldchen. 2. Ordnung. More elementary works soon followed.
While in Heidelberg he elaborated a principle, his " Uebertragungs-
princip." According to this, there corresponds to every point in a
plane a pair of points in a line, and the projective geometry of the
plane can be carried back to the geometry of points in a line.

The researches of Pliicker and Hesse were continued in England
by A. Cayley, G. Salmon, and J. J. Sylvester. It may be premised
here that among the early writers on analytical geometry in England
was James Booth (1806-1878), whose chief results are embodied in his
Treatise on Some New Geometrical Methods; and James MacCuUagh,
(1809-1846), who was professor of natural philosophy at Dublin,
and made some valuable discoveries on the theory of quadrics. The
influence of these men on the progress of geometry was insignificant,
for the interchange of scientific results between different nations was
not so complete at that tune as might have been desired. In further
illustration of this, we mention that M. Chasles in France elaborated
subjects which had previously been disposed of by J. Steiner in Ger-
many, and Steiner published researches which had been given by
1 Gustav Bauer, Ged'dchtnissrede anf Otto Hesse, Miinchen, 1882.

ANALYTIC GEOMETRY 313

Cayley, Sylvester, and Salmon nearly five years earlier. Cayley and
Salmon in 1849 determined the straight lines in a cubic surface, and
studied its principal properties, while Sylvester in 1851 discovered
the pentahedron of such a surface. Cayley extended Pliicker's equa-
tions to curves of higher singularities. Cayley's own investigations,
and those of Max Nother (1844- .) of Erlangen, G. H. Halphen,
Jules R. M. de la Gournerie (1814-1883) of Paris, A. Brill of Tubin-
gen, lead to the conclusion that each higher singularity of a curve is
equivalent to a certain number of simple singularities, — the node, the
ordinary cusp, the double tangent, and the inflection. Sylvester
studied the " twisted Cartesian," a curve of the fourth order. Georges-
Henri Halphen (1844-1889) was born at Rouen, studied at the Ecole
Polytechnique in Paris, took part in the Franco-Prussian war, then
became repetiteur and examinateur at the Ecole Polytechnique. His
investigations touched mainly the geometry of algebraic curves and
surfaces, differential invariants, the theory of E. Laguerre's invariants,
elliptic functions and their applications. A British geometrician,
Salmon, helped powerfully towards the spreading of a knowledge of
the new algebraic and geometric methods by the publication of an
excellent series of text-books (Conic Sections, Modern Higher Algebra,
Higher Plane Curves, Analytic Geometry of Three Dimensions), which
have been placed within easy reach of German readers by a free trans-
lation, with additions, made by Wilhelm Fiedler of the Polytechnicum
in Zurich. Salmon's Geometry of Three Dimensions was brought out
in the fifth and sixth editions, with much new matter, by Reginald
A. P. Rogers of Trinity College, Dublin, in 1912-1915. The next great
worker in the field of analytic geometry was Clebsch.

Rudolf Friedrich Alfred Clebsch (1833-1872) was born at Konigs-
berg in Prussia, studied at the university of that place under L. O.
Hesse, F. J. Richelot, F. Neumann. From 1858 to 1863 he held the
chair of theoretical mechanics at the Polytechnicum in Carlsruhe.
The study of Salmon's works led him into algebra and geometry. In
1863 he accepted a position at the University of Giesen, where he
worked in conjunction with Paul Gordan of Erlangen. In 1868
Clebsch went to Gottingen, and remained there until his death. He
worked successively at the following subjects: Mathematical physics,
the calculus of variations and partial differential equations of the first
order, the general theory of curves and surfaces, Abelian functions
and their use in geometry, the theory of invariants, and "Flachen-
abbildung." He proved theorems on the pentahedron enunciated
by J. J. Sylvester and J. Steiner; he made systematic use of "defi-
ciency" (Geschlecht) as a fundamental principle in the classification
of algebraic curves. The notion of deficiency was known before him
to N. H. Abel and G. F. B. Riemann. At the beginning of his career,

1 Alfred Clebsch, Vcrsuch eincr Darlegting und Wiirdigung seiner ivisscnschaftlichen
Leistungcn von cinigcn seiner Frcunde, Leipzig, 1873.

3H A HISTORY OF MATHEMATICS

Clebsch had shown how elliptic functions could be advantageously
applied to Malfatti's problem. The idea involved therein, viz. the use
of higher transcendentals in the study of geometry, led him to his great-
est discoveries. Not only did he apply Abelian functions to geometry,
but conversely, he drew geometry into the service of Abelian functions.

Clebsch made liberal use of determinants. His study of curves and
surfaces began with the determination of the points of contact of lines
which meet a surface in four consecutive points. G. Salmon had
proved that these points lie on the intersection of the surface with a
derived surface of the degree ii«— 24, but his solution was given in
inconvenient form. Clebsch's investigation thereon is a most beautiful
piece of analysis.

The representation of one surface upon another (Fl'dchenabbildung) ,
so that they have a (i, i) correspondence, was thoroughly studied for
the first time by Clebsch. The representation of a sphere on a plane
is an old problem which drew the attention of Ptolemy, Gerard Mer-
cator, J. H. Lambert, K. F. Gauss, J. L. Lagrange. Its importance in
the construction of maps is obvious. Gauss was the first to represent
a surface upon another with a view of more easily arriving at its
properties. J. Pliicker, M. Chasles, A. Cayley, thus represented on a
plane the geometry of quadric surfaces; Clebsch and L. Cremona, that
of cubic surfaces. Other surfaces have been studied in the same way
by recent writers, particularly Max N other of Erlangen, Angela
Armenante (1844-1878) of Rome, Felix Klein, Georg H. L. Korndorfcr,
Ettore Caporali (1855-1886) of Naples, H. G. Zeuthen of Copenhagen.
A fundamental question which has as yet received only a partial an-
swer is this: What surfaces can be represented by a (i, i) correspond-
ence upon a given surface? This and the analogous question for
curves was studied by Clebsch. Higher correspondences between
surfaces have been investigated by A. Cayley and M. Nother. Im-
portant bearings upon geometry has Riemann's theory of birational
transformations. The theory of surfaces has been studied by Joseph
Alfred Serret (1819-1885) professor at the Sorbonne in Paris, Jean
Gaston Darboux of Paris, John Casey (1820-1891) of Dublin, William
Roberts (1817-1883) of Dublin, Heinrich Schrb'ter (1829-1892) of
Breslau, Elwin Bruno Christoffel (1829-1900), professor at Zurich,
later at Strassburg. Christoffel wrote on the theory of potential, on
minimal surfaces, on the so-called transformation of Christoffel, of
isothermic surfaces, on the general theory of curved surfaces. His
researches on surfaces were extended by Julius Weingarten (1836-1910)
of the University of Freiburg and Hans von Mangoldt of Aachen, in
1882. As we shall see more fully later, surfaces of the fourth order
were investigated by E. E. Kummer, and Fresnel's wave-surface,
studied by W. R. Hamilton, is a particular case of Kummer's quartic
surface, with sixteen double points and sixteen singular tangent planes.1
1 A. Cayley, Inaugural Address, 1883.

ANALYTIC GEOMETRY 315

Prominent in these geometric researches was Jeaji Gaston Dar-
boux (1842-1917). He was born at Nimes, founded in 1870, with the
collaboration of Guillaume Jules Hoiiel (1823-1886) of Bordeaux and
Jules Tannery, the Bulletin des sciences mathematiques et astronomiques,
and was for half a century conspicuous as a teacher. In 1900 he
became permanent secretary of the Paris Academy of Sciences, in
which position he was succeeded after his death by Emil Picard. By
his researches, Darboux enriched the synthetic, analytic and infin-
itesimal geometries, as well as rational mechanics and analysis. He
wrote Lemons sur la theorie generate des surfaces et les applications
geometriques du calcul infinitesimal, Paris, 1887-1896, and Leqons sur
les systemes orthogonaux et les coordonnees curuilignes, Paris, 1898. He
investigated triply orthogonal systems of surfaces, the deformation
of surfaces and rolling of applicable surfaces, infinitesimal deforma-
tion, spherical representation of surfaces, the development of the
moving axes of co-ordinates, the use of imaginary geometric elements,
the use of isotropic cylinders and developables; 1 he introduced
pentaspherical coordinates.

Eisenhart says: "Darboux was a strong advocate of the use of
imaginary elements in the study of geometry. He believed that their
use was as necessary in geometry as in analysis. He has been im-
pressed by the success with which they have been employed in the
solution of the problem of minimal surfaces. From the very beginning
he made use in his papers of the isotropic line, the null sphere (the
isotropic cone) and the general isotropic developable. In his first
memoir on orthogonal systems of surfaces he showed that the envelope
of the surfaces of such a system, when defined by a single equation,
is an isotropic developable. . . . Darboux gives to Edouard Com-
bescure (1819-?) the credit of being the first to apply the considera-
tions of kinematics to the study of the theory of surfaces with the
consequent use of moving co-ordinate axes. But to Darboux we are
indebted for a realization of the power of this method, and for its
systematic development and exposition. . . . Darboux's ability was
based on a rare combination of geometrical fancy and analytical
power. He did not sympathize with those who use only geometrical
reasoning in attacking geometrical problems, nor with those who feel
that there is a certain virtue in adhering strictly to analytical proc-
esses. ... In common with Monge he was not content with dis-
coveries, but felt that it was equally important to make disciples.
Like this distinguished predecessor he developed a large group of
geometers, including C. Guichard, G. Koenigs, E. Cosserat, A. De-
moulin, G. Tzitzeica, and G. Demartres. Their brilliant researches
are the best tribute to his teaching."

Proceeding to the fuller consideration of recent developments, we

1 Am. Math. Monthly, Vol. 24, 1017, p. 354. See L. P. Eisenhart's "Darboux's
Contribution to Geometry" in Bull. Am. Math. Soc., Vol. 24, 1918, p. 227.

3i6 A HISTORY OF MATHEMATICS

quote from H. F. Baker's address before the International Congress
held at Cambridge in 1912: 1 "The general theory of Higher Plane
Curves . . . would be impossible without the notion of the genus of a
curve. The investigation of Abel of the number of independent in-
tegrals in terms of which his integral sums can be expressed may thus
be held to be of paramount importance for the general theory. This
was further emphasized by G. F. B. Riemann's consideration of the no-
tion of birational transformation as a fundamental principle. After
this two streams of thought were to be seen. First R. F. A. Clebsch
remarked on the existence of invariants for surfaces, analogous to the
genus of a plane curve. This number he denned by a double integral;
it was to be unaltered by birational transformation of the surface.
Clebsch's idea was carried on and developed by M. Noether. But
also A. Brill and Noether elaborated in a geometrical form the
results for plane curves which had been obtained with transcen-
dental considerations by N. H. Abel and G. F. B. Riemann. Then
the geometers of Italy took up Noether's work \vith very remark-
able genius, and carried it to a high pitch of perfection and clear-
ness as a geometrical theory. In connection therewith there arose
the important fact, which does not occur in Noether's papers, that
it is necessary to consider a surface as possessing two genera; and
the names of A. Cayley and H. G. Zeuthen should be referred to at
this stage. But at this time another stream was running in France.
E. Picard wras developing the theory of Riemann integrals — single
integrals, not double integrals — upon a surface. How long and
laborious was the task may be judged from the fact that the publica-
tion of Picard's book occupied ten years — and may even then have
seemed to many to be an artificial and unproductive imitation of
the theory of algebraic integrals for a curve. In the light of subse-
quent events, Picard's book appears likely to remain a permanent
landmark in the history of geometry. For now the two streams,
the purely geometrical in Italy, the transcendental in France, have
united. The results appear to me at least to be of the greatest im-
portance." The work of E. Picard in question is the Theorie des
f auctions algebriques de deux variables independantes, which was brought
out in conjunction with Georges Simart between the years 1897 and
1906.

H. F. Baker proceeds to the enumeration of some individual re-
sults: Guido Castelnuovo of Rome has shown that the deficiency of
the characteristic series of a linear system of curves upon a surface can-
not exceed the difference of the two genera of the surface. Federigo
Enriques of Bologna has completed this result by showing that for an
algebraic system of curves the characteristic series is complete. Upon
this result, and upon E. Picard's theory of integrals of the second

1 Proceed, of the $th Intern. Congress, Vol. I, Cambridge, 1913, p. 49. For more
detail, consult H. B. Baker in the Proceed, of the London Math. Soc., Vol. 12, 1912.

ANALYTIC GEOMETRY 317

kind Francesco Severi of Padua has constructed a proof that the
number of Picard integrals of the first kind upon a surface is equal to
the difference of the genera. The names of M. G. Humbert of Paris
and of G. Castelnuovo also arise here. Picard's theory of integrals of
the third kind has given rise in F. Seven's hands to the expression of
any curve lying on a surface linearly in terms of a finite number of
fundamental curves. Enriques showed that the system of curves cut
upon a plane by adjoint surfaces of order n— 3, when n is the order of
the fundamental surface, if not complete, has a deficiency not ex-
ceeding the difference of the genera of the surface. Severi has given
a geometrical proof that this deficiency is equal to the difference of
the genera, a result previously deduced by E. Picard, with transc<*n-
dental considerations, from the assumption of the number of Picard
integrals of the first kind. F. Enriques and G. Castlenuovo have shown
that a surface which possesses a system of curves for which what may
be called the canonical number, 2 TT — 2 — w, where TT is the genus of the
curve and n the number of intersections of two curves of the system,
is negative, can be transformed bi rationally to a ruled surface. On
the analogy of the case of plane curves, and of surfaces in three di-
mensions, it appears very natural to conclude that if a rational re-
lation, connecting, say, m+i variables, can be resolved by substi-
tuting for the variables rational functions of m others, then these m
others can be so chosen as to be rational functions of the m+ 1 original
variables. F. Enriques has recently given a case, with #2=3, for
which this is not so. To this summary of results, given by H. F.
Baker, should be added that he himself has made contributions,
particularly on a cubic surface and the curves which lie thereon.
In reducing singularities, the Italians and French use methods of
projecting from space of higher dimension which were perhaps first
used in 1887 by W. K. Clifford.

A publication of wide scope on collineations and correlations is
Die Lehre von den geometrischen Verwandtschaften, in four volumes,
1908 — ?, being written by Rudolf Sturm (1841 — ) of the University
of Breslau.

The theory of straight lines upon a cubic surface was first studied
by A. Cayley and G. Salmon * in 1849. Cayley pointed out that there
was a definite number of such lines, while Salmon found that there
were exactly 27 of them. "Surely with as good reason," says J. J.
Sylvester, "as had Archimedes to have the cylinder, cone and sphere
engraved on his tombstone might our distinguished countrymen leave
testamentary directions for the cubic eikosiheptagram to be engraved
on theirs." Nor would such engraving be impossible, for in 1869
Christian Wiener made a model of a cubic surface showing 27 real
lines lying upon it. J. Steiner, in 1856, studied the purely geometric

1 These historic data are taken from A. Henderson, The Twenty-seven Lines upon
he Cubic Surface, Cambridge, 1911, which gives bibliography and details.

3i8 A HISTORY OF MATHEMATICS

theory of cubic surfaces. This was done later also by L. Cremona and
R. Sturm, between whom in 1866 the "Steiner prize" was divided.
An elegant notation was invented by Andrew Hart, but the notation
which has met with general adoption was advanced by L. Schlafli of
Bern in 1858 ; it is that of the double six. Schlafli's double six theorem,
proved by him and by many others since, is as follows: "Given five
lines a, b, c, d, e which meet the same straight line X; then may any
four of the five lines be intersected by another line. Suppose that
A, B, C, D, E are the other lines intersecting (b, c, d, e), (c, d, e, a),
(d, e, a, b), (e, a, b, c), and (a, b, c, d) respectively. Then A, B, C, D, E
will all be met by one other straight line x."

L. Schlafli first considered a division of the cubic surfaces into
species, in regard to the reality of the 27 lines. His final classification
was adopted by A. Cayley. In 1872 R. F. A. Clebsch constructed a
model of the diagonal surface with 27 real lines, while F. Klein "es-
tablished the fact that by the principle of continuity all forms of real
surfaces of the third order can be derived from the particular surface
having four conical points;" he exhibited a complete set of models
of cubic surfaces at the World's Fair in Chicago in 1894. In 1869
C. F. Geiser showed that " the projection of a cubic surface from a point
upon it, on a plane of projection parallel to the tangent plane at that
point, is a quartic curve; and that every quartic curve can be generated
in this way." "The theory of varieties of the third order," says A.
Henderson, "that is to say, curved geometric forms of three dimen-
sions contained in a space of four dimensions, has been the subject
of a profound memoir by Corrado Segre (1887) of Turin. The depth
and fecundity of this paper is evinced by the fact that a large pro-
portion of the propositions upon the plane quartic and its bitangents,
Pascal's theorem, the cubic surface and its 27 lines, Kummer's sur-
face and its configuration of sixteen singular points and planes, and on
the connection between these figures, are derivable from propositions
relating to Segre's cubic variety, and the figure of six points or spaces
from which it springs. Other investigators into the properties of this
beautiful and important locus in space of four dimensions and some of
its consequences are G. Castelnuovo and H. W. Richmond."

In 1869 C. Jordan first proved "that the group of the problem of
the trisection of hyperelliptic functions of the first order is isomorphic
with the group of the equation of the 27th degree, on which the 27
lines of the general surface of the third degree depend." In 1887 F.
Klein sketched the effective reduction of the one problem to the other,
while H. Maschke, H. Burkhardt, and A. Witting completed the work
outlined by Klein. The Galois group of the equation of the 27 lines
was investigated also by L. E. Dickson, F. Kiihnen, H. Weber, E.
Pascal, E. Kasner and E. H. Moore.

Surfaces of the fourth order have been studied less thoroughly than
those of the third. J. Steiner worked out properties of a surface of the

ANALYTIC GEOMETRY 319

fourth order in 1844 when he was on a journey to Italy; that surface
bears this name, and later received the attention of E. E. Kummer.
In 1850 Thomas Weddle 1 remarked that the locus of the vertex of a
quadric cone passing through six given points is a quartic surface
and not a twisted cubic as M. Chasles had once stated. A. Cayley
gave a symmetric equation of the surface in 1861. Thereupon Chasles
in 1 86 1 showed that the locus of the vertex of a cone which divides
six given segments harmonically is also a quartic surface; this more
general surface was identified by Cayley with the Jacobian of four
quadrics, the Weddle surface corresponding to the case in which the
four quadrics have six common points. Properties of the Weddle sur-
face were studied also by H. Bateman (1905). The plane section of a
Weddle surface is not an arbitrary quartic curve, but one for which an
invariant vanishes. Frank Morley proved that the curve contains an
infinity of configurations B6, where it is cut by the lines on the sur-
face.

In 1863 and 1864, E. E. Kummer entered upon an intensive study
of surfaces of the fourth order. Noted is the surface named after him
which has 16 nodes. The various shapes it can assume have been
studied by Karl Rohn of Leipzig. It has received the attention of
many mathematicians, including A. Cayley, J. G. Darboux, F. Klein,
H. W. Richmond, O. Bolza, H. F. Baker, and J. I. Hutchinson.2 It has
been known for some time that Fresnel's wave surface is a case of
Kummer's sixteen nodal quartic surface; also it is known that the
surface of a dynamical medium possessing certain general properties
is a type of Kummer's surface which can he derived from Fresnel's
surface by means of a homogeneous strain. Kummer's quartic surface
as a wave surface is treated by H. Bateman (1909). The general
Kummer's surface appears to be the wave surface for a medium of a
purely ideal character.

F. R. Sharpe and C. F. Craig of Cornell University have studied
birational transformations which leave the Kummer and Weddle
surfaces invariant, by the application of a theory due to F. Severi
(1906).

Quintic surfaces have been investigated at intervals, since 1862,
principally by L. Cremona, H. A. Schwarz, A. Clebsch, M. Noether,
R. Sturm, J. G. Darboux, E. Caporali, A. Del Re, E. Pascal, John E.
Hill and A. B. Basset. No serious attempt has been made to enumer-
ate the different forms of these surfaces.

Ruled surfaces with isotropic generators have been considered by
G. Monge, J. A. Serret, S. Lie and others. L. P. Eisenhart of Princeton
determines such a surface by the curve in which it is cut by a plane
and the directions of the projections on the plane of the generators

1 Camb. &• Dublin Math. Jour., Vol. 5, 1850, p. 69.

2 Consult R. W. H. T. Hudson (1876-1904), Kummer's Quartic Surface, Cam-
bridge, 1905.

320 A HISTORY OF MATHEMATICS

of the surface. In this way a ruled surface of this type is determined
by a set of lineal elements, in a plane, depending on one parameter.

While the classification of cubic curves was given by I. Newton
and their general theory was well under way two centuries ago, the
theory of quartic curves was not pursued vigorously until the time
of J. Steiner and L. O. Hesse. Neglecting the classification of quartic
curves due to L. Euler, G. Cramer and E. Waring, new classifications
have been made, either according to their genus (Geschlecht) 3, 2, i, o,
or according to topologic considerations studied by A. Cayley in
1865, H. G. Zeuthen (1873), Christian Crone (1877) and others.
J. Steiner in 1855 and L. O. Hesse began researches on the 28 double
tangents of a general quartic; 24 inflections were found, of which
G. Salmon conjectured and H. G. Zeuthen proved that at most 8 are
real. An enumeration (containing nearly 200 graphs) of the funda-
mental forms of quartic curves "when projected so as to cut the line
infinity the least possible number of times" was given in 1896 by Ruth
Gentry (1862-1917), then of Bryn Mawr College.

Curves of the fourth order have received attention for many years.
More recently a good deal has been written on special curves of the
fifth order by Frank Morley, Alfred B. Basset, Virgil Snyder, Peter
Field, and others.

Gino Loria of the University of Genoa, who has written extensively
on the history of geometry, and the history of curves in particular,
has advanced a theory of panalgebraic curves, which are in general
transcendental curves. By definition, a panalgebraic curve must
satisfy a certain differential equation. A book of reference on curves
was published by Gomes Teixtira in 1905 at Madrid under the title
Tratado de las curvas especiales notables.

The infinitesimal calculus was first applied to the determination
of the measure of curvature of surfaces by J. Lagrange, L. Euler, and
Jean Baptiste Marie Meusnier (1754-1793) of Paris, noted for his
military as well as scientific career. Meusnier's theorem, relating to
curves drawn on an arbitrary surface, was extended by S. Lie and in
1908 by E. Kasner. The researches of G. Monge and E. P. C. Dupin
were eclipsed by the work of K. F. Gauss, who disposed of this difficult
subject in a way that opened new vistas to geometricians. His treat-
ment is embodied in the Disquisitiones generates circa superficies curccs
(1827) and Untersuchungen ilber Gegenstande der hoJicren Geodasie of
1843 and 1846. In 1827 he established the idea of curvature as it is
understood to-day. Both before and after the time of Gauss various
definitions of curvature of a surface had been advanced by L. Euler,
Meusnier, Monge, and Dupin, but these dkj not meet with general
adoption. From Gauss' measure of curvature flows the theorem of
Johann August Grunert (1797-1872), professor in Greifswald, and
founder in 1841 of the Archiv der Mathematik und Physik, that the
arithmetical mean of the radii of curvature of all normal sections

ANALYTIC GEOMETRY 321

through a point is the radius of a sphere which has the same measure
of curvature as has the surface at that point. Gauss's deduction of
the formula of curvature was simplified through the use of deter-
minants by Heinrich Richard Baltzer (1818-1887) of Giessen.1 Gauss
obtained an interesting theorem that if one surface be developed
(abgewickelf) upon another, the measure of curvature remains un-
altered at each point. The question whether two surfaces having
the same curvature in corresponding points can be unwound, one
upon the other, was answered by F. Minding in the affirmative only
when the curvature is constant. Surfaces of constant and negative
curvature were called pseudo-spherical surfaces by E. Beltrami in
1868, in order, as he says, "to avoid circumlocution." The case of
variable curvature is difficult, and was studied by F. Minding, Joseph
Liouville (1809-1882) of the Polytechnic School in Paris, Ossian
Bonnet (1819-1892) of Paris. Gauss's measure of curvature, expressed
as a function of curvilinear co-ordinates, gave an impetus to the study
of differential-invariants, or differential-parameters, which have been
investigated by C. G. J. Jacobi, C. Neumann, Sir James Cockle
(1819-1895) of London, G. H. Halphen, and elaborated into a general
theory by E. Beltrami, S. Lie, and others. Beltrami showed also the
connection between the measure of curvature and the geometric axioms.
In 1899 Claude Guichard of Rennes announced two theorems relating
to a quadric of revolution which marked a new epoch in the theory
of deformation of surfaces. Researches along this line by Guichard
and Luigi Bianchi of Pisa are embodied in the second edition of
Bianchi's Lezioni di geometria differenziale, Pisa, 1902. Another
treatise on metric differential geometry was brought out in 1908 by
Reinhold v. Lilienthal of the University of Miinster. Not only does
he give geometric interpretations of the first and second derivatives
by means of the tangent and the circle of curvature, but he revives
a notion due to Abel Transon (1805-1876) of Paris which gives a
geometric interpretation of the third derivative in terms of the ab-
berancy of a curve and the axis of abberancy. A still later work is the
Treatise on Differential Geometry of Curves and Surfaces (1909) by
L. P. Eisenhart of Princeton which possesses the interesting feature
of movable axes (the so-called "moving trihedrals" used extensively
by J. G. Darboux), applied to twisted curves as well as surfaces; he
gives the four transformations of surfaces of constant curvature,
due to N. Hatzidakis of Athens, L. Bianchi of Pisa, A. V. Backlund
of Lund, and S. Lie. Eisenhart developed a theory of transformations
of a conjugate system of curves on any surface into conjugate systems
on other surfaces, and also of transformations of conjugate nets on
two-dimensional spreads in space of any order.2

1 August Haas, Versuch einer Darstellung der Geschichte des Krilmmnngsmasses,
Tubingen, 1881.

2 Bull. Am. Math. Soc., Vol. 24, 1917, p. 68.

322 i

The metric part of differential geometry occupied the attention
of mathematicians since the time of G. Monge and K. F. Gauss, and
has reached a high degree of perfection. Less attention has been
given until recently to projective differential geometry, particularly
to the differential geometry of surfaces. G. H. Halphen started with
the equation y=f(x) of a curve and determined functions of y, dy/dx,
dty'dx2, etc., which are left invariant when x and y are subjected to
a general projective transformation. His early formulation of the
problem is unsymmetrical and unhomogeneous. Using a certain
system of partial differential equations and the geometrical theory
of semi-co variants, E. J. Wilczynski obtained homogeneous forms,
such forms being deduced later also by Halphen.1 Wilczynski treats
of the projective differential geometry of curves and ruled surfaces,
these surfaces being prerequisite for his theory of space curves. Wil-
czynski treated ruled surfaces by a system of two linear homogeneous
differential equations of the second order. The method was extended
to five-dimensional space by E. B. Stouffer of the University of Kan-
sas.2 Developable surfaces were studied by W. W. Denton of the
University of Illinois. Belonging to projective differential geometry
are J. G. Darboux's conjugate triple systems which are generalized
notions of the orthogonal triple systems. The projective differential
geometry of triple systems of surfaces, of one-parameter families of
space curves and conjugate nets on a curved surface, and allied topics,
were studied by Gabriel Marcus Green (1891-1919), of Harvard
University.

Differential projective geometry of hyperspace was greatly advanced
by C. Guichard who introduced two elements depending on two va-
riables; they are the reseau and the congruence. Differential geometry
of hyperspace was greatly enriched since 1906 by Corrado Segre of
Turin, and by other geometers of Italy, particularly Gino Fano of
Turin and Federigo Enriques of Bologna; 3 also by A. Ranum of
Cornell, C. H. Sisam then of Illinois and C. L. E. Moore of the Massa-
chusetts Institute of Technology.

The use of vector analysis in differential geometry goes back to
H. G. Grassmann and W. R. Hamilton, to their successors P. G. Tait,
C. Maxwell, C. Burali-Forti, R. Rothe and others. These men have
introduced the terms "grad," "div," "rot." A geometric study of
trajectories with the aid of analytic and chiefly contact transforma-
tions was made by Edward Kasner of Columbia University in his
Princeton Colloquium lectures of 1909 on the "differential-geometric
aspects of dynamics."

1 See E. J. Wilczynski in New Haven Colloquium, 1906. New Haven, 1910, p. 156;
also his Projective Differential Geometry of Curves and Ruled Surfaces, Leipzig, 1906.

2 Bull. Am. Math. Soc., Vol. 18, p. 444.

* See Enrico Bompiani in Proceed. $lh Intern. Congress, Cambridge, Cambridge,
1913, Vol. II, p. 22.

ANALYTIC GEOMETRY 323

Analysis Situs

Various researches have been brought under the head of "analysis
situs." The subject was first investigated by G. W. Leibniz, and
was later treated by L. Euler who was interested in the problem to
cross all of the seven bridges over the
Pregel river at Konigsberg without passing
twice over any one, then by K. F.' Gauss,
whose theory of knots (Verschlingungen)
has been employed more recently by Jo-

hann Benedict Listing (1808-1882) of Gottingen, Oskar Simony
of Vienna, F. Dingeldey of Darmstadt, and others in their "topologic
studies." P. G. Tait was led to the study of knots by Sir William
Thomson's theory of vortex atoms. Through Rev. T. P. Kirkman
who had studied the properties of polyhedra, Tait was led to study
knots also by the polyhedral method; he gave the number of forms
of knots of the first ten orders of knottiness. Higher orders were
treated by Kirkman and C. N. Little. Thomas Penyngton Kirkman l
(1806-1895) was born at Bolton, near Manchester. During boyhood
he was forced to follow his father's business as dealer in cotton and
cotton waste. Later he tore away, entered the University of Dublin,
then became vicar of a parish in Lancashire. As a mathematician
he was almost entirely self-taught. He wrote on pluquaternions in-
volving more imaginaries than i, j, k, on group theory, on mathe-
matical mnemonics producing what De Morgan called "the most
curious crocket I ever saw," on the problem of the "fifteen school
girls" who walk out three abreast for seven days, where it is required
to arrange them daily so that no two shall walk abreast more than
once. This problem was studied also by A. Cayley and Sylvester,
and is related to researches of J. Steiner.

Another unique problem was the one on the coloring of maps, first
mentioned by A. F. Mobius in 1840 and first seriously considered by
Francis Guthrie and A. De Morgan. How many colors are necessary
to draw any map so that no two countries having a line of boundary
in common shall appear in the same color? Four different colors
are found experimentally to be necessary and sufficient, but the proof
is difficult. A. Cayley in 1878 declared that he had not succeeded
in obtaining a general proof. Nor have the later demonstrations by
A. B. Kempe, P. G. Tait, P. J. Heawood of the University of Durham,
W. E. Story of Clark University, and J. Peterson of Copenhagen
removed the difficulty.2 Tait's proof leads to the interesting con-
clusion that four colors may not be sufficient for a map drawn on a
multiply-connected surface like that of an anchor ring. Further
studies of maps on such surfaces, and of the problem in general, are

1 A. Macfarlane, Ten British Mathematicians, 1916, p. 122.

2 W. Ahrens, U nlcrhallungen und Spiclc, 1901, p. 340.

324 i

due to O. Veblen (1912) and G. D. Birkhoff (1913). On a surface of
genus zero "it is not known whether or not only four colors always
suffice." A similar question considers the maximum possible number
of countries, when every country touches every other along a line.
Lothar Heffter wrote on this conundrum, in 1891 and again in later
articles, as did also A. B. Kempe and others. In the hands of Riemann
the analysis situs had for its object the determination of what remains
unchanged under transformations brought about by a combination
of infinitesimal distortions. In continuation of his work, Walter Dyck
of Munich wrote on the analysis situs of three-dimensional spaces.
Researches of this sort have important bearings in modern mathe-
matics, particularly in connection with correspondences and differ-
ential equations.1

Intrinsic Co-ordinates

As a reaction against the use of the arbitrary Cartesian and polar
co-ordinates there came the suggestion from the philosophers K. C. F.
Krause (1781-1832), A. Peters (1803-1876) that magnitudes inherent
to a curve be used, such as s, the length of arc measured from a fixed
point, and (p, the angle which the tangent at the end of s makes with
a fixed tangent. William Whewell (1794-1866) of Cambridge, the
author of the History of the Inductive Sciences, 1837-1838, introduced in
1849 the name "intrinsic equation" and pointed out its use in study-
ing successive evolutes and involutes. The method was used by Wil-
liam Wahon (1813-1901) of Cambridge, J. J. Sylvester in 1868, J.
Casey in 1866, and others. Instead of using s and <p, other writers
have introduced the radius of curvature p, and have used either s
and p, or (f> and p. The co-ordinates (<p, p) were employed by
L. Euler and several nineteenth century mathematicians, but alto-
gether the co-ordinates (s, p) have been used most. The latter were
used by L. Euler in 1741, by Sylvestre Francois Lacroix (1765-1843),
by Thomas Hill (1818-1891, who was at one tune president of Harvard
College), and in recent years especially by Ernesto Cesaro of the
University of Naples who published in 1896 his Geometria intrinseca
which was translated into German in 1901 by G. Kowalewski under
the title, Vorlesungen uber natiirliche Geometric.'2' Researches along
this line are due also to Amedee Mannheim (1831-1906) of Paris, the
designer of a well-known slide rule.

The application of intrinsic or natural co-ordinates to surfaces is
less common. Edward Kasner 3 said in 1904 that in the " theory of
surfaces, natural co-ordinates may be introduced so as to fit into the

1 See J. Hadamard, Four Lectures on Mathematics delivered at Columbia University
in 1911, New York, 1915, Lecture III.

2 Our information is drawn from E. Wolffing's article on "Natiirliche Koor-
dinaten" in Bibliotheca mathematica, 3. S., Vol. I, 1900, pp. 142-159.

3 Bidl. Am. Math. Soc., Vol. n, 1905, p. 303.

ANALYTIC GEOMETRY 325

so-called geometry of a flexible but inextensible surface, originated by
K. F. Gauss, in which the criterion of equivalence is applicability or,
according to the more accurate phraseology of A. Voss, isometry.
Intrinsic co-ordinates must then be invariant with respect to bend-
ing. . . . The simplest example of a complete isometric group is
the group typified by the plane, consisting of all the developable sur-
faces. In this case the equations of the group may be obtained ex-
plicitly, in terms of eliminations, differentiations and quadratures. . . .
Until the year 1866, not a single case analogous to that of the de-
velopable surfaces was discovered. Julius Weingarten, by means of
his theory of evolutes, then succeeded in determining the complete
group of the catenoid and of the paraboloid of revolution, and, some
twenty years later, a fourth group defined in terms of minimal sur-
faces. During the past decade, the French geometers have concen-
trated their efforts in this field mainly on the arbitrary paraboloid
(and to some extent on the arbitrary quadric). The difficulties even
in this extremely restricted and apparently simple case are great,
and are only gradually being conquered by the use of almost the whole
wealth of modern analysis and the invention of new methods which
undoubtedly have wider fields of application. The results obtained
exhibit, for example, connections with the theories of surfaces of
constant curvature, isometric surfaces, Backlund transformations,
and motions with two degrees of freedom. The principal workers
are J. G. Darboux, E. J. B. Goursat, L. Bianchi, A. L. Thybaut, E.
Cosserat, M. G. Servant, C. Guichard, and L. Raffy."

Definition of a Curve

The theory of sets of points, originated by G. Cantor, has given
rise-to new views on the theory of curves and on the meaning of con-
tent. .What is a curve? Camilla Jordan in his Cours dj analyse defined
it tentatively as a "continuous line." W. H. Young and Grace
Chisholm Young in their Theory of Sets of Points, 1906, p. 222, define
a "Jordan curve" as "a plane set of points which can be brought into
co~nTirmous (i, i) correspondence with the points of a closed segment
(a, z) of a straight line." A circle is a closed Jordan curvey^Jepdan
asked the question, whether it was possible for a curve to fill up a
space. G. Peano answered that a "continuous line" may do so and
constructed in Math. Annalen, Vol. 36, 1890, the so-called "space-
filling curve" (the "Peano curve") to fortify his assertion. His mode
of construction has been modified in several ways since. The most
noted of these are due to E. H. Moore * and D. Hilbert. In-t9i&xR,-L.
Moore- of- the University of Pennsylvania proved that every two points
of a continuous curve, no matter how crinkly, can be joined by a
simple continuous arc that lies wholly in the curve. As it does not

1 Trans. Am. Math. Soc., Vol. I, IQOC, pp. 72-90.

326 A HISTORY OF MATHEMATICS

seem desirable to depart from our empirical notions so far as to allow
the term curve to be applied to a region, more restricted definitions
of it become necessary. C. Jordan demanded that a curve x=<p(t),
y=ty(f) should have no double points in the interval a^t<b. Schon-
flies regards a curve as the frontier of a region. O. Veblen defines it
in terms of order and linear continuity. W. H. Young and Grace C.
Young in their Theory of Sets of Points define a curve as a plane set
of points, dense nowhere in the plane and bearing other restrictions,
yet such that it may consist of a net-work of arcs of Jordan curves.

Other curves of previously unheard of properties were created as
the result of the generalization of the function concept. The con-

«=oo
tinuous curve represented by y=2 bn cos Tr(anx), where a is an even

n=o

integer >i, b a real positive number <i, was shown by Weierstrass l
to possess no tangents at any of its points when the product ab exceeds

etc.

a certain limit; that is, we have here the startling phenomenon of a
continuous function which has no derivative. As Christian Wiener
explained in 1881, this curve has countless oscillations within every
finite interval. An intuitively simpler curve was invented by Helge
v. Koch of the University of Stockholm in 1904 (Ada math., Vol. 30,
1906, p. 145) which is constructed by elementary geometry, is con-
tinuous, yet has no tangent at any of its points; the arc between any
two of its points is infinite in length. While this curve has been repre-
sented analytically, no such representation has yet been found for
the so-called H-curve of Ludwig Boltzmann (1844-1906) of Vienna,
in Math. Annalen, Vol. 50, 1898, which is continuous, yet tangentless.
The adjoining figure shows its construction. Boltzmann used it to
visualize theorems in the theory of gases.

Fundamental Postulates

The foundations of mathematics, and of geometry in particular,
received marked attention in Italy. In 1889 G. Peano took the novel

1 P. du Bois-Reymond " Versuch einer Klassification der willkiirlichen Funk-
tionen reeller Argumente," Crelle, Vol. 74, 1874, p. 29.

ANALYTIC GEOMETRY 327

view that geometric elements are mere things, and laid down the
principle that there should be as few undefined symbols as possible.
In 1897-9 his pupil Mario Fieri (1860-1904) of Catania used only two
undefined symbols for projective geometry and but two for metric
geometry. In 1894 Peano considered the independence of axioms.
By 1897 the Italian mathematicians had gone so far as to make it a
postulate that points are classes. These fundamental features elab-
orated by the Italian school were embodied by David Hilbert of
Gottingen, along with important novel considerations of his own, in
his famous Grundlagen der Geometric, 1899. A fourth enlarged edition
of this appeared in 1913. E. B. Wilson says in praise of Hilbert: "The
archimedean axiom, the theorems of B. Pascal and G. Desargues, the
analysis of segments and areas, and a host of things are treated either
for the first time or in a new way, and with consummate skill. We
should say that it was in the technique rather than in the philosophy
of geometry that Hilbert created an epoch." 1 In Hilbert's space of
1899 are not all the points which are in our space, but only those that,
starting from two given points, can be constructed with ruler and
compasses. In his space, remarks Poincare, there is no angle of 10°.
So in the second edition of the Grundlagen, Hilbert introduced the
assumption of completeness, which renders his space and ours the
same. Interesting is Hilbert's treatment of non-Archimedean geom-
etry where all his assumptions remain true save that of Archimedes,
and for which he created a system of non-Archimedean numbers.
This non-Archimedean geometry was first conceived by Giuseppe
Veronese (1854-1917), professor of geometry at the University of
Padua. Our common space is only a part of non-Archimedean
space. Non- Archimedean theories of proportion were given in 1902 by
A. Kneser of Breslau and in 1904 by P. J. Mollerup of Copenhagen.
Hilbert devoted in his Grundlagen a chapter to Desargues' theorem.
In 1902 F. R. Moulton of Chicago outlined a simple non-Desarguesian
plane geometry.

In the United States, George Bruce Halsted based his Rational
Geometry, 1904, upon Hilbert's foundations. A second, revised edition
appeared in 1907. One of Hilbert's pupils, Max W. Dehn, showed
that the omission of the axiom of Archimedes (Eudoxus) gives rise
to a semi-Euclidean geometry in which similar triangles exist and their
sum is two right angles, yet an infinity of parallels to any straight
line may be drawn through any given point.

Systems of axioms upon which to build projective geometry were
first studied more particularly by the Italian school — G. Peano, M.
Pieri, Gino Fano of Turin. This subject received the attention also
of Theodor Vahlen of Vienna and Friedrich Schur of Strassburg.
Axioms of descriptive geometry have been considered mainly by

1 Bull. Am. Math. Soc., Vol. n, 1904, p. 77. Our remarks on the Italian school
are drawn from Wilson's article.

328 A HISTORY OF MATHEMATICS

Italian and American mathematicians, and by D. Hilbert. The
introduction of order was achieved by G. Peano by taking the class
of points which lie between any two points as the fundamental idea,
by G. Vailati and later by B. Russell, on the fundamental conception
of a class of relations or class of points on a straight line, by O. Veblen
(1904) on the study of the properties of one single three-term relation
of order. A. N. Whitehead * refers to O. Veblen's method: "This
method of conceiving the subject results in a notable simplification,
and combines advantages from the two previous methods." While
D. Hilbert has six undefined terms (point, straight line, plane, between,
parallel, congruent) and twenty-one assumptions, Veblen gives only
two undefined terms (point, between) and only twelve assumptions.
However, the derivation of fundamental theorems is somewhat
harder by Veblen's axioms. R. L. Moore showed that any plane
satisfying Veblen's axioms I-VIII, XI is a number-plane and con-
tains a system of continuous curves such that, with reference to these
curves regarded as straight lines, the plane is an ordinary Euclidean
plane.

In 1907, Oswald Veblen and J. W. Young gave a completely in-
dependent set of assumptions for projective geometry, in which points
and undefined classes of points called lines have been taken as the
undefined elements. Eight of these assumptions characterize general
projective spaces; the addition of a ninth assumption yields properly
projective spaces.2

Axioms for line geometry based upon the "line" as an undefined
element and "intersection" as an undefined relation between un-
ordered pairs of elements, were given in 1901 by M. Fieri of Catania,
and in simpler form, in 1914 by E. R. Hedrick and L. Ingold of the
University of Missouri.

Text-books built upon some such system of axioms and possessing
great generality and scientific interest have been written by Federigo
Enriques of the University of Bologna in 1898, and by O. Veblen
and J. W. Young in 1910.

Geometric Models

Geometrical models for advanced students began to be manu-
factured about 1879 by the firm of L. Brill in Darmstadt. Many
of the early models, such as Kummer's surface, twisted cubics, the
tractrix of revolution, were made under the direction of F. Klein and
Alexander von Brill. Since about 1890 this firm developed into that
of Martin Schilling (1866-1908) of Leipzig. The catalogue of the
firm for 1911 described some 400 models. Since 1905 the firm of
B. G. Teubner in Leipzig has offered models designed by Hermann

1 The Axioms of Descriptive Geometry, Cambridge, 1907, p. 2.
1 Bull. Am. Math. Soc., Vol. 14, 1908, p. 251.

ALGEBRA 329

Wiener, many of which are intended for secondary instruction. Val-
uable in this connection is the Katalog mathematischer und mathe-
matisch-physikalischer Modelle, Apparate und Instrumente by Walter
v. Dyck, professor in Munich. At the Napier tercentenary celebra-
tion in Edinburgh, in 1914, Crum Brown exhibited models of various
sorts, including models of cubic and quartic surfaces, interlacing sur-
faces, regular solids and related forms, and thermodynamic models;
D. M. Y. Sommerville displayed models of the projection, on three-
dimensional space, of a four-dimensional figure; Lord Kelvin's tide-
calculating machine illustrated the combination of simple harmonic
motions.1

Algebra

The progress of algebra in recent times may be considered under
three principal heads: the study of fundamental laws and the birth
of new algebras, the growth of the theory of equations, and the de-
velopment of what is called modern higher algebra.

The general theory of ab, where both a and b are complex numbers,
was outlined by L. Euler in 1749 in his paper, Recherches sur les ratines
imaginaries des equations, but it failed to command attention. At
the beginning of the nineteenth century the theory of the general
power was elaborated in Germany, England, France and the Nether-
lands. In the early history of logarithms of positive numbers it was
found surprising that logarithms were defined independently of ex-
ponents. Now we meet a second surprise in finding that the theory
of ab is made to depend upon logarithms. Historically the logarithmic
concept is the more primitive. The general theory of a6 was developed
by Martin Ohm (1792-1872), professor in Berlin and a brother of
the physicist to whom we owe "Ohm's law." Martin Ohm is the
author of a much criticised series of books, Versuch eines wllkommen
consequenten Systems der Mathematik, Nlirnberg, 1822-1852. Our
topic was treated in the second volume, dated 1823, second edition
1829. After having developed the Eulerian theory of logarithms Ohm
takes up ax, where a=p+qi, #= a+ffi. Assuming ez as always single-
valued and letting v=\/p2+q2, log a=Lv+(=*=2mir+<f>)i, he takes

0* = e*loga = eal>-/3 (db2m7r + 0). |cos [^ Lv+ 0. (=b 2 m 7T+ <£)]+*.

sin [j3Lv+a(±2w7r+0)] }, where m=o, +i, + 2, . . . and L. signifies
the tabular logarithm. Thus the general power has an infinite number
of values, but all are of the form a+bi. Ohm shows (i) that all of the
values (infinite in number) are equal when x is an integer, (2) that there
are n distinct values when x is a real, rational fraction, (3) that some of
the values are equal, though the number of distinct values is infinite,
when x is real but irrational, (4) that the values are all distinct when x is
imaginary. He inquires next, how the formulas (A) ax.av=a*+y, (B)a*

1 Consult E. M. Horsburgh Handbook of the Exhibition of Napier relics, etc., 1914,
p. 302.

330 A HISTORY OF MATHEMATICS

+av=a*-y, (C) ax.bx=(ab)x, (D) ax+bx=(a+b)x, (E) (ax)y=axv apply
to the general exponent ax, and finds that (A), (B) and (E) are incom-
plete equations, since the left members have "many, many more"
values than the right members, although the right-hand values (infinite
in number) are all found among the "infinite times infinite" values on
the left; that (C) and (D) are complete equations for the general case.
A failure to recognize that equation E is incomplete led Thomas
Clausen (1801-1885) of Altona to a paradox (Crelle's Journal, Vol. 2,
1827, p. 286) which was stated by E. Catalan in 1869 in more con-
densed form, thus: e^iri=e2nirij where m and n are distinct integers.
Raising both sides to the power 4, there results the absurdity,
e — m* = e ~ nir. Ohm introduced a notation to designate some particular
value of ax, but he did not introduce especially the particular value
which is now called the "principal value." Otherwise his treatment
of the general power is mainly that of the present tune, except, of
course, in the explanation of the irrational. From the general power
Ohm proceeds to the general logarithm, having a complex number
as its base. It is seen that the Lulerian logarithms served as a step-
ladder leading to the theory of the general power; the theory of the
general power, in turn, led to a more general theory of logarithms
having a complex base.

The Philosophical Transactions (London, 1829) contain two articles
on general powers and logarithms — one by John Graves, the other
by John Warren of Cambridge. Graves, then a young man of 23,
was a class-fellow of William Rowan Hamilton in Dublin. Graves
became a noted jurist. Hamilton states that reflecting on Graves's
ideas on imaginaries led him to the invention of quaternions. Graves
obtains log i = (2w/7rf)/(I+2w7rf)- Thus Graves claimed that gen-
eral logarithms involve two arbitrary integers, m and ra', instead
of simply one, as given by Euler. Lack of explicitness involved
Graves in a discussion with A. De Morgan and G. Peacock, the out-
come of which was that Graves withdrew the statement contained
in the title of his paper and implying an error in the Eulerian theory,
while De Morgan admitted that if Graves desired to extend the idea
of a logarithm so as to use the base e^ + ^mvi^ there was no error in-
volved in the process. Similar researches were carried on by A. J. H.
Vincent at Lille, D. F. Gregory, De Morgan, W. R. Hamilton and
G. M. Pagani (1796-1855), but their general logarithmic systems,
involving complex numbers as bases, failed of recognition as useful
mathematical inventions.1 We pause to sketch the life of De Morgan.

Augustus De Morgan (1806-1871) was born at Madura (Madras),
and educated at Trinity College, Cambridge. For the determination
of the year of his birth (assumed to be in the nineteenth century) he
proposed the conundrum, "I was x years of age hi the year x2." His

1 For references and fuller details see F. Cajori in Am. Math. Monthly, Vol. 20,
1913, pp. 175-182.

ALGEBRA 331

scruples about the doctrines of the established church prevented him
from proceeding to the M. A. degree, and from sitting for a fellowship.
It is said of him, "he never voted at an election, and he never visited
the House of Commons, or the Tower, or Westminster Abbey." In
1828 he became professor at the newly established University of
London, and taught there until 1867, except for five years, from 1831-
1835. He was the first president of the London Mathematical Society
which was founded in 1866. De Morgan was a unique, manly char-
acter, and pre-eminent as a teacher. The value of his original work lies
not so much in increasing our stock of mathematical knowledge as in
putting it all upon a more logical basis. He felt keenly the lack of close
reasoning in mathematics as he received it. He said once: "We know
that mathematicians care no more for logic than logicians for mathe-
matics. The two eyes of exact science are mathematics and logic:
the mathematical sect puts out the logical eye, the logical sect puts
out the mathematical eye; each believing that it can see better with
one eye than with two." De Morgan analyzed logic mathematically,
and studied the logical analysis of the laws, symbols, and operations
of mathematics; he wrote a Formal Logic as well as a Double Algebra,
and corresponded both with Sir William Hamilton, the metaphysician,
and Sir William Rowan Hamilton, the mathematician. Few con-
temporaries were as profoundly read in the history of mathematics
as was De Morgan. No subject was too insignificant to receive his
attention. The authorship of "Cocker's Arithmetic" and the work
of circle-squarers was investigated as minutely as was the history of
the calculus. Numerous articles of his lie scattered in the volumes of
the Penny and English Cyclopaedias. In the article " Induction (Mathe-
matics)," first printed in 1838, occurs, apparently for the first time,
the name "mathematical induction"; it was adopted and popularized
by I. Todhunter, in his Algebra. The term " induction" had been used
by John Wallis in 1656, in his Arithmetica infinitorum; he used the
"induction" known to natural science. In 1686 Jacob Bernoulli
criticised him for using a process which was not binding logically and
then advanced in place of it the proof from n to «+ 1 . This is one of the
several origins of the process of mathematical induction. From Wallis
to De Morgan, the term "induction" was used occasionally in mathe-
matics, and in a double sense, (i) to indicate incomplete inductions of
the kind known in natural science, (2) for the proof from n to n+ 1 . De
Morgan's "mathematical induction" assigns a distinct name for the
latter process. The Germans employ more commonly the name " voll-
standige Induktion," which became current among them after the use
of it by R. Dedekind in his Was sind und was sollen die Zahlen, 1887.
De Morgan's Differential Calculus, 1842, is still a standard work, and
contains much that is original with the author. For the Encyclopedia
Mctropolitana he wrote on the Calculus of Functions (giving principles
of symbolic reasoning) and on the theory of probability. In the Cal-

332 A HISTORY OF MATHEMATICS

culus of Functions he proposes the use of the slant line or "solidus"
for printing fractions in the text ; this proposal v.us adopted by G. G.
Stokes in 1880. l Cayley wrote Stockes, "I think the 'solidus' looks
very well indeed . . . ; it would give you a strong claim to be President
of a Society for the prevention of Cruelty to Printers." :

Celebrated is De Morgan's Budget of Paradoxes, London, 1872, a
second edition of which was edited by David Eugene Smith in 1915.
De Morgan published memoirs "On the Foundation of Algebra"
in the Trans, of the Cambridge Phil. Soc., 1841, 1842, 1844 and 1847.

The ideas of George Peacock and De Morgan recogni e the possi-
bility of algebras which differ from ordinary algebra. Such algebras
were indeed not slow in forthcoming, but, like non-Euclidean geometry,
some of them were slow in finding recognition. This is true of H. G.
Grassmann's, G. Bellavitis's, and B. Peirce's discoveries, but W. R.
Hamilton's quaternions met with immediate appreciation in England.
These algebras offer a geometrical interpretation of imaginaries.

William Rowan Hamilton (1805-1865) was born of Scotch parents
in Dublin. His early education, carried on at home, was mainly in
languages. At the age of thirteen he is said to have been familiar with
as many languages as he had lived years. About this time he came
across a copy of I. Newton's Universal Arithmetic. After reading that,
he took up successively analytical geometry, the calculus, Newton's
Principia, Laplace's Mccanique Celeste. At the age of eighteen he pub-
lished a paper correcting a mistake in Laplace's work. In 1824 he
entered Trinity College, Dublin, and in 1827, while he was still an
undergraduate, he was appointed to the chair of astronomy. C. G. J.
Jacobi met Hamilton at the meeting of the British Association at
Manchester in 1842 and, addressing Section A, called Hamilton "le
Lagrange de votre pays." Hamilton's early papers were on optics.
In 1832 he predicted conical refraction, a discovery by aid of mathe-
matics which ranks with the discovery of Neptune by U. J. J. Le
Verrier and J. C. Adams. Then followed papers on the Principle of
Varying Action (1827) and a general method of dynamics (1834-1835).
He wrote also on the solution of equations of the fifth degree, the
hodograph, fluctuating functions, the numerical solution of differential
equations.

The capital discovery of Hamilton is his quaternions, in which his
study of algebra culminated. In 1835 he published in the Transactions
of the Royal Irish Academy his Theory of Algebraic Couples. He re-
garded algebra "as being no mere art, nor language, nor primarily

1 G. G. Stokes, Math, and Phys. Papers, Vol. I, Cambridge, 1880, p. vii; see also
J. Larmor, Memoir and Scie. Corr. of G. G. Stokes, Vol. I, 1907, p. 397.

2 An earlier use of the solidus in designating fractions occurs in one of the very
first text books published in California, viz., the Definition de las prind pales
operaciones de arismttica by Henri Cambuston, 26 pages printed at Monterey in
1843. The solidus appears slightly curved.

ALGEBRA 333

a science of quantity, but rather as the science of order of progres-
sion." Time appeared to him as the picture of such a progression.
Hence his definition of algebra as "the science of pure time." It was
the subject of years' meditation for him to determine what he should
regard as the product of each pair of a system of perpendicular directed
lines. At last, on the i6th of October, 1843, while walking with his
wife one evening, along the Royal Canal in Dublin, the discovery of
quaternions flashed upon him, and he then engraved with his knife
on a stone in Brougham Bridge the fundamental formula iz=jz=k2=
ijk= — i. At the general meeting of the Irish Academy, a month
later, he made the first communication on quaternions. An account
of the discovery was given the following year in the Philosophical
Magazine. Hamilton displayed wonderful fertility in their develop-
ment. His Lectures on Quaternions, delivered in Dublin, were printed
in 1852.

In 1858 P. G. Tait was introduced to Hamilton and a correspond-
ence was carried on between them which brought Hamilton back to
the further development of quaternions along the lines of quaternion
differentials, the linear vector function and A. Fresnel's wave surface,
and led him to prepare the Elements of Quaternions, 1866, which he
did not live to complete. - Only 500 copies were printed. A new
edition has been published recently by Charles Jasper Joly (1864-
1906), his successor in Dunsink Observatory. Tait's own Elementary
Treatise on Quaternions was projected in 1859, but was withheld from
publication until Hamilton's work should appear; it was finally pub-
lished in 1867. P. G. Tait's chief accomplishment was the develop-
ment of the operator v> which was done in the later, greatly enlarged
editions.1 Tait submitted his quaternionic theorems to the judgment
of Clerk Maxwell, and Maxwell came to recognize the power of the
quaternion calculus in dealing with physical problems. "Tait brought
out the real physical significance of the quantities Sv^", Vv°", V**.
Maxwell's expressive names, Convergence (or Divergence) and Curl,
have sunk into the very heart of electromagnetic theory." 2 In 1913
J. B. Shaw generalized the Hamiltonian v for space of n dimensions,
which may be either flat or curved. Related memoirs are due to G.
Ricci (1892), T. Levi-Civita (1900), H. Maschke, and L. Ingold (1910).
Quaternions were greatly admired in England from the start, but on
the Continent they received less attention. P. G. Tait's Elementary
Treatise helped powerfully to spread a knowledge of them in England.
A. Cayley, W. K. Clifford, and Tait advanced the subject somewhat
by original contributions. But there has been little progress in recent
years, except that made by J. J. Sylvester in the solution of quaternion
equations, nor has the application of quaternions to physics been as

1 C. G. Knott, Life and Scientific. Work of Peter Gulhrie Tait, Cambridge, 1911,
pp. 143, 148.

2 C. G. Knott, op. cil., p. 167.

334 A HISTORY OF MATHEMATICS

extended as was predicted. The change in notation made in France
by Jules Hoiiel and by C. A. Laisant has been considered in England
as a wrong step, but the true cause for the lack of progress is perhaps
more deep-seated. There is indeed great doubt as to whether the
quaternionic product can claim a necessary and fundamental place
in a system of vector analysis. Physicists claim that there is a loss
of naturalness in taking the square of a vector to be negative.

Widely different opinions have been expressed on the value of
quaternions. While P. G. Tait was an enthusiastic champion of this
science, his great friend, William Thomson (Lord Kelvin), declared
that they, "though beautifully ingenious, have been an unmixed
evil to those who have touched them in any way, including Clerk
Maxwell." l A. Cayley, writing to Tait in 1874, said, "I admire the
equation d<r=uqdpq~l extremely — it is a grand example of the pocket
map." Cayley admitted the conciseness of quaternion formulas, but
they had to be unfolded into Cartesian form before they could be
made use of or even understood. Cayley wrote a paper "On Co-or-
dinates versus Quaternions" in the Proceedings of the Royal Society
of Edinburgh, Vol. 20, to which Tait replied "On the Intrinsic Nature
of the Quaternion Method."

In order to meet more adequately the wants of physicists, /. W.
Gibbs and A. Macfarlane have each suggested an algebra of vectors
with a new notation. Each gives a definition of his own for the
product of two vectors, but in such a way that the square of a vector
is positive. A third system of vector analysis has been used by Oliver
Heaviside in his electrical researches.

What constitutes the most desirable notation in vector analysis
is still a matter of dispute. Chief, among the various suggestions, are
those of the American school, started by J. W. Gibbs and those of the
German-Italian school. The cleavage is not altogether along lines
of nationality. L. Prandl of Hanover said in 1904: "After long delib-
eration I have adopted the notation of Gibbs, writing a . b for the
inner (scalar), and axb for the outer (vector) product. If one ob-
serves the rule that in a multiple product the outer product must be
taken before the inner, the inner product before the scalar, then one
can write with Gibbs a . bxc and ab. c without giving rise to doubt
as to the meaning." 2

In the following we give German-Italian notations first, the equiv-
alent American notation (Gibbs') second. Inner product a | b, a. b;
vector-product | ab, axb ; also abc, a . b xc ; ab | c, (ax b) xc ; ab | cd,
(axb).(cxd); ab2, (axb)2; ab.cd, (axb)x(cxd). R. Mehmke
said in 1904: "The notation of the German-Italian school is far pref-
erable to that of Gibbs not only in logical and methodical, but also
in practical respects."

1 S. P. Thompson, Life of Lord Kelvin, IQIO, p. 1138.

2 Jahrcsb. d. d. Math. Vcreinig., Vol. 13, p. 39.

ALGEBRA 335

In 1895 P. Molenbroek of The Hague and S. Kimura, then at Yale
University, took the first steps in the organization of an International
Association for Promoting the Study of Quaternions and Allied
Systems of Mathematics. P. G. Tait was elected the first president,
but could not accept on account of failing health. Alexander Mac-
farlane (1851-1913) of the University of Edinburgh, later of the
University of Texas and of Lehigh University, served as secretary of
the Association and was its president at the time of his death.

At the international congress held in Rome in 1908 a committee
was appointed on the unification of vectorial notations but at the
time of the congress held in Cambridge in 1912 no definite conclusions
had been reached.

Vectorial notations were subjects of extended discussion in L'En-
seignement mathematique, Vols. 11-14, 1909-1912, between C. Burali-
Forti of Turin, R. Marcolongo of Naples, G. Comberiac of Bourges,
H. C. F. Timerding of Strassburg, F. Klein of Gottingen, E. B.
Wilson of Boston, G. Peano of Turin, C. G. Knott of Edinburgh,
Alexander Macfarlane of Chatham in Canada, E. Carvallo of Paris,
and E. Jahnke of Berlin. In America the relative values of notations
were discussed in 1916 by E. B. Wilson and V. C. Poor.

We mention two topics outside of ordinary physics in which vector
analysis has figured. The generalization of A. Einstein, known as
the principle of relativity, and its interpretation by H. Minkowski,
have opened new points of view. Some of the queer consequences of
this theory disappear when kinematics is regarded as identical with
the geometry of four-dimensional space. H. Minkowski and, following
him, Max Abraham, used vector analysis in a limited degree, Min-
kowski usually preferring the matrix calculus of A. Cayley. A more
extended use of vector analysis was made by Gilbert N. Lewis of the
University of California who introduced in his extension to four di-
mensions some of the original features of H. G. Grassmann's system.

A "dyname" is, according to J. Pliicker (and others) a system of
forces applied to a rigid body. The English and French call it a
"torsor." In 1899 this subject was treated by the Russian A. P.
Kotjelnikoff under the name of projective theory of vectors. In 1903
E. Study of Greifswald brought out his book, Geometric der Dynamen,
in which a line-geometry and kinematics are elaborated, partly by
the use of group theory, which are carried over to non-Euclidean
spaces; Study claims for his system somewhat greater generality
than is found in Hamilton's quaternions and W. K. Clifford's bi-
quarternions.

Hermann Giinther Grassmann (1809-1877) was born at Stettin,
attended a gymnasium at his native place (where his father was
teacher of mathematics and physics), and studied theology in Berlin
for three years. His intellectual interests were very broad. He started
as a theologian, wrote on physics, composed texts for the study of

336 A HISTORY OF MATHEMATICS

German, Latin, and mathematics, edited a political paper and a mis-
sionary paper, investigated phonetic laws, wrote a dictionary to the
Rig-Veda, translated the Rig-Veda in verse, harmonized folk songs
in three voices, carried on successfully the regular work of a teacher
and brought up nine of his eleven children — all this in addition to the
great mathematical creations which we are about to describe. In
1834 he succeeded J. Steiner as teacher of mathematics in an industrial
school in Berlin, but returned to Stettin in 1836 to assume the duties
of teacher of mathematics, the sciences, and of religion in a school
there.1 Up to this time his knowledge of mathematics was pretty
much confined to what he had learned from his father, who had
written two books on "Raumlehre" and "Grossenlehre." But now
he made his acquaintance with the works of S. F. Lacroix, J. L. La-
grange, and P. S. Laplace. He noticed that Laplace's results could
be reached in a shorter way by some new ideas advanced in his father's
books, and he proceeded to elaborate this abridged method, and to
apply it in the study of tides. He was thus led to a new geometric
analysis. In 1840 he had made considerable progress in its develop-
ment, but a new book of Schleiermacher drew him again to theology.
In 1842 he resumed mathematical research, and becoming thoroughly
convinced of the importance of his new analysis, decided to devote
himself to it. It now became his ambition to secure a mathematical
chair at a university, but in this he never succeeded. In 1844 ap-
peared his great classical work, the Lineale Ausdehnungslehre, which
was full of new and strange matter, and so general, abstract, and out
of fashion in its mode of exposition, that it could hardly have had
less influence on European mathematics during its first twenty years,
had it been published in China. K. F. Gauss, J. A. Grunert, and A. F.
Mobius glanced over it, praised it, but complained of the strange
terminology and its " philosophische Allgemeinheit." Eight years
afterwards, C. A. Bretschneider of Gotha was said to be the only
man who had read it through. An article in Crelle's Journal, in
which Grassmann eclipsed the geometers of that time by constructing,
with aid of his method, geometrically any algebraic curve, remained
again unnoticed. Need we marvel if Grassmann turned his attention
to other subjects, — to Schleiermacher's philosophy, to politics, to
philology? Still, articles by him continued to appear in Crelle's
Journal, and in 1862 came out the second part of his Ausdehnungslehre.
It was intended to show better than the first part the broad scope of
the Ausdehnungslehre, by considering not only geometric applica-
tions, but by treating also of algebraic functions, infinite series, and
the differential and integral calculus. But the second part was no
more appreciated than the first. At the age of fifty-three, this won-
derful man, with heavy heart, gave up mathematics, and directed his
energies to the study of Sanskrit, achieving in philology results which
1 Victor Schlegel, Hermann Grassmann, Leipzig, 1878.

ALGEBRA 337

were better appreciated, and which vie in splendor with those in
mathematics.

Common to the Ausdehnungslehre and to quaternions are geometric
addition, the function of two vectors represented in quaternions by
Sa(3 and Fa/3, and the linear vector functions. The quaternion is
peculiar to W. R. Hamilton, while with Grassmann we find in addition
to the algebra of vectors a geometrical algebra of wide application,
and resembling A. F. Mobius's Barycentrische Cdcul, in which the
point is the fundamental element. Grassmann developed the idea
of the "external product," the "internal product," and the "open
product." The last we now call a matrix. His Ausdehnungslehre
has very great extension, having no limitation to any particular
number of dimensions. Only in recent years has the wonderful rich-
ness of his discoveries begun to be appreciated. A second edition of
the Ausdehnungslehre of 1844 was printed in 1877. C. S. Peirce gave
a representation of Grassmann's system in the logical notation, and
E. W. Hyde of the University of Cincinnati wrote the first text-book
on Grassmann's calculus in the English language.

Discoveries of less value, which in part covered those of Grassmann
and Hamilton, were made by Barre de Saint-Venant (1797-1886),
who described the multiplication of vectors, and the addition of vectors
and oriented areas; by A. L. Cauchy, whose "clefs algebriques" were
units subject to combinatorial multiplication, and were applied by
the author to the theory of elimination in the same way as had been
done earlier by Grassmann; by Giusto Bellavitis (1803-1880), who
published in 1835 and 1837 in the Annali delle Scienze his calculus of
jequipollences. Bellavitis, for many years professor at Padua, was
a self-taught mathematician of much power, who in his thirty-eighth
year laid down a city office in his native place, Bassano, that he
might give his time to science.

The first impression of H. G. Grassmann's ideas is marked in the
writings of Hermann Hankel, who published in 1867 his Vorlesungen
iiber die Complexen Zahlen. Hankel, then decent in Leipzig, had
been in correspondence with Grassmann. The "alternate numbers"
of Hankel are subject to his law of combinatorial multiplication. In
considering the foundations of Algebra Hankel affirms the principle of
the permanence of formal laws previously enunciated incompletely
by G. Peacock. His Complexe Zahlen was at first little read, and we
must turn to Victor Schlegel (1843-1905) of Hagen as the successful
interpreter of Grassmann. Schlegel was at one time a young col-
league of- Grassmann at the Marienstifts-Gymnasium in Stettin. En-
couraged by R. F. A. Clebsch, Schlegel wrote a System der Raumlehre,
1872-1875, which explained the essential conceptions and operations
of the Ausdehnungslehre.

Grassmann's ideas spread slowly. In 1878 Clerk Maxwell wrote
P. G. Tait: "Do you know Grassmann's Ausdehnungslehre? Spottis-

338 A HISTORY OF MATHEMATICS

woode spoke of it in Dublin as something above and beyond 4nions.
I have not seen it, but Sir W. Hamilton of Edinburgh used to say
that the greater the extension the smaller the intention."

Multiple algebra was powerfully advanced by B. Peirce, whose
theory is not geometrical, as are those of W. R. Hamilton and H. G.
Grassmann. Benjamin Peirce (1809-1880) was born at Salem, Mass.,
and graduated at Harvard College, having as undergraduate carried
the study of mathematics far beyond the limits of the college course.
When N. Bowditch was preparing his translation and commentary
of the Mecanique Celeste, young Peirce helped in reading the proof-
sheets. He was made professor at Harvard in 1833, a position which
he retained until his death. For some years he was in charge of the
Nautical Almanac and superintendent of the United States Coast
Survey. He published a series of college text-books on mathematics,
an Analytical Mechanics, 1855, and calculated, together with Sears
C. Walker of Washington, the orbit of Neptune. Profound are his
researches on Linear Associative Algebra. The first of several papers
thereon was read at the first meeting of the American Association
for the Advancement of Science in 1864. Lithographed copies of a
memoir were distributed among friends in 1870, but so small seemed
to be the interest taken in this subject that the memoir was not
printed until 1881 (Am. Jour. Math., Vol. IV, No. 2). Peirce works
out the multiplication tables, first of single algebras, then of double
algebras, and so on up to sextuple, making in all 162 algebras, which
he shows to be possible on the consideration of symbols A, B, etc.,
which are linear functions of a determinate number of letters or units
i, j, k, I, etc., with coefficients which are ordinary analytical magni-
tudes, real or imaginary, — the letters i,j, etc., being such that every
binary combination i2, ij, ji, etc., is equal to a linear function of the
letters, but under the restriction of satisfying the associative law.1
Charles S. Peirce, a son of Benjamin Peirce, and one of the foremost
writers on mathematical logic, showed that these algebras were all
defective forms of quadrate algebras which he had previously dis-
covered by logical analysis, and for which he had devised a simple
notation. Of these quadrate algebras quaternions is a simple example;
nonions is another. C. S. Peirce showed that of all linear associative
algebras there are only three in which division is unambiguous. These
are ordinary single algebra, ordinary double algebra, and quaternions,
from which the imaginary scalar is excluded. He showed that his
father's algebras are operational and matricular. Lectures on multiple
algebra were delivered by J. J. Sylvester at the Johns Hopkins Uni-
versity, and published in various journals. They treat largely of the
algebra of matrices.

While Benjamin Peirce's comparative anatomy of linear algebras
was favorably received in England, it was criticised in Germany as
1 A. Cayley, Address before British Association, 1883.

ALGEBRA 339

being vague and based on arbitrary principles of classification. Ger-
man writers along this line are Edtiard Study and Georg W. Scheffers.
An estimate of B. Peirce's linear associative algebra was given in
1902 by H. E. Hawkes,1 who extends Peirce's method and shows its
full power. In 1898 Elie Cartan of the University of Lyon used the
characteristic equation to develop several general theorems; he ex-
hibits the semi-simple, or Dedekind, and the pseudo-mil, or nilpotent,
sub-algebras; he shows that the structure of every algebra may be
represented by the use of double units, the first factor being quad-
rate, the second non-quadrate. Extensions of B. Peirce's results were
made also by Henry Taber. Olive C. Hazlett gave a classification of
nilpotent alegbras.

As shown above, C. S. Peirce advanced this algebra by using the
matrix theory. Papers along this line are due to F. G. Frobenius and
J. B. Shaw. The latter "shows that the equation of an algebra de-
termines its quadrate units, and certain of the direct units; that the
other units form a nilpotent system which with the quadrates may
be reduced to certain canonical forms. The algebra is thus made a
sub-algebra under the algebra of the associative units used in these
canonical forms. Frobenius proves that every algebra has a Dede-
kind sub-algebra, whose equation contains all factors in the equation
of the algebra. This is the semi-simple algebra of Cartan. He also
showed that the remaining units form a nilpotent algebra whose units
may be regularized" (J. B. Shaw). More recently, J. B. Shaw has
extended the general theorems of linear associative algebras to such
algebras as have an infinite number of units.

Besides the matrix theory, the theory of continuous groups has been
used in the study of linear associative algebra. This isomorphism
was first pointed out by H. Poincare (1884); the method was followed
by Georg W. Scheffers who classified algebras as quaternionic and
non-quaternionic and worked out complete lists of all algebras to
order five. Theodor Molien, in 1893, then in Dorpat, demonstrated
"that quaternionic algebras contain independent quadrates, and that
quaternionic algebras can be classified according to non-quaternionic
types " (J. B. Shaw). An elementary exposition of the relation between
linear algebras and continuous groups was given by L. E. Dickson 2 of
Chicago. This relation "enables us to translate the concepts and
theorems of the one subject into the language of the other subject.
It not only doubles our total knowledge, but gives us a better insight
into either subject by exhibiting it from a new point of view." The
theory of matrices was developed as early as 1858 by A. Cayley in an
important memoir which, in the opinion of J. J. Sylvester, ushered in

1 H. E. Hawkes in Am. Jour. Math., Vol. 24, 1902, p. 87. We are using also
J. B. Shaw's Synopsis of Linear Associative Algebra, Washington, D. C., 1007,
Introduction. Shaw gives bibliography.

2 Bull. Am. Math. Soc., Vol. 22, 1915, p. 53.

340 A HISTORY OF MATHEMATICS

the reign of Algebra the Second. W. K. Clifford, Sylvester, H. Taber,
C. H. Chapman, carried the investigations much further. The origi-
nator of matrices is really W. R. Hamilton, but his theory, published
in his Lectures on Quaternions, is less general than that of Cayley.
The latter makes no reference to Hamilton.

The theory of determinants 1 was studied by Hoene Wronski (1778-
1853), a poor Polish enthusiast, living most the time in France, whose
egotism and wearisome style tended to attract few followers, but who
made some incisive criticisms .bearing on the philosophy of mathe-
matics.2 He studied four special forms of determinants, which were
extended by Heinrich Ferdinand Scherk (1798-1885) of Bremen and
Ferdinand Schweins (1780-1856) of Heidelberg. In 1838 Liouville
demonstrated a property of the special forms which were called
"wronskians" by Thomas Muir in 1881. Determinants received the
attention of Jacques P. M. Binet (1786-1856) of Paris, but the great
master of this subject was A. L. Cauchy. In a paper (Jour, de Vecole
Polyt., IX., 16) Cauchy developed several general theorems. He in-
troduced the name determinant, a term used by K. F. Gauss in 1801 in
the functions considered by him. In 1826 C. G. J. Jacobi began using
this calculus, and he gave brilliant proof of its power. In 1841 he
wrote extended memoirs on determinants in Crelle's Journal, which
rendered the theory easily accessible. In England the study of linear
transformations of quantics gave a powerful impulse. A. Cayley de-
veloped skew-determinants and Pfaffians, and introduced the use of
determinant brackets, or the familiar pair of upright lines. The more
general consideration of determinants whose elements are formed from
the elements of given determinants was taken up by J. J. Sylvester
(1851) and especially by L. Kronecker who gave an elegant theorem
known by his name.3 Orthogonal determinants received the atten-
tion of A. Cayley in 1846, in the study of nz elements related to each
other by %n(n+i) equations, also of L. Kronecker, F. Brioschi and
others. Maximal values of determinants received the attention of
J. J. Sylvester (1867), and especially of J. Hadamard (1893) who
proved that the square of a determinant is never greater than the
norm-product of the lines.

Anton Puchta (1851-1903) of Czernowitz in 1878 and M. Noether
in 1880 showed that a symmetric determinant may be expressed as
the product of a certain number of factors, linear in the elements.
Determinants which are formed from the minors of a determinant
were investigated by J. J. Sylvester in 1851, to whom we owe the

1 Thomas Muir, The Theory of Determinants in the Historical Order of Develop-
ment [Vol. I], 2nd Ed., London, 1906; Vol. II, Period 1841 to 1860, London, 1911.
Muir was Superintendent-General of Education in Cape Colony.

2 On Wronski, see J. Bertrand in Journal des Savants, 1897, and in Revue des
Deux-Mondes, Feb., 1897. See also L' Intermediate des Mathematiciens, Vol. 23,
1916, pp. 113, 164-167, 181-183.

3 E. Pascal, Die Determinanlen, transl. by H. Leitzmann, 1900, p. 107.

ALGEBRA 341

"umbral notation," by W. Spottiswoode in 1856, and later by G.
Janni, M. Reiss (1805-1869), E. d'Ovidio, H. Picquet, E. Hunyadi
(1838-1889) E. Barbier, C. A. Van Velzer, E. Netto, G. Frobenius,
and others. Many researches on determinants appertain to special
forms. "Continuants" are due to J. J. Sylvester; "alternants," origi-
nated by A. L. Cauchy, have been developed by C. G. J. Jacobi,
Nicolo Trudi (1811-1884) of Naples, H. Nagelsbach, and G. Garbieri;
" axisymmetric determinants," first used by Jacobi, have been studied
by V. A. Lebesgue, J. J. Sylvester, and L. 0. Hesse; "circulants" are
du to Eugene Charles Catalan (1814-1894) of Liege, William Spottis-
woode (1825-1883) of Oxford, J. W. L. Glaisher, and R. F. Scott;
for " centro-symmetric determinants" we are indebted to G. Zehfuss
of Heidelberg. V. Nachreiner and S. Giinther, both of Munich,
pointed out relations between determinants and continued fractions;
R. F. Scott uses H. Hankel's alternate numbers in his treatise. A
class of determinants which have the same importance in linear inte-
gral equations as do ordinary determinants for linear equations with
n unknowns was worked out by E. Fredholm (Acta math., 1903) and
again by D. Hilbert who reaches them as limiting expressions of or-
dinary determinants.

An achievement of considerable significance was the introduction
in 1860 of infinite determinants by Eduard Furstenau in a method
of approximation to the roots of algebraic equations. Determinants of
an infinite order were used by Theodor Kotteritzsch of Grimma in
Saxony, in two papers on the solution of an infinite system of linear
equations (Zeitsch.f. Math. u. Physik, Vol. 14, 1870). Independently,
infinite determinants were introduced in 1877 by George William
Hill of Washington in an astronomical paper (Collected Works, Vol.
I, 1905, p. 243). In 1884 and 1885 H. Poincare called attention to
these determinants as developed by Hill and investigated them further.
Their theory was elaborated later by Helge von Koch and Erhard
Schmidt (1908).

• In recent years the theory of the solution of a system of linear
equations has been presented in an elegant form by means of what
is known as the rank of a determinant. In particular, G. A. Miller
has thus developed a necessary and sufficient condition that a given
unknown in a consistent system of linear equations have only one
value while some of the other unknowns may assume an infinite
number of values.1

Text-books on determinants were written by W. Spottiswoode
(1851), F. Brioschi (1854), R. Baltzer (1857), S. Gunther (1875),
G. J. Dostor (1877), R. F. Scott (1880), T. Muir (1882), P. H. Hanus
(1886), G. W. H. Kowalewski (1909).

The symbol n! for "factorial n," now universally used in algebra,
is due to Christian Kramp (1760-1826) of Strassburg, who used it in
1 Am. Math. Monthly, Vol. 17, into, p. 137.

342 A HISTORY OF MATHEMATICS

1808. The symbol = to express identity was first used by G. F. B.
Riemann.1

Modern higher algebra is especially occupied with the theory of
linear transformations. Its development is mainly the work of A.
Cayley and J. J. Sylvester.

Arthur Cayley (1821-1895), born at Richmond, in Surrey, was
educated at Trinity College, Cambridge. He came out Senior Wrang-
ler in 1842. He then devoted some years to the study and practice
of law. While a student at the bar he went to Dublin and, alongside
of G. Salmon, heard W. R. Hamilton's lectures on quaternions. On
the foundation of the Sadlerian professorship at Cambridge, he ac-
cepted the offer of that chair, thus giving up a profession promising
wealth for a very modest provision, but which would enable him to give
all his time to mathematics. Cayley began his mathematical publica-
tions in the Cambridge Mathematical Journal while he was still an
undergraduate. Some of his most brilliant discoveries were made
during the time of his legal practice. There is hardly any subject
in pure mathematics which the genius of Cayley has not enriched, but
most important is his creation of a new branch of analysis by his
theory of invariants. Germs of the principle of invariants are found
in the writings of J. L. Lagrange, K. F. Gauss, and particularly of
G. Boole, who showed, in 1841, that invariance is a property of dis-
criminants generally, and who applied it to the theory of orthogonal
substitution. Cayley set himself the problem to determine a priori
what functions of the coefficients of a given equation possess this prop-
erty of invariance, and found, to begin with, in 184$, that the so-
called "hyper-determinants" possessed it. G. Boole made a number
of additional discoveries. Then J. J. Sylvester began his papers in the
Cambridge and Dublin Mathematical Journal on the Calculus of Forms.
After this, discoveries followed in rapid succession. At that tune Cay-
ley and Sylvester were both residents of London, and they stimulated
each other by frequent oral communications. It has often been dif-
ficult to determine how much really belongs to each. In 1882, when
Sylvester was professor at the Johns Hopkins University, Cayley
lectured there on Abelian and theta functions.

Of interest is Cayley's method of work. A. R. Forsyth describes it
thus: "When Cayley had reached his most advanced generalizations
he proceeded to establish them directly by some method or other,
though he seldom gave the clue by which they had first been obtained:
a proceeding which does not tend to make his papers easy reading. . . .
His literary style is direct, simple and clear. His legal training had
an influence, not merely upon his mode of arrangement but also upon
his expression; the result is that his papers are severe and present a
curious contrast to the luxuriant enthusiasm which pervades so many

1 L. Kronecker, Vorlesungen iiber Zaklentheorie, 1901, p. 86.

ALGEBRA 343

of Sylvester's papers." Curiously, Cayley took little interest in
quaternions.

James Joseph Sylvester (1814-1897) was born in London. His
father's name was Abraham Joseph; his eldest brother assumed in
America the name of Sylvester, and he adopted this name too. About
the age of 16 he was awarded a prize of $500 for solving a question
in arrangements for contractors of lotteries in the United States.2
In 1831 he entered St. John's College, Cambridge, and came out
Second Wrangler in 1837, George Green being fourth. Sylvester's
Jewish origin incapacitated him from taking a degree. From 1838 to
1840 he was professor of natural philosophy at what is now University
College, London; in 1841 he became professor of mathematics at the
University of Virginia. In a quarrel with two of his students he slightly
wounded one of them with a metal pointed cane, whereupon he re-
turned hurriedly, to England. In 1844 he served as an actuary; in
1846 he became a student at the Inner Temple and was called to the
bar in 1850. In 1846 he became associated with A. Cayley; often
they walked round the Courts of Lincoln's Inn, perhaps discussing
the theory of invariants, and Cayley (says Sylvester) "habitually
discoursing pearls and rubies." Sylvester resumed mathematical
research. He, Cayley and William Rowan Hamilton entered upon
discoveries in pure mathematics that are unequalled irf Great Britain
since the time of I. Newton. Sylvester made the friendship of G. Sal-
mon whose books contributed greatly to bring the results of Cayley
and Sylvester within easier reach of the mathematical public. From
1855 to 1870 Sylvester was professor at the Royal Military Academy
at Woolwich, but showed no great efficiency as an elementary teacher.
There are stories of his housekeeper pursuing him from home carrying
his collar and necktie. From 1876 to 1883, he was professor at the
Johns Hopkins University, where he was happy in being free to teach
whatever he wished in the way he thought best. He became the first
editor of the American Journal of Mathematics in 1878. In 1884 he
was elected to succeed H. J. S. Smith in the chair of Savilian professor
of geometry at Oxford, a chair once occupied by Henry Briggs, John
Wallis and Edmund Halley.

Sylvester sometimes amused himself writing poetry. His Laws of
Verse is a curious booklet. At the reading, at the Peabody Institute
in Baltimore, of his Rosalind poem, consisting of about 400 lines all
rhyming with "Rosalind," he first read all his explanatory footnotes,
so as not to interrupt the poem ; these took one hour and a-half . Then
he read the poem itself to the remnant of his audience.

1 Proceed. London Royal Society, Vol. 58, 1895, pp. 23, 24.

2 H. F. Baker's Biographical Notice in The Collected Math. Papers of J. J. Syl-
vester, Vol. IV, Cambridge, 1912. We have used also P. A. MacMahon's notice in
Proceed. Royal Soc. of London, Vol. 63, 1898, p. ix. For Sylvester's activities in
Baltimore, see Fabian Franklin in Johns Hopkins Univ. Circulars, June, 1897; F.

344 A HISTORY OF MATHEMATICS

Sylvester's first papers were on Fresnel's optic theory, 1837. Two
years later he wrote on C. Sturm's memorable theorem. Sturm
once told him that the theorem originated in the theory of the com-
pound pendulum. Stimulated by A. Cayley he made important in-
vestigations on modern algebra. He wrote on elimination, on trans-
formation and canonical forms, in which the expression of a cubic
surface by five cubes is given, on the relation between the minor de-
terminants of linearly equivalent quadratic functions, in which the
notion of invariant factors is implicit, while in 1852 appeared the
first of his papers on the principles of the calculus of forms. In a
reply that he made in 1869 to Huxley who had claimed that mathe-
matics was a science that knows nothing of observation, induction,
invention and experimental verification, Sylvester narrated his per-
sonal experience: "I discovered and developed the whole theory of
canonical binary forms for odd degrees, and, as far as yet made out,
for even degrees too, at one evening sitting, with a decanter of port
wine to sustain nature's flagging energies, in a back office in Lincoln's
Inn Fields. The work was done, and well done, but at the usual
cost of racking thought — a brain on fire, and feet feeling, or feelingless,
as if plunged in an ice-pail. That night we slept no more." His reply
to Huxley is interesting reading and bears strongly on the qualities
of mental activity involved in mathematical research. In 1859 he
gave lectures on partitions, not published until 1897. He wrote on
partitions again in Baltimore. In 1864 followed his famous proof
of Newton's rule. A certain fundamental theorem in invariants
which had formed the basis of an important section of A. Cayley's
work, but had resisted proof for a quarter of a century was demon-
strated by Sylvester in Baltimore. Noteworthy are his memoirs on
Chebichev's method concerning the totality of prime numbers within
certain limits, and his latent roots of matrices. His researches on
invariants, theory of equations, multiple algebra, theory of numbers,
linkages, probability, constitute important contributions to mathe-
matics. His final studies, entered upon after his return to Oxford,
were on reciprocants or functions of differential coefficients whose
form is unaltered by certain linear transformations of the variables,
and a generalization of the theory of concomitants. In 1911, G.
Greenhill told reminiscently x how Sylvester got everybody interested
in reciprocants, "now clean forgotten"; "One day after, Sylvester
was noticed walking alone, addressing the sky, asking it: 'Are Recipro-
cants Bosh? Berry of King's says the Reciprocant is all Bosh!'
There was no reply, and Sylvester himself was tiring of the subject,
and so Berry escaped a castigation. But recently I had occasion
from the Aeronautical point of view to work out the theory of a Vortex

Cajori, Teaching and History of Mathematics in the United States, Washington, 1890,
pp. 261-272.

1 Mathematical Gazette, Vol. 6, 1912, p. 108.

ALGEBRA 345

inside a Polygon, an eddy whirlwind such as Chavez had to encounter
in the angle of the precipices, flying over the Simplon Pass. The
analysis in some cases seemed strangely familiar, and at last I recog-
nized the familiar Reciprocant. . . . Difference in Similarity and
Similarity in Difference has been called the motto of our science." x

In the American Journal of Mathematics are memoirs on binary and
ternary quantics, elaborated partly with aid of F. Franklin, then
professor at the Johns Hopkins University. The theory of reciprocants
is more general than one on differential invariants by G. H. Halphen
(1878), and has been developed further by J. Hammond of Cambridge,
P. A. McMahon of Woolwich, A. R. Forsyth now of London, and
others. Sylvester playfully lays claim to the appellation of the Mathe-
matical Adam, for the many names he has introduced into, mathe-
matics. Thus the terms invariant, discriminant, Hessian, Jacobian,
are his. That not only elementary pupils, but highly trained math-
ematicians as well, may be attracted or repelled by the kind of symbols
used, is illustrated by the experience of K. Weierstrass who related
that he followed Sylvester's papers on the theory of algebraic forms
very attentively until Sylvester began to employ Hebrew characters.
That was more than he could stand and after that he quit him.2

The great theory of invariants, developed in England mainly by
A. Cayley and J. J. Sylvester, came to be studied earnestly in Ger-
many, France, and Italy. Ch. Hermite discovered evectants, and
the theorem of reciprocity named after him, whereby " to every co vari-
ant of degree n in the coefficients of the quantic of order m, there
corresponds a covariant of degree m in the coefficients of a quantic of
order n. He discovered the skew invariant of the quintic, which was
the first example of any skew invariant. He discovered the linear co-
variants belonging to quantics of odd order greater than 3, and he ap-
plied them to obtain the typical expression of the quantic in which the
coefficients are invariants. He also invented the associated covariants
of a quantic; these constitute the simplest set of algebraically complete
systems as distinguished from systems that are linearly complete." 3

In Italy, F. Brioschi of Milan and Fad de Bruno (1825-1888) con-
tributed to the theory of invariants, the latter writing a text-book
on binary forms (1876), which ranks by the side of G. Salmon's treatise
and those of R. F. A. Clebsch and P. Gordan.

Francesco Brioschi (1824-1897) in 1852 became professor of applied
mathematics at the University of Pavia and in 1862 was commissioned
by the government to organize the Institute tecnico superiore at
Milan, where he filled until his death the chair of hydraulics and

1 Mathematical Gazette, Vol. 6, 1912, p. 108.

2E. Lampe (1840-1918), in Naturwissenschajtliche Rundschau, Bd. 12, 1897,
p. 361; quoted by R. E. Moritz, Memorabilia Mathematica, 1914, p. 180.

3 Proceed, of the Roy. Soc. of London, Vol. 45, 1905, p. 144. Obituary notice of
Hermite by A. R. Forsyth.

346 A HISTORY OF MATHEMATICS

analysis. With Abbe Barnaba Tortolini (1808-1874) he founded in
1858 the Annali di matematica pura ed applicata. Among his pupils
at Pavia were L. Cremona and E. Beltrami. V. Volterra narrates *
how F. Brioschi in 1858, with two other young Italians, Enrico Bettl
(1823-1892), later professor at the University of Pisa, and Felice
Casorati (1835-1890), later professor at the University of Pavia,
started on a journey to enter into relations with the foremost mathe-
maticians of France and Germany. "The scientific existence of Italy
as a nation" dates from this journey. "It is to the teaching, labors,
and devotion of these three, to their influence in the organization of
advanced studies, to the friendly scientific relations that they insti-
tuted between Italy and foreign countries, that the existence of a
school of analysts in Italy is due."

In Germany the early theory of invariants, as developed by Cayley,
Sylvester, and Salmon in England, Hermite in France and F. Brioschi
in Italy, did not draw attention until 1858 when Siegfried Heinrich
Aronhold (1819-1884) of the technical high school in Berlin pointed
out that Hesse's theory of ternary cubic forms of 1844 involved in-
variants by which that theory could be rounded out. F. G. Eisenstein
and J. Steiner had also given early publication to isolated develop-
ments involving the invariantal idea. In 1863 Aronhold gave a
systematic and general exposition of invariant theory (Crelle, 62).
He and Clebsch used a notation of their own, the symbolical notation,
different from Cayley's, which was used in the further developments
of the theory in Germany. Great developments were started about
1868, when R. F. A. Clebsch and P. Gordan wrote on types of binary
forms, L. Kronecker and E. B. Christoffel on bilinear forms, F. Klein
and S. Lie on the invariant theory connected with any group of linear
substitutions. Paul Gordan (1837-1912) was born at Erlangen and
became professor there. He produced papers on finite groups, par-
ticularly on the simple group of order 168 and its associated curve
y3z+z3x+x3y=o. His best known achievement is the proof of the
existence of a complete system of concomitants for any given binary
form.2 While Clebsch aimed in his researches to devise methods by
which he could study the relationships between invariantal forms
(Formenverwandtschaf t) , the chief aim of Aronhold was to examine
the equivalence or the linear transformation of one form into another.3
Investigations along this line are due to E. B. Christoffel, who showed
that the number of arbitrary parameters contained in the substitution
coefficients equals that of the absolute invariants of the form, K.
Weierstrass who gave a general treatment of the equivalence of two

1 Bull. Am. Math. Soc., Vol. 7, 1900, p. 60.

2 Nature, Vol. 90, 1913, p. 597.

3 We are using Franz Meyer, " Bericht iiber den gegenwartigen Stand der Tn-
variantentheorie " in the Jahresb. d. d. Math. Vereinigung, Vol. I, 1890-91,
pp. 79-292. See p. 99.

ALGEBRA 347

linear systems of bilinear and quadratic forms, L. Kronecker who ex-
tended the researches of Weiers trass and had a controversy with C.
Jordan on certain discordant results, J. G. Darboux who in 1874 gave
a general and elegant derivation of theorems due to K. Weierstrass
and L. Kronecker, G. Frobenius who applied the transformation of
bilinear forms to "Pfaff's problem": To determine when two given
linear differential expressions of n terms can be converted one into
the other by subjecting the variables to general point transformations.
The study of invariance of quadratic and bilinear forms from the
stand-point of group theory was pursued by H. Werner (1889), S.
Lie (1885) and W. Killing (1890). Finite binary groups were exam-
ined by H. A. Schwarz (1871), and Felix Klein. Schwarz is led to the
problem, to find "all spherical triangles whose symmetric repetitions
on the surface of a sphere give rise to a finite number of ^spherical
triangles differing in position," and deduces the forms belonging
thereto. Without a knowledge of what Schwarz and W. R. Hamilton
had done, Klein was led to a determination of the finite binary linear
groups and their forms. Representing transformations as motions
and adopting Riemann's interpretation of a complex variable on a
spherical surface, F. Klein sets up the groups of those rotations which
bring the five regular solids into coincidence with themselves, and the
accompanying forms. The tetrahedron, octahedron and icosahedron
lead respectively to 12, 24 and 60 rotations; the groups in question
were studied by Klein. The icosahedral group led to an icosahedral
equation which stands in intimate relation with the general equation
of the fifth degree. Klein made the icosahedron the centre of his theory
of the quintic as given in his Vorlesungen uber das Ikosaeder und die
Auflosung der Gleichungen fun/ten Grades, Leipzig, 1884.

Finite substitution groups and their forms, as related to linear
differential equations, were investigated by R. Fuchs (Crelle, 66, 68)
in 1866 and later. If the equation has only algebraic integrals, then
the group is finite, and conversely. Fuchs's researches on this topic
were continued by C. Jordan, F. Klein, and F. Brioschi. Finite ternary
and higher groups have been studied in connection with invariants by
F. Klein who in 1887 made two such groups the basis for the solution
of general equations of the sixth and seventh degrees. In 1886 F. N.
Cole, under the guidance of Klein, had treated the sextic equation in
the Am. Jour, of Math., Vol. 8. The second group used by Klein was
studied with reference to the 140 lines in space, to which it leads, by
H. Maschke in 1890.

The relationship of invariantal forms, the study of which was
initiated by A. Cayley and J. J. Sylvester, received since 1868 em-
phasis in the writings of R. F. A. Clebsch and P. Gordan. Gordan
proved in Crelle, Vol. 69, the finiteness of the system for a single
binary form. This is known as "Gordan's theorem." Even in the
later simplified forms the proof of it is involved, but the theorem

348 A HISTORY OF MATHEMATICS

yields practical methods to determine the existing systems. G.
Peano in 1881 generalized the theorem and applied it to the "cor-
respondences" represented by certain double-binary forms. In 1890
D. Hilbert, by using only rational processes, demonstrated the finite-
ness of the system of invariants arising from a given series of any forms
in n variables. A modification of this proof which has some advantages
was given by W. E. Story of Clarke University. Hilbert's research
bears on the number of relations called syzygies, a subject treated be-
fore this time by A. Cayley, C. Hermite, F. Brioschi, C. Stephanos
of Athens, J. Hammond, E. Stroh, and P. A. MacMahon.

The symbolic notation in the theory of invariants, introduced by
S. H. Aronhold and R. F. A. Clebsch, was developed further by P. Gor-
dan, E. Stroh, and E. Study in Germany. English writers endeavored
to make the expressions in the theory of forms intuitively evident by
graphic representation, as when Sylvester in 1878 uses the atomic
theory, an idea applied further by W. K. Clifford. The symbolic
method in the theory of invariants has been used by P. A. MacMahon
in the article "Algebra" in the eleventh edition of the Encyclopaedia
Britannica, and by J. H. Grace and A. Young in their Algebra of In-
variants, Cambridge, 1903. Using a method in C. Jordan's great
memoirs on invariants, these authors are led to novel results, notably
to "un exact formula for the maximum order of an irreducible co-
variant of a system of binary forms." A complete syzygetic theory
of the absolute orthogonal concomitants of binary quantics was con-
structed by Edwin B. Elliott of Oxford by a method that is not sym-
bolic, while P. A. MacMahon in 1905 employs a symbolic calculus in-
volving imaginary umbrae for similar purposes. While the theory
of invariants has played an important role in modern algebra and
analytic projective geometry, attention has been directed also to its
employment in the theory of numbers. Along this line are the re-
searches of L. E. Dickson in the Madison Colloquium of 1913.

The establishment of criteria by means of which the irreducibility
of expressions in a given domain may be ascertained has been inves-
tigated by F. T. v. Schubert (1793), K. F. Gauss, L. Kronecker, F. W.
P. Schonemann, F. G. M. Eisenstein, R. Dedekind, G. Floquet, L.
Konigsberger, E. Netto, O. Perron, M. Bauer, W. Dumas and H.
Blumberg. The theorem of Schonemann and Eisenstein declares that
if the polynomial xn+c\ xH~l+ . . . +cn with integral coefficients is
such that a prime p divides every coefficient c\, . . , cn, but p2 does
not go into cn, then the polynomial is irreducible in the domain of
rational numbers. This theorem may be regarded as the nucleus of
the work of the later authors. Floquet and Konigsberger do not
limit themselves to polynomials, but consider also linear homogeneous
differential expressions. Blumberg gives a general theorem which
practically includes all earlier results as special cases.1

1 For bibliography see Trans. Am. Math. Soc., Vol. 17, 1916, pp. 517-544.

ALGEBRA 349

Theory of Equations and Theory of Groups

A notable event was the reduction of the quintic equation to the
trinomial form, effected by George Birch Jerrard ( ?-i863) in his
Mathematical Researches (1832-1835). Jerrard graduated B. A. at
Trinity College, Dublin, in 1827. It was not until 1861 that it became
generally known that this reduction had been effected as early as
1786 by Erland Samuel Bring (1736-1798), a Swede, and brought
out in a publication of the University of Lund. Both Bring and Jer-
rard used the method of E. W. Tschirnhausen. Bring never claimed
that his transformation led to the general algebraic solution of the
quintic, but Jerrard persisted in making such a claim even after
N. H. Abel and others had offered proofs establishing the impossibility
of a general solution. In 1836, William R. Hamilton made a report
on the validity of Jerrard's method, and showed that by his process
the quintic could be transformed to any one of the four trinomial
forms. Hamilton defined the limits of its applicability to higher equa-
tions. J. J. Sylvester investigated this question, What is the lowest
degree an equation can have in order that' it may admit of being
deprived of i consecutive terms by aid of equations not higher than
ith degree. He carried the investigation as far as 2=8, and was led
to a series of numbers which he named "Hamilton's numbers." A
transformation of equal importance to Jerrard's is that of Sylvester,
who expressed the quintic as the sum of three fifth-powers. The
covariants and invariants of higher equations have been studied
much in recent years.

In the theory of equations J. L. Lagrange, J. R. Argand, and K. F.
Gauss furnished proof to the important theorem that every algebraic
equation has a real or a complex root. N. H. Abel proved rigorously
that the general algebraic equation of the fifth or of higher degrees
cannot be solved by radicals (Crelle, I, 1826). Before Abel, an Italian
physician, Paolo Ruffini (1765-1822), had printed a proof of the in-
solvability. It appears in his book, Teoria generate delle equazioni,
Bologna, 1799, and in later articles on this subject. Ruffini's proof
was criticised by his countryman, G. F. Malfatti. L. N. M. Carnot,
A. M. Legendre, and S. D. Poisson, in a report of 1813 on a paper of
A. L. Cauchy, had occasion to refer to Ruffini's proof as "fondee sur
des raisonnemens trop vagues, et n'ait pas ete generalement admise." *
N. H. Abel remarked that Ruffini's reasoning did not always seem
rigorous. But Cauchy in 1821 wrote to Ruffini that he had "demontre
completement 1'insolubilite algebrique des equations generates d' un
degre superieur au quatrieme." 2 J. Hecker showed in 1886 that
Ruffini's proof was sound in general outline, but faulty in some of
the detail.3 E. Bortolotti in 1902 stated that Ruffini's proof, as given

1 E. Bortolotti, Carteggio di Paolo Ruffini, Roma, 1006, p. 32.

2E. Bortolotti, Influenza dclf opera mat. di /'. Rii/luil, 1902, p. 34.

3 J. Hecker, Uebcr Ruffini's Beweis (Dissertation), Bonn, 1886.

350 A HISTORY OF MATHEMATICS

in 1813 in his book Reflessioni intorno alia soluzione dell' equazioni
algebraiche, was substantially the same as that given later by Pierre
Laurent Wantzel 1 (1814-1848), but only the second part of Wantzel's
simplified proof resembles Ruffini's; the first part is modelled after
Abel's. Wantzel, by the way, deserves credit for having given the
first rigorous proofs (Liouville, Vol. 2, 1837, p. 366.) of the im-
possibility of the trisection of any given angle by means of ruler and
compasses, and of avoiding the "irreducible case" in the algebraic
solution of irreducible cubic equations. Wantzel was repetiteur at
the Polytechnic School in Paris. As a student he excelled both in
mathematics and languages. Saint- Venant said of him: "Ordinarily
he worked evenings, not lying down until late; then he read, and took
only a few hours of troubled sleep, making alternately wrong use of
coffee and opium, and taking his meals at irregular hours until he
was married. He put unlimited trust in his constitution, very strong
by nature, which he taunted at pleasure by all sorts of abuse. He
brought sadness to those who mourn his premature death."

Ruffini's researches on equations are remarkable as containing
anticipations of the algebraic theory of groups.2 Ruffini's "per-
mutation" corresponds to our term "group." He divided groups into
"simple" and "complex," and the latter into intransitive, transitive
imprimitive, and transitive primitive groups. He established the
important theorem for which the name "Ruffini's theorem" has
been suggested,3 that a group does not necessarily have a subgroup
whose order is an arbitrary divisor of the order of the group. The
collected works of Ruffini are published under the auspices of the
Circolo Matematico di Palermo; the first volume appeared in 1915
with notes by Ettore Bortolotti of Bologna. A transcendental solu-
tion of the quintic involving elliptic integrals was given by Ch. Her-
mite (Compt. Rend., 1858, 1865, 1866). After Hermite's first publica-
tion, L. Kronecker, in 1858, in a letter to Hermite, gave a second
solution in which was obtained a simple resolvent of the sixth degree.

Abel's proof that higher equations cannot always be solved alge-
braically led to the inquiry as to what equations of a given degree
can be solved by radicals. Such equations are the ones discussed by
K. F. Gauss in considering the division of the circle. Abel advanced
one step further by proving that an irreducible equation can always
be solved in radicals, if, of two of its roots, the one can be expressed
rationally in terms of the other, provided that the degree of the equa-
tion is prime; if it is not prime, then the solution depends upon that
of equations of lower degree. Through geometrical considerations,

1 E. Bortolotti, Influenza, etc., 1902, p. 26. Wantzel's proof is given in Nouvelles
Annales Mathematiqucs, Vol. 4, 1845, pp. 57-65. See also Vol. 2, pp. 117-127. The
second part of Wantzel's proof, involving substitution-theory, is reproduced in J. A.
Serret's Algebre sup&ieure.

2 H. Burkhardt, in Zeilscltr. f. Mathemalik u. Physik, Suppl., 1892.

3 G. A. Miller, in Bibllothcca malhematica, 3. F. Vol. 10, 1909-1910, p. 318.

ALGEBRA 351

L. O. Hesse came upon algebraically solvable equations of the ninth
degree, not included in the previous groups. The subject was power-
fully advanced in Paris by the youthful Evariste Galois (1811-1832). *
He was born at Bourg-la-Reine, near Paris. He began to exhibit
most extraordinary mathematical genius after his fifteenth year.
His was a short,^ sad and pestered life. He was twice refused ad-
mittance to the Ecole Polytechnique, on account of inability to meet
the (to him) trivial demands of examiners who failed to recognize
his genius. He entered the Ecole Normale in 1829, then an inferior
school. Proud and arrogant, and unable to see the need of the cus-
tomary detailed explanations, his career in that school was not smooth.
Drawn into the turmoil of the revolution of 1830, he was forced to
leave the Ecole Normale. After several months spent in prison, he
was killed in a duel over a love affair. Ordinary text-books he dis-
posed of as rapidly as one would a novel. He read J. L. Lagrange's
memoirs on equations, also writings of A. M. Legendre, C. G. J.
Jacobi, and N. H. Abel. As early as the seventeenth year he reached
results of the highest importance. Two memoirs presented to the
Academy of Sciences were lost. A brief paper on equations in the
Bulletin de Ferussac, 1830, Vol. XIII, p. 428. gives results which seem
to be applications of a general theory. The night before the duel he
wrote his scientific testament in the form of a letter to Auguste Cheva-
lier, containing a statement of the mathematical results he had reached
and asking that the letter be published, that "Jacobi or Gauss pass
judgment, not on their correctness, but on their importance." Two
memoirs found among his papers were published by J. Liouville in
1846. Further manuscripts were published by J. Tannery at Paris
in 1908. As a rule Galois did not fully prove his theorems. It was
only with difficulty that Liouville was able to penetrate into Galois'
ideas. Several commentators worked on the task of filling out the
lacunae in Galois' exposition. Galois was the first to use the word
"group" in a technical sense, in 1830. He divided groups into simple
and compound, and observed that there is no simple group of any
composite order less than 60. The word "group" was used by A.
Cayley in 1854, by T. F. Kirkman and J. J. Sylvester in i86o.2 Galois
proved the important theorem that every invariant subgroup gives
rise to a quotient group which exhibits many fundamental properties
of the group. He showed that to each algebraic equation corresponds
a group of substitutions which reflects the essential character of the
equation. In a paper published in 1846 he established the beautiful
theorem: In order that an irreducible equation of prime degree be
solvable by radicals, it is necessary and sufficient that all its roots be

1 See life by Paul Dupuy in Annales de Vtcola normale suptrieure, 3. S., Vol. XTTT,
1896. See also E. Picard, Ocuircs math, d' fivariste Galois, Paris, 1897; J. Pierpont,
Bulletin Am. Math. Soc., 2. S., Vol. IV, 1898, pp. 332-340.

2 G. A. Miller in Am. Math. Monthly, Vol. XX, 1913, p. 18.

352 A HISTORY OF MATHEMATICS

rational in any two of them. Galois' use of substitution groups to
determine the algebraic solvability of equations, and N. H. Abel's
somewhat earlier use of these groups to prove that general equations
of degrees higher than the fourth cannot be solved by radicals, fur-
nished strong incentives to the vigorous cultivation of group theory.
It was A. L. Cauchy who entered this field next. To Galois are due
also some valuable results in relation to another set of equations,
presenting themselves in the theory of elliptic functions, viz., the
modular equations. To Cauchy has been given the credit of being
the founder of the theory of groups of finite order,1 even though funda-
mental results had been previously reached by J. L. Lagrange, Pietro
Abbati (1786-1842), P. Ruffini, N. H. Abel, and Galois. Cauchy's
first publication was in 1815, when he proved the theorem that the
number of distinct values of a non-symmetric function of degree n
cannot be less than the largest prime that divides n, without becom-
ing equal to 2. Cauchy's great researches on groups appeared in his
Exercises dj analyse et de physique mathematique, 1844, and in articles
in the Paris Comptes Rendus, 1845-1846. He did not use the term
"group," but he uses (x y z u v w) and other devices to denote sub-
stitutions, uses the terms "cyclic substitution," " order of a substitu-
tion," "identical substitution," "transposition," "transitive," "in-
transitive." In 1844 he proved the fundamental theorem (stated
but not proved by E. Galois) which is known as "Cauchy's theorem":
Every group whose order is divisible by a given prime number p must
contain at least one subgroup of order p. This theorem was later
extended by L. Sylow. A. L. Cauchy was the first to enumerate the
orders of the possible groups whose degrees do not exceed six, but this
enumeration was incomplete. At times he fixed attention on prop-
erties of groups without immediate concern as regards applications,
and thereby took the first steps toward the consideration of abstract
groups. In 1846 J. Liouville made E. Galois' researches better known
by publication of two manuscripts. At least as early as 1848 J. A.
Serret taught group theory in Paris. In 1852, Enrico Betti of the
University of Pisa published in the Annali of B. Tortolini the first
rigorous exposition of Galois' theory of equations that made the
theory intelligible to the general public. The first account of it given
in a text-book on algebra is in the third edition of J. A. Serret's Alge-
bre, 1866.

In England the earliest studies in group theory are due to Arthur
Cayley and William R. Hamilton. In 1854 A. Cayley published a
paper in the Philosophical Magazine which is usually accepted as
founding the theory of abstract groups, although the idea of abstract
groups occurs earlier ,in the papers of A. L. Cauchy, and Cayley's

1 Our account of Cauchy's researches on groups is drawn from the article of G. A.
Miller in Ribliothcca Ma'thcmatica, Vol. X, 1909-1910, pp. 317-329, and that of
Josephine E. Burns in Am. Math. Monthly, Vol. XX, 1913, pp. 141-148.

ALGEBRA 353

article is not entirely abstract. Formal definitions of abstract groups
were not given until later, by L. Kronecker (1870), H. Weber (1882),
and G. Frobenius (1887). The transition from substitution groups
to abstract groups was gradual.1 It may be recalled here that, before
1854, there were two sources from which the theory of groups of finite
order originated. In the writings of J. Lagrange, P. Ruffini, N. H.
Abel, and E. Galois it sprang from the theory of algebraic equations.
A second source is the theory of numbers; the group concept is funda-
mental in some of L. Euler's work on power residues and in some of
the early work of K. F. Gauss. It has been pointed out more recently
that the group idea really underlies geometric transformations and
is implied in Euclid's demonstrations.2 Abstract groups are con-
sidered apart from any of their applications.

A. Cayley illustrates his paper of 1854 by means of the laws of
combination of quaternion imaginaries, quaternions having been
invented by William R. Hamilton eleven years previously. In 1859
Cayley pointed out that the quaternion units constitute a group of
order 8, now known as the quaternion group, when they are multiplied
together.3 William R. Hamilton, without using the technical lan-
guage of group theory, developed in 1856, in his study of a new system
of roots of unity, the properties of the groups of the regular solids,
as generated by two operators or elements, and he proved that these
groups may be completely defined by the orders of their two generating
operators and the order of their product.

E. Picard puts the matter thus: "A regular polyhedron, say an
icosahedron, is on the one hand the solid that all the world knows; it
is also, for the analyst, a group of finite order, corresponding to the
divers ways of making the polyhedron coincide with itself. The in-
vestigation of all the types of groups of motion of finite order interests
not alone the geometers, but also the crystallographers; it goes back
essentially to the study of groups of ternary linear substitutions of
determinant +i, and leads to the thirty-two systems of symmetry
of the crystallographers for the particular complex."

In 1858 the Institute of France offered a prize for a research on
group theory which, though not awarded, stimulated research. In
1859 Emile Leonard Mathieu (1835-1890) of the University of Nancy
wrote a thesis on substitution groups, while in 1860 Camille Jordan
(1838- ) of the £cole Polytechnique in Paris contributed the first
of a series 'of papers which culminated in his great Traite des substitu-
tions, 1870. Jordan received his doctorate in 1861 in Paris; he is editor
of the Journal de mathematiques pures el appliquees. His first paper
on groups gives the fundamental theorem that the total number of

1 G. A. Miller, in Bibliotheca Mathcmatica, 3. Ed., Vol. XX, 1909-1910, p. 326.
We are making much use of Miller's historical sketch.

2 H. Poincare1 in Monist, Vol. 9, 1898, p. 34.

3 G. A. Miller in Bibliotheca Malhematica, Vol. XI, 1910-1911, pp. 314-315.

354 A HISTORY OF MATHEMATICS

substitutions of n letters which are commutative with every substitu-
tion of a regular group G on the same n letters constitute a group
which is similar to G. To Jordan is due the fundamental concept of
class of a substitution group and he proved the constancy of the
factors of composition. He also proved that there is a finite number
of primitive groups whose class is a given number greater than 3,
and that the necessary and sufficient condition that a group be solvable
is that its factors of composition are prime numbers.1 Prominent
among C. Jordan's pupils is Edmont Maillet (1865- ), editor of
L'Intermediaire des mathematiciens , who has made extensive contribu-
tions.

In Germany L. Kronecker and R. Dedekind were the earliest to
become acquainted with the Galois theory. Kronecker refers to it
in an article published in 1853 in the Berichte of the Berlin Academy.
Dedekind lectured on it in Gottingen in 1858. In 1879-1880 E. Netto
gave lectures in Strassburg. His Substitutionstheorie, 1882, was trans-
lated into Italian in 1885 by Giuseppe Battaglini (1826-1894) of the
University of Rome, and into English in 1892 by F. N. Cole, then at
Ann Arbor. The book placed the subject within easier reach of the
mathematical public.

In 1862-1863 Ludwig Sylow (1832-1918) gave lectures on substitu-
tion groups in Christiania, Norway, which were attended by Sophus
Lie. Extending a theorem given nearly thirty years earlier by A. L.
Cauchy, Sylow obtained the theorem known as "Sylow's the9rem":
Every group whose order is divisible by pm, but not by pm+l, p being a
prime number, contains i+kp subgroups of order pm. About twenty
years later this theorem was extended still further by Georg Frobenius
(1849-1917) of the University of Berlin, to the effect that the number
of subgroups is kp+i, k being an integer, even when the order of the
group is divisible by a higher power of p than pm. Sophus Lie took
a very important step by the explicit application of the group concept
to new domains and the creation of the theory of continuous groups.
Marius Sophus Lie (1842-1899) 2 was born in Nordf jordeide in Nor-
way. In 1859 he entered the University of Christiania, but not until
1868 did this slowly developing youth display marked interest in
mathematics. The writings of J. V. Poncelet and J. Pliicker awakened
his genius. In the winter of 1869-1870 he met Felix Klein in Berlin
and they published some papers of joint authorship. The summer of
1870 they were together in Paris where they were in close touch with
C. Jordan and J. G. Darboux. It was then that Lie discovered his
contact-transformation which changes the straight lines of ordinary
space over into spheres. This led him to a general theory of trans-
formation. At the outbreak ,of the Franco-Prussian war, F. Klein

1 G. A. Miller in Bibliotheca maihemalica, 3. S., Vol. X, 1909-1910, p. 323.

2 F. Engel in Bibliotheca mathematica, 3. S., Vol. I, 1900, pp. 166-204; M- Nother
in Math. Annalen, Vol. 53, pp. 1-41.

ALGEBRA 355

left Paris; S. Lie started to travel afoot through France into Italy,
but was arrested as a spy and imprisoned for a month until Darboux
was able to secure his release. In 1872 he was elected professor at the
University of Christiania, with all his time available for research. In
1871-1872 he entered upon the study of partial differential equations
of the first order, and in 1873 he arrived at the theory of transforma-
tion groups, according to which finite continuous groups are applied
to infinitesimal transformations. He considered a very general and
important kind of transformations called contact-transformations,
and their application in the theory of partial differential equations
of the first and second orders. As his group theory and theories of
integration met with no appreciation, he returned in 1876 to the study
of geometry — minimal surfaces, the classification of surfaces according
to the transformation group of their geodetic lines. The starting of
a new journal, the Archil} for Mathematik og Naturvidenskab, in 1876,
enabled him to publish his results promptly. G. H. Halphen's pub-
lications of 1882 on differential invariants induced Lie to direct at-
tention to his own earlier researches and their greater generality. In
1884 Friedrich Engel was induced by F. Klein and A. Mayer to go to
Christiania to assist Lie in the preparation of a treatise, the Theorie
der Transformations gruppen, 1888-1893. Lie accepted in 1886 a
professorship at the University of Leipzig. In 1889-1890 over-work
led to insomnia and depression of spirits. While he soon recovered
his power for work, he ever afterwards was over-sensitive and mis-
trustful of his best friends. With the aid of Engel he published in
1891 a memoir on the theory of infinite continuous transformation
groups. In 1898 he returned to Norway where he died the following
year. Lie's lectures on Dijferentialgleichungen, given in Leipzig, were
brought out in book form by his pupil, Georg Schejfers, in 1891. In
1895 F. Klein declared that Lie and H. Poincare were the two most
active mathematical investigators of the day. The following quota-
tion from an article written by Lie in 1895 indicates how his whole
soul was permeated by the group concept:1 "In this century the
concepts known as substitution and substitution group, transforma-
tion and transformation group, operation and operation group,
invariant, differential invariant, and differential parameter, appear
continually more clearly as the most important concepts of mathe-
matics. While the curve as the representation of a function of a
single variable has been the most important object of mathematical
investigation for nearly two centuries from Descartes, while on the
other hand, the concept of transformation first appeared in this
century as an expedient in the study of curves and surfaces, there
has gradually developed in the last decad.es a general theory of trans-
formations whose elements are represented by the transformation

1 Berichle d. Kocnigl. Saechs. Geselhchaft, 1895; translated by G. A. Miller in Am
Math. Monthly, Vol. Ill, 1896, p. 296.

356 A HISTORY OF MATHEMATICS

itself while the scries of transformations, in particular the transforma-
tion groups, constitute the object."

In close association with S. Lie in the advancement of group theory
and its applications was Felix Klein (1849- ). He was born at
Diisseldorf in Prussia and secured his doctorate at Bonn in 1868. After
studying in Paris, he became privat-docent at Gottingen in 1871,
professor at Erlangen in 1872, at the Technical High School in Mu-
nich in 1875, at Leipzig in 1880 and at Gottingen in 1886. He has been
active not only in the advancement of various branches of mathemat-
ics, but also in work of organization. Famous for laying out lines of
research is his Erlangen paper of 1872, Vergleichende Betrachtungen
iiber neuere geometrische Forschungen. He became member of the com-
mission on the publication of the Encyklopadie der mathematischen
Wissenschaften and editor of the fourth volume on mechanics, also
editor of Mathematische Annalen, 1877, and in 1908 president of the
International Commission on the Teaching of Mathematics. As an
inspiring lecturer on mathematics he has wielded a wide influence
upon German and American students. About 1912 he was forced by
ill-health to discontinue his lectures at Gottingen, but in 1914 the ex-
citement of the war roused him to activity, much as J. Lagrange was
aroused at the outbreak of the French Revolution, and Klein resumed
lecturing. He has constantly emphasized the importance of both
schools of mathematical thought, namely, the intuitional school, and
the school that rests everything on abstract logic. In his opinion,
"the intuitive grasp and the logical treatment should not exclude,
but should supplement each other."

S. Lie's method of treating differential invariants was further in-
vestigated by K. Zorawski-in Acta Math., Vol. XVI, 1892-1893. In
1902 C. N. Haskins determined the number of functionally independ-
ent invariants of any order, while A. R. Forsyth obtained the invariants
for ordinary Euclidean space. Differential parameters have been
investigated by J. Edmund Wright of Bryn Mawr College.1 Lie's
theory of invariants of finite continuous groups was attacked on logi-
cal grounds by E. Study of Bonn, in 1908. The validity of this criti-
cism was partly admitted by F. Engel.

Another method of treating differential invariants, originally due
to E. B. Christoffel, has been called by G. Ricci and T. Levi-Civita
of Padua "covariant derivation," (Mathematische Annalen, Vol. 54,
1901). A third method was introduced by H. Maschke 2 who used a
symbolism similar to that for algebraic invariants.

Henry W. Stager published in 1916 A Sylow Factor Table for the first
Twelve Thousand Numbers: For every number up to 1200 the divisors
of the form />(£/>+ 1) are given, where p is a prime greater than 2 and

1 We are using J. E. Wright's Invariants of Quadratic Differential Forms, 1908,
pp. 5-8.

z Trans. Am. Math. Soc., Vol. i, 1900, pp. 197-204.

ALGEBRA 357

k is a positive integer. These divisors aid in the determination of
the number of Sylow subgroups.

Solvable (H. Weber's " metacyclic ") groups have been studied by
G. Frobenius who proved that every group of composite order that
is not divisible by the square of a prime number must be compound,
and that all these groups are solvable, for the orders of their self-
conjugate subgroups and of their quotient groups cannot be divisible
by the square of a prime number.1 The study of solvable groups has
been pursued also by I.. Sylow, W. Burnside, R. Dedekind (who inves-
tigated what he called the Hamiltonian group), and G. A. Miller who
with Frobenius developed about 1893-1896 elegant methods for prov-
ing the solvability of a given group. In 1895 O- Holder enumerated all
the insolvable groups whose order does not exceed 479. In 1898 G. A.
Miller gave the numbers of all primitive solvable groups whose degree
is less than 25, also the number of insolvable groups which may be
represented as substitution groups whose degree is less than 12. G. A.
Miller (1899) and Umberto Scarpio (1901) of Verona considered
properties of commutators and commutator subgroups, and proved
that the question of solvability can be decided by means of commuta-
tor subgroups.2 Commutator groups have been studied also by W. B.
Fite and Ernst Wendt. The characteristics of non-abelian groups were
investigated since 1896 by G. Frobenius in Berlin, the characteristics
of abelian groups having been already employed by J. Lagrange and
P. Dirichlet. Characteristics of solvable groups were studied in
1901 by Frobenius. An enumeration of abstract groups was made in
1901 by R. P. Le Vavasseur of Toulouse. The list of intransitive
substitution groups of degree eleven was shown by G. A. Miller and
G. H. Ling in 1901 to include 1492 distinct substitution groups,
which is about 500 more than the number of degree ten. H. L. Rietz
proved that a primitive group of degree n and order g contains more
than g/x + i substitutions of degree less than n, x being the number of
transitive constituents in a maximal subgroup of degree n—i. This
result is closely related to investigations of C. Jordan, A. Bochert,
and E. Maillet on the class of a primitive group.3

The definition of a group was simplified in 1902 by E. V. Huntington
of Harvard University. He pointed out that the usual definition,
as given for instance in H. Weber's Algebra, contains several redun-
dancies, that only three postulates (four for finite groups) are neces-
sary, the independence of which he established.4 Later discussions of
definitions are due to Huntington and E. H. Moore.

1 See G. A. Miller's "Report of Recent Progress in the Theory of Groups of a
Finite Order" in Bull. Am. Math. Soc., Vol. 5, 1899, pp. 227-249, which we are
using.

2G. A. Miller, "Second Report on Recent Progress in the Theory of Groups of
Finite Order" in Bull. Am. Math. Soc., Vol. 9, 1902, p. 108.

3 Loc. tit., p. 118.

4 Bull. Am. Math. Soc., Vol. 8, 1902, p. 296.

35» A HISTORY OF MATHEMATICS

L. E. Dickson 1 said in 1900: "When a problem has been exhibited
in group phraseology, the possibility of a solution of a certain char-
acter or the exact nature of its inherent difficulties is determined by a
study of the group of the problem. ... As the chemist analyzes
a compound to determine the ultimate elements composing it, so the
group-theorist decomposes the group of a given problem into a chain
of simple groups. . . . Much labor has been expended in the de-
termination of simple groups. For continuous groups of a finite
number of parameters, the problem has been completely solved by
W. Killing and E. J. Cartan (1894), with the result that all such simple
groups, aside from five isolated ones, belong to the systems investi-
gated by Sophus Lie, viz., the general projective group, the pro-
jective group of a linear complex, and the projective group leaving
invariant a non-degenerate surface of the second order. The cor-
responding problem for infinite continuous groups remains to be
solved. With regard to finite simple groups, the problem has been
attacked in two directions. O. Holder,2 F. N. Cole,3 W. Burnside,4
G. H. Ling, and G. A. Miller have shown that the only simple groups
of composite orders less than 2000 are the previously known simple
groups of orders 60, 168, 360, 504, 660, 1092. On the other hand,
various infinite systems of finite simple groups have been determined.
The cyclic groups of prime orders and the alternating group of n
letters (n>4) have long been recognized as simple groups. The other
known systems of finite simple groups have been discovered in the
study of linear groups. Four systems were found by C. Jordan,
(Traite des substitutions} in his study of the general linear, the abelian,
and the two hypoabelian groups, the field of reference being the set
of residues of integers with respect to a prime modulus p. Generaliza-
tions may be made by employing the Galois field of order pn (desig-
nated GF [pn\), composed of the pn Galois complexes formed with a
root of a congruence of degree n irreducible modulo p. Groups of
linear substitutions in a Galois field were studied by E. Betti, E.
Mathieu, and C. Jordan; but the structure of such groups has been'
determined only in the past decade. The simplicity of the group of
unary linear fractional substitutions in a Galois field was first proved
by E. H. Moore (Bulletin Am. Math. Soc., Dec., 1893) and shortly
afterward by W. Burnside. The complete generalization of C. Jor-
dan's four systems of simple groups and the determination of three
new triply-infinite systems have been made by the writer" (i. e. by

1 See L. E. Dickson in Compte vendn du II. Congr. intern., Paris, 1900. Paris,
1902, pp. 225, 226.

2 O. Holder proved in Math. Annalen, 1892, that there are only two simple groups
of composite order less than 200. viz., those of order 60 and 168.

3 F. N. Cole in Am. Jour. Math., 1893 found that theie could be only three such
groups between orders 200 and 661, viz., of orders 360, 504, 660.

4 W. Burnside showed that there was only one simple group of composite order
between 661 and 1092.

ALGEBRA 359

L. E. Dickson in 1896). Aside from the cyclic and alternating groups,
the known systems of finite simple groups have been derived as quo-
tient-groups in the series of composition of certain linear groups.
Miss I. M. Schottenfels of Chicago showed that it is possible to con-
struct two simple groups of the same order.

The determination of the smallest degree (the "class") of any of the
non-identical substitutions of primitive groups which do not include
the alternating group was taken up by C. Jordan and has been called
"Jordan's problem." It was continued by Alfred Bochert of Breslau
and E. Maillet. Bochert proved in 1892: If a substitution group of
degree n does not include the alternating group and is more than
simply transitive, its class exceeds \n — i, if it is more than doubly
transitive its class exceeds %n — i, and if it is more than triply transi-
tive its class is not less than %n—i. E. Maillet showed that when
the degree of a primitive group is less than 202 its class cannot be ob-
tained by diminishing the degree by unity unless the degree is a power
of a prime number. In 1900 W. Burnside proved that every transitive
permutation group in p symbols, p being prime, is either solvable
or doubly transitive.

As regards linear groups, G. A. Miller wrote in 1899 as follows:
"The linear groups are of extreme importance on account of their
numerous direct applications. Every group of a finite order can
clearly be represented in many ways as a linear substitution group
since the ordinary substitution (permutation) groups are merely
very special cases of the linear groups. The general question of rep-
resenting such a group with the least number of variables seems to
be far from a complete solution. It is closely related to that of de-
termining all the linear groups of a finite order that can be represented
with a small number of variables. Klein was the first to determine all
the finite binary groups (in 1875) while the ternary ones were con-
sidered independently by C. Jordan (1880) and H. Valentiner (1889).
The latter discovered the important group of order 360 which was
omitted by Jordan and has recently been proved (by A. Wiman of
Lund) simply isomorphic to the alternating group of degree 6. H.
Maschke has considered many quaternary groups and established,
in particular, a complete form system of the quaternary group of
51840 linear substitutions." Heinrich Maschke (1853-1908) was born
in Breslau, studied in Berlin under K. Weierstrass, E. E. Kummer,
and L. Kronecker, later in Gottingen under H. A. Schwarz, J. B.
Listing, and F. Klein. He entered upon the study of group theory
under Klein. In 1891 he came to the United States, worked a year
with the Weston Electric Co., then accepted a place at the University
of Chicago.

Linear groups of finite order, first treated by Felix Klein, were
later used by him in the extension of the Galois theory of algebraic
equations, as seen in his Ikosaeder. As stated above, Klein's de-

360 A HISTORY OF MATHEMATICS

termination of the linear groups in two variables was followed by
groups in three variables, developed by C. Jordan and H. Valentiner
(1889), and by groups of any number of variables, treated by C.
Jordan. Special linear groups in four variables were discussed by E.
Goursat (1889) and G. Bagnera of Palermo (1905). The complete
determination of the groups in four variables, aside from intransitive
and monomial types, was carried through by H. F. Blichfeldt of Le-
land Stanford University.1 Says Blichfeldt: "There are, in the main,
four distinct principles employed in the determination of the groups
in 2, 3 or 4 variables: (a) the origin'al geometrical process of Klein . . . ;
(b) the processes leading to a diophantine equation, which may be
approached analytically (C. Jordan . . .), or geometrically (H. Valen-
tiner, G. Bagnera, H. H. Mitchell) ; (c) a process involving the relative
geometrical properties of transformations which represent ' homologies '
and like forms (H. Valentiner, G. Bagnera, H. H. Mitchell . . . ; (d)
a process developed from the properties of the multipliers of the trans-
formations, which are roots of unity (H. F. Blichfeldt). A new prin-
ciple has been added recently by L. Bieberbach, though it had already
been used by H. Valentiner in a certain form. . . . Independent of
these principles stands the theory of group characteristics, of which
G. Frobenius is the discoverer."

There is a marked difference between finite groups of even and of
odd order.2 As W. Burnside points out, the latter admit no self-
inverse irreducible representation, except the identical one; all irre-
ducible groups of odd order in 3, 5 or 7 symbols are soluble. G. A.
Miller proved in 1901 that no group of odd order with a conjugate
set of operations containing fewer than 50 members could be simple.
W. Burnside proved in 1901 that transitive groups of odd order whose
degree is less than 100 are soluble. H. L. Rietz in 1904 extended this
last result to groups whose degrees are less than 243. W. Burnside
has shown that the number of prime factors in the order of a simple
group of odd order cannot be less than 7 and that 40,000 is a lower
limit for the order 'of a group of odd degree, if simple. These results
suggest that, perhaps, simple groups of odd order do not exist. Recent
researches on groups, mainly abstract groups, are due to L. E. Dickson,
Le Vavasseur, M. Potron, L. I. Neikirk, G. Frobenius, H. Hilton,
A. Wiman, J. A. de Seguier, H. W. Kuhn, A. Loewy, H. F. Blichfeldt,3
W. A. Manning, and many others. Extensive researches on abstract
groups have been carried on by G. A. Miller of the University of
Illinois. In 1914 he showed, for instance, that a non-abelian group
can have an abelian group of isomorphisms by proving the existence

1 We are using H. F. Blichfeldt's Finite Collineation Groups, Chicago, 1917,
pp. I74-I77-

2 W. Burnside, Theory of Groups of Finite Order, 2. Ed., Cambridge, 1911, p. 503.

3 Consult G. A. Miller's "Third Report on Recent Progress in the Theory of
Groups of Finite Order" in Bull. Am. Math. Soc., Vol. 14, 1907, p. 124.

ALGEBRA 361

ol this relation in a group of order 64. He proved the existence of a
group of order p9, p being any prime number whatever, whose group
of isomorphisms has an order which is a power 1 of p. He proved also
the existence of a group G of order 128 which admits of an outer
isomorphism which changes each conjugate set of operations into
itself. Among other results due to G. A. Miller are these: The number
of independent generators of every prime power group is an invariant
of the group; a necessary and sufficient condition that a solvable
group is a direct product of a Sylow subgroup and another subgroup
is that its group of inner isomorphisms involves the corresponding
Sylow subgroup as a factor of a direct product, whenever it involves
such a subgroup.2

A work which embodied modern researches in algebra was the
Lchrbuch der Algebra, issued by H. Weber in 1895-1896 in two volumes,
and in three volumes in the revised edition of 1898 and 1899. Hein-
rich Weber (1842-1913) was born in Heidelberg and studied at
Heidelberg, Leipzig, and Konigsberg. Since 1869 he was successively
professor at Heidelberg, Konigsberg, Berlin, Marburg, Gottingen
and (since 1895) at Strassburg. He was editor of Riemann's Collected
Works (1876; 2. ed. 1892). He carried on researches in algebra, theory
of numbers, theory of functions, mechanics and mathematical physics.
In 1911 he mourned the loss of a gifted daughter who had trans-
lated Poincare's Valeur de la science and other French books into
German.

The symmetric functions of the sums of powers of the roots of an
equation, studied by I. Newton and E. Waring, was considered more
recently by K. F. Gauss, A. Cayley, J. J. Sylvester, and F. Brioschi.
Cayley gives rules for the "weight" and "order" of symmetric func-
tions.

The theory of elimination was greatly advanced by J. J. Sylvester,
A. Cayley, G. Salmon, C. G. J. Jacobi, L. O. Hesse, A. L. Cauchy,
E. Brioschi, and P. Gordan. Sylvester gave the dialytic method
(Philosophical Magazine, 1840), and in 1852 established a theorem
relating to the expression of an eliminant as a determinant. A. Cayley
made a new statement of Bezout's method of elimination and estab-
lished a general theory of elimination (1852).

Contributions to the theory of equations, based on Descartes' rule
of signs and especially on its application to infinite series were made
by Edmond Laguerre (1834-1886), professor in the College de France
in Paris. An upper limit for the number of real roots of a polynomial
with real coefficients, f(x), in an interval (o, a) results from the ap-
plication of the rule of signs to a product f2(x)=fi(x) /(#) developed
in a power series which converges for | x \<a, but diverges for x=a.
In particular, he proved that if z in eixf(x) is taken sufficiently large,

1 Bull. Am. Math. Soc., Vol. 20, 1914, pp. 310, 311.

2 Ibid, Vol. 18, 1912, p. 440.

362 A HISTORY OF MATHEMATICS

then the exact number of positive roots is ascertainable from the
variations of sign in the series. Michel Fekete and Georg Polya,
both of Budapest, use/(a;)/(i— x)n for the same purpose.1

The theory of equations commanded the attention of Leopold
Kronecker (1823-1891). He was born in Liegnitz near Breslau,
studied at the gymnasium of his native town under Kummer, later
in Berlin under C. G. J. Jacobi, J. Steiner, and P. Dirichlet, then in
Breslau again under E. E. Kummer. Though for eleven years after
1844 engaged in business and the care of his estates, he did not neglect
mathematics, and his fame grew apace. In 1855 he went to Berlin
where he began to lecture at the University in 1861. He was a very
stimulating and interesting lecturer. Kummer, K. Weierstrass, and
L. Kronecker constitute the triumvirate of the second mathematical
school in Berlin. This school emphasized severe rigor in demonstra-
tions. L. Kronecker dwelt intensely upon arithmetization wrhich
repressed as far as possible all space representations and rested solely
upon the concept of number, particularly the positive integer. He
displayed manysided talent and extraordinary ability to penetrate
new fields of thought. "But," says G. Frobenius,2 "conspicuous as
his achievements are in the different fields of number research, he
does not quite reach up to A. L. Cauchy and C. G. J. Jacobi in analy-
sis, nor to B. Riemann and Weierstrass in function- theory, nor to
Dirichlet and Kummer in number- theory." Kronecker's papers on
algebra, the theory of equations and elliptic functions proved to be
difficult reading. A more complete and simplified exposition of his
results was given by R. Dedekind and H. Weber. "Among the finest
of Kronecker's achievements," says Fine,3 "were the connections
which he established among the various disciplines in which he worked:
notably that between the theory of quadratic forms of negative deter-
minant and elliptic functions, through the singular moduli which give
rise to the complex multiplication of the elliptic functions, and that
between the theory of numbers and algebra, by his arithmetical
theory of the algebraic equation." He held to the view that the theory
of fractional and irrational numbers could be built upon the integral
numbers alone. "Die ganze zahl," said he, "schuf der liebe Gott,
alles Uebrige ist Menschenwerk." Later he even denied the existence
of irrational numbers. He once paradoxically remarked to Linde-
mann: "Of what use is your beautiful research on the number TT?
Why cogitate over such problems, when really there are no irrational
numbers whatever? "

In 1890-1891 L. Kronecker developed a theory of the algebraic equa-
tion with numerical coefficients, which he did not live to publish.
From notes of Kronecker's lectures, H. B. Fine of Princeton prepared

1 Bull. Am. Math. Soc., Vol. 20, 1913, p. 20.

1 G. Frobenius, Getfdchtnissrede auf Leopold Kronecker, Berlin, 1893, p. i.

3 Bull. Am. Math. Soc., Vol. I, 1892, p. 175.

ALGEBRA 363

an address in 1913 giving Kronecker's unpublished results.1 "All
who have read Kronecker's later writings," says Fine, "are familiar
with his contention that the theory of the algebraic equation in its
final form must be based solely on the rational integer, algebraic
numbers being excluded and only such relations and operations being
admitted as can be expressed in finite terms by means of rational
numbers and therefore ultimately by means of integers. These lec-
tures of 1890-91 are chiefly concerned with the development of such
a theory, and in particular with the proof of two theorems which
therein take the place of the fundamental theorem of algebra as
commonly stated."

Solution of Numerical Equations

Jacques Charles Franqois Sturm (1803-1855), a native of Geneva,
Switzerland, and the successor of Poisson in the chair of mechanics
at the Sorbonne, published in 1829 his celebrated theorem determining
the number and situation of real roots of an equation comprised
between given limits. De Morgan has said that this theorem "is the
complete theoretical solution of a difficulty upon which energies of
every order have been employed since the time of Descartes." Sturm
explains in that article that he enjoyed the privilege of reading Four-
ier's researches while they were still in manuscript and that his own
discovery was the result of the close study of the principles set forth
by Fourier. In 1829 Sturm published no proof. Proofs were given
in 1830 by Andreas von Ettinghausen (1796-1878) of Vienna, in
1832 by Charles Choquet et Mathias Mayer in their Algebre, and in
1835 by Sturm himself. According to J. M. C. Duhamel, Sturm's
discovery was not the result of observation, but of a well-ordered
line of thought as to the kind of function that would meet the re-
quirements. According to J. J. Sylvester, the theorem "stared him
(Sturm) in the face in the midst of some mechanical investigations
connected with the motion of compound pendulums." Duhamel and
Sylvester both state that they received their information from Sturm
directly. Yet their statements do not agree. Perhaps both statements
are correct, but represent different stages in the evolution of the dis-
covery in Sturm's mind.2

By the theorem of Sturm one can ascertain the number of complex
roots, but not their location. That limitation was removed in a bril-
liant research by another great Frenchman, A. L. Cauchy. He dis-
covered in 1831 a general theorem which reveals the number of roots,
whether real or complex, which lie within a given contour. This
theorem makes heavier demands upon the mathematical attainments

1 Bull. Am. Math. Soc., Vol. 20, 191.4, p. 339.

2 Consult also M. Bdcher, "The published and unpublished Work of Charles
Sturm on algebraic and differential Equations" in Bull. Am. Math. Soc., Vol. 18,
1912, pp. 1-18.

364 A HISTORY OF MATHEMATICS

of the reader, and for that reason has not the celebrity of Sturm's
theorem. But it enlisted the lively interest of men like Sturm, J.
Liouville, and F. Moigno.

A remarkable article was published in 1826 by Germinal Dandelin
(1794-1847) in the memoirs of the Academy of Sciences of Brussels.
He gave the conditions under which the Newton-Raphson method of
approximation can be used with security. In this part of his research
he was anticipated by both Mourraille and J. Fourier. In another
part of his paper (the second supplement) he is more fortunate; there
he describes a new and masterly device for approximating to the roots
of an equation, which constitutes an anticipation of the famous
method of C. H. Graffe. We must add here that the fundamental
idea of Graff e's method is found even earlier, in the Miscellanea
analytica, 1762, of Edward Waring. If a root lies between a and b,
a—b<i, and a is on the convex side of the curve, then Dandelin
puts x=a+y and transforms the equation into one whose root y is
small. He then multiplies f(y) by /( — y) and obtains, upon writing
y2=z, an equation of the same degree as the original one, but whose
roots are the squares of the roots of the equation f(y)=o. He remarks
that this transformation may be repeated, so as to get the fourth,
eighth, and higher powers, whereby the moduli of the powers of the
roots diverge sufficiently to make the transformed equation separable
into as many polygons as there are roots of distinct moduli. He ex-
plains how the real and imaginary roots can be obtained. Dandelin's
research had the misfortune of being buried in the ponderous tomes
of a royal academy. Only accidentally did we come upon this antici-
pation of the method of C. H. Graffe. Later the Academy of Sciences
of Berlin offered a prize for the invention of a practical method of
computing imaginary roots. The prize was awarded to Carl Heinrich
Graffe (1799-1873), professor of mathematics in Zurich, for his paper,
published in 1837 in Zurich, entitled, Die Auflosung der hoheren
numerischen Gleichungen. This contains the famous " Graffe method,"
to which reference has been made. Graffe proceeds from the same
principle as did Moritz Abraham Stern (1807-1894), of Gottingen in
the method of recurrent series* and as did Dandelin. By the process
of involution to higher and higher powers, the smaller roots are caused
to vanish in comparison to the larger. The law by which the new
equations are constructed is exceedingly simple. If, for example,
the coefficient of the fourth term of the given equation is a3, then
the corresponding coefficient of the first transformed equation is
a?<— 20004+20105— 2ae. In the computation of the new coefficients,
Graffe uses logarithms. By this remarkable method all the roots,
both real and imaginary, are found simultaneously, without the
necessity of determining beforehand the number of real roots and
the location of each root. The discussion of the case of equal imaginary
roots, omitted by Graffe, was taken up by the astronomer J. F. Encke

ALGEBRA 365

in 1841. A simplified exposition of the Dandelin-Graffe method was
given by Emmanuel Carvallo in 1896; it resembles in some parts that
of Dandelin, although Carvallo had not seen Dandelin's paper. For
didactic purposes, an able explanation is given in Gustav Bauer's
Vorlesungen iiber Algebra, 1903.

In 1860 E. Fiirstenau expressed any definite real root of an alge-
braic equation with numerical or literal coefficients, in terms of its
coefficients, through the aid of infinite determinants, a kind of de-
terminant then used for the first time. In 1867 he extended his results
to imaginary roots. The approximation is made to depend upon the
fact used by Daniel Bernoulli, L. Euler, J. Fourier, M. A. Stern, G.
Dandelin, and C. H. Graffe, that high powers of the smaller roots are
negligible in comparison with high powers of the greater roots. E.
Flirstenau's process was elaborated by E. Schroder (1870), Siegmund
Giinther, (1874), and Hans Naegelsbach (1876).

Worthy of notice is "Weddle's method" of solving numerical
equations, devised by Thomas Weddle (1817-1853) of Newcastle in
England, in 1842. It is kindred to that of W. G. Homer. The suc-
cessive approximations are effected by multiplications instead of
additions. The method is advantageous when the degree of the
equation is high and some of the terms are missing. It has received
some attention in Italy and Germany. In 1851 Simon Spitzer ex-
tended it to the computation of complex roots.

The solution of equations by infinite series which was a favorite
subject of research during the eighteenth century (Thomas Simpson,
L. Euler, J. Lagrange, and others), received considerable attention
during the nineteenth. Among the early workers were C. G. J.
Jacobi (1830), W. S. B. Woolhouse (1868), O. Schlomilch (1849),
but none of their devices were satisfactory to the practical computer.
Later writers aimed at the simultaneous calculation of all the roots
by infinite series. This was achieved for a three-term equation by
R. Dietrich in 1883 and by P. Nekrasoff in 1887. For the general
equation it was accomplished in 1895 by Emory McClintock (1840-
1916), an actuary in New York, who was president of the American
Mathematical Society from 1890 to 1894. He used a series derived
by his Calculus of Enlargement, but which may be derived also by
applying "Lagrange's series." 'A prominent part in McClintock's
treatment is his theory of "dominant" coefficients, which theory lacks
precision, inasmuch as no criterion is given to ascertain whether a
coefficient is dominant or not, which is both necessary and sufficient.
Preston A. Lambert of Lehigh University used Maclaurin's series in
1903; in 1908 he paid special attention to convergency conditions,
pointing out that the conditions for a /-term equation can be set up
when those of a (t— i)-term equation are known. In Italy Lambert's
papers were studied in 1906 by C. Rossi and in 1907 by Alfredo
Capelli (1855-1910) of Naples. These recent researches of American

366 A HISTORY OF MATHEMATICS

and Italian mathematicians have placed the determination of real
and imaginary roots of numerical equations by the method of infinite
series within reach of the practical computer. The methods them-
selves indicate the number of real and imaginary roots, so that one
can dispense with the application of Sturm's theorem here just as
easily as one can in the Dandelin-Graffe method. Considerable at-
tention has been given to the solution of special types — trinomial
equations — by G. Dandelin (1826), K. F. Gauss (1840, 1843), J.
Bella vitis (1846), Lord John M'Laren (1890). The last three used
logarithms of sums and differences, which were first suggested by
G. Z. Leonelli in 1802 and are often called "Gaussian logarithms."
The extension of the Gaussian method to quadrinomials was under-
taken by S. Gundelfinger in 1884 and 1885, Carl Faerber in 1889, and
Alfred Wiener in 1886. The extension of the Gaussian method to
any equation was taken up by R. Mehmke, professor in Darmstadt,
who published in 1889 a logarithmic-graphic method of solving nu-
merical equations, and in 1891 a more nearly arithmetical method of
solution by logarithms. The method is essentially a mixture of the
Newton-Raphson method and the regula falsi, as regards its theoretical
basis. Well known is R. Mehmke's article on methods of computation
in the Encyklopadie der mathematischen Wissenschaften, Vol. i, p. 938.

Magic Squares and Combinatory Analysis

The latter part of the nineteenth century witnesses a revival of
interest in methods of constructing magic squares. Chief among the
writers on this subject are J. Horner (1871), S. M. Drach (1873),
Th. Harmuth (1881), W. W. R. Ball (1893); E. Mafflet (1894), E. M.
Laquiere (1880), E. Lucas (1882), E. McClintock (I897).1 Magic
squares of the "diabolic" type, as Lucas calls them, are designated
" pandiagonal " by McClintock. These and similar forms are called
"Nasik squares" by A. H. Frost. An interesting book, Magic Squares
and Cubes, Chicago, 1908, was prepared by the American electrical
engineer, W. S. Andrews. Still more recent is the Combinatory Anal-
ysis, Vol. I, Cambridge, 1915, Vol. II, 1916, by P. A. MacMahon,
which touches the subject of magic squares. Says MacMahon: "In
fact, the whole subject of Magic Squares and connected arrange-
ments of numbers appears at first sight to occupy a position which is
completely isolated from other departments of pure mathematics.
The object of Chapters II and III is to establish connecting links
where none previously existed. This is accomplished by selecting
a certain differential operation and a certain algebraical function,"
I, p. VIII.

" The ' Probleme des Rencontres ' . . . can be discussed in the same
manner. The reader will be familiar with the old question of the

1 Encyclopedic des sciences malhim. T. I, Vol. 2, 1906, pp. 67-75.

ANALYSIS 367

letters and envelopes. A given number of letters are written to dif-
ferent persons and the envelopes correctly addressed but the letters
are placed at random in the envelopes. The question is to find the
probability that not one letter is put into the right envelope. The
enumeration connected with this probability question is the first
step that must be taken in the solution of the famous problem of the
Latin Square," I, p. IX.

The problem of the Latin Square: "The question is to place n dif-
ferent letters a, b, c, . . . in each row of a square of «2 compartments
in such wise that, one letter being in each compartment, each column
involves the whole of the letters. The number of arrangements is
required. The question is famous because, from the time of Euler
to that of Cayley inclusive, its solution was regarded as being beyond
the powers of mathematical analysis. It is solved without difficulty
by the method of differential operators of which we are speaking.
In fact it is one of the simplest examples of the method which is shewn
to be capable of solving questions of a much more recondite charac-
ter." 1

The extension of the principle of magic squares of the plane to three-
dimensional space has commanded the attention of many. Most
successful in this field were the Austrian Jesuit Adam Adamandus
Kochansky (1686), the Frenchman Josef Sauveur (1710), the Germans
Th. Hugel, (1850) and Hermann Scheffler (1882).

In Vol. II, Major MacMahon gives a remarkable group of identities
discovered by S. Ramanujan of Cambridge which have applications
in the partitions of numbers, but have not yet been established by
rigorous demonstration.

Analysis

Under this head we find it convenient to consider the subjects of
the differential and integral calculus, the calculus of variations, in-
finite series, probability, differential equations and integral equations.

An early representative of the critical and philosophical school of
mathematicians of the nineteenth century was Bernard Bolzano
(1781-1848), professor of the philosophy of religion at Prague. In
1816 he gave a proof of the binomial formula and exhibited clear
notions on the convergence of series. He held advanced views on
variables, continuity and limits. He was a forerunner of G. Cantor.
Noteworthy is his posthumous tract, Paradoxien des Unendlichen
(Preface, 1850), edited by his pupil, Fr. Prihonsky. Bolzano's writings
were overlooked by mathematicians until H. Hankel called attention
to them. "He has everything," says Hankel, "that can place him
in this respect [notions on infinite series] on the same level with Cauchy,
only not the art peculiar to the French of refining their ideas and
communicating them in the most appropriate and taking manner.
1 P. A. MacMahon, Combinatoty Analysis, Vol. I, Cambridge, 1915, p. ix.

368 A HISTORY OF MATHEMATICS

So it came about that Bolzano remained unknown and was soon for-
gotten." H. A. Schwarz in 1872 looked upon Bolzano as the inventor
of a line of reasoning further developed by K. Weierstrass. In 1881
O. Stolz declared that all of Bolzano's writings are remarkable
"inasmuch as they start with an unbiassed and acute criticism of the
contributions of the older literature." 1

A reformer of our science who was eminently successful in reaching
the ear of his contemporaries was Cauchy.

Augustin-Louis Cauchy2 (1789-1857) was born in Paris, and re-
ceived his early education from his father. J. Lagrange and P. S.
Laplace, with whom the father came in frequent contact, foretold
the future greatness of the young boy. At the £cole Centrale du
Pantheon he excelled in ancient classical studies. In 1805 he entered
the Ecole Polytechnique, and two years later the Ecole des Fonts et
Chaussees. Cauchy left for Cherbourg in 1810, in the capacity of
engineer. Laplace's Mecanique Celeste and Lagrange's Fonctions
Analytiques were among his book companions there. Considerations
of health induced him to return to Paris after three years. Yielding
to the persuasions of Lagrange and Laplace, he renounced engineering
in fayor of pure science. We find him next holding a professorship at
the Ecole Polytechnique. On the expulsion of Charles X, and the
accession to the throne of Louis Philippe in 1830, Cauchy, being
exceedingly conscientious, found himself unable to take the oath de-
manded of him. Being, in consequence, deprived of his positions, he
went into voluntary exile. At Fribourg in Switzerland, Cauchy re-
sumed his studies, and in 1831 was induced by the king of Piedmont
to accept the chair of mathematical physics, especially created for him
at the University of Turin. In 1833 he ob'eyed the call of his exiled
king, Charles X, to undertake the education of a grandson, the Duke
of Bordeaux. This gave Cauchy an opportunity to visit various parts
of Europe, and to learn how extensively his works were being read.
Charles X bestowed upon him the title of Baron. On his return to
Paris in 1838, a chair in the College de France was offered to him,
but the oath demanded of him prevented his acceptance. He was
nominated member of the Bureau of Longitude, but declared ineligible
by the ruling power: During the political events of 1848 the oath was
suspended, and Cauchy at last became professor at the Polytechnic
School. On the establishment of the second empire, the oath was re-
instated, but Cauchy and D. F. J. Arago were exempt from it. Cauchy
was a man of great piety, and in two of his publications staunchly de-
fended the Jesuits.

Cauchy was a prolific and profound mathematician. By a prompt
publication of his results, and the preparation of standard text-books,
he exercised a more immediate and beneficial influence upon the great

1 Consult H. Bergman, Das Philosophischc Wcrk Bernard Bolzanos, Halle, 1909.

2 C. A. Valson, La Vie el les Iravaux du Baron Cauchy, Paris, 1868.

ANALYSIS 369

mass of mathematicians than any contemporary writer. He was one
of the leaders in infusing rigor into analysis. His researches extended
over the field of series, of imaginaries, theory of numbers, differential
equations, theory of substitutions, theory of functions, determinants,
mathematical astronomy, light, elasticity, etc., — covering pretty
much the whole realm of mathematics, pure and applied.

Encouraged by P. S. Laplace and S. D. Poisson, Cauchy published in
1821 his Cours d' 'Analyse de VEcole Roy ale Poly technique, a work of
great merit. Had it been studied more diligently by writers of text-
books, many a lax and loose method of analysis long prevalent in ele-
mentary text-books would have been discarded half a century earlier.
With him begins the process of "arithmetization." He made the
first serious attempt to give a rigorous proof of Taylor's theorem.
He greatly improved the exposition of fundamental principles of the
differential calculus by his mode of considering limits and his new
theory on the continuity of functions. Before him, the limit concept
had been emphasized in France by D'Alembert, in England by I.
Newton, J. Jurin, B. Robins, and C. Maclaurin. The method of
Cauchy was accepted with favor by J. M. C. Duhamel, G. J. Hoiiel,
and others. In England special attention to the clear exposition of
fundamental principles was given by A. De Morgan ^jCauchy re-
introduced the concept of an integral of a function as the limit of a
sum, a concept originally due to G. W. Leibniz, but for a time dis-
placed by L. Euler's integral defined as the result of reversing differen-
tiation.

Calculus of Variations

A. L. Cauchy made some researches on the calculus of variations.
This subject had long remained in its essential principles the same as
when it came from the hands of J. Lagrange. More recent studies per-
tain to the variation of a double integral when the limits are also vari-
able, and to variations of multiple integrals in general. Memoirs were
published by K. F. Gauss in 1829, S. D. Poisson in 1831, and Michel
Ostrogradski (1801-1861) of St. Petersburg in 1834, without, however,
determining in a general manner the number and form of the equations
which must subsist at the limits in case of a double or triple integral.
In 1837 C. G. J. Jacobi published a memoir, showing that the difficult
integrations demanded by the discussion of the second variation, by
which the existence of a maximum or minimum can be ascertained,
are included in the integrations of the first variation, and thus are
superfluous. This important theorem, presented with great brevity
by C. G. J. Jacobi, was elucidated and extended by V. A. Lebesgue,
C. E. Delaunay, Friedrich Eisenlohr (1831-1904), Simon Spitzer (1826-
1887) of Vienna, L. O. Hesse, and R. F. A. Clebsch. A memoir by
Pierre Frederic Sarrus (1798-1861) of the University of Strasbourg on
the question of determining the limiting equations which must be com-

370 A HISTORY OF MATHEMATICS

bined with the indefinite equations in order to determine completely
the maxima and minima of multiple integrals, was awarded a prize by
the French Academy in 1845, honorable mention being made of a
paper by C. E. Delaunay. P. F. Sarrus's method was simplified by
A. L. Cauchy. In 1852 Gaspare Mainardi (1800-1879) of Pavia at-
tempted to exhibit a new method of discriminating maxima and
minima, and extended C. G. J. Jacobi's theorem to double integrals.
Mainardi and F. Brioschi showed the value of determinants in ex-
hibiting the terms of the second variation. In 1861 Isaac Todhunter
(1820-1884) of St. John's College, Cambridge, published his valuable
work on the History of the Progress of the Calculus of Variations, which
contains researches of his own. In 1866 he published a most important
research, developing the theory of discontinuous solutions (discussed
in particular cases by A. M. Legendre), and doing for this subject what
P. F. Sarrus had done for multiple integrals.

The following are the more important older authors of systematic
treatises on the calculus of variations, and the dates of publication:
Robert Woodhouse, Fellow of Caius College, Cambridge, 1810;
Richard Abbatt in London, 1837; John Hewitt Jellett (1817-1888),
once Provost of Trinity College, Dublin, 1850; Georg Wilhelm Strauch
(1811-1868), of Aargau in Switzerland, 1849; Francois Moigno (1804-
1884) of Paris, and Lorentz Leonard Lindelof (1827-1908) of the
University of Helsingfors, in 1861; Lewis Buffett Carll in 1881.
Carll (1844-1918), was a blind mathematician, graduated at Co-
lumbia College in 1870 and in 1891-1892 was assistant in mathematics
there.

That, of all plane curves of given length, the circle includes a maxi-
mum area, and of all closed surfaces of given area, the sphere encloses
a maximum volume, are theorems considered by Archimedes and
Zenodorus, but not proved rigorously for two thousand years until
K. Weierstrass and H. A. Schwarz. Jakob Steiner thought he had
proved the theorem for the circle. On a closed plane curve different
from a circle four non-cyclic points can be selected. The quadrilateral
obtained by successively joining the sides has its area increased when
it is so deformed (the lunes being kept rigid) that its vertices are
cyclic. Hence the total area is increased, and the circle has the maxi-
mum area. Oskar Perron of Tubingen pointed out in 1913 by an ex-
ample the fallacy of this proof: Let us "prove" that i is the largest of
all positive integers. No such integer larger than i can be the maxi-
mum, for the reason that its square is larger than itself. Hence, i
must be the maximum. Steiner's "proof" does not prove that among
all closed plane curves of given length there exists one whose area is a
maximum.1 K. Weierstrass gave a simple general existence theorem
applicable to the extremes of continuous (stetige) functions. The max-
imal property of the sphere was first proved rigorously in 1884 by
1 See W. Blaschke in Jahresb. d. denlsch. Afath. Vercinig., Vol. 24, 1915, p. 195.

ANALYSIS 371

H. A. Schwarz by the aid of results reached by K. Weierstrass in the
calculus of variations. Another proof, based on geometrical theorems,
was given in 1901 by Hermann Minkowski.

The subject of minimal surfaces, which had received the attention of
J. Lagrange, A. M. Legendre, K. F. Gauss and G. Monge, in later
time commanded the special attention of H. A. Schwarz. The blind
physicist of the University of Gand, Joseph Plateau (1801-1883), in
1873 described a way of presenting these surfaces to the eye by means
of soap bubbles made of glycerine water. Soap bubbles tend to be-
come as thick as possible at every point of their surface, hence to make
their surfaces as small as possible. More recent papers on minimal
surfaces are by Harris Hancock of the University of Cincinnati.

Ernst Pascal of the University of Pavia expressed himself in 1897
on the calculus of variations as follows:1 "It may be said that this de-
velopment [the finding of the differential equations which the unknown
functions in a problem must satisfy] closes with J. Lagrange, for the
later analysts turned their attention chiefly to the other, more dif-
ficult problems of this calculus. The problem is finally disposed of,
if one considers the simplicity of the formulas which arise; wholly dif-
ferent is this matter, if one considers the subject from the standpoint
of rigor of derivation of the formulas and the extension of the domain
of the problems to which these formulas are applicable. This last
is what has been done for some years. It has been found necessary to
prove certain theorems which underlie those formulas and which the
first workers looked upon as axioms, v/hich they are not." This new
field was first entered by I. Todhunter, M. Ostrogradski, C. G. J.
Jacobi, J. Bertrand, P. du Bois-Reymond, G. Erdmann, R. F. A.
Clebsch, but the incisive researches which mark a turning-point in the
history of the subject are due to K. Weierstrass. As an illustration
of Weierstrass' s method of communicatiing many of his mathematical
results to others, we quote the following from O. Bolza: 2 "Unfortu-
nately they [results on the calculus of variations] were given by Weier-
strass only in his lectures [since 1872], and thus became known only
very slowly to the general mathematical public. . ." . Weierstrass's
results and methods may at present be considered as generally known,
partly through dissertations and other publications of his pupils,
partly through A. Kneser's Lehrbuch der Variationsrechnung (Braun-
schweig, 1900), partly through sets of notes (' Ausarbeitungen ') of
which a great number are in circulation and copies of which are ac-
cessible to every one in the library of the Mathematische Verein
at Berlin, and in the Mathematische Lesezimmer at Gottingen.
Under these circumstances I have not hesitated to make use of Weier-
strass's lectures just as if they had been published in print." Weier-
strass applied modern requirements of rigor to the calculus of varia-

1 E. Pascal, Die Variationsrechnung, iibers. v. A. Schepp, Leipzig, 1899, p. §.,

2 O. Bolza, Lectures on the Calculus of Variations, Chicago, 1904, pp. ix, xi.

372 A HISTORY OF MATHEMATICS

tions in the study of the first and second variation. Not only did he
give rigorous proofs for the first three necessary conditions and for the
sufficiency of these conditions for the so-called "weak" extremum,
but he also extended the theory of the first and second variation to
the case where the curves under consideration are given in parameter
representation. He discovered the fourth necessary condition and a
sufficiency proof for a so-called "strong" extremum, which gave for
the first time a complete solution by means of a new method, based on
the so-called *' Weierstrass's construction." l Under the stimulus of
Weierstrass, new developments, were made by A. Kneser, then of
Dorpat, whose theory is based on the extension of certain theorems
on geodesies to extremals in general, and by David Hilbert of Got-
tingen, who gave an "a priori existence proof for an extremum of a
definite integral — a discovery of far-reaching importance, not only for
the Calculus of Variations, but also for the theory of differential
equations and the theory of functions " (O. Bolza). In 1909 Bolza
published an enlarged German edition of his calculus of variations,
including the results of Gustav v. Escherich of Vienna, the Hilbert
method of proving Lagrange's rule of multipliers (multiplikator-regel),
and the J. W. Lindeberg of Helsingfors treatment of the isoperimetric
problem. About the same time appeared J. Hadamard's Calcul des
variations recuellies par M. Frechet, Paris, 1910. Jacques Hadamard
(1865- ) was born at Versailles, is editor of the A nnales scientifiques
de I'ccole normale superieure. ^In 1912 he was appointed professor of
mathematical analysis at the Ecole Fob/technique of Paris as successor
to Camille Jordan. In the above mentioned book he regards the cal-
culus of variations as a part of a new and broader "functional calculus,"
along the lines followed also by V. Volterra in his functions of lines.
This functional calculus was initiated by Maurice Frechet of the Uni-
versity of Poitiers in France. The authors include also researches by
W. F. Osgood. Other prominent researches on the calculus of varia-
tions are due to J. G. Darboux, E. Goursat, E. Zermelo, H. A. Schwarz,
H. Hahn, and to the Americans H. Hancock, G. A. Bliss, E. R. Hedrick,
A. L. Underbill, Max Mason. Bliss and Mason systematically ex-
tended the Weierstrassian theory of the calculus of variations to
problems in space.

In 1858 David Bierens de Haan (1822-1895) °f Leiden published his
Tables d'Integrales Definies. A revision and the consideration of the
underlying theory appeared in 1862. It contained 8339 formulas.
A critical examination of the latter, made by E. W. Sheldon in 1912,
showed that it was "remarkably free from error when one imposes
proper limitations upon constants and functions, not stated by Haan.

The lectures on definite integrals, delivered by P. G. L. Dirichlet
in 1858, were elaborated into a standard work in 1871 by Gustav
Ferdinand Meyer of Munich.

1 This summary is taken from O. Bolza, op. cit., Preface.

ANALYSIS 373

Convergence of Series

The history of infinite series illustrates vividly the salient feature
of the new era which analysis entered upon during the first quarter
of this century. I. Newton and G. W. Leibniz felt the necessity of
inquiring into the convergence of infinite series, but they had no
proper criteria, excepting the test advanced by Leibniz for alternating
series. By L. Euler and his contemporaries the formal treatment of
.series was greatly extended, while the necessity for determining the
convergence was generally lost sight of. L. Euler reached some very
pretty results on infinite series, now well known, and also some very
absurd results, now quite forgotten. The faults of his time found
their culmination in the Combinatorial School in Germany, which
has now passed into oblivion. This combinatorial school was founded
by Carl Friedrich Hindenburg (1741-1808) of Leipzig whose pupils
filled many of the German University chairs during the first decennium
of the nineteenth century. The first important and strictly rigorous,
investigation of infinite series was made by K. F. Gauss in connection
with the hypergeometric series. This series, thus named by J. Wallis,
had been treated by L. Euler in 1769 and 1778 from the triple stand-
point of a power-series, of the integral of a certain linear differential
equation of the second order, and of a definite integral. The criterion
developed by K. F. Gauss settles the question of convergence of the
hypergeometric series in every case which it is intended to cover, and
thus bears the stamp of generality so characteristic of Gauss's writings.
Owing to the strangeness of treatment and unusual rigor, Gauss's
paper excited little interest among the mathematicians of that time.

More fortunate in reaching the public was A. L. Cauchy, whose
Analyse Algebrique of 1821 contains a rigorous treatment of series.
All series whose sum does not approach a fixed limit as the number
of terms increases indefinitely are called divergent. Like Gauss, he
institutes comparisons with geometric series, and finds that series
with positive terms are convergent or not, according as the nth root
of the nth term, or the ratio of the (w+i)th term and the nth term,
is ultimately less or greater than unity. To reach some of the cases
where these expressions become ultimately unity and fail, Cauchy
established two other tests. He showed that series with negative
terms converge when the absolute values of the terms converge, and
then deduces G. W. Leibniz's test for alternating series. The product
of two convergent series was not found to be necessarily convergent.
Cauchy's theorem that the product of two absolutely convergent
series converges to the product of the sums of the two series was
shown half a century later by F. Mertens of Graz to be still true if,
of the two convergent series to be multiplied together, only one is
absolutely convergent.

The most outspoken critic of the old methods in series was N. H.

374 A HISTORY OF MATHEMATICS

Abel. His letter to his friend B. M. Holmboe (1826) contains severe
criticisms. It is very interesting reading, even to modern students.
In his demonstration of the binomial theorem he established the
theorem that if two series and their product series are all convergent,
then the product series will converge towards the product of the
sums of the two given series. This remarkable result would dispose
of the whole problem of multiplication of series if we had a universal
practical criterion of convergency for semi-convergent series. Since
we do not possess such a criterion, theorems have been recently es-
tablished by A. Pringsheim of Munich and A. Voss now of Munich
which remove in certain cases the necessity of applying tests of con-
vergency to the product series by the application of tests to easier
related expressions. A. Pringsheim reaches the following interesting
conclusions: The product of two conditionally convergent series can
never converge absolutely, but a conditionally convergent series, or
even a divergent series, multiplied by an absolutely convergent series,
may yield an absolutely convergent product.

The researches of N. H. Abel and A. L. Cauchy caused a considerable
stir. We are told that after a scientific meeting in which Cauchy
had presented his first researches on series, P. S. Laplace hastened
home and remained there in seclusion until he had examined the
series in his Mecanique Celeste. Luckily, every one was found to be
convergent! We must not conclude, however, that the new ideas
at once displaced the old. On the contrary, the new views were
generally accepted only after a long struggle. As late as 1844 A. De
Morgan began a paper on "divergent series" in this style: "I believe
it will be generally admitted that the heading of this paper describes
the only subject yet remaining, of an elementary character, on which
a serious schism exists among mathematicians as to the absolute
correctness or incorrectness of results."

First in time in the evolution of more delicate criteria of convergence
and divergence come the researches of Josef Ludwig Raabe (1801-
1859) of Zurich, in Crelle, Vol. IX; then follow those of A. De Morgan
as given in his calculus. A. De Morgan established the logarithmic
criteria which were discovered in part independently by J. Bertrand.
The forms of these criteria, as given by J. Bertrand and by Ossian
Bonnet, are more convenient than De Morgan's. It appears from
N. H. Abel's posthumous papers that he had anticipated the above-
named writers in establishing logarithmic criteria. It was the opin-
ion of Bonnet that the logarithmic criteria never fail; but P. Du
Bois-Reymond and A. Pringsheim have each discovered series demon-
strably convergent in which these criteria fail to determine the con-
vergence. The criteria thus far alluded to have been called by Pring-
sheim special criteria, because they all depend upon a comparison of
the nth term of the series with special functions an, nx, n(\og ti)x, etc.
Among the first to suggest general criteria, and to consider the subject

ANALYSIS 375

from a still wider point of view, culminating in a regular mathematical
theory, was E. E. Kummer. He established a theorem yielding a test
consisting of two parts, the first part of which was afterwards found
to be superfluous. The study of general criteria was continued by
Ulisse Dini (1845-1918) of Pisa, P. Du Bois-Reymond, G. Kohn of
Vienna, and A. Pringsheim. Du Bois-Reymond divides criteria into
two classes: criteria of the first kind and criteria of the second kind, ac-
cording as the general nth term, or the ratio of the («+i)th term and
the nth term, is made the basis of research. E. E. Kummer's is a
criterion of the second kind. A criterion of the first kind, analogous
to this, was invented by A. Pringsheim. From the general criteria
established by Du Bois-Reymond and Pringsheim respectively, all
the special criteria can be derived. The theory of Pringsheim is very
complete, and offers, in addition to the criteria of the first kind and
second kind, entirely new criteria of a third kind, and also generalized
criteria of the second kind, which apply, however, only to series with
never increasing terms. Those of the third kind rest mainly on the
consideration of the limit of the difference either of consecutive terms
or of their reciprocals. In the generalized criteria of the second kind
he does not consider the ratio of two consecutive terms, but the ratio
of any two terms however far apart, and deduces, among others, two
criteria previously given by Gustav Kohn and W. Ermakoff respec-
tively.

It is a strange vicissitude that divergent series, which early in the
nineteenth century were supposed to have been banished once for
all from rigorous mathematics, should at its close be invited to return.
In 1886 T. J. Stieltjes and H. Poincare showed the importance to
analysis of the asymptotic series, at that time employed in astronomy
alone. In other fields of research G. H. Halphen, E. N. Laguerre, and
T. J. Stieltjes have encountered particular examples in which, a whole
series being divergent, the corresponding continued fraction was
convergent. In 1894 H. Pade now of Bordeaux, established the possi-
bility of defining, in certain cases, a function by an entire divergent
series. This subject was taken up also by J. Hadamard in 1892,
C. E. Fabry in 1896 and M. Servant in 1899. Researches on divergent
series have been carried on also by H. Poincare, E. Borel, T. J. Stieltjes,
E. Cesaro, W. B. Ford of Michigan and R. D. Carmichael of Illi-
nois. Thomas-Jean Stieltjes (1856-1894) was born in Zwolle in
Holland, came in 1882 under the influence of Ch. Hermite, became a
French citizen, and later received a professorship at the University
of Toulouse. Stieltjes was interested not only in divergent and con-
ditionally convergent series, but also in G. F. B. Riemann's £ function
and the theory of numbers.

Difficult questions arose in the study of Fourier's series.1 A. L.

1 Arnold Sachse, Vcrsuch einer Geschichtc. dcr Darstellung willkurlicher Funk-
tionen einer variablen durch ttigonomelrische Reihen, C-ottingen, 1879.

376 A HISTORY OF MATHEMATICS

Cauchy was the first who felt the necessity of inquiring into its con-
vergence. But his mode of proceeding was found by P. G. L. Dirichlet
to be unsatisfactory. Dirichlet made the first thorough researches
on this subject (Crelle, Vol. IV). They culminate in the result that
whenever the function does not become infinite, does not have an
infinite number of discontinuities, and does not possess an infinite
number of maxima and minima, then Fourier's series converges toward
the value of that function at all places, except points of discontinuity,
and there it converges toward the mean of the two boundary values.
L. Schlafli of Bern and P. Du Bois-Reymond expressed doubts as to
the correctness of the mean value, which were, however, not well
founded. Dirichlet's conditions are sufficient, but not necessary.
Rudolf Lipschitz (1832-1903), of Bonn, proved that Fourier's series
still represents the function when the number of discontinuities is
infinite, and established a condition on which it represents a function
having an infinite number of maxima and minima. Dirichlet's belief
that all continuous functions can be represented by Fourier's series
at all points was shared by G. F. B. Riemann and H. Hankel, but
was proved to be false by Du Bois-Reymond and H. A. Schwarz.
A. Hurwitz showed how to express the product of two ordinary Fourier
series in the form of another Fourier series. W. W. Kiistermann
solved the analogous problem for double Fourier series in which a
relation involving Fourier constants figures vitally. For functions
of a single variable an analogous relation is due to M. A. Parseval
and was proved by him under certain restrictions on the nature of
convergence of the Fourier series involved. In 1893 de la Vallee
Poussin gave a proof requiring merely that the function and its square
be integrable. A. Hurwitz in 1903 gave further developments. More
recently the subject has commanded general interest through the re-
searches of Frigyes Riesz and Ernst Fischer (Riesz-Fischer theorem).1
Riemann inquired what properties a function must have, so that
there may be a trigonometric series which, whenever it is convergent,
converges toward the value of the function. He found necessary
and sufficient conditions for this. They do not decide, however,
whether such a series actually represents the function or not. Rie-
mann rejected Cauchy's definition of a definite integral on account of
its arbitrariness, gave a new definition, and then inquired when a
function has an integral. His researches brought to light the fact
that continuous functions need not always have a differential coeffi-
cient. But this property, which was shown by K. Weierstrass to be-
long to large classes of functions, was not found necessarily to exclude
them from being represented by Fourier's series. Doubts on some of
the conclusions about Fourier's series were thrown by the observation,
made by Weierstrass, that the integral of an infinite series can be
shown to be equal to the sum of the integrals of the separate terms
1 Summary taken from Bull. Am. Math. Soi., Vol. 22, 1915, p. 6.

ANALYSIS 377

only when the series converges uniformly within the region in question.
The subject of uniform convergence was first investigated in 1847 by
G. G. Stokes of Cambridge and in 1848 by Philipp Ludwig v. Seidel
(1821-1896). Seidel had studied under F. W. Bessel, C. G. J. Jacobi,
J. F. Encke, and P. G. L. Dirichlet. He became professor at the
University of Munich in 1855. Later his lecturing and scientific
activity were stopped by a disease of his eyes. Uniform convergence
assumed great importance in K. Weierstrass' theory of functions. It
became necessary to prove that a trigonometric series representing
a continuous function converges uniformly. This was done by Rein-
rich Eduard Heine (1821-1881), of Halle. Later researches on Four-
ier's series were made by G. Cantor and Paul Du Bois-Reymond
(1831-1889), professor at the technical high school in Charlottenburg.
Less stringent than that of uniform convergence is U. Dini's def-
inition * of "simple uniform convergence," which is as follows: The
series is said to be simply uniformly convergent in the interval (a, b)
when corresponding to every arbitrarily chosen positive number <r
as small as we please and to every integer m', only one or several
integers m exist which are not less than m', and are such that, for all
the values of x in the interval (a, b), the | Rm(x) \ are <$. Still another
kind of convergence, the "uniform convergence by segments," some-
times called "sub-uniform convergence," was introduced in 1883 by
Cesare Arzela (1847-1912) of the University of Bologna. He advanced
the theory of functions of real variables and generalized a theorem of
U. Dini on the necessary and sufficient conditions for the continuity
of the sum of a convergent series of continuous functions.

Probability and Statistics

As compared with the vast development of other mathematical
branches, the theory of probability has made insignificant progress
since the time of P. S. Laplace. Jakob Bernoulli's Theorem which
had received the careful attention of De Moivre, J. Stirling, C. Ma-
claurin, and L. Euler, was considered especially by P. S. Laplace who
made an inverse application of it, assuming that an event had been
observed to happen m times and to fail n times in /z trials, and then
deducing the initially unknown probability of its happening at each
trial. The result thus obtained did not agree accurately with the
results gotten by the use of Bayes' Theorem. The subject was in-
vestigated by S. D. Poisson in his Recherches sur la probabilite, Paris,
1837, who obtained consonant results after carrying the approxima-
tions, in the use of Bayes' Theorem, to a higher degree. An endeavor
to remove the obscurities in which Bayes' Theorem seemed involved
was made by Poisson, and by A. De Morgan in his Theory of Probabil-

1 Grundlagcn f. c. Theorlc dcr Funclionen, by J. Liiroth u. A. Schepp, Leipzig,
1892, p. 137.

378 A HISTORY OF MATHEMATICS

ities* through the use of illustrations with urns that were exactly
alike and contained black and white balls in different numbers and
different ratios, the observed event being the drawing of a white ball
from any one of the urns. For the same purpose Johannes von Kries,
in his Prinzipien der Wahrscheinlichkeitsrechnung, Freiburg i. B., 1886,
used as an illustration six equal cubes, of which one had the + sign
on one side, another had it on two sides, a third on three sides and
finally the sixth on all six sides. All other sides were marked with a o.
Nevertheless, objections to certain applications by Bayes' Theorem
have been raised by the Danish actuary J. Bing in the Tidsskrift for
Matematik, 1879, by Joseph Bertrand in his Calcul des probabilitcs,
Paris, 1889, by Thorwald Nicolai Thiele (1838-1910) of the observa-
tory at Copenhagen in a work published at Copenhagen in 1889 (an
English edition of which appeared under the title Theory of Observa-
tions, London, 1903) by George Chrystal (1851-1911) of the Univer-
sity of Edinburgh, and others.2 As recently as 1908 the Danish
philosophic writer Kroman has come out in defence of Bayes. Thus
it appears that, as yet, no unanimity of judgment has been reached
in this matter. In determining the probability of alternative causes
deduced from observed events there is often need of evidence other
than that which is afforded by the observed event. By inverse prob-
ability some logicians have explained induction. For example, if a
man, who has never heard of the tides, were to go to the shore of the
Atlantic Ocean and witness on m successive days the rise of the sea,
then, says Adolphe Quelelet of the observatory at Brussels, he would

be entitled to conclude that there was a probability equal to

m+2

that the sea would rise next day. Putting m=o, it is seen that this
view rests upon the unwarrantable assumption that the probability
of a totally unknown event is |, or that of all theories proposed for
investigation one-half are true. William Stanley Jevons (1835-1882)
in his Principles of Science founds induction upon the theory of inverse
probability, and F. Y. Edgeworth also accepts it in his Mathematical
Psychics. Daniel Bernoulli's "moral expectation," which was elab-
orated also by Laplace, has received little attention from more recent
French writers. Bertrand emphasizes its impracticability; Poincare,
in his Calcul des probabilites, Paris, 1896, disposes of it in a few words.3
The only noteworthy recent addition to probability is the subject
of "local probability," developed by several English and a few Amer-
ican and French mathematicians. G. L. L. Buffon's needle problem
is the earliest important problem on local probability; it received the

1 Encyclopedia Metrop. II, 1845.

2 We are using Emanuel Czuber's Entwickelung der Wahrscheinlichkeitslheorie in
the Jahresb. d. deutsch. Mathematiker-Vereinigung, 1899, pp. 93-105; also Arne
Fisher, The Mathematical Theory of Probabilities, New York, 1915, pp. 54-56.

*E. Czuber, op. cit., p. 121.

ANALYSIS 379

consideration of P. S. Laplace, of Emile Barbier in the years 1860
and 1882, of Morgan W. Crofton (1826-1915) of the military school
at Woolwich, who in 1868 contributed a paper to the London Phil-
osophical Transactions, Vol. 158, and in 1885 wrote the article "Prob-
ability" in the Encyclopedia Brittannica, ninth edition. The name
"local probability" is due to Crofton. Through considerations of
local probability he was led to the evaluation of certain definite in-
tegrals.

Noteworthy is J. J. Sylvester's four point problem: To find the
probability that four points taken at random within a given boundary,
shall form a re-entrant quadrilateral. Local probability was studied
in England also by A. R. Clarke, H. McColl, S. Watson, J. Wolsten-
holme, W. S. Woolhouse; in France also by C. Jordan and E. Lemoine;
in America by E. B. Seitz. Rich collections of problems on local prob-
ability have been published by Emanuel Czuber of Vienna in his
Geometrische W ahrscheinlichkeiten und Mittelwerte, Leipzig, 1884, and
by G. B. M. Zerr in the Educational Times, Vol. 55, 1891, pp. 137-
192. The fundamental concepts of local probability have received the
special attention of Ernesto Cesaro (1859-1906) of Naples.1

Criticisms occasionally passed upon the principles of probability
and lack of confidence in theoretical results have induced several
scientists to take up the experimental side, which had been emphasized
by G. L. L. Buffon. Trials of this sort were made by A. De Morgan,
W. S. Jevons, L. A. J. Quetelet, E. Czuber, R. Wolf, and showed a
remarkably close agreement with theory. In Buffon's needle prob-
lem, the theoretical probability involves TT. This and similar expres-
sions 2 have been used for the empirical determination of TT. Attempts
to place the theory of probability on a purely empirical basis were
made by John Stuart Mill (1806-1873), John Venn (1834- ) and
G. Chrystal. Mill's induction method was put on a sounder basis
by A. A. Chuproff in a brochure, Die Statistik als Wissenschaft. Em-
pirical methods have commanded the attention of another Russian,
v. Bortkievicz.

In 1835 and 1836 the Paris Academy was led by S. D. Poisson's
researches to discuss the topic, whether questions of morality could be
treated by the theory of probability. M. H. Navier argued on the
affirmative, while L. Poinsot and Ch. Dupin denied the applicability
as "une sorte d'aberration de 1'esprit;" they declared the theory
applicable only to cases where a separation and counting of the cases
or events was possible. John Stuart Mill opposed it; Joseph L. F.
Bertrand (1822-1900), professor at the College de France in Paris
and J. v. Kries are among more recent writers on this topic.1

1 See Encydop&lie des sciences malh. I, 20 (1906), p. 23.

2 E. Czuber, op. cit., pp. 88-91.

3 Consult E. Czuber, op. cit., p. 141; J. S. Mill, System of Logic, New York, 8th
Ed., 1884, Chap. 18, pp. 379-387; J. v. Rrics, op. cit., pp. 253-259.

380 A HISTORY OF MATHEMATICS

Among the various applications of probability the one relating to
verdicts of juries, decisions of courts and results of elections is specially
interesting. This subject was studied by Marquis de Condorcet,
P. S. Laplace, and S. D. Poisson. To exhibit Laplace's method of
determining the worth of candidates by combining the votes, M. W.
Crofton employs the fortuitous division of a straight line. This in-
volves, however, an a priori distribution of values covering evenly
the whole range from o to 100. Experience shows that the normal
law of error exhibits a more correct distribution. On this point Karl
Pearson produced a most important research.1 He took a random
sample of n individuals from a population of N members and derived
an expression for the average difference in character between the pth
and the (/?+i)th individual when the sample is arranged in order of
magnitude of the character. H. L. Moore of Columbia University
has attempted to trace Pearson's theory in the statistics relating to
the efficiency of wages (Economic Journal, Dec., 1907).

Early statistical study was carried on under the name of ''political
arithmetic" by such writers as Captain John Graunt pf London (1662)
and J. P. Siissmiich, a Prussian clergyman (1788). Application of the
theory of probability to statistics was made by Edmund Halley,
Jakob Bernoulli, A. De Moivre, L. Euler, P. S. Laplace, and S. D.
Poisson. The establishment of official statistical societies and statis-
tical offices was largely due to the influence of the Belgian astronomer
and statistician, Adolphe Quetelet (1796-1874) of the observatory
at Brussels, "the founder of modern statistics." Quetelet's "average
man" in whom "all processes correspond to the average results
obtained for society," who "could be considered as a type of the
beautiful," has given rise to much critical discussion by Harold
Westergaard (1890), J. Bertillon (1896), A. de Foville in his "homo
medius" of 1907, Joseph Jacobs in his "the Middle American"
and "the Mean Englishman."2 Quetelet's visit in England led
to the organization, in 1833, of the statistical section of the British
Association for the Advancement of Science, and in 1835 of the Statis-
tical Society of London. Soon after, in 1839, was formed the Amer-
ican Statistical Society. Quetelet's best researches on the application
of probability to the physical and social sciences are given in a series
of letters to the duke of Saxe-Coburg and Gotha, Lettres sur la theorie
des probabilites, Brussels, 1846. He laid emphasis on the "law of
large numbers," which was advanced also by the Frenchman S. D.
Poisson and discussed by the German W. Lexis (1877), the Scandi-
navians H. Westergaard and Carl Charlier, and the Russian Pafnuli
Liwowich Chebichev (1821-1894) of the University of Petrograd. To
Chebichev we owe also an interesting problem : A proper fraction being

1 "Note on Francis Galton's Problem," Biomelrica, Vol. I, pp. 390-309.

2 See Franz Zizek's Statistical Averages, transl. by W. M. Persons, New York,

P- 374-

ANALYSIS 381

chosen at random, what is the probability that it is in its lowest
terms?

Of the different kinds of averages, A. De Morgan concluded that
the arithmetic mean represents a priori the most probable values.
J. W. L. Glaisher took exception to this. G. T. Fechner investigated
the cases where the "median" (which has the central position in a
series of items arranged according to size) may be used profitably.
The "mode," an average introduced by K. Pearson in 1895, and
used by G. Udny Yule, has been applied in Germany and Austria
to the fixing of workingmen's insurance. The theory of averages has
been studied by the aid of the calculus of probabilities by W. Lexis,
F. Y. Edgeworth of Oxford, H. Westergaard, L. von Bortkewich, G.
T. Fechner, J. von Kries, E. Czuber of Vienna, E. Blaschke, F. Galton,
K. Pearson, G. U. Yule, and A-. L. Bowley of London. A few
writers take the ground that it is not only unnecessary to employ prob-
ability in founding statistical theory, but that it is inadvisable to do so.
Among such writers are G. F. Knapp and A. M. Guerry.1 The Russian
actuary Jastremski in 1912 applied the Lexian dispersion theory to the
testing of the influence of medical selection in life insurance. Other
recent publications of note are by Lexis' pupil L. von Bortkewich and
by Harold Westergaard of Copenhagen. Early theories of popula-
tion were involved in much confusion. E. Halley and some eighteenth
century writers proceeded on the assumption of a stationary popu-
lation. L. Euler adopted the hypothesis that the yearly births
progress in a geometric series. This was combatted in 1839 by L.
Moser, while G. F. Knapp in 1868 represented the number of births
and deaths as a continuous function of the time and of the age, re-
spectively. He made use of graphic representation. G. Zeuner in
1869 introduced additional geometric and analytic aids. In 1874
Knapp made still further modifications, allowing for discontinuous
changes, such as were studied also by W. Lexis, in his Theorie der
Bevolkerungsstatistik, Strassburg, 1875. Formal theories of popula-
tion and the determination of mortality were investigated also by
K. Becker in 1867 and 1874, and by Th. Wittstein, about 1881. In
1877 W. Lexis introduced the idea of " dispersion "and "normal dis-
persion." Wilhelm Lexis (1837-1914) became in 1872 professor at
Strassburg, in 1884 at Breslau and in 1887 at Gottingen. In 1893 he
was drawn into the service for the German government.

The application of statistical method to biology was begun by
Sir Francis Galton (1822-1911), "a born statistician." Important
is his Natural Inheritance, 1889, in which he uses the method of per-
centiles, with the quartile deviation as the measure of dispersion.2
Two other Englishmen entered this field of research, Karl Pearson
of University College and W. F. R. Weldon. Pearson developed

1 Encyklop'adie d. Math. Wissensch, T D 4.1, p. 822.

2 G. Udny Yule, Theory of Statistics, 2. Ed., London, 1912, p. 154.

382 A HISTORY OF MATHEMATICS

general and adequate mathematical methods for the analysis of
biological statistics. To him are due the terms "mode," "standard
deviation" and "coefficient of variation." Before him the "normal
curve" of errors had been used exclusively to describe the distribution
of chance events. This curve is symmetrical, but natural phenomena
sometimes indicate an asymmetrical distribution. Accordingly
Pearson, in his Contributions to the Theory of Evolution, 1899, de-
veloped skew frequency curves. About 1890 the Georg Mendel law
of inheritance became generally known and caused some modification
in the application of statistics to heredity. Such a readjustment was
effected by the Danish botanist W. Johannsen.1

The first study of the most advantageous combinations of data
of observation is due to Roger Cotes, in the appendix to his Harmonia
mensurarum, 1722, where he assigns weights to the observations.
Trie use of the arithmetic mean was advocated by Thomas Simpson
in a paper "An attempt to show the advantage arising by taking the
mean of a number of observations, in practical astronomy," 2 also by
J. Lagrange in 1773 and by Daniel Bernoulli in 1778. The first
printed statement of the principle of least squares was made in 1806
by A. M. Legendre, without demonstration. K. F. Gauss had used
it still earlier, but did not publish it until 1809. The first deduction
of the law of probability of error that appeared in print was given in
1808 by Robert Adrain in the Analyst, a journal published by himself
in Philadelphia. Of the earlier proofs given of this law, perhaps the
most satisfactory is that of P. S. Laplace. K. F. Gauss gave two
proofs. The first rests upon the assumption that the arithmetic
mean of the observations is the most probable value. Attempts to
prove this assumption have been made by Laplace, J. F. Encke (1831),
A. De Morgan (1864), G. V. Schiaparelli, E. J. Stone (1873), and A.
Ferrero (1876). Valid criticisms upon some of these investigations were
passed by J. W. L. Glaisher.3 The founding of the Gaussian proba-
bility law upon the nature of the observed errors was attempted by
F. W. Bessel (1838), G. H. L. Hagen (1837), J. F. Encke (1853), P. G.
Tait (1867), and M. W. Crofton (1870). That the arithmetic mean,
taken as the most probable value, is not under all circumstances
compatible with the Gaussian probabilty law has been shown by
Joseph Bertrand in his Calcul des probabilites (1889), and by others.4
The development of the theory of least squares along practical lines
is due mainly to K. ,F. Gauss, J. F. Encke, P. A. Hansen, Th. Gallo-
way, J. Bienayme, J. Bertrand, A. Ferrero, P. Pizzetti.6 Simon

1 Quart. Pub. Am. Slat. Ass'n, N. S., Vol. XIV, 1914, p. 45.
8 Miscellaneous Tracts, London, 1757.

3 Land. Astr. Soc. Mem. 39, 1872, p. 75; for further references, see Cyklop'ddie d.
Math. Wiss., I Da, p. 772.

4 Consult E. L. Dodd, "Probability of the Arithmetic Mean, etc.," Annals of
Mathematics, 2. S., Vol. 14, 1913, p. 186.

8 E. Czuber, op. cit., p. 179.

ANALYSIS 383

Ncwcomb of Washington advanced a "generalized theory of the
combination of observations so as to obtain the best result," 1 when
large errors arise more frequently than is allowed by the Gaussian
probability law. The same subject was treated by R. Lehmann-
Filhes in Astronomische Nachrichten, 1887.

A criterion for the rejection of doubtful observations 2 was given
by Benjamin Peirce of Harvard. It was accepted by the American
astronomers B. A. Gould (1824-1896), W. Chauvenet (1820-1870),
and J. Winlock (1826-1875), but was criticised by the English as-
tronomer G. B. Airy. The prevailing feeling has been that there
exists no theoretical basis upon which such criterion can be rightly
established.

The application of probability to epidemiology was first considered
by Daniel Bernoulli and has more recently commanded the atten-
tion of the English statisticians William Farr (1807-1883), John
Brownlee, Karl Pearson, and Sir Ronald Ross. Pearson studied
normal and abnormal frequency curves. Such curves have been
fitted to epidemics by J. Brownlee in 1906, S. M. Greenwood in 1911
and 1913, and Sir Ronald Ross in 1916. 3

Some interest attaches to the discussion of whist from the stand-
point of the theory of probability, as is contained in William Pole's
Philosophy of Whist, New York and London, 1883. The problem
is a generalization of the game of "treize" or "recontre," treated
by Pierre R. de Montmort in 1708.

Differential Equations. Difference Equations

Criteria for distinguishing between singular solutions and particular
solutions of differential equations of the first order were advanced
by A. M. Legendre, S. D. Poisson, -S. F. Lacroix, A. L. Cauchy, and
G. Boole. After J. Lagrange, the c-discriminant relation commanded
the attention of Jean Marie Constant Duhamel (1797-1872) of Paris,
C. L. M. H. Navier, and others. But the entire theory of singular
solutions was re-investigated about 1870 along new paths by J. G.
Darboux, A. Cayley, E. C. Catalan, F. Casorati, and others. The
geometric side of the subject was considered more minutely and the
cases were explained in which Lagrange's method does not yield
singular solutions. Even these researches were not altogether satis-
factory as they did not furnish necessary and sufficient conditions
for singular solutions which depend on the differential equation alone
and not in anyway upon the general solution. Returning to more
purely analytical considerations and building on work of Ch. Briot
and J. C. Bouquet of 1856, Carl Schmidt of Giessen in 1884, H. B.
Fine of Princeton in 1890, and Meyer Hamburger (1838-1903) of

1 Am. Jour. Math., Vol. 8, 1886, p. 343.

2 Gould Astr. Jour., IE, 1852.

3 Nature, Vol. 97, 1916, p. 243.

384 A HISTORY OF MATHEMATICS

Berlin brought the problem to final solutions. Active in this line
were also John Muller Hill and A. R. Forsyth.1

The first scientific treatment of partial differential equations was
given by J. Lagrange and P. S. Laplace. These equations were in-
vestigated in more recent time by G. Monge, J. F. Pfaff, C. G. J.
Jacobi, Emile Bour (1831-1866) of Paris, A. Weiler, R. F. A. Clebsch,
A. N. Korkine of St. Petersburg, G. Boole, A. Meyer, A. L. Cauchy,
J. A. Serret, Sophus Lie, and others. In 1873 their reseaches, on
partial differential equations of the first order, were presented in
text-book form by Paul Mansion, of the University of Gand. Pro-
ceeding to the consideration of some detail, we remark that the keen
researches of Johann Friedrich Pfaff (1765-1825) marked a decided
advance. He was an intimate friend of K. F. Gauss at Gottingen.
Afterwards he was with the astronomer J. E. Bode. Later he became
professor at Helmstadt, then at Halle. By a peculiar method, Pfaff
found the general integration of partial differential equations of the
first order for any number of variables. Starting from the theory of
ordinary differential equations of the first order in n variables, he
gives first their general integration, and then considers the integra-
tion of the partial differential equations as a particular case of the
former, assuming, however, as known, the general integration of
differential equations of any order between two variables. His re-
searches led C. G. J. Jacobi to introduce the name "Pfaffian prob-
lem." From the connection, observed by W. R. Hamilton, between
a system of ordinary differential equations (in analytical mechanics)
and a partial differential equation, C. G. J. Jacobi drew the conclu-
sion that, of the series of systems whose successive integration Pfaff's
method demanded, all but the first system were entirely superfluous.
R. F. A. Clebsch considered Pfaff's problem from a new point of view,
and reduced it to systems of simultaneous linear partial differential
equations, which can be established independently of each other with-
out any integration. Jacobi materially advanced the theory of dif-
ferential equations of the first order. The problem to determine un-
known functions in such a way that an integral containing these func-
tions and their differential coefficients, in a prescribed manner, shall
reach a maximum or minimum value, demands, in the first place,
the vanishing of the first variation of the integral. This condition
leads to differential equations, the integration of which determines the
functions. To ascertain whether the value is a maximum or a mini-
mum, the second variation must be examined. This leads to new and
difficult differential equations, the integration of which, for the simpler
cases, was ingeniously deduced by C. G. J. Jacobi from the integra-
tion of the differential equations of the first variation. Jacobi's
solution was perfected by L. O. Hesse, while R. F. A. Clebsch extended

1 We have used S. Rothenberg, "Geschichte . . . der singularep Losungen" in
Abh. z. Cesch. d. Math. Wissensch. (M. Cantor), Heft XX, 3. Leipzig, 1908.

ANALYSIS 385

to the general case Jacobi's results on the second variation. A. L.
Cauchy gave a method of solving partial differential equations of the
first order having any number of variables, which was corrected and
extended by J. A. Serret, J. Bertrand, O. Bonnet in France, and
Wassili Grigorjewich Imshenetski (1832-1892) of the University of
Charkow in Russia. Fundamental is the proposition of Cauchy that
every ordinary differential equation admits in the vicinity of any non-
singular point of an integral, which is synectic within a certain circle
of convergence, and is developable by Taylor's theorem. Allied to the
point of view indicated by this theorem is that of G. F. B. Riemann,
who regards a function of a single variable as defined by the position
and nature of its singularities, and who has applied this conception
to that linear differential equation of the second order, which is satis-
fied by the hypergeometric series. This equation was studied also
by K. F. Gauss and E. E. Kummer. Its general theory, when no re-
striction is imposed upon the value of the variable, has been considered
by J. Tannery, of Paris, who employed L. Fuchs' method of linear
differential equations and found all of Kummer's twenty-four inte-
grals of this equation. This study has been continued by Edouard
Goursat (1858- ), professor of mathematical analysis in the Uni-
versity of Paris. ' His activities have been in the theory of functions,
pseudo- and hyper-elliptic integrals, differential equations, invariants
and suHaces. Jules Tannery (1848-1910) became professor of me-
chanics at the Sorbonne in 1875, and sub-director at the Fxole Normale
in Paris in 1884. His researches have been in the field of analysis and
the theory of functions.

As outlined by A. R. Forsyth l in 1908, the status of partial differen-
tial equations is briefly as follows: Since the posthumous publication,
in 1862, of C. G. J. Jacobi's treatment of partial differential equations
of the first order involving only one dependent variable, or a system
of such equations, it may be said that we have a complete method
of formal integration of such equations. In the formal integration of a
partial differential equation of the second or higher orders new dif-
ficulties are encountered. Only in rare instances is direct integration
possible. The known normal types of integrals even for such equa-
tions of only the second order are few in number. The primitive may
be given by means of a single relation between the variables, or by
means of a number of equations involving eliminable parameters (such
as the customary forms, due to A. M. Legendre, G. Monge, or K.
Weierstrass, of the primitive of minimal surfaces), or by means of a
relation involving definite integrals arising in problems in physics.
"With all these types of primitives," says A. R. Forsyth,2 "it being
assumed that immediate and direct integration is impossible — , a

1 A. R. Forsyth in Alii, del IV. Congr. Intern., Roma, 1008. Vol. I, Roma, 1909,
p. oo.

2 A. R. Forsyth, loc. til., p. 90. We are summarizing part of this article.

386 A HISTORY OF MATHEMATICS

primitive is obtained by the use of processes, that sometimes are frag-
mentary in theory, usually are tentative in practice and nearly always
are indirect in the sense that they are compounded of a number of
formal operations having no organic relation with the primitive.
In such circumstances '. . . is the primitive completely comprehen-
sive of all the integrals belonging to the equation? " A. M. Ampere in
1815 propounded a broad definition of a general integral — one in
which the only relations, which subsist among the variables and the
derivatives of the dependent variable and which are free from the
arbitrary elements in the integral, are constituted by the differential
equation itself and by equations deduced from it by differentiation.
This definition is incomplete on various grounds. E. Goursat gave in
1898 a simple instance to show that an integral satisfying all of Am-
pere's requirements was not general. A second definition of a general
integral was given in 1889 by J. G. Darboux, based on A. L. Cauchy's
existence-theorem: An integral is general when the arbitrary ele-
ments which it contains can be specialized in such a way as to provide
the integral established in that theorem. This definition, according
to A. R. Forsyth, calls for a more careful discussion of obvious and
latent singularities.

There are three principal methods of proceeding to the construc-
tion of an integral of partial differential equations of the second order,
which lead to success in special cases. One method given by P. S.
Laplace in 1777 applies to linear equations with two independent
variables. It can be used for equations of order higher than the
second. It has been developed by J. G. Darboux and V. G. Imshenet-
ski, 1872. A second method, originated by A. M. Ampere, while
general in spirit and in form, depends upon individual skill unassisted
by critical tests. Later researches along this line are due to E. Borel
(1895) and E. T. Whittaker (1903). A third method is due to J. G.
Darboux and includes, according to A. R. Forsyth's classification,
the earlier work of Monge and G. Boole. As first given by J. G. Dar-
boux in 1870, it applied only to the case of two independent variables,
but it has been extended to equations of more than two independent
variables and orders higher than the second; it is not universally
effective. "Such then," says Forsyth, "are the principal methods
hitherto devised for the formal integration of partial equations of the
second order. They have been discussed by many mathematicians
and they have been subjected to frequent modifications in details:
but the substance of the processes remains unaltered."

Instances are known in ordinary linear equations when the primitives
can be expressed by definite integrals or by means of asymptotic
expansions, the theory of which owes much to H. Poincare. Such
instances within the region of partial equations are due to E. Borel.

G. F. B. Riemann had remarked in 1857 that functions expressed
by K. F. Gauss' hypergeometric series F (a, /3, y, .r), which satisfy

ANALYSIS 387

a homogeneous linear differential equation of the second order with
rational coefficients, might be utilized in the solution of any linear dif-
ferential equation.1 Another mode of solving such equations was due
to Cauchy and was extended by C. A. A. Briot and J. C. Bouquet) and
consisted in the development into power series. The fertility of the
conceptions of G. F. B. Riemann and A. L. Cauchy with regard to
differential equations is attested by the researches to which they have
given rise on the part of Lazarus Fuchs (1833-1902) of Berlin. Fuchs
was born in Moschin, near Posen, and became professor at the Uni-
versity of Berlin in 1884. In 1865 L. Fuchs combined the two methods
in the study of linear differential equations: One method using power-
series, as elaborated by A. L. Cauchy, C. A. A. Briot, and J. C. Bou-
quet; the other method using the hypergeometric series as had been
done by G. F. B. Riemann. By this union Fuchs initiated a new
theory of linear differential equations.2 Cauchy's development into
power-series together with the calcul des limites, afforded existence
theorems which are essentially the same in nature as those relating to
differential equations in general. The singular points of the linear
differential equation received attention also from G. Frobenius in
1874, G. Peano in 1889, M. Bdcher in 1901. A second approach to
existence theorems was by successive approximation, first used in
1864 by J. Caque, then by L. Fuchs in 1870, and later by H. Poin-
care and G. Peano. A third line, by interpolation, is originally due to
A. L. Cauchy and received special attention from V. Volterra in 1887.
The general theory of linear differential equations received the atten-
tion of L. Fuchs, and of a large number of workers, including C. Jordan,
V. Volterra, and L. Schlesinger. Singular places where the solutions
are not indeterminate were investigated by J. Tannery, L. Schlesinger,
G. J. Wallenberg, and many others. Ludwig Wilhelm Thome (1841-
1910) of the University of Greifswald, discovered in 1877 what he
called normal integrals. Divergent series which formally satisfy dif-
ferential equations, first noticed by C. A. A. Briot and J. C. Bouquet
in 1856, were first seriously considered by H. Poincare in 1885 who
pointed out that such series may represent certain solutions asymp-
totically. Asymptotic representations have been examined by A.
Kneser (1896), E. Picard (1896), J. Horn (1897), and A. Hamburger
(1905). A special type of linear differential equation, the "Fuchsian
type," with coefficients that are single- valued (eindeutig), and the
solutions of which have no points of indeterminateness, was investi-
gated by Fuchs, and it was found that the coefficients of such an equa-
tion are rational functions of x. Studies based on analogies of linear
differential equations with algebraic equations, first undertaken by

1 We are using L. Schlesinger, Enttvickelung d. Theorie d. linearen Differenlial-
glcirhitngcn seit 1865, Leipzig and Berlin, 1909.

- We are using here a report by L. Schlesinger in Jahresb. d. d. Math. Vereinigting,
Vol. 18, 1909, pp. 133-260.

388 A HISTORY OF MATHEMATICS

N. H. Abel, J. Liouville and C. G. J. Jacobi, were pursued later by P.
Appell (1880), by E. Picard who worked under the influence of S.
Lie's theory of transformation groups, and by an army of workers in
France, England, Germany, and the United States. The consideration
of differential invariants enters here. Lame's differential equation,
considered by him in 1857, was taken up by Ch. Hermite in 1877 and
soon after in still more generalized form by L. Fuchs, F. Brioschi,
E. Picard, G. M. Mittag-Leffler and F. Klein.

The analogies of linear differential equations with algebraic func-
tions, problems of inversion and uniformization, as well as questions
involving group theory received the attention of the analysts of the
second half of the century.

The theory of invariants associated with linear differential equations
as developed by Halphen and by A. R. Forsyth is closely connected
with the theory of functions and of groups. Endeavors have thus
been made to determine the nature of the function defined by a dif-
ferential equation from the differential equation itself, and not from
any analytical expression of the function, obtained first by solving
the differential equation. Instead of studying the properties of the
integrals of a differential equation for all the values of the variable,
investigators at first contented themselves with the study of the prop-
erties in the vicinity of a given point. The nature of the integrals
at singular points and at ordinary points is entirely different. Charles
Auguste Albert Briot (1817-1882) and Jean Claude Bouquet (1819-1885)
both of Paris, studied the case when, near a singular point, the dif-
ferential equations take the form (x— xo)-i~= I (xy). L. Fuchs gave

the development in series of the integrals for the particular case of
linear equations. H. Poincare did the same for the case when the
equations are not linear, as also for partial differential equations of
the first order. The developments for ordinary points were given
by A. L. Cauchy and Sophie Kovalevski (1850-1891). Madame
Kovalevski was born at Moscow, was a pupil of K. Weierstrass and
became professor of Analysis at Stockholm.

Henri Poincare (1854-1912) was born at Nancy and commenced
his studies at the Lycee there. While taking high rank as a student,
he did not display exceptional precocity. He attended the ficole
Polytechnique and the Ecole Nationale Superieure des Mines in Paris,
receiving his doctorate from the University of Paris in 1879. He be-
came instructor in mathematical analysis at the University of Caen.
In 1881 he occupied the chair of physical and experimental mechanics
at the Sorbonne, later the chair of mathematical physics and, after
the death of F. Tisserand, the chair of mathematical astronomy and
celestial mechanics. Although he did not reach old age, he published
numerous books and more than 1500 memoirs. Probably neither

ANALYSIS 389

A. L. Cauchy nor even L. Euler equalled him in the quantity of scien-
tific productions. P. Painleve said that everyone of his many papers
carried the mark of a lion. Poincare wrote on mathematics, physics,
astronomy, and philosophy. No other scientist of his day was able to
work in such a wide range of subjects. Many consider him the great-
est mathematician of his time. Each year he lectured on a different
subject; these lectures were reported and published by his former
students. In this manner were brought out works on capillarity,
elasticity, Newtonian potential, vortices, the propagation of heat,
thermodynamics, light, electric oscillations, electricity and optics,
Hertzian oscillations, mathematical electricity, kinematics, equili-
brium of fluid masses, celestial mechanics, general astronomy, prob-
ability. His popular works on the philosophy of science, La science et
Ihypothese (1902), La valeur de la science (1905), Science et methode
(1908) have been translated into German, in part also into Spanish,
Hungarian, and Japanese. An English translation by George Bruce
Halsted appeared in one volume in 1913.

Our numerous references to Poincare will indicate that he wrote
on nearly every branch of pure mathematics. Says F. R. Moulton: *
"The importance of his papers can be inferred from the enormous
number of references to his theorems in all modern treatises, espe-
cially on the various branches of analysis. The emphasis on analysis
does not mean that he neglected geometry, analysis situs, groups,
number theory, or the foundations of mathematics, for he illuminated
all these subjects and others; but it is placed there because this do-
main includes his researches on differential equations, dating from his
doctor's dissertation to very recent times, his contributions to the
theory of functions, and his discovery of fuchsian and theta-fuchsian
functions. His command of the powerful methods of modern analysis
was positively dazzling." As to his method of work E. Borel says:
"The method of Poincare is essentially active and constructive. He
approaches a question, acquaints himself with its present condition
without being much concerned about its history, finds out immediately
the new analytical formulas by which the question can be advanced,
deduces hastily the essential results, and then passes to another ques-
tion. After having finished the writing of a memoir, he is sure to
pause for a while, and to think out how the exposition could be im-
proved; but he would not, for a single instance, indulge in the idea of
devoting several days to didactic work. Those days could be better
utilized in exploring new regions." Poincare tells how he came to
make his first mathematical discoveries: "For a fortnight I labored
to demonstrate that there could exist no function analogous to those
that I have since called the fuchsian functions. I was then very ig-

1 Popular Astronomy, Vol. 20, 1912. We are usint? also Ernest Lebon, Henri
Poincare, Biagrapkie, Paris, 1900; "Jules Henri Poincare" in Nature, Vol. 90,
London, 1912, p. 353; George Sarton, "Henri Poincare" in del et Terre, 1913.

390 A HISTORY OF MATHEMATICS

norant. Every day I seated myself at my work table and spent an
hour or two there, trying a great many combinations, but I arrived at
no result. One night when, contrary to my custom, I had taken black
coffee and I could not 'sleep, ideas surged up in crowds. I felt them as
they struck against one another until two of them stuck together, so
to speak, to form a stable combination. By morning I had established
the existence of a class of fuchsian functions, those wrhich are derived
from the hypergeometric series. I had merely to put the results in
shape, which only took a few hours." 1

Poincare enriched the theory of integrals. The attempt to express
integrals by developments that are always convergent and not limited
to particular points in a plane necessitates the introduction of new
transcendents, for the old functions permit the integration of only a
small number of differential equations. H. Poincare tried this plan
with linear equations, which were then the best known, having been
studied in the vicinity of given points by L. Fuchs, L. W. Thome, G.
Frobenius, H. A. Schwarz, F. Klein, and G. H. Halphen. Confining
himself to those with rational algebraical coefficients, H. Poincare was
able to integrate them by the use of functions named by him Fuch-
sians.2 He divided these equations into "families." If the integral
of such an equation be subjected to a certain transformation, the
result wrill be the integral of an equation belonging to the same family.
The new transcendents have a great analogy to elliptic functions;
while the region of the latter may be divided into parallelograms, each
representing a group, the former may be divided into curvilinear
polygons, so that the knowledge of the function inside of one polygon
carries with it the knowledge of it inside the others. Thus H. Poin-
care arrives at what he calls Fuchsian groups. He found, moreover,
that Fuchsian functions can be expressed as the ratio of two trans-
cendents (theta-fuchsians) in the same way that elliptic functions can
be. If, instead of linear substitutions with real coefficients, as em-
ployed in the above groups, imaginary coefficients be used, then dis-
continuous groups are obtained, which he called Kleinians. The ex-
tension to non-linear equations of the method thus applied to linear
equations was begun by L. Fuchs and H. Poincare.

Much interest attaches to the determination of those linear differ-
ential equations which can be integrated by simpler functions, such
as algebraic, elliptic, or Abelian. This has been studied by C. Jordan,
P. Appell of Paris, and H. Poincare.

Paul Appell (1855- ) was born in Strassburg. After the an-
nexation of Alsace to Germany in 1871, he emigrated to Nancy to
escape German citizenship. Later he studied in Paris and in 1886

1 H. Poincar6, The Foundations of Science, transl. by G. B. Halsted, The Science
Press, New York and Garrison, N. Y., 1913, p. 387.

* Henri Poincare, Notice sur Ics Travaux Scientifiques de Henri Poincare, Paris,
1886, p. 9.

ANALYSIS 391

became professor of mechanics there. His researches are in analysis,
function theory, infinitesimal geometry and rational mechanics.

Whether an ordinary differential equation has one or more solutions
which satisfy certain terminal or boundary conditions, and, if so, what
the character of these solutions is, has received renewed attention
the last quarter century by the consideration of finer and more remote
questions.1 Existence theorems, oscillation properties, asymptotic
expressions, development theorems have been studied by David Hil-
bert of Gottingen, Maxime Bocher of Harvard, Max Mason of the
University of Wisconsin, Mauro Picone of Turin, R. M. E. Mises of
Strassburg, H. Weyl of Gottingen and .especially by George D. Birk-
hoff of Harvard. Integral equations have been used to some extent
in boundary problems of one dimension; "this method would seem,
however, to be chiefly valuable in the cases of two or more dimensions
where many of the simplest questions are still to be treated."

A standard text-book on Differential Equations, including original
matter on integrating factors, singular solutions, and especially on
symbolical methods, was prepared in 1859 by G. Boole.

A Treatise on Linear Differential Equations (1889) was brought out
by Thomas Craig of the Johns Hopkins University. He chose the
algebraic method of presentation followed by Ch. Hermite and H.
Poincare, instead of the geometric method preferred by F. Klein and
H. A. Schwarz. A notable work, the Traite a" Analyse, 1891-1896,
was published by Emile Picard of Paris, the interest of which was made
to centre in the subject of differential equations. A second edition
has appeared.

Simple difference equations or "finite differences" were studied by
eighteenth century mathematicians. When in 1882 H. Poincare de-
veloped the novel notion of asymptotic representation, he applied it
to linear difference equations. In recent years a new type of problem
has arisen in connection with them. It looks now as if the continuity
of nature, which has been for so long assumed to exist, were a fiction
and as if discontinuities represented the realities. "It seems almost
certain that electricity is done up in pellets, to which we have given
the name of electrons. That heat comes in quanta also seems prob-
able." '•' Much of theory based on the assumption of continuity may
be found to be mere approximation. Homogeneous linear difference
equations, not intimately bound up with continuity, were taken up in-
dependently by investigators widely apart. In 1909 Niels Erik Nor-
lund of the University of Lund in Sweden, Henri Galbrun of I'ficole
Normale in Paris and, in 1911, R. D. Carmichael of the University of
Illinois entered this field of research. Carmichael used a method of
successive approximation and an extension of a contour integral due to

1 See a historical summary by Maxime Bocher in Proceed, of the $ln i ntcrn. Con-
gress, Cambridge, 1912, Vol. I, Cambridge, 1913, p. 163.

2 R. D. Carmichael in Science, N. S. Vol. 45, 1917, p. 472.

392 A HISTORY OF MATHEMATICS

C. Guichard. G. D. Birkhoff of Harvard made important contribu-
tions showing the existence of certain intermediate solutions and of the
principal solutions. The asymptotic form of these solutions is de-
termined by him throughout the complex plane. The extension to
non-homogeneous equations of results reached for homogeneous ones
has been made by K. P. Williams of the University of Indiana.1

Integral Equations, Integra-differential Equations, General Analysis,
Functional Calculus

The mathematical perplexities which led to the invention of integral
equations were stated by J. Hadamard 2 in 1911 as follows: "Those
problems (such as Dirichlet's) exercised the sagacity of geometricians
and were the object of a great deal of important and well-known work
through the whole of the nineteenth century. The very variety of
ingenious methods applied showed that the question did not cease to
preserve its rather mysterious character. Only in the last years of
the century were we able to treat it with some clearness and under-
stand its true nature. . . . Let us therefore inquire by what device
this new view of Dirichlet's problem was obtained. Its peculiar and
most remarkable feature consists in the fact that the partial differential
equation is put aside and replaced by a new sort of equation, namely,
the integral equation. This new method makes the matter as clear
as it was formerly obscure. In many circumstances in modern analysis,
contrary to the usual point of view, the operation of integration proves
a much simpler one than the operation of derivation. An example of
this is given by integral equations where the unknown function is
written under such signs of integration and not of differentiation. The
type of equation which is thus obtained is much easier to treat than
the partial differential equation. The type of integral equations
corresponding to the plane Dirichlet problem is

(i) 0(*) - X f B0(y) K (x,y) dy=f(x\

J A

where 0 is the unknown function of x in the interval (A, B),/and K
are known functions, and X is a known parameter. The equations
of the elliptic type in many-dimensional space give similar integral
equations, containing however multiple integrals and several inde-
pendent variables. Before the introduction of equations of the above
type, each step in the study of elliptic partial differential equations
seemed to bring with it new difficulties. . . . [But] an equation such
as (i) . . . gives all the required results at once and for all the pos-
sible types of such problems. . . . Previously, in the calculation of

1 Trans. Am. Math. Soc., Vol. 14, 1913, p. 209.

1 J. Hadamard, Four Lectures on Mathematics delivered at Columbia'University in
, New York, 1915, pp. 12-15.

ANALYSIS 393

the resonance of a room filled with air, the shape of the resonator
had to be quite simple, which requirement is not a necessary one for
the case where integral equations are employed. We need only make
the elementary calculation of the function K and apply to the function
so calculated the general method of resolution of integral equations."

The new departure in analysis was made in 1900 by Eric Ivar
Fredholm (1866- ), a native of Stockholm, who in 1898 was decent
at the University of Stockholm and later became connected with the
imperial bureau of insurance. In a paper,1 " Sur une nouvelle methode
pour la resolution du probleme de Dirichlet," 1900, he studied an
integral equation from the point of view of an immediate generaliza-
tion of a system of linear equations. Integral equations bear the
same relation to the integral calculus as differential equations do to
the differential calculus. Before this time certain integral equations
had received the attention of N. H. Abel, J. Liouville, and Eugene
Rouche (1832-1910) of Paris, but were quite neglected.

Abel had in 1823 proposed a generalization of the tautochrone
problem, the solution of which involved an integral equation that
has since been designated as of the first kind. Liouville in 1837
showed that a particular solution of a linear differential equation of
the second order could be found by solving an integral equation,
now designated as of the second kind. A method of solving integral
equations of the second kind was given by C. Neumann (1887). The
term " integral equation" is due to P. du Bois-Reymond (Crelle,
Vol. 103, 1888, p. 228) who exemplified the danger of making pre-
dictions by the declaration that "the treatment of such equations
seems to present insuperable difficulties to the analysis of to-day."
The recent theory of integral equations owes its origin to specific
problems in mechanics and mathematical physics. Since 1900 these
equations have been used in the study of existence theorems in the
theory of potential; they were employed in 1904 by W. A. Stekloff
and D. Hilbert in the consideration of boundary values and in matters
relating to Fourier series, by Henri Poincare in the study of tides and
Hertzian waves. Linear integral equations present many analogies
with linear algebraic equations. While E. I. Fredholm used the theory
of algebraic equations merely to suggest theorems on integral equa-
tions, which were proved independently, D. Hilbert in his early work
on this subject followed the process of taking limits in the results of
algebraic theory. Hilbert has introduced the term "kernel" of linear
integral equations of the first and second kind. The theory of integral
equations has been advanced by Erhard Schmidt of Breslau and
Vito Volterra of Rome. Systematic treatises on integral equations
have been prepared by Maxime Bocher of Harvard (1909), Gerhard
Kowalewski of Prag (1909), Adolf Kneser of Breslau (1911), T.
Lalesco of Paris (1912), H. B. Heywood, and M. Frechet (1912).
lQfversigt af akadcmicns forhandlingar 57, Stockholm, 1900.

394 A HISTORY OF MATHEMATICS

Maxime Bocher (1867-1918) was born in Boston and graduated
at Harvard in 1888. After three years of study at Gottingen he re-
turned to Harvard where he was successively instructor, assistant-
professor and professor of mathematics. He was president of the
American Mathematical Society in 1909-1910. Among his works are
Relhenentwickdungen der Potenlial-theorie, 1891, enlarged in 1894, and
Lemons sur les Melhodes de Sturm, containing the author's lectures de-
livered at the Sorbonne in 1913-1914.

A. Voss in 1913 stresses the value of integral equations thus: l
"In the last ten years . . . the theory of integral equations has
attained extraordinary importance, because through them problems
in the theory of differential equations may be solved which previously
could be disposed of only in special cases. We abstain from sketching
their theory, which makes use of infinite determinants that belong
to linear equations with an infinite number of unknowns, of quadratic
forms with infinitely many variables, and which has succeeded in
throwing new light upon the great problems of pure and applied
mathematics, especially of mathematical physics."

Important advances along the line of a "general analysis" and its
application to a generalization of the theory of linear integral equa-
tions have been made since 1906 by E. H. Moore of the University
of Chicago.2 From the existence of analogies in different theories he
infers the existence of a general theory comprising the analogous
theories as special cases. He proceeds to a "unification," resulting,
first, from the recent generalization of the concept of independent
variable effected by passing from the consideration of variables defined
for all points in a given interval to that of variables defined for all
points in any given set of points lying in the range of the variable,
secondly, from the consideration of functions of an infinite as well
as a finite number of variables, and, thirdly, from a still further gen-
eralization which leads him to functions of a "general variable."
E. H. Moore's general theory includes as special cases the theories of
E. I. Fredholm, D. Hilbert, and E. Schmidt. G. D. Birkhoff in 1911
presented the following birds'-eye view of recent movements: 3 "Since
the researches of G. W. Hill, V. Volterra, and E. I. Fredholm in the
direction of extended linear systems of equations, mathematics has
been in the way of great development. That attitude of mind which
conceives of the function as a generalized point, of the method of
successive approximation as a Taylor's expansion in a function va-
riable, of the calculus of variations as a limiting form of the ordinary
algebraic problem of maxima and minima is now crystallizing into a
new branch of mathematics under the leadership of S. Pincherle, J.

1 A. Voss, Ueber das Wesen d. Math., 1913, p. 63.

1 See Proceed. 5th Intern. Congress of Mathematicians, Cambridge, 1913, Vol. I,
p. 230.

3 Bull. Am. Math. Soc., Vol. 17, 1911, p. 415.

ANALYSIS 395

Hadamard, D. Hilbert, E. H. Moore, and others. For this field
Professor Moore proposes the term 'General Analysis,' defined as
'the theory of systems of classes of functions, functional operations,
etc., involving at least one general variable on a general range.' He
has fixed attention on the most abstract aspect of this field by con-
sidering functions of an absolutely general variable. The nearest
approach to a similar investigation is due to M. Frechet (Paris thesis,
1906), who restricts himself to variables for which the notion of a
limiting value is valid." Researches along the line of E. H. Moore's
"General Analysis" are due to A. D. Pitcher of Adelbert College and
E. W. Chittenden of the University of Illinois. In his "General
Analysis" Moore defines "complete independence" of postulates
which has received the further attention of E. V. Huntington, R. D.
Beetle, L. L. Dines, and M. G. Gaba.

V. Volterra discusses integro-differential equations which involve
not only the unknown functions under signs of integration but also
the unknown functions themselves and their derivatives, and shows
their use in mathematical physics. G. C. Evans of the Rice Institute
extended A. L. Cauchy's existence theorem for partial differential
equations to integro-differential equations of the "static type" hi
which the variables of differentiation are different from those of in-
tegration. Mixed linear integral equations have been discussed by
W. A. Hurwitz of Cornell University.

The study of integral equations and the theory of point sets has led
to the development of a body of theory called functional calculus.
One part of this is the theory of the functions of a line. As early as
1887 Vito Volterra of the University of Rome developed the funda-
mental theory of what he called functions depending on other func-
tions and functions of curves. Any quantity which depends for its
value on the arc of a curve as a whole is called a function of the line.
The relationships of functions depending on other functions are
called "fonctionelles" by J. Hadamard in his Leqons sur le calcul des
•variations, 1910, and "functional" by English writers. Functional
equations and systems of functional equations have received the atten-
tion of Griffith C. Evans of the Rice Institute, Luigi Sinigallia of
Pavia, Giovanni L. T. C. Giorgi of Rome, A. R. Schweitzer of Chicago,
Eric H. Neville of Cambridge, and others. Neville solves the race-
course puzzle of covering a circle by a set of five circular discs. Says
G. B. Mathews: x "We must express our regret that English math-
ematics is so predominantly analytical. Cannot some one, for in-
stance, give us a truly geometrical theory of J. V. Poncelet's poristic
polygons, or of von Staudt's thread-constructions for conicoids?" In
the theory of functional equations, "a single equation or a system of
equations expressing some property is taken as the definition of a
class of functions whose characteristics, particular as well as collective,

1 Nature, Vol. 97, 1916, p. 398.

396 A HISTORY OF MATHEMATICS

are to be developed as an outcome of the equations" (E. B. Van
Vleck).

An important generalization of Fourier series has been made,
"and we have a great class of expansions in the so-called orthogonal
and biorthogonal functions arising in the study of differential and
integral equations. In the field of differential equations the most
important class of these functions was first defined in a general and
explicit manner (in 1907) by . . . G. D. Birkhoff of Harvard Univer-
sity; and their leading fundamental properties were developed by
him." 1 In boundary value problems of differential equations which
are not self-adjoint, biorthogonal systems of functions play the same
r61e as the orthogonal systems do in the self-adjoint case. Anna J.
Pell established theorems for biorthogonal systems analogous to those
of F. Riesz and E. Fischer for orthogonal systems.

Theories of Irrationals and Theory of Aggregates

The new non-metrical theories of the irrational were called forth
by the demands for greater rigor. The use of the word "quantity"
as a geometrical magnitude without reference to number and also
as a number which measures some magnitude was disconcerting, es-
pecially as there existed no safe ground for the assumption that the
same rules of operation applied to both. The metrical view of number
involved the entire theory of measurement which assumed greater
difficulties with the advent of the non-Euclidean geometries. In at-
tempts to construct arithmetical theories of number, irrational num-
bers were a source of trouble. It was not satisfactory to operate with
irrational numbers as if they were rational. What are irrational
numbers? Considerable attention was puid to the definition of them
as limits of certain sequences of rational numbers. A. L. Cauchy in
his Cours d'Analyse, 1821, p. 4, says "an irrational number is the
limit of diverse fractions which furnish more and more approximate
values of it." Probably Cauchy was satisfied of the existence of
irrationals on geometric grounds. If not, his exposition was a rea-
soning in a circle. To make this plain, suppose we have a develop-
ment of rational numbers and we desire to define limit and also irra-
tional number. With Cauchy we may say that "when the successive
values attributed to a variable approach a fixed value indefinitely
so as to end by differing from it as little as is wished, this fixed value
is called the limit of all the others." Since we are still confined to
the field of rational numbers, this limit, if not rational, is non-existent
and fictitious. If now we endeavor to define irrational number as a
limit, we encounter a break-down in our logical development. It
became desirable to define irrational number arithmetically without
reference to limits. This was achieved independently and at almost

1 R. D. Carmichael in Science, N. S., Vol. 45, 1917, p. 471.

ANALYSIS 397

the same time by four men, Charles Meray (1835-1911), K. Weier-
strass, R. Dedekind, and Georg Cantor. Meray 's first publication was
in the Revue des societes savantes: sc. math. (2) 4, 1869, p. 284; his later
publications were in 1872, 1887, 1894. Meray was born at Chalons
in France and was professor at the University of Dijon. The earliest
publication of K. Weierstrass's presentation was made by H. Kossak
in 1872. In the same year it was published in Crelle's Journal, Vol. 74,
p. 174, by E. Heine who had received it from Weierstrass by oral
communication. R. Dedekind's publication is Stetigkeit und irra-
tionale Zahlen, Braunschweig, 1872. In 1888 appeared his Was sind
und was sollen die Zahlen? Richard Dedekind (1831-1916) was bora
in Braunschweig, studied at Gottingen and in 1854 became privat-
docent there. From 1858 to 1862 he was professor at the Polytech-
nicum in Zurich as the successor of J. L. Raabe, and from 1863 to
1894 professor at the Technical High School in Braunschweig. He
worked on the theory of numbers. The substance of his Stetigkeit und
Irrationale Zahlen, was worked out by him before he left Zurich. He
worked also in function theory. Georg Cantor's first printed statement
was in Mathem. Annalen, Vol. 5, 1872. p. I23.1 Georg Cantor (1845-
1918) was born at Petrograd, lived from 1856 to 1863 in South Ger-
many, studied from 1863 to 1869 at Berlin where he came under the
influence of Weierstrass. While in Berlin he once defended the re-
markable thesis: In mathematics the art of properly stating a question
is more important than the solving of it (In re mathematica ars pro-
ponendi quaestionem pluris facienda est quam solvendi). He became
privatdocent at Halle in 1869, extraordinary professor in 1872 and
ordinary professor in 1879. In recent years he has suffered from ill
health, taking the form of mental disturbances. When emerging
from such attacks, his mind is said to be most productive of scientific
results. Nearly all his papers are on the development of the theory,
of aggregates. It had been planned to hold on March 3, 1915, an
international celebration of his seventieth birthday, but on account
of the war, only a few German friends gathered at Halle to do him
honor.

In the theories of Ch. Meray and G. Cantor the irrational number
is obtained by an endless sequence of rational numbers a\, a2, a3, . . .
which have the property | an—am \ <€, provided n and m are sufficiently
great. The method of K. Weierstrass is a special case of this. R.
Dedekind defined every "cut" in the system of rational numbers to
be a number, the "open cuts" constituting the irrational numbers.
To G. Cantor and Dedekind we owe the important theory of the linear
continuum which represents the culmination of efforts which go back
to the church fathers of the Middle Ages and the writings of Aristotle.
By this modem continuum, "the notion of number, integral or frac-

1 For details see Encydoptdie des sciences mathimatiques, Tome I, Vol. I, 1904.
§§ 6-8, pp. I47-155-

398 A HISTORY OF MATHEMATICS

tional, has been placed upon a basis entirely independent of measur-
able magnitude, and pure analysis is regarded as a scheme which deals
with number only, and has, per se, no concern with measurable quan-
tity. Analysis thus placed upon an arithmetical basis is characterized
by the rejection of all appeals to our special intuitions of space, time
and motion, in support of the possibility of its operations" (E. W.
Hobson). The arithmetization of mathematics, which was in progress
during the entire nineteenth century, but mainly during the time of
Ch. Meray, L. Kronecker, and K. Weierstrass, was characterized by
E. W. Hobson in 1902 in the following terms: l "In some of the text-
books in common use in this country, the symbol °o is still used as if
it denoted a number, and one in all respects on a par with the finite
numbers. The foundations of the integral calculus are treated as if
Riemann had never lived and worked. The order in which double
limits are taken is treated as immaterial, and in many other respects
the critical results of the last century are ignored. . . .

"The theory of exact measurement in the domain of the ideal ob-
jects of abstract geometry is not immediately derivable from intuition,
but is now usually regarded as requiring for its development a previous
independent investigation of the nature and relations of number.
The relations of number having been developed on an independent
basis, the scheme is applied by the help of the principle of congruency,
or other equivalent principle, to the representation of extensive or
intensive magnitude. . . . This complete separation of the notion
of number, especially fractional number, from that of magnitude,
involves, no doubt, a reversal of the historical and psychological
orders. . . . The extreme arithmetizing school, of which, perhaps,
L. Kronecker was the founder, ascribes reality, whatever that may
mean, to integral numbers only, and regards fractional numbers as
possessing only a derivative character, and as being introduced only
for convenience of notation. The ideal of this school is that every
theorem of analysis should be interpretable as giving a relation be-
tween integral numbers only. . . .

"The true ground of the difficulties of the older analysis as regards
the existence of limits, and in relation to the application to measur-
able quantity, lies in its inadequate conception of the domain of
number, in accordance with which the only numbers really defined
were rational numbers. This inadequacy has now been removed by
means of a purely arithmetical definition of irrational numbers, by
means of which the continuum of real numbers has been set up as
the domain of the independent variable in ordinary analysis. This
definition has been given in the main in three forms — one by E. Heine
and G. Cantor, the second by R. Dedekind, and the third by K. Weier-
strass. Of these the first two are the simplest for wrorking purposes,
and are essentially equivalent to one another; the difference between
1 Proceed-. London Math. Soc., Vol. 35, 1902, pp. 117-139; see p. 118.

ANALYSIS 399

them is that, while Dedekind defines an irrational number by means
of a section of all the rational numbers, in the Heine-Cantor form of
definition a selected convergent aggregate of such numbers is em-
ployed. The essential change introduced by this definition of irra-
tional numbers is that, for the scheme of rational numbers, a new
scheme of numbers is substituted, in which each number, rational or
irrational, is defined and can be exhibited in an indefinitely great
number of ways, by means of a convergent aggregate of rational
numbers. . . . By this conception of the domain of number the root
difficulty of the older analysis as to the existence of a limit is turned,
each number of the continuum being really defined in such a way that
it itself exhibits the limit of certain classes of convergent sequences. . . .
It should be observed that the criterion for the convergence of an
aggregate is of such a character that no use is made in it of infinitesi-
mals, definite finite numbers alone being used in the tests. The old
attempts to prove the existence of limits of convergent aggregates
were, in default of a previous arithmetical definition of irrational
number, doomed to inevitable failure. ... In such applications of
analysis — as, for example, the rectification of a curve — the length of
the curve is defined by the aggregate formed by the lengths of a proper
sequence of inscribed polygons. ... In case the aggregate is not
convergent, the curve is regarded as not rectifiable. . . .

" It has in fact been shown that many of the properties of functions,
such as continuity, differentiability, are capable of precise definition
when the domain of the variable is not a continuum, provided, how-
ever, that domain is perfect; this has appeared clearly in the course
of recent investigations of the properties of non-dense perfect aggre-
gates, and of functions of a variable whose domain is such an aggre-
gate."

In 1912 Philip E. B. Jourdain of Fleet, near London, characterized
theories of the irrational substantially as follows:1 "Dedekind's
theory had not for its object to prove the existence of irrationals:
it showed the necessity, as Dedekind thought, for the mathematician
to create them. In the idea of the creation of numbers, Dedekind
was followed by O. Stolz; but H. Weber and M. Pasch showed how
the supposition of this creation could be avoided: H. Weber defined
real numbers as sections (Schnitte) in the series of rationals; M. Pasch
(like B. Russell) as the segments which generate these sections. In
K. Weierstrass' theory, irrationals were defined as classes of rationals.
Hence B. Russell's objections (stated in his Principles of Mathematics,
Cambridge, 1903,^. 282) do not hold against it, nor does Russell
seem to credit Weierstrass and Cantor with the avoidance of quite
the contradiction that they did avoid. The real objection to Weier-
strass' theory, and one of the objections to G. Cantor's theory, is

1 P. E. B. Jourdain, "On Isoid Relations and Theories of Irrational Number" in
International Congress of Mathematicians, Cambridge, 1912.

400 A HISTORY OF MATHEMATICS

that equality has to be re-defined. In the various arithmetical theories
of irrational numbers there are three tendencies: (a) the number is
defined as a logical entity — a class or an operation — , as with K.
Weierstrass, H. Weber, M. Pasch, B. Russell, M. Fieri; (b) it is
"created," or, more frankly, postulated, as with R. Dedekind, 0.
Stolz, G. Peano, and Ch. Meray; (c) it is defined as a sign (for what,
is left indeterminate), as with E. Heine, G. Cantor, H. Thomae, A.
Pringsheim. ... In the geometrical theories, as with Paul du Bois-
Reymond, a real number is a sign for a length. In B. Russell's theory
it appears to be equally legitimate to define a real number in various
ways."

The theory of aggregates (Mengenlehre, theorie des ensembles,
theory of sets) owes its development to the endeavor to clarify the
concepts of independent variable and of function. Formerly the
notion of an independent variable rested on the naive concept of the
geometric continuum. Now the independent variable is restricted "
to some aggregate of values or points selected out of the continuum.
The term function was destined to receive various definitions. J.
Fourier advanced the theorem that an arbitrary function can be
represented by a trigonometric series. P. G. L. Dirichlet looked upon
the general functional concept as equivalent to any arbitrary table of
values. When G. F. B. Riemann gave an example of a function ex-
pressed analytically which was discontinuous at each rational point,
the need of a more comprehensive theory became evident. The first
attempts to meet the new needs were made by Hermann Hankel and
Paul du Bois-Reymond. The Allgemeine Funktionentheorie of du
Bois-Reymond brilliantly sets forth the problems in philosophical
form, but it remained for Georg Cantor to advance and develop the
necessary ideas, involving a treatment of infinite aggregates. Even
though the infinite had been the subject of philosophic contemplation
for more than two thousand years, G. Cantor hesitated for ten years
before placing his ideas before the mathematical public. The theory
of aggregates sprang into being, as a science, when G. Cantor intro-
duced the notion of "enumerable" aggregates.1 G. Cantor began his
publications in 1870; in 1883 he published his Grundlagen einer all-
gemeinen Mannichfaltigkeitslehre. In 1895 and I&97 appeared in
Mathematische Annalen his Beitrage zur Begrundung der transfiniten
Mengenlehre. ,2 These researches have played a most conspicuous role
not only in the march of mathematics toward logical exactitude, but
also in the realm of philosophy.

G. Cantor's theory of the continuum was used by P. Tannery in
1885 in the search for a profounder view of Zeno's arguments against

1 A. Schoenflies, Entwickelung der Mengenlehre und ihrer Anwndungen, gemeinsam
mil Hans Hahn hcrausgegeben , Leipzig u. Berlin, 1913, p. 2.

2 Translated into English by Philip E. B. Jourdain and published by the Open
Court Publ. Co., Chicago, 1915.

ANALYSIS 401

motion. Paul Tannery (1843-1904), a brother of Jules Tannery,
attended the ficole Polytechnique in Paris and then entered the state
corps of manufacturing engineers. He devoted his days to business
and his evenings to the study of the history of science. From 1892
to 1896 he held the chair of Greek and Latin philosophy at the College
de France, but later he failed to receive the appointment to the chair
of the history of the sciences, although he was the foremost French
historian of his day. He was a deep student of Greek scientists, par-
ticularly of Diophantus. Other historical periods were taken up after-
wards, particularly that of R. Descartes and P. Fermat. His re-
searches, consisting mostly of separate articles, are being republished
in collected form.

In 1883 G. Cantor stated that every set and in particular the con-
tinuum can be well-ordered. In 1904 and 1907 E. Zermelo gave
proofs of this theorem, but they have not been generally accepted and
have given rise to much discussion. G. Peano objected to Zermelo's
proof because it rests on a postulate ("Zermelo's principle") expressing
a property of the continuum. In 1907 E. Zermelo formulated that
postulate thus: "A set S which is divided into subsets A, B, C, . . .,
each containing at least one element, but containing no elements in
common, contains at least one subset S, which has just one element in
common with each of the subsets A, B, C, . . ." (Math. Ann., Vol.
56, p. no). To this G. Peano objects that one may not apply an
infinite number of times an arbitrary law by virtue of which one cor-
relates to a class some member of that class. E. B. Wilson comments
on this: "Here are two postulates by two different authorities; th?
postulates are contradictory, and each thinker is at liberty to adopt
whichever appears to him the more convenient." Zermelo's postulate,
before it had been formulated, was tacitly assumed in the researches
of R. Dedekind, G. Cantor, F. Bernstein, A. Schonflies, J. Konig, and
others. Zermelo's proof was rejected by H. Poincare, E. Borel, R.
Baire, V. A. Lebesgue.1 A third proof that every set can be well-
ordered was given in 1915 by Friedrich M. Hartogs of Munich. P.
E. B. Jourdain of Fleet did not consider this proof altogether satis-
factory. In 1918 he gave one of his own (Mind, 27, 386-388) and
declared, " thus any aggregate can be well ordered, Zermelo's ' axiom '
can be proved quite generally, and Hartog's work is completed."

In 1897 Cesare Burali-Forti of Turin pointed out the following para-
dox: The series of ordinal numbers, which is well-ordered, must have
the greatest of all ordinal numbers as its order type. Yet the type of
the above series of ordinal numbers, when followed by its type, must
be a greater ordinal number, for /3+i is greater than j3. Therefore, a
well-ordered series of ordinal numbers containing all ordinal numbers
itself defines a new ordinal number not included in the original series.

Another paradox, due to Jules Antoine Richard of Chateauroux
1 See Bulletin Am. Math. Soc., Vol. 14, p. 438.

402 A HISTORY OF MATHEMATICS

in 1906, relates to the aggregate of decimal fractions between o and i
which can be defined by a finite number of words; a new decimal frac-
tion can be defined, which is not included in the previous ones.

Bertrand Russell discovered another paradox, given in his Prin-
ciples of Mathematics, 1903, pp. 364-368, 101-107), which is stated by
Philip E. B. Jourdain thus: "If w is the class of all those terms x such
that x is not a member of x, then, if w is a member of w , it is plain that
w is not a member of w; while if u> is not a member of w, it is equally
plain that w is a member of w." 1 These paradoxes are closely allied
to the "Epimenides puzzle": Epimenides was a Cretan who said that
all Cretans were liars. Hence, if his statement was true he was a liar.
H. Poincare and B. Russell attribute the paradoxes to the open and
clandestine use of the word "all." The difficulty lies in the definition
of the word "Menge."

Noteworthy among the attempts to place the theory of aggregates
upon a foundation that will exclude the paradoxes and antinomies that
had arisen, was the formulation in 1907 of seven restricting axioms
by E. Zermelo in Math. Annalen, 65, p. 261.

Julius Konig (1849-1913), the Hungarian mathematician, in his
Neue Grundlagen der Logik, Arithmetik und Mengenlehre, 1914, speaks
of E. Zermelo 's axiom of selection (Auswahlaxiom) as being really
a logical assumption, not an axiom in the old sense, whose freedom
from contradiction must be demonstrated along with the other
axioms. He takes pains to steer clear of the antinomies of B. Russell
and C. Burali-Forti. For a discussion of the logical and philosophical
questions involved in the theory of aggregates, consult the second
edition of E. Borel's Lemons sur la theorie des fonctions, Paris, 1914,
note IV, which gives letters written by J. Hadamard, E. Borel, H.
Lebesgue, R. Baire, touching the validity of Zermelo's demonstration
that the linear continuum is well-ordered. A set of axioms of ordinal
magnitude was given by A. B. Frizell in 1912 at the Cambridge
Congress.

In the treatment of the infinite there are two schools. Georg Cantor
proved that the continuum is not denumerable; J. A. Richard, con-
tending that no mathematical entity exists that is not definable in a
finite number of words, argued that the continuum is denumerable.
H. Poincare claimed that this contradiction is not real, since J. A.
Richard employs a non-predicative definition.2 H. Poincare,3 in
discussing the. logic of the infinite, states that, according to the first
school, the pragmatists, the infinite flows out of the finite; there is an
infinite, because there is an infinity of possible finite things. Accord-
ing to the second school, the Cantorians, the infinite precedes the
finite; the finite is obtained by cutting off a small piece of the infinite.

1 P. E. B. Jourdain, Contributions, Chicago, 1915, p. 206.

2 Bull. Am. Math. Soc., Vol. 17, 1911, p. 193.

3 Scientia, Vol. 12, 1912, pp. i-n.

ANALYSIS 403

For pragmatists a theorem has no meaning unless it can be verified;
they reject indirect proofs of existence; hence they reply to E. Zermelo
who proves that space can be converted into a well-ordered aggregate
(wohlgeordnete Menge): Fine, convert it! We cannot carry out this
transformation because the number of operators is infinite. For
Cantorians mathematical things exist independently of man who may
think about them; for them cardinal number is no mystery. On the
other hand, pragmatists are not sure that any aggregate has a cardinal
number, and when they say that the Machtigkeit of the continuum
is not that of the whole numbers, they mean simply that it is im-
possible to set up a correspondence between these two aggregates,
which could not be destroyed by the creation of new points in space.
If mathematicians are ordinarily agreed among themselves, it is
because of confirmations which pass final judgment. In the logic of
infinity there are no confirmations.

L. E. J. Brouwer of the University of Amsterdam, expressing views
of G. Mannoury, said in 1912 that to the psychologist belongs the
task of explaining "why we are averse to the so-called contradictory
systems in which the negative as well as the positive of certain propo-
sitions are valid," that "the intuitionist recognizes only the existence
of denumerable sets" and "can never feel assured of the exactness of
mathematical theory by such guarantees as the proof of its being
non-contradictory, the possibility of defining its concepts by a finite
number of words, or the practical certainty that it will never lead to
misunderstanding in human relations." A. B. Frizell showed in
1914 that the field of denumerably infinite processes is not a closed
domain — a concept which the intuitionist refuses to recognize, but
which "need not disturb an intuitionist who cuts loose from the prin-
cipium contradictionis." More recent tendencies of research in this
field are described by E. H. Moore:2 "From the linear continuum
with its infinite variety of functions and corresponding singularities
G. Cantor developed his theory of classes of points (Punktmengenlehre)
with the notions: limit-point, derived class, closed class, perfect class,
etc., and his theory of classes in general (allgemeine Mengenlehre) with
the notions: cardinal number, ordinal number, order- type, etc. These
theories of G. Cantor are permeating Modern Mathematics. Thus
there is a theory of functions on point-sets, in particular, on perfect
point-sets, and on more general order-types, while the arithmetic
of cardinal numbers and the algebra and function theory of ordinal
numbers are under development.

"Less technical generalizations or analogues of functions of the
continuous real variable occur throughout the various doctrines and
applications of analysis. A function of several variables is a function
of a single multipartite variable; a distribution of potential or a field

1 L. E. J. Brouwer in Bull. Am. Math. Soc., Vol. 20, 1013, pp. 84, 86.

2 New Haven Colloquium, 1906, New Haven, 1910, p. 2-4.

404 A HISTORY OF MATHEMATICS

of force is a function of position on a cuve or surface or region; the
value of the definite integral of the Calculus of Variations is a func-
tion of the variable function entering the definite integral; a curvilinear
integral is a function of the path of integration; a functional operation
is a function of the argument function or functions; etc., etc.

"A multipartite variable itself is a function of the variable index
of the part. Thus a finite sequence: xi; . . .; xn, of real numbers is a
function x of the index i, viz., x(i)=xt (i=i'} . . .; «). Similarly,
an infinite sequence: x\; . . .; xn; . . ., of real numbers is a function x
of the index n, viz., x(n)=xn («=i; 2; . . .). Accordingly, w-fold
algebra and the theory of sequences and of series are embraced in
the theory of functions.

"As apart from the determination and extension of notions and
theories in analogy with simpler notions and theories, there is the
extension by direct generalization. The Cantgr movement is in this
direction. Finite generalization, from the case n=i to the case n=n,
occurs throughout Analysis, as, for instance, in the theory of func-
tions of several independent variables. The theory of functions of a
denumerable infinity of variables is another step in this direction.1
We notice a more general theory dating from the year 1906. Recog-
nizing the fundamental role played by the notion limit-element (num-
ber, point, function, curve, etc.) in the various special doctrines, M.
Frechet has given, with extensive applications, an abstract generaliza-
tion of a considerable part of Cantor's theory of classes of points and
of the theory of continuous functions on classes of points. Frechet
considers a general class P of elements p with the notion limit defined
for sequences of elements. The nature of the elements p is not speci-
fied; the notion limit is not explicitly defined; it is postulated as de-
fined subject to specified conditions. For particular applications
explicit definitions satisfying the conditions are given. . . . The
functions considered are either functions ^ of variables p of specified
character or functions /j. on ranges P with postulated features: e. g.
limit; distance; element of condensation; connection, of specified char-
acter." E. H. Moore's own form of general analysis of 1906 considers
functions n of a general variable p on a general range P, where this
general embraces every well-defined particular case of variable and
range.

Early in the development of the theory of point sets it was pro-
posed to associate with them numbers that are analogous to those
representing lengths, areas, volumes.2 On account of the great arbi-
trariness of this procedure, several different definitions of such num-
bers have been given. The earliest were given in 1882 by H. Hankel
and A. Harnack. Another definition due to G. Cantor (1884) was
generalized by H. Minkowski in 1900. More precise measures were

1 D. Hilbert, 1906, 1909.

2 Encyclopedic des sciences mathtmaliques , Tome II, Vol. i, 1912, p. 150.

ANALYSIS 405

assigned in 1893 by C. Jordan in his Cours a" analyse, which, for plane
sets, was as follows: If the plane is divided up into squares whose
sides are 5 and if 5 is the sum of the squares all interior points of which
belong to the set P, if, moreover, S+S' is the sum of all squares which
contain points of P, then, as 5 approaches zero, S and S+S' converge
to limits / and A, called the "interior" and "exterior" areas. If
I=A, then P is said to be "measurable." Examples of plane curves
whose exterior areas were not zero were given in 1903 by W. F. Osgood
and H. Lebesgue. Another definition, given by E. Borel, was gen-
eralized by H. Lebesgue so as to present fewer inconveniences than
the older ones. The existence of non-measurable sets, according to
Lebesgue's definition, was proved by G. Vitali and Lebesgue himself;
the proofs assume E. Zermelo's axiom.

Instead of sets of points, E. Borel in 1903 began to study sets of
lines or planes; G. Ascoli considered sets of curves. M. Frechet in
1906 proposed a generalization by establishing general properties
without specifying the nature of the elements, and was led to the
so-called "calcul fonctionnel," to which attention has been called
before.

Functional equations, in which the unknown elements are one or
more functions, have received renewed attention in recent years. In
the eighteenth century certain types of them were treated by D'Alem-
bert, L. Euler, J. Lagrange, and P. S. Laplace.1 Later came the
"calculus of functions," studied chiefly by C. Babbage, J. F. W.
Herschel, and A. De Morgan, which was a theory of the solution of
functional equations by means of known functions or symbols. A. M.
Legendre and A. L. Cauchy studied the functional equation f(x+y) =
f(x)+f(y); which recently has been investigated by R. Volpi, G.
Hamel, and R. Schimmack. Other equations, f(x+y)=f(x).f(y),
f(xy)-f(x).f(y}, and <p(y-\-x)+<p(y — x~) = 2<p(x).<p(y), the last being
D'Alembert's, were treated by A. L. Cauchy, R. Schimmack, and
J. Andrade. The names of Ch. Babbage, N. H. Abel, E. Schroder, J.
Farkas, P. Appell, and E. B. Van Vleck are associated with this subject.
Said S. Pincherele of Bologna in 1912: "The study of certain classes
of functional equations has given rise to some of the most important
chapters of analysis. It suffices to cite the theory of differential or
partial differential equations. . . . The theory of equations of finite
differences, simple or partial, . . . the calculus of variations, the
theory of integral equations . . . the integrodifferential equations
recently considered by V. Volterra, are as many chapters of mathe-
matics devoted to the study of functional equations."

The theory of point sets led in 1902 to a generalization of G. F. B.
Riemann's definition of a definite integral by Henri Lebesgue of
Paris. E. B. Van Vleck describes the need of this change:2 "This

1 Encyclopidie des sciences mathe'matiques, Tome IT, Vol. 5, 1912, p. 46.

2 Bull. Am. Math. Soc., Vol. 23, 1916, pp. 6, 7.

4o6 A HISTORY OF MATHEMATICS

(Riemann's integral) admits a finite number of discontinuities but
an infinite number only under certain narrow restrictions. A totally
discontinuous function — for example, one equal to zero in the rational
points which are everywhere dense in the interval of integration,
and equal to i in the rational points which are likewise everywhere
dense — is not integrable a la Riemann. The restriction became a
very hampering one when mathematicians began to realize that the
analytic world in which theorems are deducible does not consist
merely of highly civilized and continuous functions. In 1902 Lebesgue
with great penetration framed a new integral which is identical with
the integral of Riemann when the latter is applicable but is immeasur-
ably more comprehensive. It will, for instance, include the totally
discontinuous function above mentioned. This new integral of
Lebesgue is proving itself a wonderful tool. I might compare it with
a modern Krupp gun, so easily does it penetrate barriers which before
were impregnable." Instructive is also the description of this move-
ment, as given by G. A. Bliss: l "Volterra has pointed out, in the
introductory chapter of his Leqons sur les fonctions des lignes (1913),
the rapid development which is taking place in our notions of infinite
processes, examples of which are the definite integral limit, the solu-
tion of integral equations, and the transition from functions of a
finite number of variables to functions of lines. In the field of in-
tegration the classical integral of Riemann, perfected by Darboux,
was such a convenient and perfect instrument that it impressed itself
for a long time upon the mathematical public as being something
unique and final. The advent of the integrals of T. J. Stieltjes and H.
Lebesgue has shaken the complacency of mathematicians in this
respect, and, with the theory of linear integral equations, has given
the signal for a re-examination and extension of many of the types of
processes which Volterra calls passing from the finite to the infinite.
It should be noted that the Lebesgue integral is only one of the evi-
dences of this restlessness in the particular domain of the integration
theory. Other new definitions of an integral have been devised by
Stieltjes, W. H. Young, J. Pierpont, E. Bellinger, J. Radon, M.
Frechet, E. H. Moore, and others. The definitions of Lebesgue,
Young, and Pierpont, and those of Stieltjes and Hellinger, form two
rather well defined and distinct types, while that of Radon is a gen-
eralization of the integrals of both Lebesgue and Stieltjes. The
efforts of Frechet and Moore have been directed toward definitions
valid on more general ranges than sets of points of a line or higher
spaces, and which include the others for special cases of these ranges.
Lebesgue and H. Hahn, with the help of somewhat complicated
transformations, have shown that the integrals of Stieltjes and Hel-

1 G. A. Bliss, "Integrals of Lebesgue," Bull. Am. Math. Soc., Vol. 24, 1917, pp. i-
47. See also T. H. Hildebrandt, loc. cit., Vol. 24, pp. 113-144, who gives bibliogra-
phy.

ANALYSIS 407

linger are expressible as Lebesgue integrals. . . . Van Vleck has . . .
remarked that a Lebesgue integral is expressible as one of Stieltjes
by a transformation much simpler than that used by Lebesgue for
the opposite purpose, and the Stieltjes integral so obtained is readily
expressible in terms of a Riemann integral. . . . Furthermore the
Stieltjes integral seems distinctly better suited than that of Lebesgue
to certain types of questions, as is well indicated by the original
'problem of moments' of Stieltjes, or by a generalization of it which
F. Riesz has made. . . . The conclusion then seems to be that one
should reserve judgment, for the present at least, as to the final form
or forms which the integration theory is to take."

Mathematical Logic

Summarizing the history of mathematical logic, P. E. B. Jourdain
says: l "In somewhat close connection with the work of Leibniz . . .
stands the work of Johann Heinrich Lambert, who sought — not very
successfully — to develop the logic of relations. Toward the middle
of the nineteenth century George Boole independently worked out
and published his famous calculus of logic. . . . Independently of
him or anybody else, Augustus De Morgan began to work out logic
as a calculus, and later on, taking as his guide the maxim that logic
should not consider merely certain kinds of deduction but deduction
quite generally, founded all the essential parts of the logic of relations.
William Stanley Jevons criticised and popularized Boole's work; and
Charles S. Peirce (1839-1914), Mrs. Christine Ladd-Franklin, Richard
Dedekind, Ernst Schroder (1841-1902), Hermann G. and Robert
Grassmann, Hugh MacColl, John Venn, and many others, either
developed the work of G. Boole and A. De Morgan or built up systems
of calculative logic in modes which were largely independent of the
work of others. But it was in the work of Gottlob Frege, Guiseppe
Peano, Bertrand Russell, and Alfred North Whitehead, that we find
a closer approach to the lingua characteristica dreamed of by Leibniz."
We proceed to a few details.

"Pure mathematics," says B. Russell,2 "was discovered by Boole
in a work which he called The Laws of Thought (1854). . . . His
work was concerned with formal logic, and this is the same thing as
mathematics." George Boole (1815-1864) became in 1849 professor
in Queen's College, Cork, Ireland. He was a native of Lincoln, and
a self-educated mathematician of great power. In his boyhood he
studied, unaided, the classical languages.3 While teaching school he
pursued modern languages and entered upon the study of J. Lagrange

1 The Monist, Vol. 26, 1916, p. 522.

2 International Monthly, 1901, p. 83.

3 See A. Macfarlane, Ten British Mathematicians, New York, 1916. Boole's
Laws of Thought was republished in 1917 by the Open Court Publ. Co., under the
editorship of I1. E. B. Jourdain.

408 A HISTORY OF MATHEMATICS

and P. S. Laplace. His treatises on Differential Equations (1859), and
Finite Differences (1860) are works of merit.

A point of view different from that of G. Boole was taken by Hugh
MacColl (1837-1909) who was led to his system of symbolic logic by
researches on the theory of probability. While Boole used letters to
represent the times during which certain propositions are true, Mac-
Coll employed the proposition as the real unit in symbolic reasoning.1
When the variables in the Boolean algebra are interpreted as proposi-
tions, C. I. Lewis of the University of California worked out a matrix
algebra for implications.

When the investigation of the principles of mathematics became
the chief task of logical symbolism, the aspect of symbolic logic as a
calculus ceased to be of such importance. Friedricti Ludwig Gottlob
Frege (1848- ) of the University of Jena entered this field. Con-
sidering the foundations of arithmetic he inquired how far one could
go by conclusions which rest merely on the laws of general logic.
Ordinary language was found to be unequal to the accuracy required.
So knowing nothing of the work of his predecessors, except G. W.
Leibniz, he devised a symbolism and in 1879 published his Begriffs-
schrift, and in 1893 his Grundgesetze der Arithmetik. Says P. E. B.
Jourdain: "Frege criticised the notion which mathematicians denote
by the word 'aggregate' (Menge), and particularly the views of
Dedekind and Schroder. Neither of these authors distinguished the
subordination of a concept under a concept from the falling of an
object under a concept; a distinction upon which Peano rightly laid so
much stress, and which is, indeed, one of the most characteristic
features of Peano's system of ideography." Ernst Schroder of Karls-
ruhe had published in 1877 his Algebra der Logik and John Venn his
Symbolic Logic in 1881.

"Peano's first publication on mathematical logic followed the
lines of Schroder's work of 1877 very closely. An excellent exposition
in Peano's Calcolo geometrico secondo V Ausdehnungslehre di H. Grass-
mann, Turin, 1888, of the geometrical calculus of A. F. Mobius, H. G.
Grassmann, and others was preceded by an introduction treating of
the operations of deductive logic, which are very analogous to those
of ordinary algebra and of the geometrical calculus. The signs of
logic were sometimes used in the later parts of the book, though this
was not done systematically, as it was in many of Peano's later works "
(Jourdain). In 1891 appeared under G. Peano's editorship, the first
volume of the Rivista di Matematica which contains articles on mathe-
matical logic and its applications, but this kind of work was carried
on more fully in the Formulaire de mathematiques of which the first
volume was published in 1895. This was projected to be a classified
collection of mathematical truths, written wholly in Peano's symbols:

1 For details see Philip E. B. Jourdain in Quarterly Jour, of Math., Vol. 43, 1912,
p. 219.

ANALYSIS 409

it was prepared by Peano and his collaborators, C. Burali-Forti, G.
Viviania, R. Bettazzi, F. Giudice, F. Castellano, and G. Fano. "In
the later editions of the Formulaire," says P. E. B. Jourdain, "Peano
gave up all attempts to work out which are the primitive propositions
of logic; and the logical principles or theorems which are used in the
various branches of mathematics were merely collected together in
as small a space as possible. In the last edition (v., 1905), logic only
occupied 16 pages, while mathematical theories — a fairly complete
collection — occupied 463 pages. On the other hand, in the works of
Frege and B. Russell the exact enumeration of the primitive proposi-
tions of logic was always one of the most important problems." In
England mathematical logic has been strongly emphasized by Ber-
trand Russell who in 1903 published his Principles of Mathematics
and in 1910, in conjunction with A. N. Whitehead, brought out the
first volume of the Principia mathematica, a remarkable work. Rus-
sell and Whitehead follow in the main Peano in matters of notation,
Frege in matters of logical analysis, G. Cantor in the treatment of
arithmetic, and v. Staudt, M. Pasch, G. Peano, M. Fieri, and O. Veblen
in the discussion of geometry. By their theory of Logical types they
solve the paradoxes of C. Burali-Forti, B. Russell, J. Konig, J. A.
Richard, and others. Certain points in the logic of relations as given
in the Principia mathematica have been simplified by Norbert Wiener
in 1914 and 1915.

In France this subject has been cultivated chiefly by Louis Couturat
(1868-1914), who, at the time of his death in an automobile accident
in Paris, held high rank in the philosophy of mathematics and of
language. He wrote La logique de Leibniz and Les principes des mathe-
matiques, Paris, 1905. Sets of postulates for the Boolean algebra of
logic were given by A. N. Whitehead, which were simplified in 1904
by E. V. Huntington of Harvard and B. A. Bernstein of California.
Says P. E. B. Jourdain: "Frege's symbolism, though far better for
logical analysis than Peano's, for instance, is far inferior to Peano's —
a symbolism in which the merits of internationality and power of
expressing mathematical theorems are very satisfactorily attained —
in practical convenience. B. Russell, especially in the later works,
used the ideas of Frege, many of which he discovered subsequently
to, but independently of Frege, and modified the symbolism of Peano
as little as possible. Still, the complications thus introduced take
away that simple character which seems necessary to a calculus, and
which Boole and others reached by passing over certain distinctions
which a subtler logic has shown us must be made." *

In 1886 A. B. Kempe discussed the fundamental conceptions both
of symbolic logic and of geometry. Later he developed this subject
further in the study of the relations between the logical theory of

1 Philip E. B. Jourdain in Quart. Jour, of Math., Vol. 41, 1910, p. 331.

4io A HISTORY OF MATHEMATICS

classes and the geometrical theory of points. This topic received a
re-statement and an extension in 1905 from the pen of Josiah Royce
(1855-1916), professor of philosophy at Harvard University. Royce
contends that "the entire system of the relationships of the exact
sciences stands in a much closer connection with the simple principles
of symbolic logic than has thus far been generally recognized."

There exists divergence of opinion on the value of the notation of
the calculus of logic. Says A. Voss: ' "As far as I am able to survey
the practical results of mathematical logic, they run aground at every
real application, on account of the extreme complexity of its formulas;
by a comparatively large expenditure of effort they yield almost
trivial results, which, however, can be read off with absolute certainty.
Only in the discussion of purely mathematical questions, i. e. relations
between numbers, does it, in Peano's Formulaire . . ., prove itself
to be a real power, probably replaceable by no other mode of ex-
pression. By some even this is called into question."

Alessandro Padoa of the Royal Technical Institute of Genoa said
in 1912: 2 "I do not hope to suggest to you the sympathetic and touch-
ing optimism of Leibniz, who, prophesying the triumphal success of
these researches, affirmed: 'I dare say that this is the last effort of
the human mind, and, when this project shall have been carried out,
all that men will have to do will be to be happy, since they will have
an instrument that will serve to exalt the intellect not less than the
telescope serves to perfect their vision.' Although for some fifteen
years I have given myself up to these studies, I have not a hope so
hyperbolic; but I delight in recalling the candor of this master who,
absorbed in scientific and philosophic investigations, forgot that the
majority of men sought and continue to seek happiness in the feverish
conquest of pleasure, money, and honors. Meanwhile we should
avoid an excessive scepticism, because always and everywhere, there
has been an elite — to day less restricted than in the past — which was
charmed by, and delights now in, all that raises one above the con-
fused troubles of the passions, into the imperturbable immensity of
knowledge, whose horizons become the more vast as the wings of
thought become more powerful and rapid."

In 1914 an international congress of mathematical philosophers was
held in Paris, with Emil Boutroux as president. Unfortunately the
great war nipped this promising new movement in the bud. Recent
books on the philosophy of mathematics are M. Winter's La methode
dans la philosophic des mathematics, Paris, 1911, Leon Brunschvicg's
Les elapes de la philosophic mathematique, Paris, 1912, and J. B.
Shaw's Lectures on the Philosophy of Mathematics, Chicago and Lon-
don, 1918.

1 A. Voss, Ueber das Wesender Mathematik, Leipzig u. Berlin, 2. Aufl., 1913,
p. 28.

2 Bull. Am. Math. Soc., Vol. 20, 1913, p. 98.

THEORY OF FUNCTIONS 411

Theory of Functions

We begin our sketch of the vast progress in the theory of functions
by considering investigations which center about the special class
called elliptic functions. These were richly developed by N. H. Abel
and C. G. J. Jacobi.

Niels Henrik Abel (1802-1829) was born at Findoe in Norway, and
was prepared for the university at the cathedral school in Christiania.
He exhibited no interest in mathematics until 1818, when B. Holmboe
became lecturer there, and aroused Abel's interest by assigning original
problems to the class. Like C. G. J. Jacobi and many other young
men who became eminent mathematicians, Abel found the first exer-
cise of his talent in the attempt to solve by algebra the general equa-
tion of the fifth degree. In 1821 he entered the University in Chris-
tiania. The works of L. Euler, J. Lagrange, and A. M. Legendre
were closely studied by him. The idea of the inversion of elliptic
functions dates back to this time. His extraordinary success in
mathematical study led to the offer of a stipend by the government,
that he might continue his studies in Germany and France. Leaving
Norway in 1825, Abel visited the astronomer, H. C. Schumacher, in
Hamburg, and spent six months in Berlin, where he became intimate
with August Leopold Crelle (1780-1855), and met J. Steiner. En-
couraged by Abel and J. Steiner, Crelle started his journal in 1826.
Abel began to put some of his work in shape for print. His proof
of the impossibility of solving the general equation of the fifth degree
by radicals, — first printed in 1824 in a very concise form, and difficult
of apprehension, — was elaborated in greater detail, and published in
the first volume. He investigated also the question, what equations
are solvable by algebra and deduced important general theorems
thereon. These results were published after his death. Meanwhile
E. Galois traversed this field anew. Abel first used the expression,
now called the "Galois resolvent"; Galois himself attributed the idea
of it to Abel. Abel showed how to solve the class of equations, now
called " Abelian." He entered also upon the subject of infinite series
(particularly the binomial theorem, of which he gave in Crelle's
Journal a rigid general investigation), the study of functions, and of
the integral calculus. The obscurities everywhere encountered by him
owing to the prevailing loose methods of analysis he endeavored to
clear up. For a short time he left Berlin for Freiberg, where he had
fewer interruptions to work, and it was there that he made researches
on hyperelliptic and Abelian functions. In July, 1826, Abel left
Germany for Paris without having met K. F. Gauss ! Abel had sent
to Gauss his proof of 1824 of the impossibility of solving equations of
the fifth degree, to which Gauss never paid any attention. This slight,
and a haughtiness of spirit which he associated with Gauss, prevented
the genial Abel from going to Gottingen. A similar feeling was enter-

4i2 A HISTORY OF MATHEMATICS

tained by him later against A. L. Cauchy. Abel remained ten months
in Paris. He met there P. G. L. Dirichlet, A. M. Legendre, A. L.
Cauchy, and others, but was little appreciated. He had already pub-
lished several important memoirs in Crelle's Journal, but by the
French this new periodical was as yet hardly known to exist, and
Abel was too modest to speak of his own work. Pecuniary embarrass-
ments induced him to return home after a second short stay in Berlin.
At Christiania he for some time gave private lessons, and served as
decent. Crelle secured at last an appointment for him at Berlin;
but the news of it did not reach Norway until after the death of
Abel at Froland.1 Ch. Hermite is said to have remarked: "Abel a
laisse aux mathematiciens de quoi travailler pendant cent cinquante
ans."

At nearly the same time with Abel, C. G. J. Jacobi published articles
on elliptic functions. A. M. Legendre's favorite subject, so long
neglected, was at last to be enriched by some extraordinary discov-
eries. The advantage to be derived by inverting the elliptic integral
of the first kind and treating it as a function of its amplitude (now
called elliptic function) was recognized by Abel, and a few months
later also by Jacobi. A second fruitful idea, also arrived at inde-
pendently by both, is the introduction of imaginaries leading to the
observation that the new functions simulated at once trigonometric
and exponential functions. For it was shown that while trigonometric
functions had only a real period, and exponential only an imaginary,
elliptic functions had both sorts of periods. These two discoveries
were the foundations upon which Abel and Jacobi, each in his own
way, erected beautiful new structures. Abel developed the curious
expressions representing elliptic functions by infinite series or quo-
tients of infinite products. Great as were the achievements of Abel
in elliptic functions, they were eclipsed by his researches on what are
now called Abelian functions. Abel's theorem on these functions was
given by him in several forms, the most general of these being that
in his Memoir e sur une propriete generate d'une classe tres-etendue de
f auctions transcendentes (1826). The history of this memoir is inter-
esting. A few months after his arrival in Paris, Abel submitted it to
the French Academy. A. L. Cauchy and A. M. Legendre were ap-
pointed to examine it; but said nothing about it until after Abel's
death. In a brief statement of the discoveries in question, published
by Abel in Crelle's Journal, 1829, reference is made to that memoir.
This led C. G. J. Jacobi to inquire of Legendre what had become of it.
Legendre says that the manuscript was so badly written as to be
illegible, and that Abel was asked to hand in a better copy, which he

1 C. A. Bjerknes, Niels-Benrik Abel, Tableau de sa vie et de son action scientifique,
Paris, 1885. See also Abel (N . H.) Memorial publie a I'occasion du ccntenaire de sa
naissance. Kristiania [1902]; also N. H. Abel. Sa vie et son Oeuvre, par Ch. Lucas de
Pesloiian, Paris, 1906.

THEORY OF FUNCTIONS 413

neglected to do. Others have attributed this failure to appreciate
Abel's paper to the fact that the French academicians were then in-
terested chiefly in applied mathematics — heat, elasticity, electricity.
S. D. Poisson having in a report on C. G. J. Jacobi's Fundamenta nova
recalled the reproach made by J. Fourier to Abel and Jacobi of not
having occupied themselves preferably with the movement of heat,
Jacobi wrote to Legendre: "It is true that Monsieur Fourier held the
view that the principal aim of mathematics was public utility, and
the explanation of natural phenomena; but a philosopher such as he
should have known that the unique aim of science is the honor of the
human spirit, and that from this point of view a question about
numbers is as important as a question about the system of the world."
In 1823 Abel published a paper * in which he is led, by a mechanical
question including as a special case the problem of the tautochrone,
to what is now called an integral equation, on the solution of which
the solution of the problem depends. His problem was, to determine
the curve for which the time of descent is a given function of the ver-
tical height. In view of the recent developments in integral equa-
tions, Abel's problem is of great historical interest. Independently
of Abel, researches along this line were published in 1832, 1837, and
1839 by J. Liouville, who in 1837 showed that a particular solution of
a certain differential equation can be obtained by the aid of an in-
tegral equation of "the second kind," somewhat different from Abel's
equation "of the first kind."

Abel's Memoire of 1826 remained in A. L. Cauchy's hands. It was
not published until 1841. By a singular mishap, the manuscript was
lost before the proof-sheets were read.

In its form; the contents of the memoir belongs to the integral
calculus. Abelian integrals depend upon an irrational function y
which is connected with x by an algebraic equation F(x, y) =o. Abel's
theorem asserts that a sum of such integrals can be expressed by a
definite number p of similar integrals, where p depends merely on the
properties of the equation F(x, y) =o. It was shown later that p is the
deficiency of the curve F(x, y}=o. The addition theorems of elliptic
integrals are deducible from Abel's theorem. The hyperelliptic in-
tegrals introduced by Abel, and proved by him to possess multiple
periodicity, are special cases of Abelian integrals whenever />=or >3.
The reduction of Abelian to elliptic integrals has been studied mainly by
C. G. J. Jacobi, Ch. Hermite, Leo Konigsberger, F. Brioschi, E. Gour-
sat, E. Picard, and O. Bolza, then of the University of Chicago. Abel's
theorem was pronounced by Jacobi the greatest discovery of our cen-
tury on the integral calculus. The aged Legendre, who greatly ad-
mired Abel's genius, called it "monumentum aere perennius." Some
cases of Abel's theorem were investigated independently by William
Henry Fox Talbot (1800-1877), the English pioneer of photography,
1 N. H. Abel, Oeuvrcs completes, 1881, Vol. i, p. it. See also p. 97.

4i4 A HISTORY OF MATHEMATICS

who showed that the theorem is deducible from symmetric functions
of the roots of equations and partial fractions.1

Two editions of Abel's works have been published: the first by
Berndt Michael Holmboe (1795-1850) of Christiania in 1839, and the
second by L. Sylow and S. Lie in 1881. During the few years of work
allotted to the young Norwegian, he penetrated new fields of research.
Abel's published papers stimulated researches containing certain
results previously reached by Abel himself in his then unpublished
Parisian memoir. We refer to papers of Christian Jiirgensen (1805-
1861) of Copenhagen, Ole Jacob Broch (1818-1889) of Christiania,
Ferdinand Adolf Minding (1806-1885) of Dorpat, and G. Rosenhain.

Some of the discoveries of Abel and Jacobi were anticipated by
K. F. Gauss. In the Disquisitiones Arithmetics he observed that
the principles which he used in the division of the circle were ap-
plicable to many other functions, besides the circular, and particularly

r dx

to the transcendents dependent on the integral I — . .-. From this

J VI — * •

Jacobi 2 concluded that Gauss had thirty years earlier considered the
nature and properties of elliptic functions and had discovered their
double periodicity. The papers in the collected works of Gauss con-
firm this conclusion.

Carl Gustav Jacob Jacobi (1804-1851) was born of Jewish parents
at Potsdam. Like many other mathematicians he was initiated into
mathematics by reading L. Euler. At the University of Berlin, where
he pursued his mathematical studies independently of the lecture
courses, he took the degree of Ph.D. in 1825. After giving lectures
in Berlin for two years, he was elected extraordinary professor at
Konigsberg, and two years later to the ordinary professorship there.
After the publication of his Fundamenta Nova in 1829 he spent some
time in travel, meeting Gauss in Gottingen, and A. M. Legendre,
J. Fourier, S. D. Poisson, in Paris. In 1842 he and his colleague,
F. W. Bessel, attended the meetings of the British Association, where
they made the acquaintance of English mathematicians. Jacobi
was a great teacher. "In this respect he was the very opposite of his
great contemporary Gauss, who disliked to teach, and who was any-
thing but inspiring."

Jacobi's early researches were on Gauss' approximation to the
value of definite integrals, partial differential equations, Legendre's
coefficients, and cubic residues. He read Legendre's Exercises, which
give an account of elliptic integrals. When he returned the book to
the library, he was depressed in spirits and said that important books
generally excited in him new ideas, but that this time he had not
been led to a single original thought. Though slow at first, his ideas

1 G. B. Mathews in Nature, Vol. 95, 1915, p. 219.

2R. Tucker, "Carl Friedrich Gauss," Nature, April, 1877.

THEORY OF FUNCTIONS 415

flowed all the richer afterwards. Many of his discoveries in elliptic
functions were made independently by Abel. Jacobi communicated
his first researches to Crelle's Journal. In 1829, at the age of twenty-
five, he published his Fundamenta Nova Theories Functionum Ellip-
ticarum, which contains in condensed form the main results in elliptic
functions. This work at once secured for him a wide reputation. He
then made a closer study of theta-functions and lectured to his pupils
on a new theory of elliptic functions based on the theta-functions. He
developed a theory of transformation which led him to a multitude
of formulae containing q, a transcendental function of the modulus,
defined by the equation q=e~wk /k. He was also led by it to consider
the two new functions H and 0, which taken each separately with
two different arguments are the four (single) theta-functions desig-
nated by the ©i, ©2, 63, 04.1 In a short but very important memoir
of 1832, he shows that for the hyperelliptic integral of any class the
direct functions to which Abel's theorem has reference are not func-
tions of a single variable, such as the elliptic sn, en, dn, but functions
of p variables.1 Thus in the case p=2, which Jacobi especially con-
siders, it is shown that Abel's theorem has reference to two functions
\(u, v), \\(u, v), each of two variables, and gives in effect an addition-
theorem for the expression of the functions \(u-\-u', v+v'), \i(u+u',
v+v') algebraically in terms of the functions \(u, v), \\(u,v), \(u',v'),
\i(u', v'}. By the memoirs of N. H. Abel and Jacobi it may be con-
sidered that the notion of the Abelian function of p variables was
established and the addition-theorem for these functions given. Re-
cent studies touching Abelian functions have been made by K. Weier-
strass, E. Picard, Madame Kovalevski, and H. Poincare. Jacobi's
work on differential equations, determinants, dynamics, and the
theory of numbers is mentioned elsewhere.

In 1842 C. G. J. Jacobi visited Italy for a few months to recuperate
his health. At this time the Prussian government gave him a pension,
and he moved to Berlin, where the last years of his life were spent.

Among those who greatly extended the researches on functions
mentioned thus far was Charles Hermite (1822-1901), who was born
at Dieuze in Lorraine.2 He early manifested extraordinary talent for
mathematics. Neglecting the regular courses of study, he read in
Paris with greatest ardor the masterpieces of L. Euler, J. Lagrange,
K. F. Gauss, and C. G. J. Jacobi. In 1842 he entered the Ecole Poly-
technique. From birth he had suffered from an infirmity of the right
leg and had to use a cane. On this account he was declared ineligible
to any government position given to graduates of the Ecole. Hermite,
therefore, left at the end of the first year. A letter to Jacobi displayed
his mathematical genius, but the necessity of taking examinations
which he held en horreur compelled him to descend from his lofty

1 Arthur Cayley, Inaugural Address before the British Association, 1883.

1 Bull. Am. Math. Soc., Vol. 13, 1907, p. 182.

416 A HISTORY OF MATHEMATICS

mathematical speculations and take up the irksome details prepara-
tory to examinations. In 1848 he became examinateur d'admission
and repetiteur d'analyse at the ficole Polytechnique. In that position
he succeeded P. L. Wantzel. That year he married a sister of his
friend, Joseph Bertrand. In 1869, at the age of forty-seven, he became
professor and at length reached a position befitting his talents. At
the Sorbonne he succeeded J. M. C. Duhamel as professor of higher
algebra. He occupied the chair at the Ecole Polytechnique until
1876, at the Sorbonne until 1897. For many years he had been re-
garded as the venerated chief among French mathematicians. Hermite
had no fondness for geometry. His researches are confined to algebra
and analysis. He wrote on the theory of numbers, invariants and
covariants, definite integrals, theory of equations, elliptic functions
and the theory of functions. Of his collected works, or Oeuvres,
Vol. Ill appeared in 1912, edited by E. Picard. In the theory of
functions he was the foremost French writer of his day, since A. L.
Cauchy. He has given an entirely new significance to the use of
definite integrals in the theory of functions: we name the develop-
ments of the properties of the gamma-function which have been thus
initiated.

Elliptic functions, considered on the Jacobian rather than on the
Weierstrassian basis, was a favorite study of Hermite. "To him is
due the reduction of an elliptic integral to its canonical form by means
of the syzygy among the concomitants of a binary quartic. His in-
vestigations on modular functions and modular equations are of the
highest importance. It was Hermite who discovered pseudo-periodic
functions of the second kind, and developed their properties. In a
memoir that may be fairly described as classical, ' Sur quelques appli-
cations des fonctions elliptiques' in the Complex Rendus, 1877-1882,
he applied these functions to the integration of the unspecialized form
of Lame's differential equation; and elliptic functions generally were
applied in that memoir to obtain the solution of a number of physical
problems" (A. R. Forsyth).

In 1858 Hermite introduced in place of the variable q of Jacobi a
new variable <o connected with it by the equation q=etvu, so that o>=
ik' Ik, and was led to consider the functions <£(«), ^(w)> xM-1
Henry Smith regarded a theta-function with the argument equal to
zero, as a function of w. This he called an omega-function, while
the three functions </>(<•>), ^(w), x(w)> are ms modular functions.
Researches on theta-f unctions with respect to real and imaginary
arguments have been made by Ernst Meissel (1826-1895) of Kiel,
J. Thomae of Jena, Alfred Enneper (1830-1885) of Gottingen. A
general formula for the product of two theta-functions was given in
1854 by H. Schroter (1829-1892) of Breslau. These functions have

1 Arthur Cayley, Inaugural Address, 1883.

THEORY OF FUNCTIONS 417

been studied also by Cauchy, Konigsberger of Heidelberg (born 1837),
Friedrich Julius Richelot (1808-1875) of Konigsberg, Johann Georg
Rosenhain (1816-1887) of Konigsberg, Ludwig Schlafli (1814-1895)
of Bern.1

A. M. Legendre's method of reducing an elliptic differential to its
normal form has called forth many investigations, most important
of which are those of F. J. Richelot and of K. Weierstrass of Berlin.

The algebraic transformations of elliptic functions involve a relation
between the old modulus and the new one which C. G. J. Jacobi ex-
pressed by a differential equation of the third order, and also by an
algebraic equation, called by him "modular equation." The notion
of modular equations was familiar to Abel, but the development of
this subject devolved upon later investigators. These equations
have become of importance in the theory of algebraic equations, and
have been studied by Ludwig Adolph Sohncke (1807-1853) of Halle,
E. Mathieu, L. Konigsberger, E. Betti of Pisa, Ch. Hermite of Paris,
P. Joubert of Angers, Francesco Brioschi of Milan, L. Schlafli, H.
Schroter, C. Gudermann of Cleve, Carl Eduard Giitzlaff (1805-?) of
Marienwerder in Prussia.

Felix Klein of Gottingen has made an extensive study of modular
functions, dealing with a type of operations lying between the two
extreme types, known as the theory of substitutions and the theory
of invariants and covariants. Klein's theory has been presented in
book-form by his pupil, Robert Fricke. The bolder features of it
were first published in his Ikosaeder, 1884. His researches embrace
the theory of modular functions as a specific class of elliptic functions,
the statement of a more general problem as based on the doctrine
of groups of operations, and the further development of the subject
in connection with a class of Riemann's surfaces.

The elliptic functions were expressed by N. H. Abel as quotients
of doubly infinite products. He did not, however, inquire rigorously
into the convergency of the products. In 1845 A. Cayley studied
these products, and found for them a complete theory, based in part
upon geometrical interpretation, which he made the basis of the whole
theory of elliptic functions. F. G. Eisenstein discussed by purely
analytical methods the general doubly infinite product, and arrived
at results which have been greatly simplified in form by the theory of
primary factors, due to K. Weierstrass. A certain function involving
a doubly infinite product has been called by Weierstrass the sigma-
function, and is the basis of his beautiful theory of elliptic functions.
The first systematic presentation of Weierstrass' theory of elliptic
functions was published in 1886 by G. H. Halphen in his Thcorie des
fonctions elliptiques et des leurs applications. Applications of these
functions have been given also by A. G. Greenhill of London. Gener-

1 Alfred Knneper, Elliplisfhe Funklioncn, Theorlc undGeschichtc, Halle a/S, 1876.

4i8 A HISTORY OF MATHEMATICS

alizations analogous to those of Weierstrass on elliptic functions have
been made by Felix Klein on hyperelliptic functions.

Standard -works on elliptic functions have been published by C. A .
A. Briot and /. C. Bouquet (1859), by L. Konigsberger, A. Cayley,
Heinrich Durege (1821-1893) of Prague, and others.

Jacobi's work on Abelian and theta-functions was greatly extended
by Adolph Gopel (i8i2)-i847), professor in a gymnasium near Pots-
dam, and Johann Georg Rosenhain (1816-1887) of Konigsberg.
Gopel in his Theories transcendentium primi ordinis adumbratio levis
(Crelle, 35, 1847) and Rosenhain in several memoirs established each
independently, on the analogy of the single theta-functions, the func-
tions of two variables, called double theta-functions, and worked out
in connection with them the theory of the Abelian functions of two
variables. The theta-relations established by Gopel and Rosenhain
received for thirty years no further development, notwithstanding
the fact that the double theta series came to be of increasing impor-
tance in analytical, geometrical, and mechanical problems, and that
Ch. Hermite and L. Konigsberger had considered the subject of trans-
formation. Finally, the investigations of C. W. Borchardt, treating
of the representation of Kummer's surface by Gopel's biquadratic
relation between four theta-functions of two variables, and researches
of H. H. Weber, F. Prym, Adolf Krazer, and Martin Krause of Dres-
den led to broader views. Carl Wilhelm Borchardt (1817-1880) was
born in Berlin, studied under P. G. L. Dirichlet and C. G. J. Jacobi
in Germany, and under Ch. Hermite, M. Chasles, and J. Liouville
in France. He became professor in Berlin and succeeded A. L. Crelle
as editor of the Journal fur Mathematik. Much of his time was given
to the applications of determinants in mathematical research.

Friedrich Prym (1841-1915) studied at Berlin, Gottingen, and Hei-
delberg. He became professor at the Polytechnicum in Zurich, then at
Wiirzburg. His interest lay in the theory of functions. Researches
on double theta-functions, made by A. Cayley, were extended to
quadruple theta-functions by Thomas Craig (1855-1900), professor
at the Johns Hopkins University. He was a pupil of J. J. Sylvester.
While lecturing at the University he was during 1879-1881 connected
with the United States Coast and Geodetic Survey. For many years
he was an editor of the American Journal of Mathematics.

Starting with the integrals of the most general form and considering
the inverse functions corresponding to these integrals (the Abelian
functions of p variables), G. F. B. Riemann defined the theta-functions
of p variables as the sum of a />-tuply infinite series of exponentials,
the general term depending on p variables. Riemann shows that the
Abelian functions are algebraically connected with theta-functions of
the proper arguments, and presents the theory in the broadest form.1

1 Arthur Cayley, Inaugural Address, 1883.

THEORY OF FUNCTIONS 419

He rests the theory of the multiple theta-functions upon the general
principles of the theory of functions of a complex variable.

Through the researches of A. Brill of Tubingen, M. Nother of
Erlangen, and Ferdinand Lindemann of Munich, made in connection
with Riemann-Roch's theorem and the theory of residuation, there
has grown out of the theory of Abelian functions a theory of algebraic
functions and point-groups on algebraic curves.

General Theory of Functions

The history of the general theory of functions begins with the
adoption of new definitions of a function. As an inheritance from the
eighteenth century, y was called a function of x, if there existed an
equation between these variables which made it possible to calculate
y for any given value of x lying anywhere between — oo and + oo .
We have seen that L. Euler sometimes used a second, more general,
definition, which was adopted by J. Fourier and which was translated
by P. G. L. Dirichlet into the language of analysis thus: y is called a
function of x, if y possess one or more definite values for each of cer-
tain values that x is assumed to take in an interval xo to x\. In func-
tions thus defined, there need be no analytical connection between
y and x, and it becomes necessary to look for possible discontinuities.
This definition was still further emphasized and generalized later,
after the introduction of the theory of aggregates. There a function
need not be defined for each point in the continuum embracing all
real and complex numbers, nor for each point in an interval, but only
for the points x in some particular set of points. Thus, y is a function
of x, if for each point or number in any set of points or numbers x,
there corresponds a point or number in a set y.

P. G. L. Dirichlet lectured on the theory of the potential and thereby
made this theory more generally known in Germany. In 1839 K. F.
Gauss had made researches on the potential; in England George
Green had issued his fundamental memoir as early as 1828. Dir-
ichlet's lectures on the potential became known to G. F. B. Riemann
who made it of fundamental importance for the whole of mathe-
matics. Before considering Riemann we must take up A. L.
Cauchy.

J. Fourier's declaration that any given arbitrary function can be
represented by a trigonometric series led Cauchy to a new formulation
of the concepts " continuous," " limiting value " and " function." In his
Cours a" Analyse, 1821, he says: "The function f(x) is continuous
between two given limits, if for each value of x that lies between
these limits, the numerical value of the difference f(x+ a) -/(*) di-
minishes with a in such a way as to become less than every finite
number " (Chap. II, § 2). With S. F. Lacroix and A. L. Cauchy there
are indications of a tendency to free the functional concept from an ac-

420 A HISTORY OF MATHEMATICS

tual representation.1 Although in his earlier writings slow to recognize
the importance of imaginary variables, Cauchy later entered deeply
into the treatment of functions of complex variables, not in a geometri-
cal form as found in C. Wessel, J. R. Argand, and K. F. Gauss, but
rather in analytical form. He carried on integrations through imag-
inary fields. While L. Euler and P. S. Laplace had declared the order
of integration in double integrals to be immaterial, A. L. Cauchy
showed that this was true only when the expression to be integrated
does not become indeterminate in the interval (Memoire sur la theorie
des integrates definies, read 1814, printed 1825).

If between two paths of integration, in the complex plane, there
lies a pole, then the difference between the respective integrals can be-
represented by means of a "residu de la fonction" (1826), a concept of
undoubted importance known as the calculus of residues. In 1846
he showed that if X and Y are continuous functions of x and y within

a closed area, then I (Xdx+Ydy) = =i= I I ( jdxdy, where

the left integral extends over the boundary and the right integral
over the inner area of the complex plane; he considers integration
along a closed path surrounding a "pole," and later along a closed
path surrounding a line on which the function is discontinuous, as
for instance log x for x<o when the function changes by 2iri in crossing
the x-axis. The fundamental theorem of Cauchy's theory of series
was given in 1837: " A function can be expanded in an ascending power
series in x, as long as the modulus of x is less than that for which the
function ceases to be finite and continuous." In 1840 the proof of this
theorem is made to rest on the theorem of mean value. Cauchy,
J. C. F. Sturm, and J. Liouville had carried on discussions as to
whether the continuity of a function was sufficient to insure its ex-
pandibility or whether that of its derivative must be demanded as well.
In 1851 Cauchy concluded that the continuity of the derivative must
be demanded. A function f(z), which is single- valued for z=x+iy
was called by Cauchy "monotypique," later "monodrome," by Briot
and Bouquet "monotrope," by Hermite "uniforme," by the Germans
"eindeutig." Cauchy called a function "monogen" when for every
z in a region it had only one derivative value, "synectique" if it is
monodromic and monogenic and does not become infinite. Instead
of "synectique," C. A. A. Briot and J. C. Bouquet, and later French
writers say "holomorph," also "meromorph" when the function has
"poles" in the region.

Some parts of Cauchy's theory of functions were elaborated by
P. M. H. Laurent and Victor Alexandre Puiseux (1820-1883), both

1 A. Brill und M. Noether, "Entwicklung der Theorie der algebraischen Func-
tionen in alterer und neuerer Zeit," Jahresb. d. d. Math. Vereinig., Vol. 3, 1892-1893,
p. 162. We are making extensive use of this historical monograph.

THEORY OF FUNCTIONS 421

of Paris. Laurent pointed out the advantage resulting in certain
cases from a mixed expansion in ascending and descending powers of a
variable, while Puiseux demonstrated the advantage that may be
gained by the use of series involving fractional powers of the variable.
Puiseux examined many-valued algebraic functions of a complex va-
riable, their branch-points and moduli of periodicity.

We proceed to investigations made in Germany by G. F. B.
Riemann.

Georg Friedrich Bernhard Riemann (1826-1866) was bora at
Breselenz in Hanover. His father wished him to study theology, and
he accordingly entered upon philological and theological studies at
Gottingen. He attended also some lectures on mathematics. Such
was his predilection for this science that he abandoned theology.
After studying for a time under K. F. Gauss and M. A. Stern, he
was drawn, in 1847, to Berlin by a galaxy of mathematicians, in
which shone P. G. L. Dirichlet, C. G. J. Jacobi, J. Steiner, and F. G.
Eisenstein. Returning to Gottingen in 1850, he studied physics under
W. Weber, and obtained the doctorate the following year. The thesis
presented on that occasion, Grundlagen fur eine allgemeine Theorie der
Funktionen einer veranderlichen complexen Grosse, excited the admira-
tion of K. F. Gauss to a very unusual degree, as did also Riemann's
trial lecture, Ueber die Hypothesen welche der Geometric zu Grunde
liegen. Influenced by Gauss and W. Weber, physical views were the
mainspring of his purely mathematical investigations. Riemann's
Habilitationsschrift (1854, published 1867) was on the Representa-
tion of a Function by means of a Trigonometric Series, in which he
advanced materially beyond the position of Dirichlet. A. L. Cauchy
had set up criteria for the existence of a definite integral defined as
the limit of a sum, and had stated that such a limit always exists
when the function is continuous. Riemann made a startling extension
by pointing out that the existence of such a limit is not confined to
cases of continuity. Riemann's new criterion placed the definite
integral upon a foundation wholly independent of the differential
calculus and the existence of a derivative. It led to the consideration
of areas and lengths of arcs which may transcend all geometric figures
within the reach of our intuitions. Half a century later the concept
of a definite integral was still further extended by H. Lebesgue of
Paris and others. Our hearts are drawn to Riemann, an extraordina-
rily gifted but shy genius, when we read of the timidity and nervousness
displayed when he began to lecture at Gottingen, and of his jubilation
over the unexpectedly large audience of eight students at his first
lecture on differential equations.

Later he lectured on Abelian functions to a class of three only,
E. C. J. Schering, Bjerknes, and Dedekind. K. F. Gauss died in 1855,
and was succeeded by P. G. L. Dirichlet. On the death of the latter,
in 1859, Riemann was made ordinary professor. In 1860 he visited

422 A HISTORY OF MATHEMATICS

Paris, where he made the acquaintance of French mathematicians.
The delicate state of his health induced him to go to Italy three times.
He died on his last trip at Selasca, and was buried at Biganzolo.

Like all of Riemann's researches, those on functions were profound
and far-reaching. A decidedly modern tendency was his mode of in-
vestigating functions. In the words of E. B. Van Vleck: 1 "He [Rie-
mann] presents a strange antithesis to his contemporary countryman,
Weierstrass. Riemann bases the function theory upon a property
rather than upon an algorism — to wit, the possession of a differential
coefficient by the function in the complex plane. Thus at a stroke
it is freed from dependence upon a particular process like the power
series of Taylor. His celebrated memoir upon the P-function is a
characteristic development of a whole Schar (family) of functions
from their mutual relations."

G. F. B. Riemann laid the foundation for a general theory of func-
tions of a complex variable. The theory of potential, which up to
that time had been used only in mathematical physics, was applied
by him in pure mathematics. He accordingly based his theory

of functions on the partial differential equation, — H — 5=A«=o,

d*2 d;y2

which must hold for the analytical function w=u+iv of z=x+iy.
It had been proved by P. G. L. Dirichlet that (for a plane) there is
always one, and only one, function of x and y, which satisfies AM=O,
and which, together with its differential quotients of the first two
orders, is for all values of x and y within a given area one-valued and
continuous, and which has for points on the boundary of the area
arbitrarily given values.2 Riemann called this "Dirichlet's principle,"
but the same theorem was stated by Green and proved analytically by
Sir William Thomson. It follows then that w is uniquely determined
for all points within a closed surface, if u is arbitrarily given for all
points on the curve, whilst v is given for one point within the curve.
In order to treat the more complicated case where w has n values for
one value of z, and to observe the conditions about continuity, Rie-
mann invented the celebrated surfaces, known as "Riemann's sur-
faces," consisting of n coincident planes or sheets, such that the pas-
sage from one sheet to another is made at the branch-points, and that
the n sheets form together a multiply-connected surface, which can
be dissected by cross-cuts into a singly-connected surface. The n-
valued function w becomes thus a one-valued function. Aided by
researches of Jacob Liiroth (1844-1910) of Freiburg and of R. F. A.
Clebsch, W. K. Clifford brought Riemann's surface for algebraic func-
tions to a canonical form, in which only the last two of the n leaves
are multiply-connected, and then transformed the surface into the

1 Bull. Am. Math. Soc., Vol. 23, 1916, p. 8.

2O. Henrici "Theory of Functions," Nature, Vol. 43, 1891, p. 322.

THEORY OF FUNCTIONS 423

surface of a solid with p holes. This surface with p holes had been
considered before Clifford by A. Tonelli, and was probably used by
Riemann himself.1 A. Hurwitz of Zurich discussed the question, how
far a Riemann's surface is determinate by the assignment of its number
of sheets, its branch-points and branch-lines.

Riemann's theory ascertains the criteria which will determine an
analytical function by aid of its discontinuities and boundary condi-
tions, and thus defines a function independently of a mathematical
expression. In order to show that two different expressions are
identical, it is not necessary to transform one into the other, but it is
sufficient to prove the agreement to a far less extent, merely in certain
critical points.

Riemann's theory, as based on Dirichlet's principle (Thomson's
theorem), is not free from objections which have been raised by L.
Kronecker, K. Weierstrass, and others. In consequence of this,
attempts have been made to graft Riemann's speculations on the
more strongly rooted methods of K. Weierstrass. The latter developed
a theory of functions by starting, not with the theory of potential,
but with analytical expressions and operations. Both applied their
theories to Abelian functions, but there Riemann's work is more
general.2

Following a suggestion found in Riemann's Habilitationsschrift, H.
Hankel prepared a tract, Unendlich oft oscillirende und unstetige Funk-
tionen, Tubingen, 1870, giving functions which admit of an integral,
but where the existence of a differential coefficient remains doubtful.
He supposed continuous curves generated by the motion of a point
to and fro with infinitely numerous and infinitely small oscillations,
thus presenting "a condensation of singularities" at every point, but
possessing no definite direction nor differential coefficient. These
novel ideas were severely criticised, but were finally cleared up by
K. Weierstrass' well-known rigorous example of a continuous curve
totally bereft of derivatives. Hermann Hankel (1839-1873) satisfied
at Leipzig the gymnasium requirements in ancient languages by read-
ing the ancient mathematicians in the original. He studied at Leipzig
under A. F. Mobius, at Gottingen under G. F, B. Riemann, at Berlin
under K. Weierstrass and L. Kronecker. He became professor at
Erlangen and Tubingen. The interest of his lectures was enhanced
by his emphasis upon the history of his subject. In 1867 appeared his
Theorie der Complexen Zahlensysteme. His brilliant Geschichte der
Mathematik in AUerthum und Mittelalter came out in 1874 as a post-
humous publication.

Karl Weierstrass (1815-1897) was born in Ostenfelde, a village in
Westphalia. He attended a gymnasium at Paderborn where he became
interested in the geometric researches of J. Steiner. He entered the

1 Math. Annalen. Vol. 45, p. 142.

2 O. Henrici, Nature, Vol. 43, 1891, p. 323.

424 A HISTORY OF MATHEMATICS

University of Bonn as a student in law but all by himself he studied
also mathematics, particularly P. S. Laplace. Wilhelm Diesterweg
and J. Pliicker, who lectured in Bonn, did not influence him. Seeing
in a student note-book a transcript of Christof Gudermann's lectures
on elliptic transcendents, Weierstrass went in 1839 to Minister, where
he was during one semester the only student to attend Gudermann's
lectures on this topic and on analytical spherics. Christof Guderman:i
(1798-1851) whose researches on hyperbolic functions led to a func-
tion tan~1 (sink x), called the " Gudermannian," was a favorite teacher
of Weierstrass. Then he became a gymnasium teacher at Miinster,
then at Deutsch-Krone in western Prussia where he taught science,
also gymnastics and writing, and finally at Braunsberg where he
entered upon the study of Abelian functions. It is told that he missed
one morning an eight-o'clock class. The director of the gymnasium
went to his room to ascertain the cause, and found him working
zealously at a research which he had begun the evening before and
continued through the night, being unconscious that morning had
come. He asked the director to excuse his lack of punctuality to
his class, for he hoped soon to surprise the world by an important
discovery. While at Braunsberg he received an honorary doctorate
from Konigsberg for scientific papers he had published. In 1855 E. E.
Kummer went from Breslau to Berlin; he expressed it as his opinion
that the paper on Abelian functions was not sufficient guarantee that
Weierstrass was the proper man to train young mathematicians at
Breslau. So Ferdinand Joachimsthal (1818-1861) was appointed
there, but Kummer secured for Weierstrass in 1856 a position at the
Gewerbeakademie in Berlin and at the same time an Extraordinariat
at the University. The former he held until 1864 when he received an
Ordinariat at the University as successor to the aged Martin Ohm.
In that year E. E. Kummer and Weierstrass organized an official
mathematical seminar, P. G. L. Dirichlet having held before this a
private seminar. It is noteworthy that Weierstrass did not begin his
university career as a professor until his forty-ninth year, a time when
many scientists cease their creative work. K. Weierstrass, E. E.
Kummer, and L. Kronecker added lustre to the University of Berlin
which previously had been made famous by the researches of P. G. L.
Dirichlet, J. Steiner, and C. G. J. Jacobi. Especially through Weier-
strass unprecedented stress came to be put u*pon rigor of demon-
stration. The movement toward arithmetization of mathematics re-
ceived through Kronecker and Weierstrass its greatest emphasis. The
number-concept, especially that of the positive integer, was to become
the sole foundation, and the space-concept was to be rejected as a
primary concept.

As early as 1849 Weierstrass began to investigate and write on
Abelian integrals. In 1863 and 1866 he lectured on the theory of
Abelian functions and Abelian transcendents. No authorized publi-

THEORY OF FUNCTIONS 425

cation of these lectures was made in his lifetime, but they became
known in part through researches based upon them that were written
by some of his pupils, E. Netto, F. Schottky, Georg Valentin, F.
Kotter, Georg Hettner (1854-1914), and Johannes Knoblauch (1855-
1915). Hettner and Knoblauch prepared Weierstrass' lectures on the
theory of Abelian transcendents for the fourth volume of his collected
works. In 1915 appeared the fifth volume, on elliptic functions, edited
by Knoblauch. Weierstrass had selected Hettner to edit the works of
C. W. Borchardt (1888), also the last two volumes of Jacobi's works.
Knoblauch lectured at the University of Berlin since 1889, his chief
field of activity being differential geometry. Another prominent pupil
of Weierstrass was Otto Stolz (1842-1905) of the University of Inns-
bruck. The difficulty which was experienced for many years in as-
certaining what were the methods and results of Weierstrass, was
set forth by Adolf Mayer (1839-1908) of Leipzig who at one time had
put at his disposal the manuscript notes of a lecture for only twenty-
four hours. Mayer worked on differential equations, the calculus of
variations and mechanics.

In 1 86 1 Weierstrass made the extraordinary discovery of a function
which is continuous over an interval and does not possess a derivative
at any point on this interval. The function was published by P. du
Bois Reymond in Crelle's Journal, Vol. 79, 1874, p. 29. In 1835 N. I.
Lobachevski had shown in a memoir the necessity of distinguishing
between continuity and differentiability.1 Nevertheless, the mathe-
matical world received a great shock when Weierstrass brought forth
that discovery, "and H. Hankel and G. Cantor by means of their
principle of condensation of singularities could construct analytical
expressions for functions having in any interval, however small, an
infinity of points of oscillation, an infinity of points in which the dif-
ferential coefficient is altogether indeterminate, or an infinity of points
of discontinuity" (J. Pierpont). J. G. Darboux gave new examples of
continuous functions having no derivatives. Formerly it had been
generally assumed that every function had a derivative. A. M. Am-
pere was the first who attempted to prove analytically (1806) the
existence of a derivative, but the demonstration is not valid. In
treating of discontinuous functions, J. G. Darboux established rigor-
ously the necessary and sufficient condition that a continuous or dis-
continuous function be susceptible of integration. He gave fresh
evidence of the care that must be exercised in the use of series by giv-
ing an example of a series always convergent and continuous, such
that the series formed by the integrals of the terms is always con-
vergent, and yet does not represent the integral of the first series.2

Central in Weierstrass' view-point is the concept of the "analytic
function." The name, "general theory of analytic functions," says

1 G. B. Halsted's transl. of A. Vasiliev's Address on Lobachevski, p. 23.
• Police sur les Travaux Scienlijiques de M. Gaston Darboux, Paris, 1884.

426 A HISTORY OF MATHEMATICS

A. Hurvvitz,1 applies to two theories, that of A. L. Cauchy and G. F. B.
Riemann, and that of K. Weierstrass. The two emanate from dif-
ferent definitions of a function. J. Lagrange, in his Theorie des fonc-
tions analytiques had tried to prove the incorrect theorem that every
continuous (stetige) function can be expanded in a power series. K.
Weierstrass called every function "analytic" when it can be expanded
into a power series, which is the centre of Weierstrass' theory of ana-
lytic functions. All properties of the function are contained in nuce in
the power series, with its coefficients ci, c^, . . , Q», . . The behavior
of a power series on the circle of convergence C had received considera-
tion long before this time. N. H. Abel had demonstrated that the
power series having a determinate value in a point on the circle of
convergence C tends uniformly toward that value when the variable
approaches that point along a path which does not touch the circle.
If two power series involve a complex variable, whose circles of
convergence overlap, so that the two series have the same value for
every point common to the too circular areas, then Weierstrass calls
each power series a direct continuation of the other. Using several
such series, K. Weierstrass introduces the idea of a monogenic system
of power series and then gives a more general definition of analytic
function as a function which can be defined by a monogenic system
of power series. In 1872 the Frenchman Ch. Meray gave independ-
ently a similar definition. In case of a uniform (eindeutige) function,
the points in a complex plane are either within the circle of conver-
gence of the power series in the system or else they are without. The
totality of the former points constitutes the "field of continuity"
(Stetigkeitsbereich) of the function. This field constitutes an ag-
gregate of "inner" points that is dense; if this continuum is given,
then there exist always single-valued analytic functions possessing
this field of continuity, as was first proved by G. M. Mittag-Leffler,
later by C. Runge and P. Stackel. The points on the boundary of
this field, called "singular points," constitute by themselves a set
of points, by the properties of which K. Weierstrass classifies the
function (1876). This classification was studied also by C. Guichard
(1883) and by G. M. Mittag-Leffler, making use of theorems on point
sets, as developed in 1879-1885 by G. Cantor and by I. O. Bendlxson
and E. Phragmen, both of Stockholm. Thus, transfinite numbers
began to play a part in the theory of functions. Single- valued analytic
functions resolve themselves into two classes, the one class in which
the singular points form an enumerable (abzahlbares) aggregate, the
other class in which they do not.

Abel had proposed the problem, if one supposes the power series
convergent for all positive values less than r, find the limit to
which the function tends when x approaches r. The first sub-
stantial advance to a solution of Abel's problem was made in 1880
1 A. Hurwitz in Vcrh. des i. Intern. Congr., Zurich, i8g7, Leipzig, 1898, pp. 91-112.

THEORY OF FUNCTIONS 427

by G. Frobenius and in 1882 .by O. Holder, but neither of them
developed conditions that are both necessary and sufficient for
the establishment of the convergence of their expressions. Finally
in 1892 J. Hadamard obtained expressions which include those of
G. Frobenius and O. Holder and determined the conditions under
which they converge on the circle of convergence. The problem pre-
sented itself now thus: To set up analytic expressions of the complex
variable x that are linear in the constants cn and also represent the
function given by the power series, or rather a branch of this function
in a field D, in such a manner that they converge uniformly in the in-
terior of D and diverge in the exterior. The first important step
toward the resolution of this matter was taken in 1895 by E. Borel
who proved that the expression

00 co"+I

Urn 2 (c0+cix+ . . +Cjxr)e u., ..

w=00 v = Q (H-i)I

converges not only in all regular points (points reguliers) of the circle
of convergence of the power series, but even beyond that, within a
summation polygon. E. Borel held the view that his formula gave the
sum of the power series even for points where it diverges. This inter-
pretation of Borel's results was resisted by G'osta Magnus Mittag-Leffler
(1846- ) of Stockholm, the founder1 of the journal Acta mathe-
matica, and of a "Mathematical Institute" (in 1916) to further mathe-
matical research in the Scandinavian countries. Mittag-Leffler con-
ducted important researches along the above line. E. Borel's statement
implies that his formula extends the boundaries of the theory of ana-
lytic functions beyond the classic region, which is denied by Mittag-
Leffler. The latter published in 1898 studies on a problem more gen-
eral than that of Borel. If a ray ap revolves about a through an angle
27T, the variable distance ap always exceeding a fixed value /, a sur-
face is generated which Mittag-Leffler calls a star (Stern) with the
center a. A star E is called a convergence star (Konvergenzstern) for a
definite arithmetical expression, if the latter converges uniformly for
each region within E, but diverges for every outside point. He shows
that to each analytic function there corresponds a principal star, and
that there is an infinite number of arithmetical expressions for a given
star. Equivalent results were obtained by C. Runge. E. Borel gave
in 1912 an example of an analytic function which, by an extension of
the concept of a derivative so as to pass to the limit not through all
the neighboring points but only through those belonging to certain
dense aggregates, has a certain linear continuation beyond the do-
main of existence. Studies of monogenic uniform functions along
the line of E. Borel and G. M. Mittag-Leffler have been made also
by G. Vivanti, Marcel Riesz, Ivar Fredholm, and E. Phragmen.

1 See G. M. Mittag-Leffler in Atti del IV Congr. Intern, Roma, iooS. Roma, 1909,
Vol. I, p. 69. Here Mittag-Leffler gives a summary of recent results.

428 A HISTORY OF MATHEMATICS

Interesting is the manner in which K. Weierstrass in Berlin and
G. F. B. Riemann in Gottingen influenced each other. We have
seen that Weierstrass denned functions of a complex variable by the
power series and avoids geometrical means. Riemann begins with
certain differential equations in the region of mathematical physics.
In 1856 Riemann was urged by his friends to publish a resume of
his researches on Abelian functions, "be it ever so crude," because
Weierstrass was at work on the same subject. Riemann's publication
induced Weierstrass to withdraw from the press a memoir he had
presented to the Berlin Academy in 1857, because, as he himself says,
"Riemann published a memoir on the same problem which rested on
entirely different foundations from mine, and did not immediately
reveal that in its results it agreed completely with my own. The
proof of this required investigations which were not quite easy, and
took much time; after this difficulty had been removed a radical
remodelling of my dissertation seemed necessary." In 1875 Weier-
strass wrote H. A. Schwarz: "The more I ponder over the principles
of the theory of functions — and I do so incessantly — the stronger
grows my conviction that it must be built up on the foundation of
algebraical truths, and that, therefore, to employ for the truth of
simple and fundamental algebraical theorems the 'transcendental,'
if I may say so, is not the correct way, however enticing prima vista
the considerations may be by which Riemann has discovered many
of the most important properties of algebraical functions." This
refers mainly to the " Thomson-Dirichlet Principle," the validity of
which depended on a certain minimum theorem which was shown
by Weierstrass to rest upon unsound argument.

It has been objected that K. Weierstrass' definition of analytic
functions is based on power series. A. L. Cauchy's definition, which
was adopted by G. F. B. Riemann, is not open to this objection, but
labors under the burden of requiring at the start the most difficult
forms of the theory of limits. According to A. L. Cauchy a function
is analytic (his "synectic") if it possesses a single-valued differential
coefficient. Using Cauchy's integral theorem (Integralsatz) , it follows
that the synectic function admits not only of a single-valued differen-
tiation but also of a single-valued integration. Giacinto Morera
(1856-1909) of Turin showed that the synectic function might be
defined by the single- valued integration. More recent researches,
1883-1895, which aim at a rigorous exposition of A. L. Cauchy's
integral theorem, are due to M. Falk, E. Goursat, M. Lerch, C. Jor-
dan, and A. Pringsheim. Cauchy's theorem may be stated : If the func-
tion f(z) is synectic in a continuum in which every simply closed

curve forms the boundary of an area, then the integral lf(z)dz is
always zero, if it is extended over a closed curve which lies wholly

THEORY OF FUNCTIONS 429

within the continuum. Here the questions arise, what is a curve, a
closed curve, a simply closed curve?

Analytic functions of several variables were treated by C. G. J.
Jacobi in 1832, in his Consider utiones generates de transcendentibus
Abelianis, but received no attention until Weierstrass set himself the
task presented to him by the study of Abelian functions, to find a
solid foundation for functions of several variables that would corre-
spond to his treatment of functions of one variable. He obtained a
fundamental theorem on null-places; he also enunciated, without
proof, the theorem that each single-valued (eindeutig) and in a finite
region meromorphic function of several variables can be represented
as the quotient of two integral functions, i. e. of two bestandig con-
vergent power series. This theorem was proved in 1883 by H. Poin-
care for two variables and in 1895 by Pierre Cousin of Bordeaux for
n variables. Later researches are by H. Hahn (1905), P. Boutroux
(1905), G. Faber (1905), and F. Hartogs (1907).

Dirichlet's principle has repeatedly commanded attention. The
question of its rigor has been put by E. Picard as follows: * "The
conditions at the limits that one is led to assume are very different
according as it is question of an equation of which the integrals are
or are not analytic. A type of the first case is given by the problem
generalized by P. G. L. Dirichlet; conditions of continuity there play
an essential part, and, in general, the solution cannot be prolonged
from the two sides of the continuum which serves as support to the
data; it is no longer the same in the second case, where the disposition
of this support in relation to the characteristics plays the principal
role, and where the field of existence of the solution presents itself
under wholly different conditions. . . . From antiquity has been
felt the confused sentiment of a certain economy in natural phenomena ;
one of the first precise examples is furnished by Fermat's principle
relative to the economy of time in the transmission of light. Then we
came to recognize that the general equations of mechanics correspond
to a problem of minimum, or more exactly of variation, and thus we
obtained the principle of virtual velocities, then Hamilton's principle,
and that of least action. A great number of problems appeared then
as corresponding to minima of certain definite integrals. This was a
very important advance, because the existence of a minimum could
in many cases be regarded as evident, and consequently the demon-
stration of the existence of the solution was effected. This reasoning
has rendered immense services; the greatest geometers, K. F. Gauss
in the problem of the distribution of an attracting mass corresponding
to a given potential, G. F. B. Riemann in his theory of Abelian func-
tions, have been satisfied with it. To-day our attention has been
called to the dangers of this sort of demonstration; it is possible for
the minima to be simply limits and not to be actually attained by
1 Congress of Arts and Science, St. Louis, 1904, Vol. I, p. 510.

430 A HISTORY OF MATHEMATICS

veritable functions possessing the necessary properties of continuity.
We are, therefore, no longer content with the probabilities offered
by the reasoning long classic."

David Hilbert in 1899 spoke as follows: l "Dirichlet's principle
owed its celebrity to the attractive simplicity of its fundamental
mathematical idea, to the undeniable richness of its possible applica-
tions in pure and applied mathematics and to its inherent persuasive
power. But after Weierstrass' criticism of it, Dirichlet's principle
was considered as only of historical interest and discarded as a means
of solving the boundary-value problem. C. Neumann deplores that
this beautiful principle of Dirichlet, formerly used so much, has no
doubt passed away forever. Only A. Brill and M. Noether arouse
new hopes in us by giving expression to the conviction that Dirichlet's
principle, being so to speak an imitation of nature, may sometime
receive new life in modified form." Hilbert proceeds thereupon to
rehabilitate the Principle, which involves a special problem in the cal-
culus of variations. Dirichlet's procedure was briefly thus: On the
xy plane erect at the points of the boundary curve perpendiculars the
lengths of which represent the boundary values. Among the surfaces
z=f(x,y) which are bounded by the space-curve thus obtained, select the

one for which the value of the integral/(/) = i C! f|Q + (— ) j dxdy

is a minimum. As shown by the calculus of variations, that surface
is necessarily a potential surface. By reference to such a procedure
G. F. B. Riemann thought he had settled the existence of the solution
of boundary-value problems. But K. Weierstrass made it plain that
among an infinite number of values there does not necessarily exist
a minimum value; a minimum surface may therefore not exist. D.
Hilbert generalizes Dirichlet's principle in this manner: "Every prob-
lem of the calculus of variations has a solution, as soon as restricting
assumptions suitable to the nature of the given boundary conditions
are satisfied and, if necessary, the concept of the solution receives a
fitting extension." D. Hilbert shows how this may be used in finding
rigorous, yet simple, existence proofs. In 1901 it was used in disserta-
tions prepared by E. R. Hedrick and C. A. Noble.

Taking a birds' eye view of the development of the theory of func-
tions during the nineteenth century since the time of A. L. Cauchy,
James Pierpont said in 1904: 2 "Weierstrass and Riemann develop
Cauchy's theory along two distinct and original paths. Weierstrass
starts with an explicit analytic expression, a power series, and defines
his function as the totality of its analytical continuations. No appeal
is made to geometric intuition, his entire theory is strictly arithmetical.
Riemann growing up under Gauss and Dirichlet, not only relies largely

1 Jahresb. d. d. Math. Vereinig., Vol. 8, 1900, p. 185.

2 Bull. Am. Math. Soc., 2. S., Vol. n, 1904, p. 137.

THEORY OF FUNCTIONS 431

on geometric intuition, but also does not hesitate to impress mathe-
matical physics into his service. Two noteworthy features of his
theory are the many-leaved surfaces named after him, and the ex-
tensive use of conformal representation. The history of functions
as first developed is largely a theory of algebraic functions and their
integrals. A general theory of functions is only slowly evolved. For
a long time the methods of Cauchy, Riemann, and Weierstrass were
cultivated along distinct lines by their respective pupils. The schools
of Cauchy and Riemann were first to coalesce. The entire rigor
which has recently been imparted to their methods has removed all
reason for founding, as Weierstrass and his school have urged, the
theory of functions on a single algorithm, viz., the power series. We
may therefore say that at the close of the century there is only one
theory of functions, in which the ideas of its three great creators are
harmoniously united."

The study of existence theorems, particularly in the theory of alge-
braic functions and the calculus of variations, began with Cauchy.
For implicit functions he assumed that they were expressible as power
series, a restriction removed by U. Dini of Pisa. Simplifications are
due to R. Lipschitz of Bonn. Existence theorems of sets of implicit
functions were studied by G. A. Bliss of Chicago in the Princeton
Colloquium of 1909. By means of a sheet of points Bliss deduces
from an initial solution at an ordinary point a sheet of solutions
somewhat analogous to K. Weierstrass' analytical continuation of a
branch of a curve.

Accompanying and immediately following Riemann's time there
was a development of the theory of algebraic functions, that was
partly geometric in character and not purely along the line of function
theory. A. Brill and M. Noether ' in 1894 marked five directions of
advance: First, the geometrico-algebraic direction taken by G. F. B.
Riemann and G. Roch in the years 1862-1866, then by R. F. A.
Clebsch 1863 to 1865, by Clebsch and P. Gordan since 1865 and since
1871 by A. Brill and M. Noether; second, the algebraic direction,
followed by L. Kronecker and K. Weierstrass since 1860, more gen-
erally known since 1872, and in 1880 taken up by E. B. Christoffel;
third, the invariantal direction, represented since 1877 by H. Weber,
M. Noether, E. B. Christoffel, F. Klein, F. G. Frobenius, and F.
Schottky; Fourth, the arithmetical direction of R. Dedekind and H.
Weber since 1880, of L. Kronecker since 1881, of K. W. S. Hensel and
others; Fifth, the geometrical direction taken by C. Segre and G.
Castelnuovo since 1888.

Hermann Amandus Schwarz (1845- ) of Berlin a pupil of K.

Weierstrass, has given the conform representation (Abbildung) of

various surfaces on a circle. G. F. B. Riemann had given a general

theorem on the conformation of a given curve with another curve.

1 A. Brill and M. Noether, Jahrb. d. d. Math. Vcrcinigung, Vol. 3, p. 287.

432 A HISTORY OF MATHEMATICS

In transforming by aid of certain substitutions a polygon bounded
by circular arcs into another also bounded by circular arcs, Schwarz
was led to a remarkable differential equation *f(u', t} = $(u, t), where
ij>(u, f) is the expression which Cayley called the " Schwarzian deriva-
tive," and which led J. J. Sylvester to the theory of reciprocants.
Schwarz's developments on minimum surfaces, his work on hyper-
geometric series, his inquiries on the existence of solutions to important
partial differential equations under prescribed conditions, have se-
cured a prominent place in mathematical literature.

Modular functions were at first considered merely as a by-product
of elliptic functions, growing out of the study of transformations.
After the epoch-making creations of E. Galois and G. F. B. Riemann,
the subject of elliptic modular functions was developed into an in-
dependent theory, mainly by the efforts of H. Poincare and F. Klein,
which stands in close relation to the theory of numbers, algebra and
synthetic geometry. F. Klein began to lecture on this subject in
1877; researches bearing upon this were pursued also by his then
pupils W. Dyck, Joseph Gierster, and A. Hurwitz. One of the problems
of modular functions is, to determine all subgroups of the linear group
xl = (ax+(3) l(yx+8), where a, /?, 7, 8 are integers and a8-^j^o.
F. Klein's Vorlesungen uber das Ikosceder, Leipzig, 1884, is a work
along this line. As an extended continuation of that are F. Klein's
Vorlesungen uber die Theorie der dliptischen Modulfunctionen, gotten
out by Robert Fricke (Vol. i, 1890, Vol. II, 1892) and as a still
further generalization we have the theory of the general linear auto-
morphic functions, developed mainly by F. Klein and H. Poincare.
In 1897, under the joint authorship of Robert Fricke and Felix Klein,
there appeared the first volume of the Vorlesungen uber die Theorie
der Automorphen Funclionen, the second volume of which did not ap-
pear until 1912, after the theory had come under the influence of the
critical tendencies due to K. Weierstrass and G. Cantor, and after
E. Picard and H. Poincare had brought out further incisive researches.
It has been noted that F. Klein's own publications on these topics
are in the order in which the subject itself sprang into existence.
"Historically, the theory of automorphic functions developed from
that of the regular solids and modular functions. At least this is the
path which F. Klein followed under the influence of the well-known
researches of Schwarz and of the early publications of H. Poincare.
If H. Poincare brings in also other considerations, namely the arith-
metic methods of Ch. Hermite . . . and the function-theoretical
problems of Fuchs with regard to single valued inversion of the solu-
tions of linear differential equations of the second order (eindeutige
Umkehr der Losungen . . . ), these topics in turn go back to the very
regions of thought from which have grown the theories of the regular
solids and the elliptic modular functions." H. Poincare published
on this subject in Math, Annalen, Vol. 19, " Sur les fonctions uniformes

THEORY OF FUNCTIONS 433

qui se reproduisent par des substitutions lineaires," in the Ada ma-
thematica, Vol. i, a "Memoire sur les fonctions fuchsiennes," and a
procession of other papers extending over many years. Recently
active along this same line were P. Koebe and L. E. J. Brouwer.

The question what automorphic forms can be expressed analytically
by the H. Poincare . series has been investigated by Poincare himself
and also by E. Ritter and R. Fricke (1901).

After the creation of the theory of automorphic functions of a single
variable, mainly by F. Klein and H. Poincare, similar generalizations
were sought for functions of several complex variables. The pioneer
in this field was E. Picard; other workers are T. Levi-Civita, G. A.
Bliss, W. D. MacMillan, and W. F. Osgood who lectured thereon at the
Madison (Wisconsin) Colloquium in 1913. Charles Emile Picard
(1856 — ) whose extensive researches on analysis have been men-
tioned repeatedly and whose Traite d' 'Analyse is well known, was
born in Paris. He studied at the Ecole Normale where he was in-
spired by J. G. Darboux. In 1881 he married a daughter of Hermite.
Picard taught for a short_time at Toulouse. Since 1881 he has been
professor in Paris at the Ecole Normale and the Sorbonne.

Uniformization

The uniformization of an algebraic or analytic curve, that is, the
determination of such auxiliary variables which taken as independent
variables render the co-ordinates of the points of the curve single-
valued (eindeutig) analytic functions, is organically connected with
the theory of automorphic functions. It was F. Klein and H. Poin-
care who soon after 1880 developed the theory of automorphic func-
tions and introduced systematically the idea of the uniformization of
algebraic curves which G. F. B. Riemann had visualized upon the
surfaces named after him. More recent researches on uniformi ation
connect chiefly with the work of H. Poincare and are due to D. Hil-
bert (1900), W. F. Osgood, T. Broden, and A. M. Johanson. In 1907
followed important generalizations by H. Poincare and by P. Koebe
of Leipzig.1 Dirichlet's Principle, having been established upon a
sound foundation by D. Hilbert in 1901, was used as a starting point,
for the derivation of new proofs of the general principle of uniformiza-
tion, by P. Koebe of Leipzig and R. Courant of Gottingen.

Important works on the theory of functions are the Cours de Ch.
Hermite, J. Tannery's Theorie des Fonctions d'une variable seule, A
Treatise on the Theory of Functions by James Harkness and Frank Mor-
ley, and Theory of Functions of a Complex Variable by A. R. Forsyth.
A broad and comprehensive treatise is the Lchrbuch der Funktionen-
theorie by W. F. Osgood of Harvard University, the first edition of
which appeared in 1907 and the second enlarged edition in 1912.

1 P. Koebe, Alii del IV Congr., Roma, ipoS, Roma, 1909, Vol. II, p. 25.

434 A HISTORY OF MATHEMATICS

Theory of Numbers

" Mathematics, the queen of the sciences, and arithmetic, the queen
of mathematics." Such was the dictum of K. F. Gauss, who was
destined to revolutionize the theory of numbers. When asked who
was the greatest mathematician in Germany, P. S. Laplace answered,
Pfaff . When the questioner said he should have thought Gauss was,
Laplace replied, "Pfaff is by far the greatest mathematician in Ger-
many; but Gauss is the greatest in all Europe." 1 Gauss is one of
the three greatest masters of analysis, — J. Lagrange, P. S. Laplace, K.
F. Gauss. Of these three contemporaries he was the youngest. While
the first two belong to the period in mathematical history preceding
the one now under consideration, Gauss is the one whose writings may
truly be said to mark the beginning of our own epoch. In him that
abundant fertility of invention, displayed by mathematicians of the
preceding period, is combined with rigor in demonstration which is too
often wanting in their writings, and which the ancient Greeks might
have envied. Unlike P. S. Laplace, Gauss strove in his writings after
perfection of form. He rivals J. Lagrange in elegance, and surpasses
this great Frenchman in rigor. Wonderful was his richness of ideas;
one thought followed another so quickly that he had hardly time to
write down even the most meagre outline. At the age of twenty
Gauss had overturned old theories and old methods in all branches of
higher mathematics; but little pains did he take to publish his results,
and thereby to establish his priority. He was the first to observe
rigor in the treatment of infinite series, the first to fully recognize
and emphasize the importance, and to make systematic use of de-
terminants and of imaginaries, the first to arrive at the method of
least squares, the first to observe the double periodicity of elliptic
functions. He invented the heliotrope and, together with W. Weber,
the bifilar magnetometer and the declination instrument. He re-
constructed the whole of magnetic science.

Karl Friedrich Gauss 2 (1777-1855), the son of a bricklayer, was
born at Brunswick. He used to say, jokingly, that he could reckon
before he could talk. The marvellous aptitude for calculation of the
young boy attracted the attention of Johann Martin Bartels (1769-
1836), afterwards professor of mathematics at Dorpat, who brought
him under the notice of Charles William, Duke of Brunswick. The
duke undertook to educate the boy, and sent him to the Collegium
Carolinum. His progress in languages there was quite equal to that
in mathematics. In 1795 he went to Gottingen, as yet undecided
whether to pursue philology or mathematics. Abraham Gotthelf
Kastner (1719-1800), then professor of mathematics there, and now
chiefly remembered for his Geschichte der Mathematik (1796), was not

1 R. Tucker, "Carl Friedrich Gauss," Nature, Vol. 15, 1877, p. 534.

2 W. Sartorius Waltershausen, Gauss, zum Ged'dchtniss, Leipzig, 1856.

THEORY OF NUMBERS 435

a teacher who could inspire Gauss, though Kastner's German con-
temporaries ranked him high and admired his mathematical and
poetical ability. Gauss declared that Kastner was the first mathe-
matician among the poets and the first poet among the mathemati-
cians. When not quite nineteen years old Gauss began jotting down
in a copy-book very brief Latin memoranda of his mathematical dis-
coveries. This diary was published in 1901. 1 Of the 146 entries, the
first is dated March 30, 1796, and refers to his discovery of a method
of inscribing in a circle a regular polygon of seventeen sides. This dis-
covery encouraged him to pursue mathematics. He worked quite
independently of his teachers, and while a student at Gottingen made
several of his greatest discoveries. Higher arithmetic was his favorite
study. Among his small circle of intimate friends was Wolfgang
Bolyai. After completing his course he returned to Brunswick. In
1798 and 1799 he repaired to the university at Helmstadt to consult
the library, and there made the acquaintance of J. F. Pfaff, a mathe-
matician of much power. In 1807 the Emperor of Russia offered Gauss
a chair in the Academy at St. Petersburg, but by the advice of the
astonomer Olbers, who desired to secure him as director of a proposed
new observatory at Gottingen, he declined the offer, and accepted
the place at Gottingen. Gauss had a marked objection to a mathe-
matical chair, and preferred the post of astronomer, that he might
give all his time to science. He spent his life in Gottingen in the midst
of continuous work. In 1828 he went to Berlin to attend a meeting
of scientists, but after this he never again left Gottingen, except in
1854, when a railroad was opened between Gottingen and Hanover.
He had a strong will, and his character showed a curious mixture of
self-conscious dignity and child-like simplicity. He was little com-
municative, and at times morose. Of Gauss' collected works, or
Werke, an eleventh volume was planned in 1916, to be biographical and
bibliographical in character.

A new epoch in the theory of numbers dates from the publication
of his Disquisitiones Arithmetics, Leipzig, 1801. The beginning of
this work dates back as far as 1795. Some of its results had been
previously given by J. Lagrange and L. Euler, but were reached inde-
pendently by Gauss, who had gone deeply into the subject before he
became acquainted with the writings of his great predecessors. The
Disquisitiones Arithmetics was already in print when A. M. Legendre's
Theorie des N ombres appeared. The great law of quadratic reciprocity,
given in the fourth section of Gauss' work, a law which involves the
whole theory of quadratic residues, was discovered by him by in-
duction before he was eighteen, and was proved by him one year
later. Afterwards he learned that L. Euler had imperfectly enunciated
that theorem, and that A. M. Legendre had attempted to prove it,

1 Gauss' wissenschafllichc Tagebuck, 1796-1814. Mit Anraerkungen herausgege-
ben von Felix Klein, Berlin, 1901.

436 A HISTORY OF MATHEMATICS

but met with apparently insuperable difficulties. In the fifth section
Gauss gave a second proof of this "gem" of higher arithmetic. In
1808 followed a third and fourth demonstration; in 1817, a fifth and
sixth. No wonder that he felt a personal attachment to this theorem.
Proofs * were given also by C. G. J. Jacobi, F. Eisenstein, J. Liouville,
Victor Amedee Lebesgue (1791-1875) of Bordeaux, Angelo Genocchi
(1817-1889) of the University of Turin, E. E. Kummer, M. A. Stern,
Christian Zeller (1822-1899) of Markgroningen, L. Kronecker, Victor
Jacovlevich Bouniakovski (1804-1889) of Petrograd, Ernst Schering
(1833-1897) of Gottingen, Julius Peter Christian Petersen (1839-1910)
of Copenhagen, E. Busche, Th. Pepin, Fabian Franklin, J. C. .Fields,^
and others. Quadratic reciprocity "stands out not only for the in- £.#q»
fluence it has exerted in many branches, but also for the number of
new methods to which it has given birth" (P. A. MacMahon). The
solution of the problem of the representation of numbers by binary
quadratic forms is one of the great achievements of Gauss. He created
a new algorithm by introducing the theory of congruences. The fourth
section of the Disquisiliones Arithmetics, treating of congruences of
the second degree, and the fifth section, treating of quadratic forms,
were, until the time of C. G. J. Jacobi, passed over with universal
neglect, but they have since been the starting-point of a long series
of important researches. The seventh or last section, developing the
theory of the division of the circle, was received from the start with
deserved enthusiasm, and has since been repeatedly elaborated for
students. A standard work on Kreistheilung was published in 1872
by Paul Bachmann, then of Breslau.

The equation for the division of the circle and the construction of
a regular polygon of n sides, n being prime, can be solved by square
root extractions alone, always and only when n — i is a power of 2.
Hence such regular polygons can be constructed by ruler and com-
passes when the prime number n is 3, 5, 17, 257, 65,537, • • but cannot
be constructed when n is 7, n, 13, . . The results may be stated also
thus: The Greeks knew how to inscribe regular polygons whose sides
numbered 2m, 2m. 3, 2TO. 5 and 2m. 15. Gauss added in 1801 that the
construction is possible when the number of sides n is prime and of
the form 22/*+i. L. E. Dickson computed that the number of such
inscriptible polygons for n < 100 is 24, for n < 300 is 37, for n < 1000 is
52, for n ^ 100,000 is 206.

Three classical constructions of the regular inscribed polygon of
seventeen sides have been given: one by J. Serret in his Algebra, II,
§ 547, another by von Staudt in Crelle, Vol. 24, and a third by L.
Gerard in Math. Annalen, Vol. 48 (1897), using compasses only. The
analytic solution, as outlined by Gauss, was actually carried out for
the regular polygon of 257 sides by F. J. Richelot of Konigsberg in
four articles in Crelle, Vol. 9. For the polygon of 65,537 sides this
1 0. Baumgart, Ueber das Quadratische Reciprocitalsgesetz, Leipzig, 1885.

THEORY OF NUMBERS 437

was accomplished after ten years of labor by Oswald Hermes (1826-
1909) of Steglitz; his manuscript is deposited in the mathematical
seminar at Gottingen.1 Gauss had planned an eighth section of his Dis-
quisitiones Arithmeticae, which was omitted to lessen the expense of
publication. His papers on the theory of numbers were not all included
in his great treatise. Some of them were published for the first time
after his death in his collected works. He wrote two memoirs on the
theory of biquadratic residues (1825 and 1831), the second of which
contains a theorem of biquadratic reciprocity.

K. F. Gauss was led to astronomy by the discovery of the planet
Ceres at Palermo in 1801. His determination of the elements of its
orbit with sufficient accuracy to enable H. W. M. Olbers to rediscover
it, made the name of Gauss generally known. In 1809 he published
the Theoria motus corporum ccelestium, which contains a discussion
of the problems arising in the determination of the movements of
planets and comets from observations made on them under any cir-
cumstances. In.it are found four formulas in spherical trigonometry,
now usually called "Gauss' Analogies," but which were published
somewhat earlier by Karl Brandon Mollweide (1774-1825) of Leipzig,
and earlier still by Jean Baptiste Joseph Delambre (1749-1822). 2
Many years of hard work were spent in the astronomical and magnetic
observatory. He founded the German Magnetic Union, with the
object of securing continuous observations at fixed tunes. He took
part in geodetic observations, and in 1843 and 1846 wrote two me-
moirs, Ueber Gcgenstande der hoheren Geodesic. He wrote on the at-
traction of. homogeneous ellipsoids, 1813. In a memoir on capillary
attraction, 1833, he solves a problem in the calculus of variations
involving the variation of a certain double integral, the limits of in-
tegration being also variable; it is the earliest example of the solution
of such a problem. He discussed the problem of rays of light passing
through a system of lenses.

Among Gauss' pupils were Heinrich Christian Schumacher, Chris-
tian Gerling, Friedrich Nicolai, August Ferdinand Mobius, Georg
Wilhelm Struve, Johann Frantz Encke.

Gauss' researches on the theory of numbers were the starting-point
for a school of writers, among the earliest of whom was C. G. J.
Jacobi. The latter contributed to Crclle's Journal an article on cubic
residues, giving theorems without proofs. After the publication of
Gauss' paper on biquadratic residues, giving the law of biquadratic
reciprocity, and his treatment of complex numbers, C. G. J. Jacobi
found a similar law for cubic residues. By the theory of elliptical
functions, he was led to beautiful theorems on the representation of

1 A. Mitzscherling, Das Problem der Krchicilung, Leipzig u. Berlin, 1913, pp. 14,

23-

2 1. Todhunter, "Note on the History of Certain Formulae in Spherical Trigo-
nometry," Philosophical Magazine, Feb., 1873.

438 A HISTORY OF MATHEMATICS •»>

numbers by 2, 4, 6, and 8 squares. Next come the researches of P. G.
L. Dirichlet, the expounder of Gauss, and a contributor of rich results
of his own.

Peter Gustav Lejeune Dirichlet l (1805-1859) was born in Diiren,
attended the gymnasium in Bonn, and then the Jesuit gymnasium
in Cologne. In 1822 he was attracted to Paris by the names of P. S.
Laplace, A. M. Legendre, J. Fourier, D. S. Poisson, and A. L. Cauchy.
The facilities for a mathematical education there were far better
than in Germany, where K. F. Gauss was the only great figure. He
read in Paris Gauss' Disquisitiones Arithmetics, a work which he
never ceased to admire and study. Much in it was simplified by
Dirichlet, and thereby placed within easier reach of mathematicians.
His first memoir on the impossibility of certain indeterminate
tions of the fifth degree was presented to the French Academy ii
He showed that P. Fermat's equation, xn+yn=zn, cannot
w=5. Some parts of the analysis are, however, A. M. Le
Dirichlet's acquaintance with J. Fourier led him to invesj
ier's series. He became decent in Breslau in 1827. In
cepted a position in Berlin, and finally succeeded K.
Gottingen in 1855. The general principles on which depends the
average number of classes of binary quadratic forms of positive and
negative determinant (a subject first investigated by Gauss) wrere
given by Dirichlet in a memoir, Ueber die Bestimmung der mittleren
Werthe in der Zahlentheorie, 1849. More recently F. Mertens of Graz,
since 1894 of Vienna, determined the asymptotic values of several
numerical functions. Dirichlet gave some attention to prime num-
bers. K. F. Gauss and A. M. Legendre had given expressions denoting
approximately the asymptotic value of the number of primes inferior
to a given limit, but it remained for G. F. B. Riemann in his memoir,
Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse, 1859,
to give an investigation of the asymptotic frequency of primes which
is rigorous. Approaching the problem from a different direction,
P. L. Chebichev, formerly professor in the University of St. Petersburg,
established, in a celebrated memoir, Sur les N ombres Premiers, 1850,
the existence of limits within which the sum of the logarithms of the
primes P, inferior to a given number x, must be comprised.2 He
proved that, if w>3, there is always at least one prime between n
and 2n—2 (inclusive). This theorem is sometimes called "Bertrand's
postulate," since J. L. F. Bertrand had previously assumed it for
the purpose of proving a theorem in the theory of substitution groups.
This paper depends on very elementary considerations, and, in that
respect, contrasts strongly with Riemann's, which involves abstruse
theorems of the integral calculus. H. Poincare's papers, J. J. Syl-

1 E. E. Kummer, Gedachtnissrede auf Gustav Peter Lejeiine-Dirichlet, Berlin, 1860.

2 H. J. Stephen Smith "On the Present State and Prospects of some Branches of
Pure Mathematics," Proceed. London Math. Soc., Vol. 8, 1876, p. 17.

THEORY OF NUMBERS 439

vester's contraction of Chebichev's limits, with reference to the dis-
tribution of primes, and researches of J. Hadamard (awarded the
Grand prix of 1892), are among the later researches in this line.
G. F. B. Riemann had advanced six properties relating to

00 i
£ (s) = 2 — , where s= <r+ti,

n=i«s

none of which he was able to prove.1 In 1893 J. Hadamard proved
three of these, thereby establishing the existence of null-places in
Riemann's zeta-f unction; H. von Mangoldt of Danzig proved in 1895
a fourth and in 1905 a fifth of Riemann's six properties. The remain-
ing one, that the roots of £(s) in the strip 0 S <7 < i, have all the real
part £, remains unproved, though progress in the study of this case
has been made by F. Mertens and R. v. Sterneck. If x is a positive
number, and if TT(X) denotes the number of primes less than x, then
what Landau calls the "prime-number theorem" (Primzahlsatz)
states that the ratio of TT(X) to x/\og x approaches i as x increases
without end. A. M. Legendre, K. F. Gauss, and P. G. L. Dirichlet
had guessed this theorem. As early as 1737 L. Euler2 had given an
analogous theorem, that Si//> approaches log (log p), where the sum-
mation extends over all primes not greater than p. The prime-number
theorem was proved in 1896 by J. Hadamard and Charles Jean de la
Vallee Poussin of Lou vain, in 1901 by Nils Fabian Helge von Koch
of Stockholm, in 1903 by E. Landau, now of Gottingen, in 1915 by
G. H. Hardy and J. E. Littlewood of Cambridge. Hardy discovered
an infinity of zeroes of the zeta-f unction with the real part ^; E.
Landau simplified Hardy's proof.

G. F. B. Riemann's zeta-function £(s) was first studied on account
of its fundamental importance in the theory of prime numbers, but it
has become important also in the theory of analytic functions in
general. In 1909 E. Landau published his Handbuch der Lehre von
der Verteilung der Primzahlen. In 1912 he pronounced the following
four questions to be apparently incapable of answer in the present
state of the science of numbers: (i) Does w2+i for integral values of
n represent an infinite number of primes? (2) C. Goldbach's theorem:
Can prime values of p and p' be found to satisfy m=p+p' for each
even m larger than 2 ? (3) Has 2 = p — p1 an infinite number of solutions
in primes? (4) Is there between nz and (w+i)2 at least one prime for
every positive integral n?

The enumeration of prime numbers has been undertaken at differ-
ent times by various mathematicians. Factor tables, giving the least
factor of every integer not divisible by 2, 3, or 5, did not extend above
408,000 previous to the year 1811, when Ladislaus Chernac published
his Cribrum arithmeticum at Deventer in Netherlands, which gives

1 For details, consult E. Landau in Proceed, jth Intern. Congress, Cambridge, 1912,
Vol. i, 1913, p. 97.

2G. Enestrom in Bibliothcca mathematical, 3. S., Vol. 13, p. 81.

440 A HISTORY OF MATHEMATICS

factors for numbers up to 1,020,000. J. Ch. Burckhardt (1773-1815)
published factor tables in Paris, in 1817 for the numbers i to 1,020,000,
in 1814 for the numbers 1020000 to 2028000, in 1816 for the numbers
2,028,000 to 3,036,000. James Glaisher (1809-1903) published factor
tables at London, in 1879 for the numbers 3,000,000 to 4,000,000, in
1880 for numbers 4,000,000 to 5,000,000, in 1883 for the numbers
5,000,000 to 6,000,000. -Zacharias Dase (1824-1861) published factor
tables at Hamburg, in 1862 for the numbers 6,000,001 to 7,002,000, in
1863 for the numbers 7,002,001 to 8,010,000, in 1865 for the numbers
8 010,001 to 9,000,000. In 1909 the Carnegie Institution of Washing-
ton published factor tables for the first ten millions, prepared by D.
N. Lehmer of the University of California. Lehmer gives the errors
discovered in the earlier publications. Historical details about factor
tables are given by Glaisher in his Factor Table. Fourth Million, 1879.
Miscellaneous contributions to the theory of numbers were made
by A . L. Cauchy. He showed, for instance, how to find all the infinite
solutions of a homogeneous indeterminate equation of the second
degree in three variables when one solution is given. He established
the theorem that if two congruences, which have the same modulus,
admit of a common solution, the modulus is a divisor of their resultant.
Joseph Liouville (1809-1882), professor at the College de France,
investigated mainly questions on the theory of quadratic forms of two,
and of a greater number of variables. A research along a different
line proved to be an entering wedge into a subject which since has
become of vital importance. In 1844 he proved (Liouville' 's Journal,
Vol. 5) that neither e nor e2 can be a root of a quadratic equation with
rational coefficients. By the properties of convergents of a continued
fraction representing a root of an algebraical equation with rational
coefficients he established later the existence of numbers — the so-
called transcendental numbers — which cannot be roots of any such
equation. He proved this also by another method. A still different
approach is due to G. Cantor. Profound researches were instituted
by Ferdinand Gotthold Eisenstein (1823-1852), of Berlin. Ternary
quadratic forms had been studied somewhat by K. F. Gauss, but the
extension from two to three indeterminates was the work of Eisen-
stein who, in his memoir, Neue Theoreme der hoheren Arithmetic,
defined the ordinal and generic characters of ternary quadratic forms
of uneven determinant; and, in case of definite forms, assigned the
weight of any order or genus. But he did not publish demonstrations
of his results. In inspecting the theory of binary cubic forms, he was
led to the discovery of the first covariant ever considered in analysis.
He showed that the series of theorems, relating to the presentation
of numbers by sums of squares, ceases when the number of squares
surpasses eight. Many of the proofs omitted by Eisenstein were sup-
plied by Henry Smith, who was one of the few Englishmen who de-
voted themselves to the study of- higher arithmetic.1

THEORY OF NUMBERS 441

Henry John Stephen Smith l (1826-1883) was born in London,
and educated at Rugby and at Balliol College, Oxford. Before 1847
he travelled much in Europe for his health, and at one time attended
lectures of D. F. J. Arago in Paris, but after that year he was never
absent from Oxford for a single term. In 1849 he carried off at Oxford
the highest honors, both in the classics and in mathematics, thus
ranking as a "double first." There is a story that he decided between
classics and mathematics as the field for his' life- work, by tossing up a
penny. He never married and had no household cares to destroy the
needed serenity for scientific work, " excepting that he was careless in
money matters, and trusted more to speculation in mining shares
than to economic management of his income." : In 1 86 1 he was
elected Savilian professor of geometry. -His first paper on the theory
of numbers appeared in 1855. The results of ten years' study of
everything published on the theory of numbers are contained in his
Reports which appeared in the British Association volumes from 1859
to 1865. These reports are a model of clear and precise exposition
and perfection of form. They contain much original matter, but the
chief results of his own discoveries were printed in the Philosophical
Transactions for 1861 and 1867. They treat of linear indeterminate
equations and congruences, and of the orders and genera of ternary
quadratic forms. He established the principles on which the exten-
sion to the general case of n indeterminates of quadratic forms de-
pends. He contributed also two memoirs to the Proceedings of the
Royal Society of 1864 and 1868, in the second of which he remarks that
the theorems of C. G. J. Jacobi, F. Eisenstein, and J. Liouville, re-
lating to the representation of numbers by 4, 6, 8 squares, and other
simple quadratic forms are deducible by a uniform method from the
principles indicated in his paper. Theorems relating to the case of
5 squares were given by F. Eisenstein, but Smith completed the enunci-
ation of them, and added the corresponding theorems for 7 squares.
The solution of the cases of 2, 4, 6 squares may be obtained by elliptic
functions, but when the number of squares is odd, it involves processes
peculiar to the theory of numbers. This class of theorems is limited
to 8 squares, and Smith completed the group. In ignorance of Smith's
investigations, the French Academy offered a prize for the demon-
stration and completion of F. Eisenstein's theorems for 5 squares.
This Smith had accomplished fifteen years earlier. He sent in a dis-
sertation in 1882, and next year, a month after his death, the prize
was awarded to him, another prize being also awarded to H. Min-
kowsky of Bonn. The theory of numbers led Smith to the study of
elliptic functions. He wrote also on modern geometry. His succes-
sor at Oxford was J. J. Sylvester. Taking an anti-utilitarian view of

1 J. W. L. O.laisher in Monthly Notices R. Astr. Soc., Vol. 44, 1884.
- \. Macfarlanc, Ten British Mathematicians, 1916, p. 98.

442 A HISTORY OF MATHEMATICS

mathematics, Smith once proposed a toast, "Pure mathematics;
may it never be of any use to any one."

Ernst Eduard Kummer (1810-1893), professor in the University
of Berlin, is closely identified with the theory of numbers. P. G. L.
Dirichlet's work on complex numbers of the form a+ib, introduced
by K. F. Gauss, was extended by him, by F. Eisenstein, and R. Dede-
kind. Instead of the equation x*— 1=0, the roots of which yield
Gauss' units, F. Eisenstein used the equation x?—i=o and complex
numbers a+bp (p being a cube root of unity), the theory of which
resembles that of Gauss' numbers. E. E. Kummer passed to the
general case xn - 1 =o and got complex numbers of the form o.=a\A \+
a2A 2+dzA 3+ . . . , were a,- are whole real numbers, and AI roots of the
above equation. Euclid's theory of the greatest common divisor is
not applicable to such complex numbers, and their prime factors can-
not be denned in the same way as prime factors of common integers
are defined. In the effort to overcome this difficulty, E. E. Kummer
was led to introduce the conception of "ideal numbers." These
ideal numbers have been applied by G. Zolotarev of St. Petersburg
to the solution of a problem of the integral calculus, left unfinished by
Abel.1 J. W. R. Dedekind of Braunschweig has given in the second
edition of Dirichlet's Vorlesungen uber Zahlentheorie a new theory
of complex numbers, in which he to some extent deviates from the
course of E. E. Kummer, and avoids the use of ideal numbers. De-
dekind has taken the roots of any irreducible equation with integral
coefficients as the units for his complex numbers. F. Klein in 1893
introduced simplicity by a geometric treatment of ideal numbers.

Fermat's "Last Theorem," Waring's Theorem

E. E. Kummer's ideal numbers owe their origin to his efforts to
prove the impossibility of solving in integers Fermat's equation
xn+yn=zn for n>2. We premise that some progress in proving this
impossibility has been made by more elementary means. For in-
tegers x, y, z not divisible by an odd prime n, the theorem has been
proved by the Parisian mathematician and philosopher Sophie Ger-
main (1776-1831) for 'n<ioo, by Legendre for n<2oo, by E. T. Mail-
let for «<223, by Dmitry Mirimanoff for n<2$f, by L. E. Dickson
for «<7ooo.2 The method used here is due to Sophie Germain and
requires the determination of an odd prune p for which xn+yn+
z*^o (mod. p} has no solutions, each not divisible by p, and n is not
the residue modulo p of the nth power of any integer. E. E. Kum-
mer's results rest on an advanced theory of algebraic numbers which he

1 H. J. S. Smith, "On the Present State and Prospects of Some Branches of Pure
Mathematics," Proceed. London Math. Soc., Vol. 8, 1876, p. 15.

2 See L. E. Dickson in Annals of Mathematics, 2. S., Vol. 18, 1917, pp. 161-187.
See also L. E. Dickson in Alii del IV. Congr. Roma, 1908, Roma, 1909, Vol. II,
p. 172.

THEORY OF NUMBERS 443

helped to create. Once at an early period he thought that he had a com-
plete proof. He laid it before P. G. L. Dirichlet who pointed out that,
although he had proved that any number /(a), where a is a complex nlh
root of unity and n is prime, was the product of indecomposable factors,
he had assumed that such a factorization was unique, whereas this was
not true in general.1 After years of study, E. E. Kummer concluded
that this non-uniqueness of factorization was due to /(a) being too
small a domain of numbers to permit the presence in it of the'true prune
numbers. He was led to the creation of his ideal numbers, the ma-
chinery of which, says L. E. Dickson,2 is "so delicate that an expert
must handle it with the greatest care, and (is) nowadays chiefly of
historical interest in view of the simpler and more general theory of
R. Dedekind." By means of his ideal numbers he produced a proof
of Fermat's last theorem, which is not general but excludes certain
particular values of n, which values are rare among the smaller values
of n; there are no values of n below 100, for which E. E. Kummer's
proof does not serve. In 1857 the French Academy of Sciences
awarded E. E. Kummer a prize of 3000 francs for his researches on
complex integers.

The first marked advance since Kummer was made by A. Wieferich
of Munster, in Crelle's Journal, Vol. 136, 1909, who demonstrated that
if p is prime and 2^ — 2 is not divisible by />2, the equation xp+yp=zv
cannot be solved in terms of positive integers which are not mul-
tiples of p. Waldemar Meissner of Charlottenburg found that 2^ — 2
is divisible by p2 when ^=1093 and for no other prime p less than
2000. Recent advances toward a more general proof of Fermat's
last theorem have been made by D. Mirimanoff of Geneva, G. Fro-
benius of Berlin, E. Hecke of Gottingen, F. Bernstein of Gottingen,
Ph. Furtwangler of Bonn, S. Bohnicek and H. S. Vandiver of Phila-
delphia. Recent efforts along this line have been stimulated in part
by a bequest of 100,000 marks made in 1908 to the Konigliche Gesell-
schaft der Wissenschaften in Gottingen, by the mathematician F. P.
Wolfskehl of Darmstadt, as a prize for a complete proof of Fermat's
last theorem. Since then hundreds of erroneous proofs have been
published. Post-mortems over proofs which fall still-born from the
press are being held in the " Sprechsaal " of the Archiv der Mathematik
und Physik.

At the beginning of the present century progress was made in prov-
ing another celebrated theorem, known as "Waring's theorem." In
1909 A. Wieferich of Munster proved the part which says that every
positive integer is equal to the sum of not more than 9 positive cubes.
He established also, that every positive integer is equal to the sum
of not more than 37 (according to Waring, it is not more than 19)
positive fourth powers, while D. Hilbert proved in 1909 that, for

1 Festschrift z. Feierdes 100. Geburlstages fyluard Kummers, Leipzig, 1910, p. 22.

2 Bull. Am. Math. Soc., Vol. 17, 1911, p. 371.

444 A HISTORY OF MATHEMATICS

every integer n>2 (Waring had declared for every integer n>4),
each positive integer is expressible as the sum of positive nth powers,
the number of which lies within a limit dependent only upon the
value of //. Actual determinations of such upper limits have been
made by A. Hurwitz, E. T. Maillet, A. Fleck, and A. J. Kempner.
Kempner proved in 1912 that there is an infinity of numbers which
are not the sum of less than 4 . 2" positive 2nth powers, n^.2.

Other Recent Researches. Number Fields

Attracted by E. E. Kummer's investigations, his pupil, Leopold
Kronecker (1823-1891) made researches which he applied to algebraic
equations. On the other hand, efforts have been made to utilize in
the theory of numbers the results of the modern higher algebra.
Following up researches of Ch. Hermite, Paul Bachmann of Munster,
now of Weimar, investigated the arithmetical formula which gives
the automorphics of a ternary quadratic form.1 Bachmann is the
author of well-known texts on Zahlenlheorie, in several volumes, which
appeared in 1892, 1894, 1872, 1898, and 1905, respectively. The prob-
lem of the equivalence of two positive or definite ternary quadratic
forms was solved by L. Seeber; and that of the arithmetical auto-
morphics of such forms, by F. G. Eisenstein. The more difficult prob-
lem of the equivalence for indefinite ternary forms has been investi-
gated by Eduard Selling of Wtirzburg. On quadratic forms of four
or more indeterminates little has yet been done. Ch. Hermite showed
that the number of non-equivalent classes of quadratic forms having
integral coefficients and a given discriminant is finite, while Zolotarev
and Alexander Korkine (1837-1908), both of St. Petersburg, investi-
gated the minima of positive quadratic forms. In connection with
binary quadratic forms, H. J. S. Smith established the theorem that
if the joint invariant of two properly primitive forms vanishes, the
determinant of either of them is represented primitively by the dupli-
cate of the other.

The interchange of theorems between arithmetic and algebra is
displayed in the recent researches of J. W. L. Glaisher (1848- )
of Trinity College and J. J. Sylvester. Sylvester gave a Constructive
Theory of Partitions, which received additions from his pupils, F.
Franklin, now of New York city, and George Stetson Ely (P-igiS),
for many years examiner in the U. S. Patent Office.

By the introduction of "ideal numbers" E. E. Kummer took a
first step toward a theory of fields of numbers. The consideration of
super fields (Oberkorper) from which the properties of a given field
of numbers may be easily derived is due mainly to R. Dedekind and
to L. Kronecker. Thereby there was opened up for the theory of
numbers a new and wide territory which is in close connection with

1 H. J. S. Smith in Proceed. London Math. Soc., Vol. 8, 1876, p. 13.

THEORY OF NUMBERS 445

algebra and the theory of functions. The importance of this subject
in the theory of equations is at once evident if we call to mind E.
Galois' fields of rationality. The interrelation between number theory
and function theory is illustrated in Riemann's researches in which
the frequency of primes was made to depend upon the zero-places of
a certain analytical function, and in the transcendence of e and TT
which is an arithmetic property of the exponential function. In 1883-
1890 L. Kronecker published important results on elliptic functions
which contain arithmetical theorems of great elegance. The Dedekind
method of extending Kummer's results to algebraic numbers in
general is based on the notion of an ideal. A common characteristic
of Dedekind and Kroneckers procedure is the introduction of com-
pound moduli. G. M. Mathews says * that, in practice it is convenient
to combine the methods of L. Kronecker and R. Dedekind. Of
central importance are the Galoisian or normal fields, which have been
studied extensively by D. Hilbert. L. Kronecker established the
theorem that all Abelian fields are cyclotomic, which was proved also
by H. Weber and D. Hilbert. An important report, prepared by D.
Hilbert and entitled Theorie der algebraischen Zaldkorper, was pub-
lished in i894.2 D. Hilbert first develops the theory of general number-
fields, then that of special fields, viz., the Galois field, the quadratic
field, the circle field (Kreiskorper), the Kummer field. A report on
later investigations was published by R. Fueter in 191 1.3 Chief among
the workers in this subject which have not yet been mentioned are
F. Bernstein, Ph. Furtwangler, H. Minkowski, Ch. Hermite, and A.
Hurwitz. Accounts of the theory are given in H. Weber's Lehrbuch der
Algebra, 'Vol. 2 (1899), J. Sommer's Vorlesungen uber Zahlentheorie
(1907), and Hermann Minkowski's Diophantische Approximationen,
Leipzig (1907). H. Minkowski gives in geometric and arithmetic
language both old and new results. His use of lattices serves as a
geometric setting for algebraic theory and for the proof of some new
results'.

A new and powerful method of attacking questions on the theory
of algebraic numbers was advanced by Kurt Hensel of Konigsberg
in his Theorie der algebraischen Zahlen, 1908, and in his Zahlentheorie,
1913. His method is analogous to that of power series in the theory
of analytic functions. He employs expansions of numbers into power
series in an arbitrary prime number p. This theory of />-adic numbers
is generalized by him in his book of 1913 into the theory of g-adic
numbers, where g is any integer.4

The resolution of a given large number into factors is a difficult
problem which has been taken up by Paul Seelhof , Francois Edouard

1 Art. "Number" in the Enr.ydop. Britannica, nth ed., p. 857.
zJahresbericht d. d. Math. Vereinigung, Vol. 4, pp. 177-546.
3 Loc. cit., Vol. 20, pp. 1-47.
* Bull. Am. Math. Soc., Vol. 20, 1914, p. 259.

446 A HISTORY OF MATHEMATICS

Anatolc Lucas (1842-1891) of Paris, Fortune Landry (1799-?), A. J. C.
Cunningham, F. W. P. Lawrence and D. N. Lehmer.

Transcendental Numbers. The Infinite

Building on the results previously reached by J. Liouville, Ch.
Hermite proved in 1873 in the Comptes Rendus, Vol. 77, that e is
transcendental, while F. Lindemann in 1882 (Ber. Akad. Berlin)
proved that ir is transcendental. Ch. Hermite reached his result by
showing that aem+ben+cer+ . . . =o cannot subsist, where m, n, r, . . .
a, b, c, . . . are whole numbers; F. Lindemann proved that this equa-
tion cannot subsist when m, n, r, . . a, b, c . . are algebraic numbers,
that in particular, eix+i=o cannot subsist if x is algebraic. Conse-
quently TT cannot be an algebraic number. But, starting with two
points, (o, o) and (i, o), a third point (a, o) can be constructed by the
aid of ruler and compasses only when a is a certain special type of
algebraic number that is obtainable by successive square root extrac-
tions. Hence the point (IT, o) cannot be constructed, and the "quad-
rature of the circle" is impossible. The proofs of Ch. Hermite and F.
Lindemann involved complex integrations and were complicated.
Simplified proofs were given by K. Weierstrass in 1885, Th. J. Stieltjes
in 1890, D. Hilbert, A. Hurwitz, and P. Gordan in 1896 (Math. An-
nalen, Vol. 43), F. Mcrtens in 1896, Th. Vahlen in 1900, H. Weber,
F. Enriques, and E. W. Hobson in 1911. G. B. Halsted says of the
circle, "John Bolyai squared it in non-Euclidean geometry and Linde-
mann proved no man could square it in Euclidean geometry."

That there are many other transcendental numbers beside e and TT
is evident from the researches of J. Liouville, E. Maillet, G. Faber
and Aubrey J. Kempner, who give new forms of infinite series which
define transcendental numbers. Of interest are the theorems estab-
lished in 1913 by G. N. Bauer and H. L. Slobin of Minneapolis, that
the trigonometric functions and the hyperbolic functions represent
transcendental numbers whenever the argument is an algebraic num-
ber other than zero, and vice versa, the arguments are transcendental
numbers whenever the functions are algebraic numbers.1

The notions of the actually infinite have undergone radical change
during the nineteenth century. As late as 1831 K. F. Gauss expressed
himself thus: "I protest against the use of infinite magnitude as
something completed, which in mathematics is never permissible.
Infinity is merely a faqon de parlcr, the real meaning being a limit
which certain ratios approach indefinitely near, while others are per-
mitted to increase without restriction." 2 Gauss' contemporary, A. L.
Cauchy, likewise rejected the actually infinite, being influenced by

1 Rendiconli d. Circolo Malh. di Palermo, Vol. 38, 1914, p. 353,

2 C. F. Gauss, Brief on Schumacher, Werke, Bd. 8, 216; quoted from Moritz,
Memorabilia mathcmatica, 1914, p. 337.

APPLIED MATHEMATICS 447

the eighteenth century philosopher of Turin, Father Gerdil.1 In 1886
Georg Cantor occupied a diametrically opposite position, when he
said: "In spite of the essential difference between the conceptions of
the potential and the actual infinite, the former signifying a variable
finite magnitude increasing beyond all finite limits, while the latter
is a fixed, constant quantity lying beyond all finite magnitudes, it
happens only too often that the one is mistaken for the other. . . .
Owing to a justifiable aversion to such illegitimate actual infinities
and the influence of the modern epicuric-materialistic tendency, a
certain horror infmiti has grown up in extended scientific circles,
which finds its classic expression and support in the letter of Gauss,
yet it seems to me that the consequent uncritical rejection of the
legitimate actual infinite is no lesser violation of the nature of things,
which must be taken as they are." 2

In 1904 Charles Emile Picard of Paris expressed himself thus: 3
"Since the concept of number has been sifted, in it have been found
unfathomable depths; thus, it is a question still pending to know, be-
tween the two forms, the cardinal number and the ordinal number,
under which the idea of number presents itself, which of the two is
anterior to the other, that is to say, whether the idea of number prop-
erly so called is anterior to that of order, or if it is the inverse. It
seems that the geometer-logician neglects too much in these questions
psychology and the lessons uncivilized races give us; it would seem to
result from these studies that the priority is with the cardinal number."

Applied Mathematics. Celestial Mechanics

Notwithstanding the beautiful developments of celestial mechanics
reached by P. S. Laplace at the close of the eighteenth century, there
was made a discovery on the first day of the nineteenth century which
presented a problem seemingly beyond the power of that analysis.
We refer to the discovery of Ceres by Giuseppe Piazzi in Italy, which
became known in Germany just after the philosopher G. W. F. Hegel
had published a dissertation proving a priori that such a discovery
could not be made. From the positions of the planet observed by
Piazzi its orbit could not be satisfactorily calculated by the old
methods, and it remained for the genius of K. F. Gauss to devise a
method of calculating elliptic orbits which was free from the assumption
of a small eccentricity and inclination. Gauss' method was developed
further in his Thcoria Motus. The new planet was re-discovered with
aid of Gauss' data by H. W. M. Olbers, an astronomer who promoted
science not only by his own astronomical studies, but also by discern-

1 See F. Cajori, "History of Zeno s Arguments on Motion," Am. Math. Monthly,
Vol. 22, 1915, p. 114.

2 G. Cantor, Zum Problem dcs actualen Unendlichen, Nalur und 0/enbarung,
Bd. 32, 1886, p. 226; quoted from Moritz, Memorabilia mathcmalica, 1914, p. 337.

3 Congress of Arts and Science, St. Louis, 1904, Vol. I, p. 498.

448 A HISTORY OF MATHEMATICS

ing and directing towards astronomical pursuits the genius of F. W.
Bessel.

Friedrich Wilhelm Bessel ] (1784-1846) was a native of Minden in
Westphalia. Fondness for figures, and a distaste for Latin grammar
led him to the choice of a mercantile career. In his fifteenth year he
became an apprenticed clerk in Bremen, and for nearly seven years
he devoted his days to mastering the details of his business, and part
of his nights to study. Hoping some day to become a supercargo on
trading expeditions, he became interested in observations at sea.
With a sextant constructed by him and an ordinary clock he deter-
mined the latitude of Bremen. His success in this inspired him for
astronomical study. One work after another was mastered by him,
unaided, during the hours snatched from sleep. From old observa-
tions he calculated the orbit of Halley's comet. Bessel introduced
himself to H. W. M. Olbers, and submitted to him the calculation,
which Olbers immediately sent for publication. Encouraged by Ol-
bers, Bessel turned his back to the prospect of affluence, chose poverty
and the stars, and became assistant in J. H. Schroter's observatory at
Lilienthal. Four years later he was chosen to superintend the con-
struction of the new observatory at Konigsberg.2 In the absence of
an adequate mathematical teaching force, Bessel was obliged to lecture
on mathematics to prepare students for astronomy. He was relieved
of this work in 1825 by the arrival of C. G. J. Jacobi. We shall not
recount the labors by which Bessel earned the title of founder of
modem practical astronomy and geodesy. As an observer he towered
far above K. F. Gauss, but as a mathematician he reverently bowed
before the genius of his great contemporary. Of Bessel's papers, the
one of greatest mathematical interest is an " Untcrsuchung des Theils
der planctarischcn Sforungen, wclchcr aus dcr Bcwegung dcr Sonne
enlsteht" (1824), in which he introduces a class of transcendental
functions, Jn(x), much used in applied mathematics, and known as
"Bessel's functions." He gave their principal properties, and con-
structed tables for their evaluation. It has been observed that Bes-
sel's functions appear much earlier in mathematical literature.3 Such
functions of the zero order occur in papers of Daniel Bernoulli (1732)
and L. Euler on vibration of heavy strings suspended from one end.
All of Bessel's functions of the first kind and of integral orders occur
in a paper by L. Euler (1764) on the vibration of a stretched elastic
membrane. In 1878 Lord Rayleigh proved that Bessel's functions
are merely particular cases of Laplace's functions. J. W. L. Glaisher
illustrates by Bessel's functions his assertion that mathematical

1 Bessel als Bremer Ilandlungslchrling, Bremen, 1800.

2 J. Frantz, Feslrcde aus Veranlassiing von Bessel's hnndcrtj'dhrigcm Gcbiirlslag,
Konigsberg, 1884.

3 Maxime Bdcher, "A bit of Mathematical History," Bull, of the N. Y. Math. Soc.,
Vol. II, 1893, p. 107.

APPLIED MATHEMATICS 449

branches growing out of physical inquiries as a rule "lack the easy
flow or homogeneity of form which is characteristic of a mathematical
theory properly so called." These functions have been studied by
Carl Theodor Anger (1803-1858) of Danzig, Oskar Schlomilch (1823-
1901) of Dresden who was the founder in 1856 of the Zeitschrift fur
Mathematik und Physik, R. Lipschitz of Bonn, Carl Neumann of
Leipzig, Eugen Lommel (1837-1899) of Munich, Isaac Todhunter of St.
John's College, Cambridge.

Prominent among the successors of P. S. Laplace are the follow-
ing: Simeon Denis Poisson (1781-1840), who wrote in 1808 a classic
Mcmoire sur les incgalites seculaires des moyens mowuements des plan-
ties. Giovanni Antonio Amaedo Plana (1781-1864) of Turin, a nephew
of J. Lagrange, who published in 1811 a Memoria sulla teoria dell'
attrazione dcgli sferoidi cllilici, and contributed to the theory of the
moon. Peter Andreas Hansen (1795-1874) of Gotha, at one time a
clockmaker in Tondern, then H. C. Shumacher's assistant at Altona,
and finally director of the observatory at Gotha, wrote on various
astronomical subjects, but mainly on the lunar theory, which he
elaborated in his work Fundamenta nova investigations orbitce vero3 quam
Lima perlustrat (1838), and in subsequent investigations embracing
extensive lunar tables. George Biddel Airy (1801-1892), royal as-
tronomer at Greenwich, published in 1826 his Mathematical Tracts
on the Lunar and Planetary Theories. These researches were later
greatly extended by him. August Ferdinand Mobius (1790-1868)
of Leipzig wrote, in 1842, Elcmente dcr Mechanik des Himmels.
Urbain Jean Joseph Leverrier (1811-1877) of Paris wrote, the
Recherches Aslronomiques, constituting in part a new elaboration of
celestial mechanics, and is famous for his theoretical discovery of
Neptune. John Couch Adams (1819-1892) of Cambridge divided
with Leverrier the honor of the mathematical discovery of Nep-
tune, and pointed out in 1853 that Laplace's explanation of the
secular acceleration of the moon's mean motion accounted for only
half the observed acceleration. Charles Eugene Delaunay (born
1816, and drowned off Cherbourg in 1872), professor of mechanics at
the Sorbonne in Paris, explained most of the remaining acceleration of
the moon, unaccounted for by Laplace's theory as corrected by J. C.
Adams, by tracing the effect of tidal friction, a theory previously
suggested independently by Immanuel Kant, Robert Maytr, and
William Ferrcl of Kentucky. G. H. Darwin of Cambridge made
some very remarkable investigations on tidal friction.

Sir George Howard Darwin (1845-1912), a son of the naturalist
Charles Darwin, entered Trinity College, Cambridge, was Second
Wrangler in 1868, Lord Moulton being Senior Wrangler. He began
in 1875 to publish important papers on the application of the theory
of tidal friction to the evolution of the solar system. The earth-moon
system was found to form a unique example within the solar system

450 A HISTORY OF MATHEMATICS

of its particular mode of evolution. He traced back the changes in
the figures of the earth and moon, until they united into one pear-
shaped mass. This theory received confirmation in 1885 from a paper
in Ada math., Vol. 7 by H. Poincare in which he enunciates the prin-
ciple of exchange of stabilities. H. Poincare and Darwin arrived at
the same pear-shaped figure, Poincare tracing the process of evolution
forwards, Darwin proceeding backwards in time. Questions of
stability of this changing pear-shaped figure occupied Darwin's later
years. Researches along the same line were made by one of his
pupils, James H. Jeans of Trinity College, Cambridge.

About the same time that George Darwin began his researches,
George William Hill (1838-1914) of the Nautical Almanac Office
in Washington began to study the moon. Hill was born at Nyack,
New York, graduated at Rutgers College in 1859, and was an as-
sistant in the Nautical Almanac Office from his graduation till 1892,
when he resigned to pursue further the original researches which
brought him distinction. In 1877 he published Researches on Lunar
Theory, in which he discarded the usual mode of procedure in the
problem of three bodies, by which the problem is an extension of the
case of two bodies. Following a suggestion of Euler, Hill takes the
earth finite, the sun of infinite mass at an infinite distance, the moon
infinitesimal and at a finite distance. The differential equations which
express the motion of the moon under the limitations adopted are
fairly simple l and practically useful. "It is this idea of Hill's that
has so profoundly changed the whole outlook of celestial mechanics.
H. Poincare took it up as the basis of his celebrated prize essay of
1887 on the problem of three bodies and afterwards expanded his
work into the three volumes, Les mcthodes nouvelles de la mecanique
celeste" 1892-1899. It seems that at first G. H. Darwin paid little
attention to Hill's paper; Darwin often spoke of his difficulties in
assimilating the work of others. However in 1888 he recommended
to E. W. Brown, now professor at Yale, the study of Hill. Nor does
Darwin seem to have studied closely the "planetesimal hypothesis"
of T. C. Chamberlin and F. R. Moulton of the University of Chicago.
A marked contrast between G. H. Darwin and H. Poincare lay in
the fact that Darwin did not undertake investigations for their
mathematical interest alone, while H. Poincare and some of his
followers in applied mathematics "have less interest in the phenomena
than in the mathematical processes which are used by the student
of the phenomena. They do not expect to examine or predict physical
events but rather to take up the special classes of functions, differen-
tial equations or series which have been used by astronomers or phy-
sicists, to examine their properties, the validity of the arguments and
the limitations which must be placed on the results" (E. W. Brown).

1 We are using E. W. Brown's article in Scientific Papers by Sir G. H. Darwin,
Vol. V, 1916, pp. xxxiv-lv.

APPLIED MATHEMATICS 451

Prominent in mathematical astronomy was Simon Newcomb
(1835-1909), the son of a country school teacher. He was born at Wal-
lace in Nova Scotia. Although he attended for a year the Lawrence
Scientific School at Harvard University, he was essentially self-taught.
In Cambridge he came in contact with B. Peirce, B. A. Gould, J. D.
Runklc, and T. H. Safford. In 1861 he was appointed professor in
the United States Navy; in 1877 he became superintendent of the
American Ephemeris and Nautical Almanac Office. This position
he held for twenty years. During 1884-1895 he was also professor
of mathematics and astronomy at the Johns Hopkins University,
and editor of the American Journal of Mathematics. His researches
were mainly in the astronomy of position, in which line he was pre-
eminent. In the comparison between theory and observation, in
deducing from large masses of observations the results which he
needed and which would form a basis of comparison with theory,
he was a master. As a supplement to the Nautical Almanac for 1897
he published the Elements of the Four Inner planets, and the Funda-
mental . Constants of Astronomy, which gathers together Newcomb 's
life-work.1 For the unravelling of the motions of Jupiter and Saturn,
S. Newcomb enlisted the services of G. W. Hill. All the publications
of the tables of the planets, except those of Jupiter and Saturn, bear
Newcomb's name. These tables supplant those of Leverrier. S. New-
comb devoted much time to the moon. He investigated the errors in
Hansen's lunar tables and continued the lunar researches of C. E.
Delaunay. Brief reference has already been made to G. W. Hill's
lunar work and his contribution of an elegant paper on certain possible
abbreviations in the computation of the long-period of the moon's
motion due to the direct action of the planets, and made elaborate
determination of the inequalities of the moon's motion due to the
figure of the earth. He also computed certain lunar inequalities due
to the action of Jupiter.

The mathematical discussion of Saturn's rings was taken up first
by P. S. Laplace, who demonstrated that a homogeneous solid ring
could not be in equilibrium, and in 1851 by B. Peirce, who proved
their non-solidity by showing that even an irregular solid ring could
not be in equilibrium about Saturn. The mechanism of these rings
was investigated by James Clerk Maxwell in an essay to which the
Adams prize was awarded. He concluded that they consisted of an
aggregate of unconnected particles. "Thus an idea put forward as a
speculation in the seventeenth century, and afterwards in the eight-
eenth century by J. Cassini and Thomas Wright, was mathematically
demonstrated as the only possible solution." ;

The progress in methods of computing planetary, asteroidal, and
cometary orbits has proceeded along two more or less distinct lines,

1 E. W. Brown in Bull. Am. Maih. Soc., Vol. 16, 1910, p. 353.

2 W. W. Bryant, A llislory of Astronomy, London, 1907, p. 233.

452 A HISTORY OF MATHEMATICS

the one marked out by P. S. Laplace, the other by K. F. Gauss.1 La-
place's method possessed theoretical advantages, but lacked practical
applicability for the reason that in the second approximation the
results of the first approximation could be used only in part and the
computation had to be gone over largely anew. To avoid this labor
in finding asteroidal and comctary orbits, Heinrich W. M. Olbers
(1758-1840) and K. F. Gauss devised more expeditious processes for
carrying out the second approximation. The Gaussian procedure
was refined and simplified by Johann Franz Encke (1791-1865),
Francesco Carlini (1783-1862), F. W. Bessel, P. A. Hanscn, and es-
pecially by Theodor von Oppolzer (1841-1886) of Vienna whose
method has been used by practical astronomers down to the present
day. Most original among the new elaborations of Gauss' method is
that of /. Willard Gibbs of Yale, which employs vector analysis and,
though rather complicated, yields remarkable accuracy even in the
first approximation. Gibbs' procedure was modified in 1905 by J.
Frischauf of Graz. P. S. Laplace's method has attracted mathemati-
cians by its elegance. It received the attention of A. L. Cauchy,
Antoine Yvon Villarceau (1813-1883) of the Paris Observatory,
Rodolphe Radau of Paris, H. Bruns of Leipzig, and H. Poincare. Paul
Harzer of Kiel and especially Armin Otto Leuschner of the University
of California made striking advances in rendering Laplace's method
available for rapid computation. Leuschner adopts from the start
geocentric co-ordinates and considers the effects of the perturbating
body in the very first approximation; it is equally applicable to plane-
tary and to cometary orbits.2

Problem of Three Bodies

The problem of three bodies has been treated in various ways since
the time of J. Lagrangc, and some decided advance towards a more
complete solution has been made. Lagrange's particular solution
based on the constancy of the relative distances of the three bodies,
one from the other (called by L. O. Hesse the reduced problem of
three bodies) has recently been modified by Carl L. Charlier of the
observatory at Lund, in which the mutual distances are replaced
by the distances from the centre of gravity.3 This new form possesses
no marked advantage. "Theoretical interest in the Lagrangian solu-
tions has been increased," says E. O. Lovett, "by K. F. Sundman's
theorem that the more nearly all three bodies in the general problem
tend to collide simultaneously, the more nearly do they tend to as-
sume one or the other of Lagrange's configurations; . . . practical

1 We are using an article by A. Vcnturi in Rhista di Astronomia, June, 1913.

2 For a fuller historical account, see A. O. Leuschner in Science, N. S., Vol. 45,
1917, pp. 571-584-

3 We are drawing from E. O. Lovett's "The Problem of Three Bodies" in Science,
N. S., Vol. 29, 1909, pp. 81-91.

APPLIED MATHEMATICS 453

interest in them has been revived by the discovery of three small
planets, 1906 T. G., 1906 V. Y., 1907 X. M., near the equilateral tri-
angular points of the Sun-Jupiter-Asteroid system. ... R. Leh-
mann-Filhes, R. Hoppe, and Otto Dziobek, all three of Berlin, have
generalized the exact solutions to cases of more than three bodies
placed on a line or at the vertices of a regular polygon or polyhe-
dron. . . . Among the most interesting extensions of Lagrange's
theorem are those due to T. Banachievitz of Kasan and F. R. Moul-
ton." In 1912 H. Poincare indicated that on the basis of a ring
representation (but in Keplerian variables), that if a certain geometric
theorem (later established by G. D. Birkhoff of Harvard University)
were true, the existence of an infinite number of periodic solutions
would follow in the restricted proWem of three bodies. These results
were amplified by G. D. Birkhoff.1 The so-called isosceles-triangle
solutions of the problem of three bodies (periodic solutions in which
two of the masses are finite and equal, while the third body moves in
a straight line and remains equidistant from the equal bodies) received
the attention of Giulio Pavanini of Treviso in Italy in 1907, W. D.
MacMillan of Chicago in 1911, and D. Buchanan of Ontario in 1914.
G. W. Hill, in his researches on lunar theory, added in 1877 to the
Lagrangian periodic solution, which for 105 years had been the only
such solution known, another periodic solution which could serve as
a starting point for a study of the moon's orbit. Says E. O. Lovett:
"With these memoirs he broke ground for the erection of the new
science of dynamical astronomy whose mathematical foundations
were laid broad and deep by Poincare," in a research which in 1889
won a prize offered by King Oscar II, and which he developed more
fully later. The original memoir of Poincare, says Moulton, "con-
tained an error which was discovered by E. Phragmen, of Stockholm,
but it affected only the discussion of the existence of the asymptotic
solutions; and in correcting this part H. Poincare . . . confessed
fully his obligations to Phragmen. . . . There is not the slightest
doubt that in spite of it ... the prize was correctly bestowed."
The researches of G. W. Hill and H. Poincare have been continued
mainly by E. W. Brown, G. H. Darwin, F. R. Moulton, Hugo Gylden
(1841-1896) of Stockholm, P. Painleve, C. L. W. Chiirlier, S. E.
Stromgren, and T. Levi-Civita, in which questions of stability have re-
ceived much attention. The general question, whether the solar
system is stable, was affirmed by eighteenth century mathematicians;
it was rc-opcncd by K. Weierstrass who, in the last years of his life,
devoted considerable attention to it. Expressions for the co-ordinates
of the planets converge cither not at all or for only limited time. In
addition to the complex mixture of known cyclical changes, there
might, perhaps, be a small residue of change of such a nature that
the system will ultimately be wrecked. At present no rigorous answer
1 Bull. Am. Math. Soc., Vol. 20, 1914, p. 292.

454 A HISTORY OF MATHEMATICS

has been given, but "Poincare showed that solutions exist in which
the motion is purely periodic, and therefore that in them at least no
disaster of collision or indefinite departure from the central mass will
ever occur" (F. R. Moulton). A startling result was Poincare's dis-
covery that some of the series which have been used to calculate the
positions of the bodies of the solar system arc divergent. An exam-
ination of the reasons why the divergent series gave sufficiently ac-
curate results gave rise to the theory of asymptotic series now applied
to the representation of many functions. Does the ultimate diver-
gence of the series throw doubt upon the stability of the solar system?
H. Gylden thought that he had overcome the difficulty, but H. Poin-
care showed that in part it still exists. Following Poincare's lead,
E. W. Brown has formulated the sufficient conditions for stability in
the w-body problem. T. Levi-Civita worked out criteria in which
the stability is made to depend upon that of a certain point trans-
formation associated with the periodic function. He proved the
existence of zones of instability surrounding Jupiter's orbit. The
new methods in celestial mechanics have been found useful in com-
puting the perturbations of certain small planets. Material advances
in the problem of three bodies were made by Karl F. Sundman of
Helsingfors in Finland, in a memoir which received a prize of the
Paris Academy in 1913. This research is along the path first blazed
by P. Painleve, continued by T. Levi-Civita and others.

In the transformation and reduction of the three-body problem, "a
principal role has been played by the ten known integrals, namely,
the six integrals of motion of the centre of gravity, three integrals of
angular momentum, and the integral of energy. The question of
further progress in this reduction is vitally related to the non-existence
theorems of H. Bruns, H. Poincare, and P. Painleve. H. Bruns demon-
strated that the n-body problem admits of no algebraical integral
other than the ten classic ones, and H. Poincare proved the non-
existence of any other uniform analytical integral." Other researches
on these non-existence theorems are due to P. Painleve, D. A. Grave,
and K. Bohlin.

E. Picard expresses himself as follows:1 "What admirable recent
researches have best taught them [analysts] is the immense difficulty
of the problem; a new way has, however, been opened by the study
of particular solutions, such as the periodic solutions and the asymp-
totic solution which have already been utilized. It is not perhaps
so much because of the needs of practice as in order not to avow it-
self vanquished, that analysis will never resign itself to abandon, with-
out a decisive victory, a subject where it has met so many brilliant
triumphs; and again, what more beautified field could the theories
new-born or rejuvenated of the modern doctrine of functions find,
to essay their forces, than this classic problem of n bodies?"
1 Congress of Arts and Science, St. Louis, 1904, Vol. I, p. 512.

APPLIED MATHEMATICS 455

Among valuable text-books on mathematical astronomy of the
nineteenth century rank the following works: Manual of Spherical
and Practical Astronomy by William Chauvenet (1863), Practical and
Spherical Astronomy by Robert Main of Cambridge, Theoretical As-
tronomy by James C. Watson of Ann Arbor (1868), Traite ele-.nentaire
de Mecanique Celeste of H. Resal of the Ecole Polytechnique in Paris,
Cours d'Astronomie de I' Ecole Polytechnique by Faye, Traite de Mecani-
que Celeste by F. F. Tisserand, Lehrbuch der Bahnbestimmung by T.
Oppolzer, Mathematische Theorien der Planetenbewegung by O. Dziobek,
translated into English by M. W. Harrington and W. J. Hussey.

General Mechanics

During the nineteenth century we have come to recognize the ad-
vantages frequently arising from a geometrical treatment of me-
chanical problems. To L. Poinsot, M. Chasles, and A. F. Mobius we
owe the most important developments made in geometrical mechanics.
Louis Poinsot (1777-1859), a graduate of the Polytechnic School in
Paris, and for many years member 9f the superior council of public
instruction, published in 1804 his Elements de Statique. This work
is remarkable not only as being the earliest introduction to synthetic
mechanics, but also as containing for the first time the idea of couples,
which was applied by Poinsot in a publication of 1834 to the theory
of rotation. A clear conception of the nature of rotary motion was
conveyed by Poinsot's elegant geometrical representation by means
of an ellipsoid rolling on a certain fixed plane. This construction was
extended by J. J. Sylvester so as to measure the rate of rotation of the
ellipsoid on the plane.

A particular class of dynamical problems has recently been treated
geometrically by Sir Robert Stawell Ball (1840-1913) at one time
astronomer royal of Ireland, later Lowndean Professor of Astronomy
and Geometry at Cambridge. His method is given in a work entitled
Theory of Screws, Dublin, 1876, and in subsequent articles. Modern
geometry is here drawn upon, as was done also by W. K. Clifford
in the related subject of Bi-quaternions. Arthur Buchheim (1859-
1888), of Manchester showed that H. G. Grassmann's Ausdehnungs-
lehre supplies all the necessary materials for a simple calculus of screws
in elliptic space. Horace Lamb applied the theory of screws to the
question of the steady motion of any solid in a fluid.

Advances in theoretical mechanics, bearing on the integration and
the alteration in form of dynamical equations, were made since J.
Lagrange by S. D. Poisson, Sir William Rowan Hamilton, C. G. J.
Jacobi, Madame Koalevski, and others. J. Lagrange had established
the "Lagrangian form" of the equations of motion. He had given a
theory of the variation of the arbitrary constants which, however,
turned out to be less fruitful in results than a theory advanced by S. D.

456 A HISTORY OF MATHEMATICS

Poisson.1 Poisson's theory of the variation of the arbitrary constants
and the method of integration thereby afforded marked the first
onward step since J. Lagrange. Then came the researches of Sir
William Rowan Hamilton. His discovery that the integration of the
dynamic differential equations is connected with the integration of a
certain partial differential equation of the first order and second degree,
grew out of an attempt to deduce, by the undulatory theory, results
in geometrical optics previously based on the conceptions of the emis-
sion theory. The Philosophical Transactions of 1833 and 1834 contain
Hamilton's papers, in which appear the first applications to mechanics
of the principle of varying action and the characteristic function,
established by him some years previously. The object which Hamilton
proposed to himself is indicated by the title of his first paper, viz.,
the discovery of a function by means of which all integral equations
can be actually represented. The new form obtained by him for the
equation of motion is a result of no less importance than that which
was the professed object of the memoir. Hamilton's method of in-
tegration was freed by C. G. J. Jacobi of an unnecessary complica-
tion, and was then applied by him to the determination of a geodetic
line on the general ellipsoid. With aid of elliptic co-ordinates Jacobi
integrated the partial differential equation and expressed the equation
of the geodetic in form of a relation between two Abelian integrals.
C. G. J. Jacobi applied to differential equations of dynamics the theory
of the ultimate multiplier. The differential equations of dynamics are
only one of the classes of differential equations considered by Jacobi.
Dynamic investigations along the lines of J. Lagrange, Hamilton, and
Jacobi were made by J. Liouville, Adolphe Desboves, (1818-1888)
of Amiens, Serret, J. C. F. Sturm, Michel Ostrogradski, J. Bertrand,
William Fishburn Donkin (1814-1869) of Oxford, F. Brioschi, leading
up to the development of the theory of a system of canonical integrals.

An important addition to the theory of the motion of a solid body
about a fixed point was made by Madame Sophie Kovalevski (1850-
1891), who discovered a new case in which the differential equations
of motion can be integrated. By the use of theta-functions of two
independent variables she furnished a remarkable example of how
the modern theory of functions may become useful in mechanical
problems. She was a native of Moscow, studied under K. Weierstrass,
obtained the doctor's degree at Gottingen, and from 1884 until her
death was professor of higher mathematics at the University of Stockr
holm. The research above mentioned received the Bordin prize
of the French Academy in 1888, which was doubled on account of
the exceptional merit of the paper.

There are in vogue three forms for the expression of the kinetic
energy of a dynamical system: the Lagrangian, the Hamiltonian, and

1 Arthur Cayley, " Report on the Recent Progress of Theoretical Dynamics,"
Report British Ass'n for 1857, p. 7.

APPLIED MATHEMATICS 457

a modified form of Lagrange's equations in which certain velocities
are omitted. The kinetic energy is expressed in the first form as a
homogeneous quadratic function of the velocities, which are the time-
variations of the co-ordinates of the system; in the second form, as
a homogeneous quadratic function of the momenta of the system;
the third form, elaborated recently by Edward John Routh (1831-
1907) of Cambridge, in connection with his theory of "ignoration of
co-ordinates," and by A. B. Basset of Cambridge, is of importance in
hydro-dynamical problems relating to the motion of perforated solids
in a liquid, and in other branches of physics.

Practical importance has come to be attached to the principle of
mechanical similitude. By it one can determine from the performance
of a model the action of the machine constructed on a larger scale.
The principle was first enunciated by I. Newton (Principia, Bk. II,
Sec. VIII, Prop. 32), and was derived by Joseph Bertrand from the
principle of virtual velocities. A corollary to it, applied in ship-
building, is named after the British naval architect William Froude
(1810-1879), but was enunciated also by the French engineer Frederic
Reech.

The present problems of dynamics differ materially from those of
the last century. The explanation of the orbital and axial motions
of the heavenly bodies by the law of universal gravitation was the
great problem solved by A. C. Clairaut, L. Euler, D'Alembert, J.
Lagrange, and P. S. Laplace. It did not involve the consideration of
frictional resistances. In the present time the aid of dynamics has
been invoked by the physical sciences. The problems there arising
are often complicated by the presence of friction. Unlike astronomical
problems of a century ago, they refer to phenomena of matter and
motion that are usually concealed from direct observation. The great
pioneer in such problems is Lord Kelvin. While yet an undergraduate
at Cambridge, during holidays spent at the seaside, he entered upon
researches of this kind by working out the theory of spinning tops,
which previously had been only partially explained by John Hewitt
Jellett (1817-1888) of Trinity College,. Dublin, in his Treatise on the
Theory of Friction (1872), and by Archibald Smith (1813-1872).

Among standard works on mechanics of the nineteenth century are
C. G. J. Jacobi's Vorlesungen uber Dynamik, edited by R. F. A . Clebsch,
1866; G. R. Kirchho/'s Vorlesungen uber mathematische Physik, 1876;
Benjamin Peirce's Analytic Mechanics, 1855; J. I. Somojf's Theoretische
Mechanik, 1879; P. G. Tail and W. J. Steele's Dynamics of a Particle,
1856; George Minchin's Treatise on Statics; E. J. Routh' s Dynamics of
a System of Rigid Bodies; J. C. F. Sturm's Cours de Mecanique de I'Ecole
Polytechnique. George M. Minchin (1845-1914) was professor at the
Indian engineering college.

In 1898 Felix Klein pointed out the separation which existed be-
tween British and Continental mathematical research, as seen, for

458 A HISTORY OF MATHEMATICS

instance, by the contents of E. J. Routh's Dynamics, which contains
the results of twenty years of research along that line in England and,
in comparison with the German school, emphasizes a concrete and
practical treatment. To make these treasures more readily accessible
to German students, Routh's text was translated into German by
Adolf Schepp (1837-1905) of Wiesbaden hi 1898. Particularly strong
was Routh in the treatment of small oscillations of systems; the
technique of integration of linear differential equations with con-
stant coefficients is highly developed, except that, perhaps, the extent
to which the developments are valid may need closer examination.
This is done in F. Klein and A. Sommerfeld's Theorie des Kreisels,
1897-1910. This last work gives attention to the theory of the top,
the history of which reaches back to the eighteenth century.

In 1744 Serson started on a ship (that was lost), to test the prac-
ticability of the artificial horizon furnished by the polished surface
of a top. This idea has been recently revived by French navigators.1
Serson's top induced J. A. Segner of Halle in 1755 to give precision
to the theory of the spinning top, which was taken up more fully by
L. Euler in 1765 and then by J. Lagrange. L. Euler considers the
motion on a smooth horizontal plane. Later come the studies due
to L. Poinsot, S. D. Poisson, C. G. J. Jacobi, G. R. Kirchhoff, Eduard
Lottner (1826-1887) °f Lippstadt, Wilhelm Hess, Clerk Maxwell,
E. J. Routh and finally F. Klein and A. Sommerfeld. In 1914 G.
Greenhill prepared a Report on Gyroscopic Theory 2 which is of more
direct interest to engineers than is Klein and Sommerfeld's Theorie des
Kreisels, developed by the aid of the theory of functions of a complex
variable. Among recent practical applications of gyroscopic action
are the torpedo exhibited before the Royal Society of London in 1907
by Louis Brennan, also Brennan's monorail system, and the methods
of steadying ships and aircraft, devised by the American engineer
Elmer A. Sperry and by Otto Schlick in Germany.

Among the deviations of a projectile from the theoretic parabolic
path there are two which are of particular interest. One is a slight
bending to the right, in the northern hemisphere, owing to the rotation
of the earth; it was explained by S. D. Poisson (1838) and W. Ferrel
(1889). The other is due to the rotation of the projectile; it was ob-
served by I. Newton in tennis balls and applied by him to explain
certain phenomena in his corpuscular theory of light; it was known
to Benjamin Robins and L. Euler. In 1794 the Berlin academy
offered a prize for an explanation of the phenomenon, but no satis-
factory explanation appeared for over half a century. S. D. Poisson
in 1839 (Journ. ecole polyt., T. 27) studied the effect of atmospheric

1 See A. G. Greenhill in Verhandl. III. Intern. Congr., Heidelberg, 1904, Leipzig,
1905, p. 100. We are summarizing this article.

2 Advisory Committee for Aeronautics, Reports and Memoranda, No. 146, London,
1914.

APPLIED MATHEMATICS 459

friction against the rotating sphere, but finally admitted that friction
was not sufficient to explain the deviations. The difference in the
pressure of the air upon the rotating sphere also demands attention.
An explanation on this basis, which was generally accepted as valid
was given by H. G. Magnus (1802-1870) of Berlin, in Poggendorffs
Annalen, Vol. 88, 1853. In connection with golf -balls the problem
was taken up by Tait.

Peter Guthrie Tait (1831-1901) was born at Dalkeith, studied at
Cambridge and came out Senior Wrangler in 1854, which was a sur-
prise, as W. J. Steele had been generally ahead in college examinations.
From 1854-1860 Tait was professor of mathematics at Belfast, where
he studied quaternions; from 1860 to his death he held the chair of
Natural Philosophy at Edinburgh. Tait found the problem of the
flight of the golf ball capable of exact statement and approximate
solution. One of his sons had become a brilliant golfer. Tait at first
was scoffed at when he began to offer explanations of the secret of
long driving. In 1887 (Nature, 36, p. 502) he shows that "rotation"
played an important part, as established experimentally by H. G.
Magnus (1852). Says P. G. Tait: "In topping, the upper part of the
ball is made to move forward faster than does the center, consequently
the front of the ball descends in virtue of the rotation, and the ball
itself skews in that direction. When a ball is undercut it gets the
opposite spin to the last, and, in consequence, it tends to deviate up-
wards instead of downwards. The upward tendency often makes the
path of a ball (for a part of its course) concave upwards in spite of
the effects of gravity. ..." P. G. Tait explained the influence of
the underspin in prolonging not only the range but also the tune of ,
flight. The essence of his discovery was that without spin a ball
could not combat gravity greatly, but that with spin it could travel
remarkable distances. He was fond of the game while H. Helmholtz
(who was in Scotland in 1871) "could see no fun in the leetle hole."

P. G. Tait generalized in 1898 the Josephus problem and gave the
rule for n persons, certain v of which shall be left after each m th man
is picked out.

The deviations of a body falling from rest near the surface of the
earth have been considered in many memoirs from the time of P. S.
Laplace and K. F. Gauss to the present. All writers agree that the
body will deviate to the eastward with respect to the plumb-line hung
from the initial point, but there has been disagreement regarding the
deviation measured along the meridian. Laplace found no meridional
deviation, Gauss found a small deviation toward the equator. Re-
cently this problem has commanded the attention of writers in the
'United States. R. S. Woodward, president of the Carnegie Institution
in Washington, found in 1913 a deviation away from the equator.
F. R. Moulton of the University of Chicago found in 1914 a formula
indicating a southerly deviation. W. H. Rover of Washington Uni-

460 A HISTORY OF MATHEMATICS

versity in St. Louis has, since 1901, treated the subject in several
articles which indicate southerly deviations. He declares that "no
potential function is known that fits all parts of the earth," "that
the formula of Gauss, the three formulae of Comte de Sparre [Lyon,
1905], the formula of Professor F. R. Moulton, and my first formula,
are all special cases of my general formula." l

Fluid Motion

The equations which constitute the foundation of the theory of
fluid motion were fully laid down at the time of J. Lagrange, but the
solutions actually worked out were few and mainly of the irrotational
type. A powerful method of attacking problems in fluid motion is
that of images, introduced in 1843 by G. G. Stokes of Pembroke Col-
lege, Cambridge. It received little attention until Sir William Thom-
son's discovery of electrical images, whereupon the theory was ex-
tended by G. G. Stokes, W. M. Hicks, and T. C. Lewis.

George Gabriel Stokes (1819-1903) was born at Skreen, County
Sligo, in Ireland. In 1837, the year of Queen Victoria's accession, he
commenced residence at Cambridge, where he was to find his home,
almost without intermission, for sixty-six years. At Pembroke College
his mathematical abilities attracted attention and in 1841 he graduated
as Senior Wrangler and first Smith's prizeman. He distinguished
himself along the lines of applied mathematics. In 1845 he published
a memoir on "Friction of Fluids in Motion." The general motion of
a medium near any point is analyzed into three constituents — a mo-
tion of pure translation, one of pure rotation and one of pure strain.
Similar results were reached by H. Helmholtz twenty-three years
later. In applying his results to viscous fluids, Stokes was led to
general dynamical equations, previously reached from more special
hypotheses by L. M. H. Navier and S. D. Poisson. Both Stokes and
G. Green were followers of the French school of applied mathemati-
cians. Stokes applies his equations to the propagation of sound, and
shows that viscosity makes the intensity of sound diminish as the
time increases and the velocity less than it would otherwise be —
especially for high notes. He considered ine two elastic constants in
the equations for an elastic solid to be independent and not reducible
to one as is the case in Poisson's theory. Stokes' position was sup-
ported by Lord Kelvin and seems now generally accepted. In 1847
Stokes examined anew the theory of oscillatory waves. Another
paper was on the effect of internal friction of fluids on the motion of
pendulums. He assumed that the viscosity of the air was propor-
tional to the density, which was shown later by Maxwell to be erro-
neous. In 1849 he treated the ether as an elastic solid in the study of
diffraction. He favored Fresnel's wave theory of light as opposed to

1 See Washington University Studies, Vol. Ill, 1916, pp. 153-168.

APPLIED MATHEMATICS 461

the corpuscular theory supported by David Brewster. In a report
on double refraction of 1862 he correlated the work of A. L. Cauchy,
J. MacCullagh, and G. Green. Assuming that the elasticity of the
ether has its origin in deformation, he inferred that J. MacCullagh's
theory was contrary to the laws of mechanics, but recently J. Larmor
nas shown that J. MacCullagh's equations may be explained on the
supposition that what is resisted is not deformation, but rotation.
Stokes wrote on Fourier series and the discontinuity of arbitrary
constants in semi-convergent expansions over a plane. His contribu-
tions to hydrodynamics and optics are fundamental. In 1849 William
Thomson (Lord Kelvin) gave the maximum and minimum theorem
peculiar to hydrodynamics, which was afterwards extended to dynam-
ical problems in general.

A new epoch in the progress of hydrodynamics was created, in 1856,
by H. Helmholtz, who worked out remarkable properties of rotational
motion in a homogeneous, incompressible fluid, devoid of viscosity.
He showed that the vortex filaments in such a medium may possess
any number of knottings and twistings, but are either endless or the
ends are in the free surface of the medium ; they are indivisible. These
results suggested to William Thomson (Lord Kelvin) the possibility
of founding on them a new form of the atomic theory, according to
which every atom is a vortex ring in a non-frictional ether, and as
such must be absolutely permanent in substance and duration. The
vortex -atom theory was discussed by J. J. Thomson of Cambridge
(born 1856) in his classical treatise on the Motion of Vortex Rings, to
which the Adams Prize was awarded in 1882. Papers on vortex motion
have been published also by Horace Lamb, Thomas Craig, Henry A.
Rowland, and Charles Chree of Kew Observatory.

The subject of jets was investigated by H. Helinholtz, G. R. Kirch-
hoff, J. Plateau, and Lord Rayleigh; the motion of fluids in a fluid by
G. G. Stokes, W. Thomson (Lord Kelvin), H. A. Kopcke, G. Greenhill,
and H. Lamb; the theory of viscous fluids by H. Navier, S. D. Poisson,
B. de Saint- Venant, Stokes, Oskar Emil Meyer (1834-1909) of Breslau,
A. B. Stefano, C. Maxwell, R. Lipschitz, T. Craig, H. Helmholtz, and
A. B. Basset. Viscous fluids present great difficulties, because the
equations of motion have not the same degree of certainty as in per-
fect fluids, on account of a deficient theory of friction, and of the
difficulty of connecting oblique pressures on a small area with the
differentials of the velocities.

Waves in liquids have been a favorite subject with English mathe-
maticians. The early inquiries of S. D. Poisson and A. L. Cauchy
were directed to the investigation of waves produced by disturbing
causes acting arbitrarily on a small portion of the fluid. The velocity
of the long wave was given approximately by J. Lagrange in 1786 in
case of a channel of rectangular cross-section, by Green in 1839 for
a channel of triangular section, and by Philip Kelland (1810-1879)

462 A HISTORY OF MATHEMATICS

of Edinburgh for a channel of any uniform section. Sir George B.
Airy, in his treatise on Tides and Waves, discarded mere approxima-
tions, and gave the exact equation on which the theory of the long
wave in a channel of uniform rectangular section depends. But he
gave no general solutions. J. McCowan of University College at
Dundee discussed this topic more fully, and arrived at exact and com-
plete solutions for certain cases. The most important application of
the theory of the long wave is to the explanation of tidal phenomena
in rivers and estuaries.

The mathematical treatment of solitary waves was first taken up
by S. Earnshaw in 1845, then by G. G. Stokes; but the first sound
approximate theory was given by J. Boussinesq in 1871, who obtained
an equation for their form, and a value for the velocity in agreement
with experiment. Other methods of approximation were given by
. Lord Rayleigh and John McCowan. In connection with deep-water
waves, Osborne Reynolds (1842-1912) of the University of Manchester
gave in 1877 the dynamical explanation for the fact that a group of
such waves advances with only half the rapidity of the individual
waves.

The solution of the problem of the general motion of an ellipsoid
in a fluid is due to the successive labors of George Green (1833),
R. F. A. Clebsch (1856), and Carl Anton Bjerknes (1825-1903) of
Christiania (1873). The free motion of a solid in a liquid has been
investigated by W. Thomson (Lord Kelvin), G. R. Kirchhoff, and
Horace Lamb. By these labors, the motion of a single solid hi a fluid
has come to be pretty well understood, but the case of two solids in a
fluid is not developed so fully. The problem has been attacked by
W. M. Hicks.

The determination of the period of oscillation of a rotating liquid
spheroid has important bearings on the question of the origin of the
moon. G. H. Darwin's investigations thereon, viewed in the light of
G. F. B. Riemann's and H. Poincare's researches, seem to disprove
P. S. Laplace's hypothesis that the moon separated from the earth
as a ring, because the angular velocity was too great for stability;
G. H. Darwin finds no instability.

The explanation of the contracted vein has been a point of much
controversy, but has been put in a much better light by the application
of the principle of momentum, originated by W. Froude and Lord
Rayleigh. Rayleigh considered also the reflection of waves, not at
the surface of separation of two uniform media, where the transition
is abrupt, but at the confines of two media between which the transition
is gradual.

The first serious study of the circulation of winds on the earth's
surface was instituted at the beginning of the second quarter of the last
century by William C. Redfield (1789-1857), an American meteorolo-
gist and railway projector, James Pollard Espy (1786-1860) of Wash-

APPLIED MATHEMATICS 463

ington, through whose stimulus the present United States Weather
Bureau was started and Heinrich Wilhelm Dove (1803-1879) of Berlin,
followed by researches by Sir William Reid (1791-1858) a British
major-general who developed his circular theory of hurricanes while in
the West Indies, Henry Piddington (1797-1858) a British commander
in the mercantile marine who accumulated data for determining the
course of storms at sea and originated the term "cyclone," and Elias
Loomis (1811-1889) of Yale University. But the deepest insight into
the wonderful correlations that exist among the varied motions of the
atmosphere was obtained by William Ferrel (1817-1891). He was
born in Fulton County, Pa., and brought up on a farm. Though in
unfavorable surroundings, a burning thirst for knowledge spurred
the boy to the mastery of one branch after another. He attended
Marshall College, Pa., and graduated in 1844 from Bethany College.
While teaching school he became interested in meteorology and in
the subject of tides. In 1856 he wrote an article on "the winds and
currents of the ocean." The following year he became connected
with the Nautical Almanac. A mathematical paper followed in 1858
on "the motion of fluids and solids relative to the earth's surface."
The subject was extended afterwards so as to embrace the mathe-
matical theory of cyclones, tornadoes, water-spouts, etc. In 1885
appeared his Recent Advances in Meteorology. In the opinion of
Julius Hann of Vienna, Ferrel has "contributed more to the advance
of the physics of the atmosphere than any other living physicist or
meteorologist."

W. Ferrel taught that the air flows in great spirals toward the poles,
both in the upper strata of the atmosphere and on the earth's surface
beyond the 3oth degree of latitude; while the return current blows at
nearly right angles to the above spirals, in the middle strata as well
as on the earth's surface, in a zone comprised between the parallels
30° N. and 30° S. The idea of three superposed currents blowing spirals
was first advanced by James Thomson (1822-1892), brother of Lord
Kelvin, but was published in very meagre abstract.

W. Ferrel's views have given a strong impulse to theoretical re-
search in America, Austria, and Germany. Several objections raised
against his argument have been abandoned, or have been answered
by W. M. Davis of Harvard. The mathematical analysis of F. Waldo
of Cambridge, Mass., and of others, has further confirmed the accuracy
of the theory. The transport of Krakatoa dust and observations made
on clouds point toward the existence of an upper east current on the
equator, and Josef M. Pernter (1848-1908) of Vienna has mathe-
matically deduced from Fen-el's theory the existence of such a current.

Another theory of the general circulation of the atmosphere was
propounded by Werner Siemens (1816-1892) of Berlin, in which an
attempt is made to apply thermodynamics to aerial currents. Im-
portant new points of view have been introduced by H. Helmholtz,

464 A HISTORY OF MATHEMATICS .

who concluded that when two air currents blow one above the other
in different directions, a system of air waves must arise in the same
way as waves are formed on the sea. He and Anton Oberbeck (1846-
1900) of Tubingen showed that when the waves on the sea attain
lengths of from 1 6 to 33 feet, the air waves must attain lengths of from
10 to 20 miles, and proportional depths. Superposed strata would
thus mix more thoroughly, and their energy would be partly dissipated.
From hydrodynamical equations of rotation H. Helmholtz established
the reason why the observed velocity from equatorial regions is much
less in a latitude of, say, 20° or 30°, than it would be were the move-
ments unchecked. Other important contributors to the general theory
of the circulation of the atmosphere are Max Moller of Braunschweig
and Luigi de March! of the University of Pavia. The source of the
energy of atmospheric disturbances was sought by W. Ferrel and Th.
Reye in the heat given off during condensation. Max Margules of the
University of Vienna showed in 1905 that this heat energy contributes
nothing to the kinetic energy of the winds and that the source of
energy is found in the lowering of the centre of gravity of an air column
when the colder air assumes the lower levels, whereby the potential
energy is diminished and the kinetic energy increased.1 Asymmetric
cyclones have been studied especially by Luigi de Marchi of Pavia.
Anticyclones have received attention from Henry H. Clayton of the
Blue Hill Observatory, near Boston, from Julius Hann of Vienna, F.
H. Bigelow of Washington, and Max Margules of Vienna.

Sound. Elasticity

About 1860 acoustics began to be studied with renewed zeal. The
mathematical theory of pipes and vibrating strings had been elabo-
rated in the eighteenth century by Daniel Bernoulli, D'Alembert,
L. Euler, and J. Lagrange. In the first part of the present century
P. S. Laplace corrected Newton's theory on the velocity of sound in
gases; S. D. Poisson gave a mathematical discussion of torsional
vibrations; S. D. Poisson, Sophie Germain, and Charles Wheatstone
studied Chladni's figures; Thomas Young and the brothers Weber
developed the wave-theory of sound. Sir J. F. W. Herschel (1792-
1871) wrote on the mathematical theory of sound for the Encyclo-
paedia Metropolitana, 1845. Epoch-making were H. Helmholtz's
experimental and mathematical researches. In his hands and Ray-
leigh's, Fourier's series received due attention. H. Helmholtz gave
the mathematical theory of beats, difference tones, and summation
tones. Lord Rayleigh (John William Strutt) of Cambridge (born
1842) made extensive mathematical researches in acoustics as a part
of the theory of vibration in general. Particular mention may be
made of his discussion of the disturbance produced by a spherical

1 Encykhpadie der Math. Wissenschaften, Bd. VI, i, 8, 1912, p. 216.

APPLIED MATHEMATICS 465

obstacle on the waves of sound, and of phenomena, such as sensitive
flames, connected with the instability of jets of fluid. In 1877 and 1878
he published in two volumes a treatise on The Theory of Sound. Other
mathematical researches on this subject have been made in England
by William Fishburn Donkin (1814-1869) of Oxford and G. G. Stokes.
An interesting point in the behavior of a Fourier's series was brought
out in 1898 by J. W. Gibbs of Yale. A. A. Michelson and S. W. Strat-
ton at the University of Chicago had shown experimentally by their
harmonic analyses that the summation of 160 terms of the series
S( — i)in+l(sinnx)/n revealed certain unexpected small towers in the
curve for the sum, as n increased. J. W. Gibbs showed (Nature, Vol.
59, p. 606) by the study of the order of variation of n and x that these
phenomena were not due to imperfections in the machine, but were
true mathematical phenomena. They are called the " Gibbs' phenom-
enon," and have received further attention from Maxime Bdcher,
T. H. Gronwall, H. Weyl, and H. S. Carslaw.

The theory of elasticity 1 belongs to this century. Before 1800 no
attempt had been made to form general equations for the motion or
equilibrium of an elastic solid. Particular problems had been solved
by special hypotheses. Thus, James Bernoulli considered elastic
laminae; Daniel Bernoulli and L. Euler investigated vibrating rods;
J. Lagrange and L. Euler, the equilibrium of springs and columns.
The earliest investigations of this century, by Thomas Young
("Young's modulus of elasticity") in England, J. Binet in France,
and G. A. A. Plana in Italy, were chiefly occupied in extending and
correcting the earlier labors. Between 1820 and 1840 the broad out-
line of the modern theory of elasticity was established. This was ac-
complished almost exclusively by French writers, — Louis-Marie-
Henri Navier (1785-1836), S. D. Poisson, A. L. Cauchy, Mademoiselle
Sophie Germain (1776-1831), Felix Savart (1791-1841). Says H.
Burkhardt: "There are two views respecting the beginnings of the
theory of elasticity of solids, of which no dimension can be neglected:
According to one view the deciding impulse came from Fresnel's
undulatory theory of light, according to the other, everything goes
back to the technical theory of rigidity (Festigkeitstheorie), the rep-
resentative of which was at that time Navier. As always in such
cases, the truth lies in the middle: Cauchy to whom we owe primarily
the fixing of the fundamental concepts, as strain and stress, learned
from Fresnel as well as from Navier."

Simeon Denis Poisson2 (1781-1840) was born at Pithiviers. The
boy was put out to a nurse, and he used to tell that when his father
(a common soldier) came to see him one day, the nurse had gone out

1 T. Todhunter, History of the Theory of Elasticity, edited by Karl Pearson, Cam-
bridge, 1886.

2 Ch. Hermite, "Discours prononc6 devant le president de la R£publique,"
Bulletin dcs sciences mathtmaliques, XIV, Janvier, 1890.

466 A HISTORY OF MATHEMATICS

and left him suspended by a thin cord to a nail in the wall in order to
protect him from perishing under the teeth of the carnivorous and un-
clean animals that rokmed on the floor. Poisson used to add that his
gymnastic efforts when thus suspended caused him to swing back and
forth, and thus to gain an early familiarity with the pendulum, the
study of which occupied him much in his maturer life. His father
destined him for the medical profession, but so repugnant was this
to him that he was permitted to enter the Polytechnic School at the
age of seventeen. His talents excited the interest of J. Lagrange and
P. S. Laplace. At eighteen he wrote a memoir on finite differences
which was printed on the recommendation of A. M. Legendre. He
soon became a lecturer at the school, and continued through life to
hold various government scientific posts and professorships. He pre-
pared some 400 publications, mainly on applied mathematics. His
Traite de Mecanigue, 2 vols., 1811 and 1833, was long a standard work.
He wrote on the mathematical theory of heat, capillary action, proba-
bility of judgment, the mathematical theory of electricity and mag-
netism, physical astronomy, the attraction of ellipsoids, definite in-
tegrals, series, and the theory of elasticity. He was considered one
of the leading analysts of his time. The story is told that in 1802 a
young man, about to enter the army, asked Poisson to take $100 in
safe-keeping. "All right," said Poisson," set it down there and let
me work; I have much to do." The recruit placed the money-bag on
a shelf and Poisson placed a copy of Horace over the bag, to hide it.
Twenty years later the soldier returned and asked for his money,
but Poisson remembered nothing and asked angrily: "You claim
to have put the money in my hands?" "No," replied the soldier,
" I put in on this shelf and you placed this book over it." The soldier
removed the dusty copy of Horace and found the $100 where they had
been placed twenty years before.

His work on elasticity is hardly excelled by that of A. L. Cauchy,
and second only to that of B. de Saint- Venant. There is hardly a
problem in elasticity to which he has not contributed, while many of
his inquiries were new. The equilibrium and motion of a circular plate
was first successfully treated by him. Instead of the definite integrals
of earlier writers, he used preferably finite summations. Poisson 's
contour conditions for elastic plates were objected to by Gustav
Kirchhoff of Berlin, who established new conditions. But Thomson
(Lord Kelvin) and P. G. Tait in their Treatise on Natural Philosophy
have explained the discrepancy between Poisson's and Kirchhoff's
boundary conditions, and established a reconciliation between them.

Important contributions to the theory of elasticity were made by
A. L. Cauchy. To him we owe the origin of the theory of stress, and
the transition from the consideration of the force upon a molecule
exerted by its neighbors to the consideration of the stress upon a
small plane at a point. He anticipated G. Green and G. G. Stokes

APPLIED MATHEMATICS 467

in giving the equations of isotropic elasticity, with two constants.
The theory of elasticity was presented by Gabrio Piola of Italy ac-
cording to the principles of J. Lagrange's Mechanique Analytique, but
the superiority of this method over that of Poisson and Cauchy is far
from evident. The influence of temperature on stress was first in-
vestigated experimentally by Wilhelm Weber of Gottingen, and
afterwards mathematically by J. M. C. Duhamel, who, assuming
Poisson's theory of elasticity, examined the alterations of form which •
the formulae undergo when we allow for changes of temperature. W.
Weber was also the first to experiment on elastic after-strain. Other
important experiments were made by different scientists, which dis-
closed a wider range of phenomena, and demanded a more compre-
hensive theory. Set was investigated by Franz Joseph von Gerstner
(1756-1832), of Prague and Eaton Hodgkinson of University College,
London, while the latter physicist in England and Louis Joseph Vicat
(1786-1861) in France experimented extensively on absolute strength.
L. J. Vicat boldly attacked the mathematical theories of flexure be-
cause they failed to consider shear and the time-element. As a result,
a truer theory of flexure was soon propounded by B. de Saint-Venant.
J. V. Poncelet advanced the theories of resilience and cohesion.

Gabriel Lame (1795-1870) was born at Tours, and graduated at
the Polytechnic School. He was called to Russia with B. P. E. Clap-
eyron and others to superintend the construction of bridges and roads.
On his return, in 1832, he was elected professor of physics at the Poly-
technic School. Subsequently he held varioife engineering posts and
professorships in Paris. As engineer he took an active part in the con-
struction of the first railroads in France. Lame devoted his fine mathe-
matical talents mainly to mathematical physics. In four works:
Lemons sur lesjonctions inverses des transcendantes et les surfaces isother-
mes; Sur les coordonnees curvilignes et leurs diver ses applications; Sur
la theorie analytique de la chaleur; Sur la theorie mathematique de Velas-
ticite des corps solides (1852), and in various memoirs he displays fine
analytical powers; but a certain want of physical touch sometimes re-
duces the value of his contributions to elasticity and other physical
subjects. In considering the temperature in the interior of an ellip-
soid under certain conditions, he employed functions analogous to La-
place's functions, and known by the name of "Lame's functions."
A problem in elasticity called by Lame's name, viz., to investigate
the conditions for equilibrium of a spherical elastic envelope subject
to a given distribution of load on the bounding spherical surfaces, and
the determination of the resulting shifts is the only completely general
problem on elasticity which can be said to be completely solved. He
deserves much credit for his derivation and transformation of the
general elastic equations, and for his application of them to double
refraction. Rectangular and triangular membranes were shown by
him to be connected with questions in the theory of numbers. H.

468 A HISTORY OF MATHEMATICS

Burkhardt 1 is of the opinion that the importance of the classic
period of French mathematical physics, about 1810-1835, is often
undervalued, but that the direction it took finally under the leader-
ship of Lame was unfortunate. "By his (Lame's) taste for algebraic
elegance .he was misled to prefer problems which are of interest in pure
rather than applied mathematics; he went so far as to require of tech-
nical men the study of number theory, because the determination of
the simple tones of a rectangular plate with commensurable sides
calls for the solution of an indeterminate quadratic equation."

Continuing our outline of the history of elasticity, we observe that
the field of photo-elasticity was entered upon by G. Lame, F. E. Neu-
mann, and Clerk Maxwell. G. G. Stokes, W. Wertheim, R. Clausius,
and J. H. Jellett, threw new light upon the subject of " rari-constancy "
and " multi-constancy," which has long divided elasticians into two op-
posing factions. The uni-constant isotropy of L. M. H. Navier and
S. D. Poisson had been questioned by A. L. Cauchy, and was severely
criticised by G. Green and G. G. Stokes.

Barre de Saint-Venant (1797-1886), ingenieur des ponts et chaus-
sees, made it his life-work to render the theory of elasticity of prac-
tical value. The charge brought by practical engineers, like Vicat,
against the theorists led Saint-Venant to place the theory in its true
place as a guide to the practical man. Numerous errors committed
by his predecessors were removed. He corrected the theory of flexure
by the consideration of slide, the theory of elastic rods of double
curvature by the introduction of the third moment, and the theory
of torsion by the discovery of the distortion of the primitively plane
section. His results on torsion abound in beautiful graphic illustra-
tions. In case of a rod, upon the side surfaces of which no forces act,
he showed that the problems of flexure and torsion can be solved,
if the end-forces are distributed over the end-surfaces by a definite
law. R. F. A. Clebsch, in his Lehrbuch der Elasticitat, 1862, showed
that this problem is reversible to the case of side-forces without end-
forces. Clebsch 2 extended the research to very thin rods and to very
thin plates. B. de Saint-Venant considered problems arising in the
scientific design of built-up artillery, and his solution of them differs
considerably from G. Lame's solution, which was popularized by W. J.
M. Rankine, and much used by gun-designers. In Saint-Venant's
translation into French of Clebsch's Elasticitiit, he develops extensively
a double-suffix notation for strain and stresses. Though often advan-
tageous, this notation is cumbrous, and has not been generally adopted.
Karl Pearson, Galton professor of eugenics at the University of London,
in his early mathematical studies, examined the permissible limits of
the application of the ordinary theory of flexure of a beam.

1 Jahresb. d. d. Math. Vereinigung, Vol. 12, 1903, p. 564.

2 Alfred Clebsch, Versiich einer Darlegung und Wiirdigung seiner wissenschaftlichen
Leistungcn von einigen seiner Freundc, Leipzig, 1873.

APPLIED MATHEMATICS 469

The mathematical theory of elasticity is still in an unsettled con-
dition. Not only are scientists still divided into two schools of "rari-
constancy" and "multi-constancy," but difference of opinion exists
on other vital questions. Among the numerous modern writers on
elasticity may be mentioned fimile Mathieu (1835-1890), professor
at Besancon, Maurice Levy (1838-1910) of the College de France in
Paris, Charles Chree, superintendent of the Kew Observatory, A. B.
Basset, Lord Kelvin of Glasgow, J. Boussinesq of Paris, and others.
Lord Kelvin applied the laws of elasticity of solids to the investigation
of the earth's elasticity, which is an important element in the theory
of ocean-tides. If the earth is a solid, then its elasticity co-operates
with gravity in opposing deformation due to the attraction of the sun
and moon. P. S. Laplace had shown how the earth would behave if it
resisted deformation only by gravity. G. Lame had investigated how
a solid sphere would change if its elasticity only came into play.
Lord Kelvin combined the two results, and compared them with the
actual deformation. Kelvin, and afterwards G. H. Darwin, computed
that the resistance of the earth to tidal deformation is nearly as great
as though it were of steel. This conclusion was confirmed more re-
cently by Simon Newcomb, from the study of the observed periodic
changes in latitude and by others. For an ideally rigid earth the
period would be 360 days, but if as rigid as steel, it would be 441, the
observed period being 430 days.

Among the older text-books on elasticity may be mentioned the
works of G. Lame, R. F. A. Clebsch, A. Winckler, A. Beer, E. L.
Mathi^u, W. J. Ibbetson, and F. Neumann, edited by O. E. Meyer.

In recent years the modern analytical developments, particularly
along the line of integral equations, have been brought to bear on
theories of elasticity and potential. The solution of the static prob-
lem of the theory of elasticity of a homogeneous iso tropic body under
certain given surface conditions has been taken up particularly by
E. I. Fredholm of Stockholm, G. Lauricella of the University of
Catania, R. Marcolongo of Naples and Hermann Weyl of Zurich,
and by a somewhat different mode of procedure, by A. Korn of Berlin
and T. Boggio of Turin.1

Closely connected with researches on attraction and elasticity is
the development of spherical harmonics. After the initial paper of
A. M. Legendre on zonal harmonics applied by him to the study of
the attraction of solids of revolution, and after the remarkable memoir
of 1782 by P. S. Laplace who used spherical harmonics in finding the
potential of a solid nearly spherical, the first advance was made by
Olinde Rodrigues (1794-1851), a French economist and reformer,
who in 1816 gave a formula for Pn which later was derived independ-
ently by J. Ivory and C. G. J. Jacobi. The name " Kugelfunktion "
is due to K. F. Gauss. Important contributions were made in Gcr-
1 See Rendkonti del Circolo Math, di Palermo, Vol. 39, 1915, p. i.

470 A HISTORY OF MATHEMATICS

many by C. G. J. Jacobi, L. Dirichlet, Franz Ernst Neumann (1798-
1895) who was professor of physics and mineralogy in Konigsberg,
his son Carl Neumann (1832- ), Elwin Bruno Christoffel (1829-
1900) of the University of Strassburg, R. Dedekind, Gustav Bauer
(1820-1906) of Munich, Gustav Mehler (1835-1895) of Elbingin West
Prussia, and Karl Baer (1851- ) of Kiel. Especially active was
Eduard Heine (1821-1881) of the University of Halle, the author of
the Handbuch der Kugelfunktionen, 1861, 2. Ed. 1878-1881. The chief
representative in the cultivation of this subject, in Switzerland, was
L. Schlafli of the University of Bern; in Belgium, was Eugene Catalan
of the University of Liege; in Italy, was E. Beltrami; in the United
States, was W. E. Byerly of Harvard University. In France there
were S. D. Poisson, G. Lame, T. J. Stieltjes, J. G. Darboux, Ch.
Hermite, Paul Mathieu, Hermann Laurent (1841-1908), Professor at
the Polytechnic School in Paris whose researches gave rise to contests
of priority with German writers. In Great Britain spherical har-
monics received the attention of Thomson, (Lord Kelvin) and P. G.
Tait in their Natural Philosophy of 1867, and of Sir William D. Niven
of Manchester, Norman Ferrers (1829-1903) of Cambridge, E. W.
Hobson of Cambridge, A. E. H_ Love of Oxford, and others.

Light, Electricity, Heat, Potential

G. F. B. Riemann's opinion that a science of physics only exists since
the invention of differential equations finds corroboration even in this
brief and fragmentary outline of the progress of mathematical physics.
The undulatory theory of light, first advanced by C. Huygens, owes
much to the power of mathematics: by mathematical analysis its
assumptions were worked out to their last consequences. Thomas
Young1 (1773-1829) was the first to explain the principle of inter-
ference, both of light and sound, and the first to bring forward the
idea of transverse vibrations in light waves. T. Young's explanations,
not being verified by him by extensive numerical calculations, at-
tracted little notice, and it was not until Augustin Fresnel (1788-
1827) applied mathematical analysis to a much greater extent than
Young had done, that the undulatory theory began to carry convic-
tion. Some of FresnePs mathematical assumptions were not satis-
factory; hence P. S. Laplace, S. D. Poisson, and others belonging to
the strictly mathematical school, at first disdained to consider the
theory. By their opposition Fresnel was spurred to greater exertion.
D. F. J. Arago was the first great convert made by Fresnel. When
polarization and double refraction were explained by T. Young and
A. Fresnel, then P. S. Laplace was at last won over. S. D. Poisson
drew from Fresnel's formulae the seemingly paradoxical deduction

1 Arthur Schuster, "The Influence of Mathematics on the Progress of Physics,"
Nature, Vol. 25, 1882, p. 398.

APPLIED MATHEMATICS 471

that a small circular disc, illuminated by a luminous point, must cast
a shadow with a bright spot in the centre. But this was found to be
in accordance with fact. The theory was taken up by another great
mathematician, W. R. Hamilton, who from his formulae predicted
conical refraction, verified experimentally by Humphrey Lloyd. These
predictions do not prove, however, that Fresnel's formulae are correct,
for these prophecies might have been made by other forms of the
wave- theory. The theory was placed on a sounder dynamical basis
by the writings of A. L. Cauchy, J. B. Biot, G. Green, C. Neumann,
G. R. Kirchhoff, J. MacCullagh, G. G. Stokes, B. de Saint-Venant,
Emile Sarrau (1837-1904) of the Polytechnic School in Paris, Ludwig
Lorenz (1829-1891) of Copenhagen, and Sir William Thomson (Lord
Kelvin). In the wave-theory, as taught by G. Green and others, the
luminiferous ether was an incompressible elastic solid, for the reason
that fluids could not propagate transverse vibrations. But, according
to G. Green, such an elastic solid would transmit a longitudinal dis-
turbance with infinite velocity. G. G. Stokes remarked, however, that
the ether might act like a fluid in case of finite disturbances, and like
an elastic solid in case of the infinitesimal disturbances in light prop-
agation. A. Fresnel postulated the density of ether to be different in
different media, but the elasticity the same, while C. Neumann and
J. MacCullagh assumed the density uniform and the elasticity different
in all substances. On the latter assumption the direction of vibration
lies in the plane of polarization, and not perpendicular to it, as in the
theory of A. Fresnel.

While the above writers endeavored to explain all optical properties
of a medium on the supposition that they arise entirely from difference
in rigidity or density of the ether in the medium, there is another
school advancing theories in which the mutual action between the
molecules of the body and the ether is considered the main cause of
refraction and dispersion.1 The chief workers in this field were J.
Boussinesq, W. Sellmeyer, H. Helmholtz, E. Lommel, E: Ketteler,
W. Voigt, and Sir William Thomson (Lord Kelvin) in his lectures
delivered at the Johns Hopkins University in 1884. Neither this nor
the first-named school succeeded in explaining all the phenomena.
A third school was founded by C. Maxwell. He proposed the electro-
magnetic theory, which has received extensive development recently.
It will be mentioned again later. According to Maxwell's theory, the
direction of vibration does not lie exclusively in the plane of polariza-
tion, nor in a plane perpendicular to it, but something occurs in both
planes — a magnetic vibration in one, and an electric in the other.
G. F. Fitzgerald and F. T. Trouton in Dublin verified this conclusion
of C. Maxwell by experiments on electro-magnetic waves.

Of recent mathematical and experimental contributions to optics,

1 R. T. Glazebrook, "Report on Optical Theories," Report British Ass'n for 1885,
p. 213.

472 A HISTORY OF MATHEMATICS

mention must be made of Henry Augustus Rowland (1848-1901), who
was professor of physics at the Johns Hopkins University and his
theory of concave gratings, and of A. A. Michelson's work on interfer-
ence, and his application of interference methods to astronomical
measurements.

A function of fundamental importance in the mathematical theories

of electricity and magnetism is the "potential." It was first used by

J. Lagrange in the determination of gravitational attractions in 1773.

Soon after, P. S. Laplace gave the celebrated differential equation,

tfV

which was extended by S. D. Poisson by writing— 4 irk in place of
zero in the right-hand member of the equation, so that it applies not
only to a point external to the attracting mass, but to any point what-
ever. The first to apply the potential function to other than gravita-
tion problems was George Green (1793-1841). He introduced it into
the mathematical theory of electricity and magnetism. Green was a
self-educated man who started out as a baker, and at his death was
fellow of Caius College, Cambridge. In 1828 he published by private
subscription at Nottingham a paper entitled Essay on the application
of mathematical analysis to the theory of electricity and magnetism.
About 100 copies were printed. It escaped the notice even of English
mathematicians until 1846, when William Thomson (Lord Kelvin) had
it reprinted in Crelle's Journal, vols. xliv. and xlv. It contained what is
now known as "Green's theorem" for the treatment of potential.
Meanwhile all of Green's general theorems had been rediscovered by
William Thomson (Lord Kelvin), M. Chasles, J. C. F. Sturm, and
K. F. Gauss. The term potential function is due to G. Green. W. R.
Hamilton used the word force-function, while K. F. Gauss, who about
1840 secured the general adoption of the function, called it simply
potential. G. Green wrote papers on the equilibrium of fluids, the
attraction of ellipsoids, on the reflection and refraction of sound and
light. His researches bore on questions previously considered by
S. D. Poisson. K. F. Gauss proved what C. Neumann has called
"Gauss' theorem of mean value" and then considered the question of
maxima and minima of the potential.1

Large contributions to electricity and magnetism have been made
by William Thomson later Sir William Thomson and Lord Kelvin
(1824-1907). He was born at Belfast, Ireland, but was of Scotch de-
scent. He and his brother James studied in Glasgow. From there he
entered Cambridge, and was graduated as Second Wrangler in 1845.
William Thomson, J. J. Sylvester, C. Maxwell, W. K. Clifford, and J. J.
Thomson are a group of great men who were Second Wranglers at Cam-
bridge. At the age of twenty-two W. Thomson was elected professor

1 For details see Max Bacharach, Geschiclite der Potentidtheorie, Gottingen, 1883.

APPLIED MATHEMATICS 473

of natural philosophy in the University of Glasgow, a position which
he held till his death. For his brilliant mathematical and physical
achievements he was knighted, and in 1892 was made Lord Kelvin.
He was greatly influenced by the mathematical physics of J. Fourier
and other French mathematicians. It was Fourier's mathematics on
the flow of heat through solids which led him to the mastery of the
diffusion of an electric current through a wire and of the difficulties
encountered in signalling through the Atlantic telegraph. In 1845
W. Thomson visited Paris. P. S. Laplace, A. M. Legendre, J. Fourier,
Sadi Carnot, S. D. Poisson, and A. Fresnel were no longer living. W.
Thomson met J. Liouville to whom he gave the now famous memoir
of G. Green of the year 1828. He met M. Chasles, J. B. Biot, H. V.
Regnault, J. C. F. Sturm, A. L. Cauchy, and J. B. L. Foucault. A. L
Cauchy tried to convert him to Roman Catholicism. One evening.
J. C. F. Sturm called upon him in high excitement. "Vous avez le
memoire de Green," he exclaimed. The Essay was produced; Sturm
eagerly scanned its contents. "Ah! voila mon affaire," he cried,
jumping from his seat as he caught sight of the formula in which G.
Green had anticipated his theorem of the equivalent distribution.
Kelvin's researches on the theory of potential are epoch-making.
What is called "Dirichlet's principle" was discovered by him in 1848,
somewhat earlier than by P. G. L. Dirichlet. Jointly with P. G. Tait
he prepared the celebrated Treatise of Natural Philosophy, 1867.
As a mathematician he belonged most decidedly to the intuitional
school. Purists in mathematics often carped at Kelvin's " instinctive "
mathematics. "Do not imagine," he once said, "that mathematics
is hard and crabbed, and repulsive to common sense. It is merely the
etherealization of common sense." Yet even in mathematics he had
his dislikes. When in 1845 he met W. R. Hamilton at a British Asso-
ciation meeting, who then read his first paper on Quaternions, one
might have thought that W. Thomson would welcome the new
analysis: but it was not so. He did not use it. On the merits of quater-
nions he had a thirty-eight years' war with P. G. Tait.1 We owe to
W. Thomson new synthetical methods of great elegance, viz., the
theory of electric images and the method of electric inversion founded
thereon. By them he determined the distribution of electricity on a
bowl, a problem previously considered insolvable. The distribution
of static electricity on conductors had been studied before this mainly
by S. D. Poisson and G. A. A. Plana. In 1845 F. E. Neumann of
Konigsberg developed from the experimental laws of Lenz the mathe-
matical theory of magneto-electric induction. In 1855 W. Thomson
predicted by mathematical analysis that the discharge of a Leyden jar
through a linear conductor would in certain cases consist of a series of
decaying oscillations. This was first established experimentally by
Joseph Henry of Washington. William Thomson worked out the
1 S. P. Thompson, Life of William Thomson, London, 1910, pp. 452, 1136-1139.

474 A HISTORY OF MATHEMATICS

electro-static induction in submarine cables. The subject of the
screening effect against induction, due to sheets of different metals,
was worked out mathematically by Horace Lamb and also by Charles
Niven. W. Weber's chief researches were on electro-dynamics. H.
Helmholtz in 1851 gave the mathematical theory of the course of in-
duced currents in various cases. Gustav Robert Kirchhoff 1 (1824-
1887), who was professor at Breslau, Heidelberg and since 1875 at
Berlin, investigated the distribution of a current over a flat conductor,
and also the strength of current in each branch of a network of linear
conductors.

The entire subject of electro-magnetism was revolutionized by
James Clerk Maxwell (1831-1879). He was born near Edinburgh,
entered the University of Edinburgh, and became a pupil of Kelland
and Forbes. In 1850 he went to Trinity College, Cambridge, and
came out Second Wrangler, E. Routh being Senior Wrangler. Max-
well then became lecturer at Cambridge, in 1856 professor at Aber-
deen, and in 1860 professor at King's College, London. In 1865 he
retired to private life until 1871, when he became professor of physics
at Cambridge. Maxwell not only translated into mathematical lan-
guage the experimental results of Michael Faraday, but established
the electro-magnetic theory of light, since verified experimentally by
H. R. Hertz. His first researches thereon were published in 1864.
In 1871 appeared his great Treatise on Electricity and Magnetism.
He constructed the electro-magnetic theory from general equations,
which are established upon purely dynamical principles, and which
determine the state of the electric field. It is a mathematical discus-
sion of the stresses and strains in a dielectric medium subjected to
electro-magnetic forces. The electro-magnetic theory has received
developments from Lord Rayleigh, J. J. Thomson, H. A. Rowland,
R. T. Glazebrook, H. Helmholtz, L. Boltzmann, O. Heaviside, J. H.
Poynting, and others. Hermann von Helmholtz (1821-1894) was
born in Potsdam, studied medicine, was assistant at the charity hos-
pital in Berlin, then a military surgeon, a teacher of anatomy, a pro-
fessor of physiology at Konigsberg, at Bonn and at Heidelberg. In
1871 he went to Berlin as successor to Magnus in the chair of physics.
In 1887 he became director of the new Physikalisch-Technische
Reichsanstalt. As a young man of twenty-six he published the now
famous pamphlet Ueber die Erhaltung der Kraft. His work on Tonemp-
findung was written in Heidelberg. After he went to Berlin he was
engaged chiefly on inquiries in electricity and hydrodynamics. Helm-
holtz aimed to determine in what direction experiments should be
made to decide between the theories of W. Weber, F. E. Neumann,
G. F. B. Riemann, and R. Clausius, who had attempted to explain
electro-dynamic phenomena by the assumption of forces acting at a
distance between two portions of the hypothetical electrical fluid, —
1 W. Voigt, Zum Gcd'dchtniss wn G. Kirchhojf, Gottingen, 1888.

APPLIED MATHEMATICS 475

the intensity being dependent not* only on the distance, but also on
the velocity and acceleration, — and the theory of M. Faraday and
C. Maxwell, which discarded action at a distance and assumed stresses
and strains in the dielectric. His experiments favored the British
theory. He wrote on abnormal dispersion, and created analogies
between electro-dynamics and hydrodynamics. Lord Rayleigh com-
pared electro-magnetic problems with their mechanical analogues,
gave a dynamical theory of diffraction, and applied Laplace's coeffi-
cients to the theory of radiation. H. Rowland made some emenda-
tions on G. G. Stokes' paper on diffraction and considered the pro-
pagation of an arbitrary electro-magnetic disturbance and spherical
waves of light. Electro-magnetic induction has been investigated
mathematically by Oliver Heaviside, and he showed that in a cable
it is an actual benefit. O. Heaviside and J. H. Poynting have reached
remarkable mathematical results in their interpretation and develop-
ment of Maxwell's theory. Most of Heaviside's papers have been
published since 1882; they cover a wide field.

One part of the theory of capillary attraction, left defective by P. S.
Laplace, namely, the action of a solid upon a liquid, and the mutual
action between two liquids, was made dynamically perfect by K. F.
Gauss. He stated the rule for angles of contact between liquids and
solids. A similar rule for liquids was established by Franz Erpst
Neumann. Chief among more recent workers on the mathematical
theory of capillarity are Lord Rayleigh and E. Mathieu.

The great principle of the conservation of energy was established
by Robert Mayer (1814-1878), a physician in Heilbronn, and again
independently by Ludwig A. Colding of Copenhagen, J. P. Joule, and
H. Helmholtz. James Prescott Joule (1818-1889) determined ex-
perimentally the mechanical equivalent of heat. H. Helmholtz in
1847 applied the conceptions of the transformation and conservation
of energy to the various branches of physics, and thereby linked to-
gether many well-known phenomena. These labors led to the aban-
donment of the corpuscular theory of heat. The mathematical treat-
ment of thermic problems was demanded by practical considerations.
Thermodynamics grew out of the attempt to determine mathemati-
cally how much work can be gotten out of a steam engine. Sadi
Nicolas Leonhard Carnot (1796-1832) of Paris, an adherent of the
corpuscular theory, gave the first impulse to this. The principle
known by his name was published in 1824. Though the importance
of his work was emphasized by B. P. E. Clapeyron, it did not meet
with general recognition until it was brought forward by William
Thomson (Lord Kelvin). The latter pointed out the necessity of
modifying Carnot's reasoning so as to bring it into accord with the
new theory of heat. William Thomson showed in 1848 that Carnot's
principle led to the conception of an absolute scale of temperature.
In 1849 he published "an account of Carnot's theory of the motive

476 A HISTORY OF MATHEMATICS

power of heat, with numerical results deduced from Regnault's ex-
periments." In February, 1850, Rudolph Clausius (1822-1888), then
in Zurich (afterwards professor in Bonn), communicated to the Berlin
Academy a paper on the same subject which contains the Protean
second law of thermodynamics. In the same month William John
M. Rankine (1820-1872), professor of engineering and mechanics
at Glasgow, read before the Royal Society of Edinburgh a paper in
which he declares the nature of heat to consist in the rotational mo-
tion of molecules, and arrives at some of the results reached previously
by R. Clausius. He does not mention the second law of thermody-
namics, but in a subsequent paper he declares that it could be derived
from equations contained in his first paper. His proof of the second
law is not free from objections. In March, 1851, appeared a paper
of William Thomson (Lord Kelvin) which contained a perfectly
rigorous proof of the second law. He obtained it before he had seen
the researches of R. Clausius. The statement of this law, as given by
Clausius, has been much criticised, particularly by W. J. M. Rankine,
Theodor Wand, P. G. Tait, and Tolver Preston. Repeated efforts to
deduce it from general mechanical principles have remained fruitless.
The science of theormodynamics was developed with great success
by W. Thomson, Clausius, and Rankine. As early as 1852 W. Thom-
son discovered the law of the dissipation of energy, deduced at a
later period also by R. Clausius. The latter designated the non-
transformable energy by the name entropy, and then stated that the
entropy of the universe tends toward a maximum. For entropy
Rankine used the term thermodynamic function. Thermodynamic
investigations have been carried on also by Gustav Adolph Hirn (1815-
1890) of Colmar, and H. Helmholtz (monocyclic and polycyclic sys-
tems). Valuable graphic methods for the study of thermodynamic
relations were devised by J. W. Gibbs of Yale College.

Josiah Willard Gibbs (1839-1903) was born in New Haven, Conn.,
and spent the first five years after graduation mainly in mathematical
studies at Yale. He passed the winter of 1866-1867 in Paris, of 1867-
1868 in Berlin, of 1868-1869 in Heidelberg, studying physics and
mathematics. In 1871 he was elected professor of mathematical physics
at Yale. " His direct geometrical or graphical bent is shown by the at-
traction which vectorial modes of notation in physical analysis exerted
over him, as they had done in a more moderate degree over C. Max-
well." Greatly influenced by Sadi Carnot, by William Thomson (Lord
Kelvin) and especially by R. Clausius, Gibbs began in 1873 to pre-
pare papers on the graphical expression of thermodynamic relations,
in which energy and entropy appeared as variables. He discusses the
entropy-temperature and entropy-volume diagrams, and the volume-
energy-entropy surface (described in C. Maxwell's Theory of Heat}.
Gibbs formulated the energy-entropy criterion of equilibrium and
stability, and expressed it in a form applicable to complicated problems

APPLIED MATHEMATICS 477

of dissociation. That chemistry has tended to take a mathematical
turn, says E. Picard, is evident from "the celebrated memoir of J. W.
Gibbs on the equilibrium of chemical systems, so analytic in char-
acter, and where is needed some effort on the part of the chemists to
recognize, under their algebraic mantle, laws of high importance."

In 1902 appeared J. W. Gibbs' Elementary Principles in Statistical
Mechanics, developed with special reference to the rational foundation
of thermodynamics. The modern kinetic theory of gases was mainly
the work of R. Clausius, C. Maxwell, and Boltzmann. "In reading
Clausius we seem to be reading mechanics; in reading Maxwell, and
in much of L. Boltzmann's most valuable work, we seem rather to be
reading in the theory of probabilities." C. Maxwell, and L. Boltzmann
are the creators of "statistical dynamics." While they treated of
molecules of matter directly, J. W. Gibbs considers "the statistics
of a definite vast aggregation of ideal similar mechanical systems of
types completely defined beforehand, and then compares the precise
results reached in this ideal discussion with the principles of thermo-
dynamics, already ascertained in the semi-empirical manner." 1 Im-
portant works on thermodynamics were prepared by R. Clausius
in 1875, by R. Ruhlmann in 1875, and by H. Poincare in 1892.

In the study of the law of dissipation of energy and the principle
of least action, mathematics and metaphysics met on common ground.
The doctrine of least action was first propounded by P. L. M. Mau-
pertius in 1744. Two years later he proclaimed it to be a universal
law of nature, and the first scientific proof of the existence of God.
It was weakly supported by him, violently attacked by Konig of
Leipzig, and keenly defended by L. Euler. J. Lagrange's conception
of the principle of least action became the mother of analytic me-
chanics, but his statement of it was inaccurate, as has been remarked
by Josef Bertrand in the third edition of the Mecaniqite Analytique.
The form of the principle of least action, as it now exists, was given
by W. R. Hamilton, and was extended to electro-dynamics by F. E.
Neumann, R. Clausius, C. Maxwell, and H. Helmholtz. To sub-
ordinate the principle to all reversible processes, H. Helmholtz intro-
duced into it the conception of the "kinetic potential." In this form
the principle has universal validity.

An offshoot of the mechanical theory of heat is the modern kinetic
theory of gases, developed mathematically by R. Clausius, C. Maxwell,
Ludwig Boltzmann of Vienna, and others. The first suggestions of a
kinetic theory of matter go back as far as the time of the Greeks. The
earliest work to be mentioned here is that of Daniel Bernoulli, 1738.
He attributed to gas-molecules great velocity, explained the pressure
of a gas by molecular bombardment, and deduced Boyle's law as a
consequence of his assumptions. Over a century later his ideas were
taken up by J. P. Joule (in 1846), A. K. Kronig (in 1856), and R.
1 Proceed, of the Royal Soc. of London, Vol. 75, 1905, p. 293.

478 A HISTORY OF MATHEMATICS

Clausius (in 1857). J. P. Joule dropped his speculations on this
subject when he began his experimental work on heat. A. K. Kronig
explained by the kinetic theory the fact determined experimentally
by Joule that the internal energy of a gas is not altered by expansion
when no external work is done. R. Clausius took an important step
in supposing that molecules may have rotary motion, and that atoms
in a molecule may move relatively to each other. He assumed that
the force acting between molecules is a function of their distances,
that temperature depends solely upon the kinetic energy of molecular
motions, and that the number of molecules which at any moment are
so near to each other that they perceptibly influence each other is
comparatively so small that it may be neglected. He calculated the
average velocities of molecules, and explained evaporation. Objections
. to his theory, raised by C. H. D. Buy's-Ballot and by Emil Jochmann,
were satisfactorily answered by R. Clausius and C. Maxwell, except in
one case where an additional hypothesis had to be made. C. Maxwell
proposed to himself the problem to determine the average number
of molecules, the velocities of which lie between given limits. His
expression therefor constitutes the important law of distribution of
velocities named after him. By this law the distribution of molecules
according to their velocities is determined by the same formula (given
in the theory of probability) as the distribution of empirical observa-
tions according to the magnitude of their errors. The average mo-
lecular velocity as deduced by C. Maxwell differs from that of R.
Clausius by a constant factor. C. Maxwell's first deduction of this
average from his law of distribution was not rigorous. A sound deriva-
tion was given by O. E. Meyer in 1866. C. Maxwell predicted that
so long as Boyle's law is true, the coefficient of viscosity and the coeffi-
cient of thermal conductivity remain independent of the pressure.
His deduction that the coefficient of viscosity should be proportional
to the square root of the absolute temperature appeared to be at
variance with results obtained from pendulum experiments. This
induced him to alter the very foundation of his kinetic theory of gases
by assuming between the molecules a repelling force varying inversely
as the fifth power of their distances. The founders of the kinetic
'theory had assumed the molecules of a gas to be hard elastic spheres;
but Maxwell, in his second presentation of the theory in 1866, went
on the assumption that the molecules behave like centres of forces.
He demonstrated anew the law of distribution of velocities; but the
proof had a .flaw in argument, pointed out by L. Boltzmann, and
recognized by C. Maxwell, who adopted a somewhat different form
of the distributive function in a paper of 1879, intended to explain
mathematically the effects observed in Crookes' radiometer. L. Boltz-
mann gave a rigorous general proof of Maxwell's law of the distribu-
tion of velocities.
None of the fundamental assumptions in the kinetic theory of gases

APPLIED MATHEMATICS 479

leads by the laws of probability to results in very close agreement
with observation. L. Boltzmarm tried to establish kinetic theories
of gases by assuming the forces between molecules to act according to
different laws from those previously assumed. R. Clausius, C. Max-
well, and their predecessors took the mutual action of molecules in
collision as repulsive, but L. Boltzmann assumed that they may be
attractive. Experiments of J. P. Joule and Lord Kelvin seem to sup-
port the latter assumption.

Among the later researches on the kinetic theory is Lord Kelvin's
disproof of a general theorem of C. Maxwell and L. Boltzmann, as-
serting that the average kinetic energy of two given portions of a
system must be in the ratio of the number of degrees of freedom of
those portions.

In recent years the kinetic theory of gases has received less attention;
it is considered inadequate since the founding of the quantum hypothe-
sis in physics.

Relativity

Profound and startling is the "theory of relativity." On the theory
that the ether was stationary it was predicted that the time required
for light to travel a given distance forward and back would be different
when the path of the light was parallel to the motion of the earth in its
orbit from what it was when the path of the light was perpendicular.
In 1887 A. A. Michelson and E. W. Merely found experimentally that
such a difference in time did not exist. More generally, the results
of this and other experiments indicate that the earth's motion through
space cannot be detected by observations made on the earth alone.
In order to explain Michelson and Morley's negative result and at
the same time save the stationary-ether theory, H. A. Lorentz con-
structed in 1895 a " contraction hypothesis," according to which a mov-
ing solid contracts slightly longitudionally. This same idea occurred
independently to G. F. Fitzgerald. In 1904 and in his Columbia Uni-
versity Lectures Lorentz aimed to reduce the electromagnetic equations
for a moving system to the form of those that hold for a system at
rest. Instead of x, y, z, t he introduced new independent variables,

viz., x'= \y(x— vl), y=\y, z'=Xz, t'=\y(t — ^x), where y depends

upon velocity of light c and of the moving body v, and X is a numerical
coefficient such that, X=i when v=o. His fundamental equations
turned out to be invariant under this now called "Lorentz transforma-
tion." In 1906 H. Poincare made use of this transformation for the
treatment of the dynamics of the electron and also of universal gravi-
tation.1 In 1905 A. Einstein published a paper on the electrodynam-
ics of moving bodies in Annalen der Physik, Vol. 17, aiming at perfect

1 L. Silberstein, The Theory of Relativity, London, 1914, p. 87.

480 A HISTORY OF MATHEMATICS

reciprocity or equivalence of a pair of moving systems, and investi-
gating the whole problem from the bottom, carefully considering the
matter of "simultaneous" events in two distant places; he has suc-
ceeded in giving plausible support to and a striking interpretation
of Lorentz's transformations. Einstein opened the way to the modern
"theory of relativity." He developed it somewhat more fully in 1907.
A fundamental point of view in his theory was that mass and energy
are proportional. For the purpose of taking account of gravitational
phenomena, Einstein generalized his theory by assuming that mass
and weight are also proportional, so that, for example, a ray of light
is attracted by matter. The mathematical part of Einstein's theory,
as developed by M. Grossmann in 1913, employs quadratic differential
forms and the absolute calculus of Gregorio Ricci of Padua. Another
remarkable speculation was brought out in 1908 by Hermann Min-
kowski who read a lecture on Raum und Zeit, in which he maintained
that the new views of space and time, developed from experimental
considerations, are such that "space by itself and time by itself sink
into the shadow and only a kind of union of the two retains self-de-
pendence." No one notices a place, except at some particular time,
nor time except at a particular place. A system of values x, y, 2, /
he calls a "world point" (Weltpunkt) ; the life-path of a material point
in four-dimensional space is a "world line." The idea of time as a
fourth dimension had been conceived much earlier by J. Lagrange in
his Theorie des f auctions analytiques and by D'Alembert in his article
"Dimension" in Diderot's Encyclopedic,1 1754. H. Minkowski con-
siders the group belonging to the differential equation for the propaga-
tion of waves of light. Hermann Minkowski (1864-1909), was born at
Alexoten in Russia, studied at Konigsberg and Berlin, held associate
professorships at Bonn and Konigsberg and was promoted to a full
professorship at Konigsberg in 1895. In 1896 he went to the poly-
technic school at Zurich and in 1903 to Gottingen. The importance
which H. Minkowski, starting with the principle of relativity in the
form given it by Einstein, has given to the Lorentzian transforma-
tions by the introduction of a four dimensional manifoldness or space-
time-world, has been made intuitively evident by a number of writers,
particularly F. Klein (1910), L. Heffter (1912), A. Brill (1912), and
H. E. Timerding (1912). F. Klein said: "What the modern physicists
call ' theory of relativity ' is the theory of invariants of the fourth di-
mensional space-time-region x, y, z, / (Minkowski's world) in relation
to a definite group of collineations, namely the 'Lorentz-group.' '
A novel presentation aiming at great precision was given in 1914 by
Alfred A. Robb who on the idea of "conical order" and 21 postulates
builds up a system in which the theory of space becomes absorbed in
the theory of time. A philosophical discussion of relativity, mechan-

1 R. C. Archibald in Btill. Am. Math. Soc., Vol. 20, 1914, p. 410.

2 Klein in Jahresb. d. d. Math. Verein., Vol. 19, 1910, p. 287.

APPLIED MATHEMATICS 481

ics and geometrical axioms is given by Federigo Enriques of Bologna
in his Problems of Science (1906), which has been translated into Eng-
lish by Katharine Royce in 1914. F. Enriques argues that certain
optical and electro-optical phenomena seem to lead to a direct contra-
diction of the principles of classic mechanics, especially of Newton's
principle of action and reaction. "Physics," says Enriques, "instead
of affording a more precise verification of the classic mechanics, leads
rather to a correction of the principles of the latter science, taken a
priori as rigid."

The Russian mathematician, Vladimir Varicak found that the
Lobachevskian geometry presented itself as the best adapted for the
mathematical treatment of the physics of relativity. He enters upon
optical phenomena and the resolution of paradoxes due to Ehrenfest
and Bonn. Starting from this point of view, E. Borel in 1913 wras able
to deduce new consequences of the theory of relativity. One advantage
of Varicak's presentation is that it safeguards the parallelism between
the old enunciations of physics and the new. L. Rougier of Lyon *
asks the question, is then the Lobachevskian geometry physically
true and the Euclidean wrong? No. One may keep, says he, the or-
dinary geometry for the discussion of the physics of relativity, as is
done by H. A. Lorentz and A. Einstein, or one may add a fourth imag-
inary dimension to our three dimensions in the manner of H. Min-
kowski, or one may use the non-Euclidean geometries of mechanics
and electromagnetics developed by E. B. Wilson and G. N. Lewis,2
then of Boston. Each of these interpretations enjoys some particular
advantages of its own.

Nomography

The use of simple graphic tables for computation is encountered in
antiquity and the middle ages. The graphic solution of spherical
triangles was in vogue in the time of Hipparchus,3 and in the seven-
teenth century by W. Oughtred,4 for instance. Edmund Wingate's
Construction and Use of the Line of Proportion, London, 1628, de-
scribed a double scale upon which numbers are indicated by spaces
on one side of a straight line and the corresponding logarithms by
spaces on the other side of the line.5 Recently this idea has been car-
ried out by A. Tichy in his Graphische Logarithmentafeln, Vienna, 1897.
The Longitude Tables and Horary Tables of Margetts, London, 1791,
were graphical. More systematic use of this idea was made by
Pouchet in his Arithmetique lineaire, Rouen, 1795. In 1842 appeared
the Anamorphose logarithmique of the Parisian engineer Leon Lalanne

1 L ' Enseignemenl Mathematique, Vol. XVI, 1914, p. 17.

2 Proceed. Am. Acad. of Arts and Science, Vol. 48, 1912.

3 A. von Braunmiihl, Geschichte der Trigonometric, Leipzig, Vol. I, 1900, pp. 3, 10,
85, 191.

4 F. Cajori, William Oughlred, Chicago and London, 1916.

6 F. Cajori, Colorado College Publication, General Series 47, 1910, p. 182.

482 A HISTORY OF MATHEMATICS

(1811-1892) in which the distances of points from the origin are not
necessarily proportional to the actual values of the data, but may be
other functions of them, judiciously chosen. In the product 2122=23,
the variables z\ and 23 are brought in correspondence, respectively,
with the straight lines x=\og si, ;y=log z2, so that x+y=\og z3,
which represents the straight lines perpendicular to the bisectors of the
angle between the co-ordinate axes. Advances along this line were
made by J. Massau of the University of Ghent, in 1884, and E. A.
Lallemand in 1886. The Scotch Captain Patrick Weir in 1889 gave
an azimuth diagram which was an anticipation of a spherical triangle
nomogram. But the real creator of nomography is Maurice d'Ocagne
of the ficole Polytechnique in Paris, whose first researches appeared in
1891; his Traite de nomographie came out in 1899. The principle of
anamorphosis, by successive generalizations, "has led to the con-
sideration of equations representable not only by two systems of
straight lines parallel to the axes of co-ordinates and one other un-
restricted system of straight lines, but by three systems of straight
lines under no such restrictions." D'Ocagne also studied equations
representable by means of systems of circles. He has introduced the
method of collinear points by which "it has been possible to represent
nomographically equations of more than three variables, of which
the previous methods gave no convenient representation."

Mathematical Tables

The increased accuracy now attainable in astronomical and geodetic
measurements and the desire to secure more complete elimination of
errors from logarithmic tables, has led to recomputations of logarithms.
Edward Sang of Edinburgh published in 1871 a 7-place table of com-
mon logarithms of numbers to 200,000. These were mainly derived
from his unpublished 2S-place table of logarithms of primes to 10,037
and composite numbers to 20,000, and his i5-place table from 100,000
to 37o,ooo.2 In 1889 the Geographical Institute of Florence issued a
photographic reproduction of G. F. Vega's Thesaurus of 1794 (10
figures). Vega had computed A. Vlacq's tables anew, but his last
figure was unreliable. In 1891 the French Government issued 8-
place tables which were derived from the unpublished Tables du
Cadastre (i4-places, 12 correct) which had been computed near the
close of the eighteenth century under the supervision of G. Riche de
Prony. These tables give logarithms of numbers to 120,000, and of
•sines and tangents for every 10 centesimal seconds, the quadrant being
divided centesimally.3 Prony consulted A. M. Legendre and other

1 D'Ocagne in Napier Tercentenary Memorial Volume, London, 1915, pp. 279-
283. See also D'Ocagne, Le calatl simplifit, Paris, 1905, pp. 145-153.

2 E. M. Horsburgh, Napier Tercentenary Celebration Handbook, 1914, pp. 38-43.
J This and similar information is drawn from J. W. L. Glaisher in Napier Ter-
centenary Memorial Volume, London, 1915, pp. 71-73.

APPLIED MATHEMATICS 483

mathematicians on the choice of methods and formulas, and entrusted
the computation of primary results to professional calculators, while
the task of filling the rest of the columns beyond the primary results
was performed by assistants "apt merely in performing additions"
by the use of the method of differences. " It is curious," says D'Ocagne
" to note that the majority of these assistants had been recruited from
among the hair-dressers whom the abandonment of the powdered wig
in men's fashion had deprived of a livelihood."

In 1891 M. J. de M endiz&bel-Tamborrel published at Paris tables of
logarithms of numbers to 125,000 (8 places) and of sines and tangents
(7 or 8 places) for every millionth of the circumference, which were
almost wholly derived from original lo-place calculations. W. W.
Duffield published in the Report of the U. S. Coast and Geodetic Sur-
vey, 1895-1896, a lo-place table of logarithms of numbers to 100.000,
in 1910, 8-place tables of numbers to 200,000 and trigonometric tables
to every sexagesimal second were published by /. Bauschinger and
/. Peters of Strassburg. A- special machine was constructed for the
computation of these tables. In 1911 H. Andoyer of Paris published a
i4-place table of logarithms of sines and tangents to every 10 sex-
agesimal seconds. "This table was derived from a complete recalcula-
tion, made entirely by M. Andoyer himself, without any assistance,
personal or mechanical."

In recent years a demand has arisen for tables giving the natural
values of sines and cosines. In 1911 /. Peters published in Berlin such
a table, extending from o° to 90°, and carried to 21 decimals, for every
10 sexagesimal seconds (and for every second of the first six degrees).
Extensive tables of natural values, first computed by Rhaeticus and
published in 1613, were abandoned after the invention of logarithms,
but are now returning in use again, since they are better fitted for the
growing practice of calculating directly by means of machines and
without resort to logarithms.

The decimal division of angles has been agitated again in recent
years. In 1900 R. Mehmke made a report to the German Mathema-
tiker Vereinigung.1 Why are degrees preferred to radians in practical
trigonometry? Because, on account of the periodicity of the trig-
onometric functions, we frequently would have to add and subtract
TT or 27T which are irrational numbers and therefore objectionable.
The sexagesimal subdivision of the degree which resulted in great
harmony among the Babylonians who used the sexagesimal notation
of numbers and fractions, and the sexagesimal divisions of the day,
hour and minute, is less desirable now that we have the decimal
notation of numbers. There has been some difference of opinion
among advocates of the decimal system in angular measurement, what
unit should be chosen for the decimal subdivision. In 1864 Yvon

1 See Jahresb. d. d. Math. Vereinigung, Leipzig, Vol. 8, Part i, 1900, p. 139.

484 A HISTORY OF MATHEMATICS

Villarceau, at a meeting of the Bureau of Longitudes in Paris, sug-
gested the decimal subdivision of the entire circumference, while in
1896 Bouquet de la Grye preferred the semi-circumference. R. Mehmke
argues that whatever the unit may be that is subdivided, the four
arithmetical operations with angles would be materially simplified,
interpolation in the use of trigonometric tables would be easier, the
computation of the lengths of arcs would be shorter. If the right
angle is the unit that is subdivided, then the reduction of large angles
to corresponding acute angles can be effected merely by the subtrac-
tion of the integers i, 2, 3, . . The determination of supplementary or
complementary angles is less laborious. A more convenient arrange-
ment of trigonometric tables was claimed by G. J. Hoiiel and greater
comfort in taking observations was promised by J. Delambre. Never-
theless, no decimal division of angles is at the present time threatened
with adoption, not even in France.

A very specialized kind of logarithms, the so-called "Gaussian loga-
rithms," which give log (a+b} and log (a—b), when log a and log b
are known, were first suggested by the Italian physicist Guiseppe
Zecchini Leonelli (1776-1847) in his Theorie des logarithmes, Bordeaux
1803; the first table was published by K. F. Gauss in 1812 in Zach's
Monatliche Korrespondenz. It is a 5-place table. More recent tables
are the 6-place tables of Carl Bremiker (1804-1877) of the geodetic
institute of Berlin, Siegmund Gundelfinger (1846-1910) of Darmstadt,
and George William Jones (1837-1911) of Cornell University, and the
7-place table of T. Wittstein.

Proceeding to hyperbolic and exponential functions, we mention
the 7-place tables of log 10 sink x and log 10 cosh x prepared by Christoph
Gudermann of Miinster in 1832, the s-place tables by Wilhelm
Ligowski (1821-1893) of Kiel in 1890, the 5-place tables by G. F.
Becker and C. E. Van Orstrand in their Smithsonian Mathematical
Tables , 1909. Tables for sink x and cosh x were published by Ligowski
(1890), Burrau (1907), Dale, Becker, and Van Orstrand. In the Cam-
bridge Philosophical Transactions, Vol. 13, 1883, there are tables for
log ez and & by J. W. L. Glaisher, for erx by F. W. Newman.
G. F. Becker and C. E. Van Orstrand also give tables for these
functions.

f An isolated matter of interest is the origin of the term "radian,"
used with trigonometric functions. It first appeared in print on
June 5, 1873, in examination questions set by James Thomson at
Queen's College, Belfast. James Thomson was a brother of Lord
Kelvin. He used the term as early as 1871, while in 1869 Thomas
Muir, then of St. Andrew's University, hesitated between "rad,"
V "radial" and "radian." In 1874 T. Muir adopted "radian" after
\a consultation with James Thomson.1

1 Nature, Vol. 83, pp. 156, 217, 459, 460.

APPLIED MATHEMATICS 485

Calculating Machines, Planimeters, I nte graphs

The earliest calculating machine, invented by Blaise Pascal in 1641,
was designed only to effect addition. Three models of Pascal's ma-
chine are kept in the Conservatoire des arts et metiers in Paris. It
was G. W. Leibniz who conceived the idea of adapting to a machine
of this sort a mechanism capable of repeating several times, rapidly,
the addition of one and the same number, so as to effect multiplication
mechanically. Of the two Leibniz machines said to have been con-
structed, one (completed 1694?) is preserved in the library of Hanover.
This idea was re-invented and worked out for actual use in practice
in 1820 by Ch. X>. Thomas de Colmar in his Arithmometre. More
limited practical use was given to the machine of Ph. M. Hahn, first
constructed in Stuttgart in 1774.

A machine effecting multiplication, not by repeated additions, but
directly by the multiplication table, was first exhibited at the universal
exposition in Paris in 1887. This decidedly original design was the
invention of a young Frenchman, Leon Bollee, who took also a prom-
inent part in the development of the automobile. In his computing
machine there are calculating plates furnished with tongues of appro-
priate length which constitute a kind of multiplication table, acting
directly on the recording apparatus of the machine. A somewhat
simpler elaboration of the same idea is due to O. Steiger (1892) in a
machine called the millionnaire.1 In 1892 a Russian engineer, W. T.
Odhner, invented and constructed a widely used machine, called the
Brunsviga Calculator, which is of the "pin wheel and cam disc" type,
the first idea of which goes back to Polenus (1709) and to Leibniz.
Of American origin are the Burroughs Adding and Listing machine,
and the Comptometer invented about 1887 by Dorr E. Felt of Chicago.

The first idea of automatic engines calculating by the aid of func-
tional differences of various orders goes back to J. H. Miiller (1786),
but no steps towards definite plans and actual construction were
taken before the time of Babbage. Charles Babbage (1792-1871) in-
vented a machine, called a "difference-engine," about 1812. Its con-
struction was begun in 1822 and was continued for 20 years. The
British Government contributed £17,000 and Babbage himself £6000.
Through some misunderstanding with the Government, work on the
engine, though nearly finished, was stopped. Inspired by Ch. Bab-
bage's design, Georg und Eduard Scheutz (father and son) of Stock-
holm made a difference engine which was acquired by the Dudley
Observatory in Albany.

In 1833 Ch. Babbage began the design of his "analytical engine";

1 D'Ocagne in the Napier Tercentenary Memorial Volume, London, 1915, pp. 283-
285. For details, see D'Ocagne, Le calcnl simplifit, Paris, 1905, pp. 24-92; Ency-
klopddie d. Math. Wiss., Bd. I, Leipzig, 1898-1904, pp. 952-982; E. H. Horsburgh,
Napier Tercentenary Celebration Handbook, Edinburgh, 1914, "Calculating Ma-
chines" by J. W. Whipple, pp. 69-135.

486 A HISTORY OF MATHEMATICS

a small portion of it was put together before his death. This engine
was intended to evaluate any algebraic formula, for any given values
of the variables. In 1906 H. P. Babbage, a son of Charles Babbage,
completed part of the engine, and a table of 25 multiples of TT to 29
figures was published as a specimen of its work.1

Planimeters have been designed independently and in many dif-
ferent ways. It is probable that J. M. Hermann designed one in 1814.
Planimeters were devised in 1824 by Gonella in Florence, about 1827
by Johannes Oppikoffer (1783-1859) 2 of Bern and constructed by
Ernst in Paris, about 1849 by Wetli of Vienna and improved by the
astronomer Peter Andreas Hansen of Gotha, about 1851 by Edward
Sang of Edinburgh and improved by Clerk Maxwell, J. Thomson and
Lord Kelvin. All of these were rotation planimeters. Most noted of
polar planimeters are that of Jakob A msler (1823-1912) and those con-
structed by Coradi of Zurich. J. Amsler was at one time privatdocent
at the University of Zurich, later manufacturer of instruments for
precise measurements. He invented his polar planimeter in 1854; his
account of it was published in 1856.

Another interesting class of instruments, called "integraphs" has
been invented by Abdank Abakanovicz (1852-1900) in 1878 and by C.
Vernon Boys 3 in 1882. These instruments draw an "integral curve"
when a pointer is passed round the periphery of a figure whose area is
required. More recently numerous integraphs have been invented
through the researches of E. Pascal of the University of Naples. Thus
in 1911 he designed a polar integraph for the quadrature of differential
equations.

1 Napier Tercentenary Celebration Handbook, 1914, p. 127.

2 Morin, Les Appareils d 'Integration, 1913. See E. M. Horsburgh, op. cit., p. 190.

3 Boys in Phil. Mag., 1882; Abdank Abakanowicz, Les Integraphes, Paris, 1886.
See also H. S. Hele Shaw, "Graphic Methods in Mechanical Science" in Report oj
British Ass'n for 1892, pp. 373-531; E. M. Horsburgh, Handbook, pp. 194-206.

INDEX

Abacus, 7, 52, 53, 64, 65, 67, 68, 78, no, 114, 73, 75. Egyptian, 13, 14. Greek, 56-

116, 118-122

Abbati, P., 352; 253

Abbatt, R., 370

Abdank Abakanovicz, 486

Abel, N. H., 410-417; 238, 266, 278, 289,
200, 313, 316, 349-353, 374, 393, 4°5, 4*7,
426, 442; Abelian functions, 281, 313, 342,
300, 411, 412, 415, 418, 419, 421, 423, 424,
428, 429; Abelian groups, 357, 358, 360,
388; Abel's theorem, 413, 415

Abraham, M., 335

Abraham, ibn Esra, no

Abscissa, 175

Abu Kamil, 103, 104, 121

Abu'l Hasan Ali, 109

Abu'l Jud, 106, 107

Abu'l Wefa, 105; 104, 106, 107, 109

Acceleration, 172, 183

"Achilles," 23, 182

Ada erudilorum, founded, 209

Adam, Ch., 177

Adams, J. C., 449; 200, 332

Adler, A., 268

Adrain, R., 382

Aganis, 48, 184

Aggregates, theory of, 24, 66, 172, 325, 326,
390-406; Enumerable, 66, 400. See
Point-sets

Agnesi, M. G., 250; witch, 250

Agrimensores, 66

Ahmes, 7, 9-14, 16, 44, 60, 103, 123

Ahrens, W., 323

Aida Ammel, 81

Airy, J. B., 449; 383, 462

Ajima Chokuyen, 81

Akhmim papyrus, 14

Al-Battani, 105; 118

Albertus Magnus, 127

Al-Biruni, 100, 101, 105, 106

Alchazin, 107

Alcuin, 114; 116, 120

Alembert, Jean le Rond d', See D'Alembert

Alexander, C. A., 266

Alexander the Great, 7

Alexandrian School, first, 20-45; second, 45-
52

Alfonso X, 1 19

Algebra, Arabic, 102-104, i°°, 120. Chinese,

62. Hindu, 93-96, 103. Japanese, 79.
Laws of, 273. Linear associative, 285,
338, 339. Modem developments, 329-
366. Multiple, 289, 338, 339, 344. Of
Renaissance, 137-141. Origin of word,
103. Rhetorical, syncopated and sym-
bolic, in; the science of time, 333. See
Ausdehnungslehre, Covariants, Equa-
tions, Invariants, Quaternions

Algebra of logic, 285

Algorithm, 119. Origin of word, 102

Al-Hasan ibn Al-Haitam, 104, 107

Al-Kalsadi, no, in

Al-Karkhi, 106, 107

Al-Kashi, 108

Al-Khojandi, 106

Al-Khowarizmi, 102-104; 94. IQ6, 108, 118,
119

Al-Kuhi, 105-107

Allman, G. J., 16; 30

Almagest, see Ptolemy

Al-Mahani, 107

Al-Majriti, 109

Al-Nadim, 102

Al-Nirizi, 48

Al-Sagani, 105, 106

Al-Zarkali, 132

Amasis, King, 15

Amicable numbers, 56, 104, 109, 239

Ampere, A. M., 281, 386, 425

Amsler, J., 486

Amyclas, 29

Analemma, 48

Analysis, method of, 26, 27, 29. Modern,
367-411

Analysis situs, 211, 285, 323, 324

Analytic functions, 257, 258, 425-428, 439

Analytic geometry, 40, 159, 162, 163, 167,
173-184, 224, 275, 276, 293-295, 300-329

Anaxagoras, 17; 25

Anaximander, 16

Anaximenes, 17

Andoyer, H., 483

Andrade, J., 405

Andrews, W. S., 366

Angeli, Stefano delgi, 175

Anger, C. T., 449

Angle, trisection of. See Trisection

487

488

INDEX

Angle, vertical, 16

Anharmonic ratio. See Cross-ratio

Annuities, 171

Anthonisz, A., 73, 143

Antinomies, 286, 401, 402, 409

Anti-parallel, 209, 300

Antiphon, 23; 24, 51

Apices of Boethius, 52, 68, 100, 114, 116,

119, 121

Apollonian problem, 41, 144, 179, 288
Apollonius, 38-43; r, 30, 31, 33, 51, 55, 101,

104, 106, 109, 131, 142, 166, 174, 175, 181,

275

Appell, P., 300; 388, 405

Approximations to roots of equations. See
Numerical equations

Aquinas, Th, 126; 161

Arabic notation. See Hindu-arabic nu-
merals

Arabs, 90-112

Arago, D. F. J., 260, 269, 275, 368, 441, 470

Arbogaste, L., 250; 271

Arbuthnot, J., 244

Archibald, R. C., 33, 41, 246, 275, 301, 480

Archimedes, 34~3Q; i, 28, 30, 31, 33, 41, 42,
5i» 54. 73. 77, 90, loi, 104, io6, 128, 131,
i43» i5°» 171. 181, 221, 317, 370. Archi-
medean postulate, 35, 327. Archime-
dean problem, 107. Archimedian spiral,
36, 163. Cattle-problem, 59. Mensura
tion of the circle, 54. Sand counter, 54,
78,90

Archytas, 19; 20, 25, 27, 37, 53

Arenarius, 54, 78, 90

Argand, J. R., 265; 254, 349, 420

Aristaeus, 29, 39

Aristophanes, 17

Aristotle, 29; 7, 9, 15, 23, 24, 37, 51, 55, 118,
126, 129, 161, 179, 285, 397. On dyna-
mics, 171. His Physics, 23, 29

Arithmetic, Arabic, 102-104, 108, in.
Babylonian, 4-7. Chinese, 71-74, 76, 77.
Egyptian, 9-14. Greek, 18, 19, 32, 52-
62. Hindu, 85, 90-93. Japanese, 78, 79.
Middle Ages, 114. Renaissance, 125, 127,
128., Roman, 63-68

Arithmetical machines, 206, 272, 483, 485,
486

Arithmetical progression, 5, 12, 13, 58, 75,
140, 150, 185, 235

Arithmetical triangle, 76, 183, 187

Arithmetization, 362, 398, 369, 424

Armenante, A., 314

Arnauld, A., 170

Arneth, A., 97

Aronhold, S. H., 346; 348

"Arrow," 23

Aryabhafa, 85; 86, 87, 89, 92, 94-96

Arzela, C., 377

Aschieri, F.. 289, 307, 308

Ascoli, G., 405

Astroid, 269

Astrolabe, 48

Astronomy, Arabic, 102, 104, 109. Bnhy
Ionian, 7-9. Chinese, 76, 77. Greek, 16,
J9» 43. 46-48. Hindu, 83, 84, 95. Mod-
ern, 130, 131, 159, 160, 280, 289, 437, 450,
4SI-45S

Asymptotes, 40, 142, 177, 185, 188, 224

Asymptotic solutions of equations, 391, 392,

454

Asymptotic values, 438
Atabeddin Jamshid, 1 10
Athelard of Bath, 118
Atomic theory, 126
Atwood, G., 155
Aubrey, 151
Augustine, St., 67
Ausdehnungslehre, 336, 337
Axioms, Geometrical, n, 26, 31, 32, 48,

108, 184, 302, 303, 305, 308. Algebraical,

409

Babbage, C., 485; 272, 405, 486
Babbage, H. P., 486
Babylonians, 2, 4-8, 17
Bacharach, M., 472
Bachet de Meziriac, 167; 168, 170, 254
Bachmann, P., 444; 436
Backhand, A. V., 321, 325
Bacon, R., 126
Baer, K., 470
Bagnera, G., 360
Baillet, J., 14
Baire, R., 401, 402
Baker, H. F., 317, 319, 343. Quoted, 282,

283, 316

Baker, Th., 107, 203

Bakhshali arithmetic, 84, 85, 89, 91, 92
Ball, R. S., 455; 308
Ball, W. W. R , 204
Ballistic curve, 266
Baltzer, H. R., 321; 304, 341
Banachievitz, T., 453
Bang, A. S., 300
Barbier, E., 341, 379
Bar-le-Duc, Eward. See Eward de Bar-le-

Duc

Barnard, F. P., 122
Barr, A., 301

Barrow, I., 188-190; 158, 163, 192, 207, 212
Bartels, J. M., 434

Basset, A. B., 319, 320, 457, 461, 469
Bateman, H., 319
Battaglini, G., 354; 308

INDEX

489

Bauer, G., 470; 312, 365

Bauer, G. N., 446

Bauer, M., 348

Baumgart, O., 239, 436

Bauschinger, J., 483

Bayes, Th., 230, 263. Bayes's theorem, 377,

378

Bayle, P., 182
Beaumont, E. de., 266, 267
Beaune, F. de., 180; 174, 176, 209, 210;

B.'s problem, 209
Becker, G. F., 484
Becker, K., 381
Bede, the Venerable, 114; 120
Beer, A., 469
Beetle, R. D., 395
Beha-Fxldin, 106, 108, no
Bellavitis, G., 224, 292, 297, 306, 332, 337,

366

Beltrami, E., 307; 306, 321, 346, 470
Beman, W. W., 177, 265, 291
Bendixson, I. O., 426
Bennett, G. T., 301
Bentley, 226
Bergman, H., 368
Berkeley, G., 219; 228; " Analyst, 218;

Lemma, 219
Bernelinus, 116; 117
Bernoulli, Daniel (born, 1700) 220; 222, 227,

240, 242, 251, 252, 365, 378, 382, 383, 448,

464, 465, 477
Bernoulli, Jakob (James, born 1654) 220,

221; 81, 171, 173, 210, 211, 213, 220, 224,

234, 238; 331, 380, 465; Numbers of B.,

221, 238; B.'s theorem, 222, 263, 377;

Law of large numbers, 222, 380
Bernoulli, Jakob (James, born 1758) 220; 223
Bernoulli, Johann (John, born 1667) 220,

222; 57, 210, 211, 213, 216-218, 220, 221,

224, 227, 232, 234-238, 242, 251
Bernoulli, Johann (John, born 1710) 220;

223
Bernoulli, Johann (John, born 1744) 220;

223

Bernoulli, Nicolaus (born 1687) 220; 223
Bernoulli, Nicolaus (born, 1695) 220; 222,

238

Bernoullis, genealogical Table of, 220
Bernstein, B. A., 409
Bernstein, F., 401, 443, 445
Berry, A., 344
Berthollet, C. L., 270
Bertillon, J., 380
Bertini, E., 307
Bertrand, J. L. F., 379; 340, 371, 374, 378,

382, 385, 416, 438, 456. 457, 477
Bortrand's postulate, 438

Bessel, F. W., 448; 304, 311, 377, 382, 452

Bessy, Frenicle de, 169, 170

Bettazzi, R., 409

Betti, E., 346; 307, 352, 358, 417

Bpvan, B., 298

Beyer, J. H., 148

Bezout, E., 249; 235, 253, 259, 361

Bdzoutiant, 249

Bhaskara, 85; 86-88, 92, 93, 141, 142

B'anchi, L., 321, 325

Bieberbach, 360

Bienayme", J., 382

Bigelow, F. H., 464

Billingsley, H., 130

Billy, Jacobo de. See Jacobo de Billy

Binet, J. P. M., 340; 465

Bing, J, 378

Binomial coefficients, 76, 140

Binomial theorem, 178, 186, 187, 192, 205,

212, 213, 221, 222, 238, 374, 4!!

Biot, J. B., 275; 216, 262, 471, 473

Biquaternions, 307

Birational transformations, 295, 314, 316,

3i7, 319
Birch, S., 9

Birkhoff, G. D., 324, 391, 392, 394, 396, 453
Bjerknes, C. A., 462; 412, 421
Bjornbo, A. A., 45
Blasckke, E., 381
Blaschke, W., 370
Bledsoe, A. T., 173
Blichfeldt, H. F., 360. Quoted, 360
Bliss, G. A., 372, 406, 431, 433
Blumberg, H., 348
Bobillier, E., 310
Bocher, M., 394; 2, 363, 387, 391, 393, 448,

465. Quoted, 284, 286
Bochert, A., 357, 359
Bode, J. E., 384
Boethius, 67; 52, 53, 68, 113-116, n8, 119,

127

Boggio, T., 469
Bohlin, K., 454
Bohniceck, S., 443
Bois-Reymond, P. du, 377; 326, 371, 374-

376. 393, 4°o, 425
Bollee, L., 485

Boltzmann, L., 326; 176, 474, 477-479
Bolyai, J., 304; 278, 303, 305-307, 446
Bolyai, W., 303; 278, 302, 304, 435
Bolza, O., 201, 319, 372, 413. Quoted, 371
Bolzano, B., 367; 258
Bombelli, R., 135; 137, 141, 147
Bompiani, E., 322
Boncompagni, B., 178
Bond, H., 189
Bonnet, O., 321; 374, 385

490

Bonola, R., 48, 108, 184, 307

Boole, G., 407; 278, 281, 285, 342, 383, 384,

386,391,408
Booth, J., 312
Bopp, K., 181
Borchardt, C. W., 418; 425
Borda, J. C., 266
Borel, E., 375, 386, 401, 402. 404. 427, 481.

Quoted, 389
Borelli, G. A., 184
Bortkewich, L. v. See Bortkievicz
Bortkievicz, v., 379, 381
Bortolotti, E., 349, 350
Bosnians, H., 77
Bouguer P., 157, 273
Boundary-value problems, 270, 284, 391,

396, 430

Bouniakovski, V. J., 436
Bouquet, J. C., 388; 241, 383, 387, 418, 420
Bouquet de la Grye, 484
Bour, E., 384

Boussinesq, J., 462, 469, 471
Boutroux, E., 410
Boutroux, P., 429
Bouvelles, C., 162
Bowditch, N., 262; 338
Bowley, A. L., 381
Boys, V., 486
Brachistochrone, 234

Bradwardine, T., 127; 116, 128, 132, 161
Brahmagupta, 85; 71, 86, 87, 92, 94, 97, 99
Braikenridge, W., 228
Brancker, T., 140, 169
Braunmiihl, A. v. 48; 137, 235, 481
Bredon, S., 128
Bremiker, C., 484
Brennan, L., 458
Bret, J. J., 269

Bretschneider, C. A., 9; 88, 336
Brewster, D., 191, 193, 201, 461
Brianchon, C. J., 166, 275, 287, 288, 298
Briggs, H., 150-152; 155, 187, 343
Brill, A., 293, 313, 316, 328, 419, 430, 431

480

Brill L. 309, 328
Bring, E. S., 349
Brioschi, F.. 345-347! 279, 307, 340, 34*1

347, 348, 361, 370, 388, 413, 417, 456
Briot, C., 388; 241, 383, 387, 418, 420
Brisson, M. J., 265
Brocard, H., 298; 299; B. points, 299;

B. angles, 299; B. circle, 299, 300
Broch, O. J., 414
Broden, T., 433

Brouncker, W., 156; 169, 187, 188, 228
Brouwer, L. E. J., 403, 433
Brown, C., 329

Brown, E. W., 450, 451, 453, 454. Quoted,

45°

Brownlee, J. W., 383
Brunei, H. M., 301
Bruno, Fa£ de, 345
Brans, H., 452, 454
Brussels academy of sciences, 168
Bryant, W. W., 451
Bryson of Heraclea, 23; 24
Bubnow, N., 98
Buchanan, D., 453
Buchheim, A., 455; 308
Buckle, H. T., 190
Buckley, W., 147; 183
Budan, F. D., 269, 271
Buffon, Count de, 243, 244, 263, 378, 379
Biihler, 88
Bungus, P., 144
Burckhardt, J. C. H., 440
Burali-Forti, C., 289, 322, 335, 401, 402, 408
Burgess, E., 85

Biirgi, J., 152; 137, 148, 154, 178
Bttrja, A., 155, 258
Burkhardt, H., 280; 350, 252, 318. Quoted,

465, 468

Burkhardt, J. P., 262
Burmann, H., 272
Burmester, L., 297
Burns, J. E., 352
Burnside, W., 357, 358, 359, 360
Burrau, 484
Burroughs, 485
Busche, E., 436
Buteo, J., 143, 156
Butterworth, J., 298
Butzberger, F., 292
Buy's-Ballot, C. H. D., 478
Byerly, W. E., 470
Cajori, F., 3, 24, 127, 156, 174, 182, 190, 202,

224, 248, 271, 330, 344, 447
Calandri, Ph., 128
Calculate, origin of word, 64
Calculating engine. See Arithmetical ma-
chine.

Calculus, See Differential C., Integral C.
Calculus integralis, the name, 221
Calculus of functions, 405
Calculus of residues, 420
Calculus of variations, 232, 234, 251, 255,

267, 281, 291, 394, 313, 367, 369-372,

404, 405, 430, 431, 437
Calendars, 8, 66, 70, 76, 78, 114, 122, 13*,

144

Callet, F., 266
Callisthenes, 7
Cambuston, H., 332
Campano, G., 120; 142

INDEX

491

Campbell, 0., 202

Cantor, G., 307-404; 24, 67, 172, 285, 325,
367, 400, 404, 409, 426, 432, 440, 447.
Quoted, 447

Cantor, M., 6; 10, 13, 14, 42, 63, 87, 91, 96,
101, 105, 106, no, 114, 115, 117, 119, 123,
140, 211, 247, 249, 269, 384

Capella, M., 113

Capelli, A., 365

Capillarity, 264

Caporali, E., 314; 319

Caque", J., 387

Cardan, H., 134-136; 137-141, 145, 147,
170, 179, 181, 182, 184, 185

Carette, A. M., 268

Carlini, F., 452

Carll, L. B., 370

Carmichael, R. D., 391, 396. Quoted, 391

Carnot, L. N. M., 276; 46, 219, 221, 287,
302, 349,

Carnot, S., 475, 476; 473

Carra de Vaux, 98

Carslaw, H. S., 48, 108, 465

Cartan, E. J., 339, 358

Carvallo, E., 335, 365

Casey, J., 314; 324

Casorati, F., 346; 307, 383

Cassini, D., 244; 190, 222, 245, 451

Cassini, J., 244

Cassini's oval, 221, 245

Cassiodorius, 68, 113

Castellano, F., 409

Castelnuovo, G., 316; 317, 318, 431

Casting out nines, 59, 91, 103

Catalan, E. C., 341; 330, 383, 470

Cataldi, P. A., 147, 184, 254

Catenary, 183, 217

Cattle problem, 59, 60

Cauchy, A. L., 368-370; 227, 232, 238, 249,
253, 258, 265, 287, 337, 340, 341, 349,
354. 361-363, 367, 373, 374, 376, 383-
389, 395, 396, 405, 412, 4i6, 417, 419, 420,
426, 428, 430, 431, 438, 440, 446, 452, 461,
465-468, 471, 473. Cauchy's theorem on
groups, 352. Cours d'analyse, 369, 373.
Tests of convergence of series, 373

Caustic curves, 222, 225

Cavalieri, B., 161, 162; 79, 159, 162, 165,
175, 177, 190, 191, 207

Cayley, A., 342-348; 240, 278, 290, 291, 293,
295-297, 302, 308, 310, 312-314, 316-320,
323, 332-335, 338-340, 351-353, 361, 383,
415, 417, 418, 432, 456. Cayley line, 291,
Quoted, 280. Sixth memoir on quantics,
307, 308

Celestial element method, 75-80

Center of gravity, 289

Center of oscillation, 183, 227

Center of similitude of circles, 275

Centrifugal force, 172, 183, 200, 244

Cesaro, E., 324, 375, 379

Ceva, G., 277

Chamberlin, T. C., 450

Champollion, n

Chance. See Probability

Chandler, S. C., 241

Chang Ch'iu-chien, 73

Chang Chun-Ch'ing, 88

Chang T'sang, 71; 97

Characteristic triangle, 189, 207

Chapman, C. H., 340

Charlier, C., 380, 452, 453

Charpit, P., 255

Chasles, M., 292-294; 33, 39-41, 43, 162,

174, 246, 276, 287, 295, 297, 308, 310, 312,

314, 319, 418, 455, 472, 473
Chauvenet, W., 383; 455
Chebichev, P. L., 380; 301, 344, 438
Ch' 6ng Tai-wei, 76
Chernac, L., 439
Chevalier de M&-6, 170
Cheyne, G., 194
Child, J. M., 189
Ch' in Chiu-shao, 74; 75
Chinese, 71-77; 17, 84. Solution of equa-
tions, 74, 75, 271. Magic squares, 76, 77
Ching Ch' ou-ch' angTTi —
Chittenden, E. W., 395
Chiu-ckang, 71
Chladni, 464
Choquet, C., 363
Chou-pei, 71
Chree, C., 461, 469
Christina, Queen, 179
Christoffel, E. B., 314; 346, 356, 431, 470
Chrystal, G., 378; 379
Chuproff, A. A., ,379
Chuquet, N., 125, 178
Chu Shih-Chieh, 75; 76
Cipher, origin of term, 121
Circle, 20, 22, 23, 25, 42, 104, 143, 297-300,

370. Nine-point Circle, 298. Division

of, 107, 350, 414, 435, 436
Circle-squarers, i, 331. See Quadrature of

the circle.

Circular points at infinity, 282
Circumference, 297
Cissoid, 42, 51, 182
Clairaut, A. C., 244; 227, 229, 239, 242, 245,

252, 302, 457. His differential equat., 245
Clapeyron, B. P. E., 467, 475
Clarke. A. R., 379
Classes, theory of, 410
Clausen, T., 330; 22

4Q2

INDEX

Clausius, R., 476; 468, 474, 477-470

Clavius, C., 144; 47, 143, 158, 181, 184

Clayton, H. H., 464

Clebsch, R. F. A., 313, 314; 291, 296, 311,
316, 318, 319, 337, 345-348, 369, 371, 384,
422, 431, 457, 462, 468, 469

Clifford, W. K., 307; 278, 303, 308, 317, 333,
33S, 34°, 348, 422, 423, 455, 472

Cockle, J., 321

Colburn, Zerah, 169

Colding, L. A., 475

Cole, F. N., 347, 354, 3S»

Colebrooke, H. T., 85

Colla, 133, 135

Collins, J., 192, 193, 203, 209, 212-216

Colson, J., 193

Combescure, E., 315

Combinations, theory of, 128, 170, 183, 221

Combinatorial school, 231, 232

Commandinus, F., 141; 175, 184

Commercium epistolicum (Collins1) 194,
215, 216

Commercium epistolicum (Wallis') 168.

Compensation of errors, 219

Complex variables, 420, 422

Comte, A., 285

Conchoid, 42, 51, 202

Condorcet, N. C. de, 244; 252, 266, 380

Cone, 27, 33, 39, 46, 79, 141, 319

Congresses, international. See Interna-
tional c.

Conies, 27, 29, 33, 36, 38-41, 5°, Si, 88, 141,
142, 160, 165-167, 181, 184, 246; Con-
jugate diameters, 41; Foci, 40, 41, 160;
Generation of, 180, 228; Names ellipse,
parabola, hyperbola, 39; Name latus
rectum, 40

Conoid, 36

Conon, 34; 36

Conservation of areas, 240

Conservation of vis vim or energy, 183

Constructions, 2, 21, 22, 27, 47, 84, 86,
106, 297, 300, 310, 336, 436; By com-
passes only, 268; By insertion, 36; By
ruler and compasses, 124, 174, 177, 202,
292, 350, 436, 446; By ruler and fixed
circle, 291; By single opening of com-
passes, 106; Of maps, 295; Of regular
polygons, 47, 128

Contact-transformation, 354, 355

Conti, A. S., 216

Continued fractions, in, 147, 188, 246, 258,

375
Continuity, 22, 24, 29, 94, 160, 184, 185,

211, 218, 282, 283, 287, 318, 326, 367, 391,

399, 410-421, 426
Continuum, 24, 35, 126, 285, 397, 398,

400; Well-ordered, 401; Not denumerable,

402

Convergence of series. See Series
Convergence of aggregates, 398
Convergent series, use of term, 228, 237
Coolidge, J. L., 300, 308
Coordinates, 40, 42, 174, 175, 184, 211, 235,

289, 294, 310, 314, 321, 324, 482; Elliptic,

456; Generalized, 255; Homogeneous, 297;

Intrinsic, 324; Oblique, 174; Pentaspher-

ical, 315; Polar, 221; Tangential, 310;

Trilinear, 310; Movable axes, 321
Copernicus, N., 46, 130, 131
Cosserat, E., 315, 325
Cotes, R., 226; 199, 236, 382; Theorem of,

228

Counters, 122

Counting board, 75, oo, 91, 122
Courant, R., 433
Cournot, 281
Courtivron, 227
Cousin, P., 429
Cousin6ry, B. E., 296
Couturat, L., 286

Covariants, 345, 348, 349, 356, 417, 440
Cox, H., 308
Craig, C. F., 319
Craig, J., 171, 210
Craig, T., 418; 308, 391, 461
Cramer, G., 241; 175, 204, 223, 320; C.

paradox, 228

Crelle, A., 411; 289, 200, 298, 299, 418
Cremona, L., 295, 296; 278, 287, 291, 307,

314, 318, 319, 346.; C. transformation,

295

Crew, H., 172

Crofton, M. W., 379; 380, 382
Crone, C., 320

Cross-ratio, 166, 289, 293, 294, 297, 308
Crozet, C., 276
Ctesibius, 43

Cube, duplication of, See Duplication
Cube root, 71, 74, 123
Cubic curves, 204, 228, 229, 244, 249, 295,

320
Cubic equations, 74, 107, no, in, 124, 133-

138, 140, 142, 177, 247, 350
Culmann, K., 296; 294, 297
Cuneiform writing, 4, 7, 8
Cunningham, A. J. C., 446
Curtze, M., 296; 73, 123, 170
Curvature, theory of, 275, 296, 320, 321
Curves, 163, 202, 204, 206, 207, 209, 224,

226, 228, 235, 244, 250, 275, 295, 318-320,

321; Algebraic, 302, 419; Ballistic, 266;

Catenary, 183, 217; Caustic curves, 222,

225; Class of curves, 288; Conchoid, 42,

INDEX

493

51, 202; Courbe du diable, 341; Cubics,
204, 228, 229, 244, 249, 295, 320; Defi-
ciency, 313, 317; Definition of curve, 325,
326; Elastic curve, 221, 291; Fourth or-
der, 310, 313; Fourteenth order, 312;
Genus, 316, 320; Hippopede, 42; Logarith-
mic curve, 156, 183, 236; Logarithmic
spiral, 156, 221; Loxodromic, 142, 221;
Multiple points, 224, 229; Of pursuit, 273;
Of swiftest descent, 222; Order of, 288;
Panalgebraic, 320; Peanp curve, 325;
Polar curve, 307; Prony curve, 301;
Quartic, 188, 241, 245, 319, 320; Quintic,
241; Rectification of, 181, 182, 221, 224,
225; Second degree, 200; Space curves,
So, 295, 322; Third degree, 310, 312;
Third order, 310, 312; Three-bar-curve,
301; Transcendental, 21, 320; Twisted,
321; Versiera, 250; Witch of Agnesi, 250;
Without tangents, 326

Cusanus, N., 143

Cyclic method (Hindu), 95, 96

Cycloid, 162, 165, 166, 177, 181-184, 188,
210, 217

Cyzicenus, 29

Czuber, E., 378, 379, 381, 382

Dale, 484

D'Alembert, J., 241-245; 220, 233, 237, 251,
252, 257-259, 269, 306, 369, 405, 457, 464.
480; D. principle, 242

Damascius, 51; 32, 101

Dandelin, 364; 365

Darboux, J. G., 315; 301, 310, 314, 319, 321,
322, 325, 347, 354, 355, 372, 383, 386, 406,
425, 433, 470; Quoted, 276, 279, 287-289,
292

Darwin, G. H., 449; 450, 453, 462, 469

Dase, Z., 440

Davies, T. S., 271, 298

Da Vinci, Leonardo; See Vinci, da

Davis, E. W., 308

Davis, W. M., 463

De Beaune, F., 180; 174, 176

Decimal fractions, 5, 119, 147, 148

Decimal point, 148

Decimal system, 4-6, n, 72, 88

Decimal weights and measures, 148, 256;
Dec. subdivision of degree, 148, 484;
Centesimal subdivision of degree, 152,
482-484

Dedekind, J. W. R., 397-3991 32, 35, 172,
285, 331, 339, 348, 354. 357, 362, 400, 401,
407, 421, 431, 442, 445, 470

Definite integrals. See Integrals

Degree, decimal subdivision of, 148, 483,
484; Centesimal subdivision of, 152,
482-484; Sexagesimal, 6, 152, 483

De Gua, J. P., 224; 175

Dehn, M. W., 327

De Lahire, 166; 141, 167, 170, 222, 273, 288

Delamain, R., 158, 159

Delambre, J. B. J., 437; 484

Delaunay, C. E., 449; 369, 370, 451

Delboeuf, J., 302

Del Ferro, S., 133; 134

Delian problem, 21, 27

Del Pezzo, P., 307

Del Re, A., 319

Demartres, G., 315

Democritus of Abdera, 25; 15

De Moivre, A., 229; 222, 224, 377, 380;
De Moivre's problem, 230

De Morgan, A., 330; 32, 159, 194, 196, 212,
215, 233, 250, 271, 273, 278, 323, 331, 369,
374, 377, 379, 381, 382, 405, 407; Quoted,
i, 2, 57, 95, 149. 172, 213, 217, 263, 363;
Budget of Paradoxes, 332

Demotic writing, n

Demoulin, A., 315

Den ton, W. W., 322

Derivatives, method of, 258

Desargues, G., 166; 141, 146, 164, 167, 174,
273f 285, 287; Theorem of, 285, 327

Desboves, A., 456

Descartes, R., 173-184; 2, 40, 50, 107, 141,
146, 156, 162-167, 172, 181, 100, 192, 200,
205, 207, 209, 239, 240, 273, 276, 310,
355, 361, 363, 4°i: Folium of, 177, 229;
Ovals, 176; Rule of Signs, 178, 179, 201,
224, 248

Descriptive geometry, 246, 274-276, 296,
297, 308, 327; Shades and shadows, 297

DeSluse. SeeSluse

De Sparre, 459

Determinants, 80, 211, 249, 254, 264, 266,
312, 314, 340-342, 362, 370, 434; Name
"determinant," 340; Skew, 340; Pfaf-
fians, 340; Infinite, 341, 394

Devanagari numerals, 100, 101

De Witt, J., 180

Dichotomy, 23

Dickson, L. E., 318, 339, 348, 357-359, 360,
442; Quoted, 443

Differentiability, 376, 399, 423, 425

Differential Calculus, 3, 41, 163, 191, 196,
201, 208-210, 276, 393, 320, 367-369.
Controversy on invention of, 212-218;
Japanese, 79

Differential coefficient, first use of word, 272

Differential equations, 164, 195, 196, 208,
an, 222-225, 227, 238, 239, 243, 245, 254,
255, 263, 264, 282, 324, 332, 367, 371, 372,
373, 383-391, 394, 396, 405, 417, 432, 450,
456; Hyper-geometric, 282; Linear, 238,

494

INDEX

263, 391, 393. See Partial differential
equations, Singular solutions, Differential
calculus, Integral calculus, Three bodies
(problem of).

Differential geometry, 306, 315, 321, 322

Differential invariants, 345, 355, 356, 388

Dingeldey, F., 323

Djni, 375; 279, 377, 431

Dinostratus, 27; 21

Diocles, 42

Diodorus, 9, 34

Diogenes Laertius, 9, 16

Dionysodorus, 45

Diophantine analysis, 62, 8r, 95, 168

Diophantus, 60-62; 45, 48, 51, 87, 93-95,
101, 103, 105, 106, in, 135, 167, 168,
401

Directrix, 50

Dirichlet, P. G. L., 438; 168, 170, 270, 278,
357, 362, 372, 376, 377, 392, 400, 412, 418,
419, 421, 422, 424, 429, 439, 442, 443, 470;
D. principle, 284, 392, 422, 428, 429, 430,

433, 473

Discriminant (name) 345
Distance, 308

Divergent series, 228, 237, 238, 242
Division of circle, 107, 350, 414, 435, 436
Division of numbers, 7, 73, 117, 119
Diwani-numerals, 100
D'Ocagne, M., 482; 483, 485
Dodd, E. L., 382
Dodgson, C. L., 302
Dodson, J., 155
Donkin, W. F., 456; 465
Dositheus, 34
Dostor, G. J., 341

Double false position. See False position
Dove, H. W., 463
D'Ovidio, E., 308, 341
Drach, S. M., 366
Drobisch, M. W., 224
Dronke, A., 311

Duality, principle of, 288, 200, 294, 3 10
Dubois-Aym6, 273
Duffield, W. W., 483

Duhamel, J. M. C., 383; 363, 369, 416, 467
Duhem, P., 127; 128
Duhring, E., 183
Duillier. See Fatio de Duillier
Dulaurens, F., 225
Dumas, W., 348
Duns Scotus, 126
Duodecimals, 63, 64, 117, 119
Dupin, C., 275; 296, 320, 379; D. theorem,

275

Duplication of a cube, 2, 19, 20, 21, 27, 38,
42, 142, 177, 202, 246

Dupuy, P., 351

Durege, H., 311; 418

Durer, A., 141; 170, 145

Dyck, W., 324, 329, 432

Dyname, 335

Dynamics, 171, 172, 183, 223, 255, 307, 322,

332, 477- See Potential
Dziobek, O., 453, 455
Earnshaw, S., 462

Earth, figure and size of, 102, 229, 281
Earth, rigidity of, 240, 469
Ecole normale founded, 256
Ecole polytechnique founded, 256
Eddy, H. T., 296
Edgeworth, F. Y., 378, 381
Edleston, J., 212, 216
Eells, W. C., 70
Egyptians, 0-15, 17-19
Ehrenfest, 481

^Einstein, A., 335, 470. 480. 481
"ElSCnharT, L. P., 310, 321; Quoted, 315
Eisenlohr, A., 9, 10
Eisenlohr, F., 369.
Eisenstein, F. G., 440; 346, 348, 417, 421,

436, 441, 442, 444
Elastic curve, 221, 296
Elasticity, 460, 464-470
Eliminant, 249
Elimination, 311, 312, 361
Elizabeth, Princess, 179
Elliott, E. B., 348
Ellipsoid, attraction of, 229, 244, 263, 266,

267, 273, 293, 437
Elliptic functions, 225, 232, 239, 291, 313,

314, 362, 390, 441, 414-417, 434, 437;

Addition-theorem, 291; Double periodi-
city, 414

Elliptic integrals, 239, 258, 266, 267, 414
Ellis, A. J., 155
Ellis, L., 152
Ely, G. S., 444
Emsh, A., 300
Emmerich, A., 300
Encke, J. F., 452; 364, 377, 382, 437
Encyclopedic des sciences math., 280
Encyklopadie d. math. Wiss., 280
Enestrom, G., 128, 140, 148, 158, 173, 174,

179, 184, 221, 223, 225, 233, 235, 239,

439

Engel, F., 184, 354, 355, 35$
Enneper, A., 416, 417
Enriques, F., 316, 317, 322, 328, 446, 481
Entropy, 476

Enumerative geometry, 292, 293, 295
Envelopes, theory of, 211
Epicycloids, 141, 166, 224
Epimenides puzzle, 402

495

Epping, J., 8

Equations, theory of, 138, 130, 156, 201, 249,
253, 254, 264, 344, 347, 340-366; Abelian,
411; Cubic, 74, 107, no, in, 124,
133-137, 140, i"77, 247; irreducible case,
I3S, 138, 142,' 3So; Every e. has a
root, 3, 237, 253, 349; Functional, 395,
405; Indeterminate, 60, 73, 74, 94-96, 106,
124, 167, 441; Linear, 13, 75, 95, 103, 211,
393, 394; Modular, 352, 416, 417; Nega-
tive roots of, 176; Of squared differences,
249, 254; Resultant of, 249; Rule of signs,
178, 179, 201, 224, 248; Quadratic, 13,
575 74, 75, 94, 95, i°3, 106, 107, 138;
Quartic, 61, 107, 75, 177, 235, 135, 138;
Resolvents, 138; Quintic, 253, 332, 349,
350, 411; solution by elliptic integrals,
350. See Differential e., Integral e.,
numerical e

Equipollences, 337

Eratosthenes, 38; 21, 30, 34, 58. His
"sieve," 58

Erdmann, G., 371

Ermakoff, W., 375

Errard de Bar-le-Duc, I., 158

Escherich, G., v., 372

Espy, J. P., 462

Ether, theory of, 460, 461

Ettinghausen, A. v., 363

Euclid, 29-34; 18, 22, 25, 29, 39, 46, 53, 59,
101, 123, 184, 268, 353, 442

Euclid's Elements, 15, 18, 19, 27, 28-35, 44,
47, So, 51, 57, 58, 67, 86, 104, 108, 118-
120, 125, 129, 130, 142, 148, 165, 167, 192,
205, 226, 302, 303, 307; Euclid's Data, 33;
Elements in China, 77; Algebra, 61

Eudemian summary, 15, 16, 18, 26, 28, 30

Eudemus, 15, 19, 38, 39, 57

Eudoxus, 28; 15, 25, 27, 30, 31, 32, 35, 42,

327
Euler, L., 232-242; 62, 95, 143, 144, 152,

158; 175, 190, 220, 222, 223, 22S-227, 231,

232, 245-247, 249, 251-254, 257, 260, 264,
270, 275, 297, 320, 322, 324, 329, 330,
353, 365, 369, 373, 377, 380, 381, 389, 4°5,
411, 419, 420, 435, 439, 448, 450, 457, 458,
464, 465, 477; Euler's Algebra, 233;Analy-
sis situs, 323; Euler line, 298; Infinite
series, 373; Institutiones calculi diff., 233,
239; Institutiones calculi int., 233, 239;
Integrating factors, 239; Introductio in
analysin, 227, 233, 241; Magic squares,
170; Mechanka, 240; Methodus inveniendi
lineas curuas, 234; Method of elimina-
tion, 25; Number-theory, 168-170, 239;
Polyedra, 240; Quadratic reciprocity, 239;
Symmetric functions, 235; Theoria mo-

luum lunae, 234; Theoria motuum plane-

tarum, 234

Eutocius, 51; 38, 44, 53, 54
Evans, G. C., 395
Evolutes, 41, 183
Exchequer, 122
Exhaustion, method and process of, 23, 24,

28, 31, 35, 36, 41, 109, 160, 161
Exhaustion, origin of name, 181
Exponential calculus, 222
Exponents, 140, 148, 149, 178, 187, 235;

Fractional, 148,185, 238, 247; Imaginary,

225; Literal, 192; Negative, 185, 238
Faber, G., 429, 446
Fabri, H., 206
Fabry, C. E., 375
Faerber, C., 366
Fagnano, Count de, 225; 239
Falk, M., 428
Falling bodies, 171, 183
False position, 12, 13, 91, 93, 103, 137, 366;

Double, 44, 103, no, 123
Fano, G., 322, 327, 409 •
Faraday, M., 474, 475
Farkas, J., 405
Farr, W., 383
Fatio de Duillier, 214
Faye, H., 455
Fechner, G. T., 381
Fekete, M., 362
Felt, D. E., 485
Fenn, J., 302
Fermat, P. de, 163-170; 142, 146, 147, 162,

174-177, 180-182, 180-191, 239, 250, 276,

401, 438

Fermat's theorem, 169, 239, 254
Fermat's last theorem, 106, 168, 239, 254,

442, 443

Ferrari, L., 135; 134, 139, 253
Ferrel, W., 463; 449, 458, 464
Ferrero, A., 382
Ferrers, N., 470

Ferro, S. del. See Del Ferro, S.
Ferroni, P., 221
Feuerbach, K. W., 298
Fibonacci. See Leonardo of Pisa
Fiedler, W., 297; 313
Field, P., 320
Fields, J. C., 436

Fifteen school girls, problem of, 323
Finck, Th., 151

Fine, H. B., 362, 383; Quoted, 362
Finger symbolism, 63, 65, 68, 114
Finite differences, 224, 226, 230, 238, 258,

264, 405, 408, 466
Fink, K., 291
Fischer, E., 376, 306

496

INDEX

Fisher, A., 378

Fiske, T. S., 279

Fite, W. B., 357

Fitzgerald, G. F., 471, 479

Fleck, A., 444

Floquet, G., 348

Floridas, 133, 134

Flower, R., 155

Fluents, 193, 194, 195, 200, 213

Fluxions, 150, 192-197, 200, 210, 213, 228,
247; Controversy on invention, 212-218;
Berkeley's attack on, 219, 220; Compen-
sation of errors in, 219

Folium of Descartes, 177, 229

Foncenex, D. le., 237; 257

Fonctionelles, 395, 405

Fontaine, A., 239; 242

Ford, W. B., 375

Forsyth, A. R., 279, 345, 356, 384, 385, 386,
388, 433; Quoted, 281, 416, 342

Foster, S., 158

Foucault, J. B. L., 473

Fourier, J., 269-271; 164, 234, 242, 247, 281,
363, 364, 365, 40°, 413. 4i8, 410, 438, 473;
Analyse des equations, 269, 284; Fourier's
series, 242, 270, 283, 375~377, 393, 39O,
461, 465; Fourier's theorem, 269, 284,
438; TMorie analytique de la chalcur, 270,
271

Fourth dimension, 184, 256, 318, 335, 480

Foville, A. de, 380

Fractions: Duodecimal, 63, 64, 117, IIQ;
Partial, 211; Rational, 22; Sexagesimal,
5, 54, 483; Unit-fractions, 12, 14, 44, 71,
123; Continued, in, 147, 188, 246, 258,
375; Roman, 64; Chinese, 71; Decimal,
5, 119, 147, 148; Fractional line, 123

Francesca, Pier della, 128

Franklin, B., 170

Franklin, Christine Ladd, 407

Franklin, F., 343, 345, 436, 444

Frantz, J., 448

Frechet, M., 372, 393, 394, 404-406

Fredholm, E. I., 393; 394, 341, 427, 469

Frege, G., 408; 286, 407, 409

Frenicle de Bessy, 169, 170

Fresnel, A. J., 470, 471; 183, 275, 311, 314,
319, 333, 344, 46°, 465, 473

Frezier, A. F., 274

Fricke, R., 417, 432, 433

Friedlein, G., 64; 65

Frischauf, J., 452

Frizell, A. B., 402, 403

Frobenius, F. G., 354; 339, 341, 347, 353,
357, 360, 362, 387, 390, 427, 431, 443;
Quoted, 362

Frost, A. H., 366

Froude, W., 457; 462

Fuchs, L., 387; 385, 388, 390, 433

Fuchs, R., 279, 347

Fueter, R., 445

Fujita Sadasuke, 81

Functional calculus, 392, 395

Functionals, 395

Functions, 127, 211, 234, 238, 258, 270, 284,
388, 389, 400, 411-433, 44S, 446; Abelian
f., 281, 313, 342, 390, 411, 412, 415, 418,
419, 421, 423, 424; Algebraic, f., 295, 418,.,
43°, 43U Analytic, f., 257, 258, 425-428,
439; Arbitrary, f., 242, 251, 252, 258, 270,
419; Automorphic f., 432; Bessel f., 448;
Beta f., 234; Calculus of, 331, 332; Com-
plex variables, 420, 422; Definition of, 270,
326, 400, 419; Fuchsian f., 389, 390;
Gamma f., 234, 416; Hyperbolic f., 246,
424, 484; Hyperelliptic f., 281, 411, 418;
Modular f., 416, 417, 432; Multiply-
periodic f., 283; Non-differentiable f., 326;
F. on point sets, 403, 404; Orthogonal f.,
396; Potential f., 284, 422; Sigma f., 417;
Symmetric f., 293, 361, 414; Theta, 342,
415,416,418. See Elliptic functions
Trigonometric f., 234, 236; Zeta f., 439.

Funicular polygons, 296

Fiirstenau, E., 341, 365

Furtwangler, P. H., 443, 445

Fuss, P. H., 157, 237, 249

Gaba, M. G., 395

Galbrun, H., 391

Galileo, 171, 172; 37, 80, 130, 146, 159, 161,
162, 170, 179, 223

Galloway, T., 382

Galois. E., 351; 352, 353, 354, 358, 411, 432,
445; G. resolvent, 411; G. group, 318

Gallon, F., 381; 380

Garbieri, G., 341

Gardiner, 235

Gauss, K. F., 434-439; 3, 6, 62, 146, 169,
184, 231, 232, 235, 237, 238, 248, 253, 265,
278, 281, 284, 289, 295, 314, 320, 322, 325,
336, 340, 342, 348-351, 353, 361, 366,
369, 371, 373, 382, 430, 438-440, 442, 44"-
448, 452, 459, 460, 469, 472, 475, 484;
Disquisitiones arithmelicae, 435-437; Non-
euclidean geometry, 303-306; Theoria
molus, 437, 447

Gay de Vernon, S. F., 274

Gay-Lussac, 275

Geber, 109

Gehrke, J., 300

Geiser, C. F., 318

Gellibrand, H., 151-152

Geminus, 44; 39, 42, 45, 47, 48

General Analysis, 392, 394, 395

INDEX

497

Genocchi, A., 436

Gentry, R., 320

Geodesies, 234, .267, 372

Geometrical progressions, 5, 7, 13, 140, 150,
154, 185, 235

Geometrographics, 300

Geometry, Analytic, 40 159, 162, 163, 167,
173 184, 224, 275, 276, 293-295, 309-
329; Analytic geometry, rivalry with
synthetic, 288; Analysis situs, 211, 285,
323, 324; Arabic, 104; Babylonian, 6, 7;
Chinese, 76; Descriptive, 246, 274, 175;
Differential, 306, 315, 321, 322; Egyptian,
o-n; Enumerative, 292, 293; Geometro-
graphics, 300; Greek, 15-52; Hindu, 83-
88; Ideal elements in, 288; Knots, 322;
Models, 328, 329; Non-archimedian, 327;
Non-desarguesian, 327; Non-euclidean,
32, 302-309; Of position, 276, 297; Of
n dimensions, 184, 256, ,293, 306, 308, 318,
321, 322, 333, 337, 480; Projective, 276,
285, 292-294, 297, 308, 327, 328; Roman,
65; Shades and shadows, 297; Synthetic,
166, 167, 286-309

Gerard, L., 436

Gerard of Cremona, 132

Gerbert, 115; 116-120

Gerdil, 447

Gergonne, J. D., 288, 289; 166, 287, 290, 310

Gergonne's Annales, 273, 288

Gerhardt, C. I., 208, 210, 212, 214, 215, 217

Gerling, C., 347

Germain, S., 442; 464, 465

Gerson, Levi ben, 128

Gerstner, F. J. v., 467

Ghetaldi, M., 174

Gibbs, J. W., 476, 477; 278, 282, 289, 334,
452, 465; Gibbs phenomena, 465

Gierster, J., 432

Giorgi, G. L. T. C., 395

Girard, A., 156; 148, 158, 202

Giudice, F., 408

Glaisher, J., 440

Glaisher, J. W. L., 153, 155, 341, 381, 382,
441, 444, 448, 482, 484

Glazebrook, R. T., 471, 474

Globular projection, 167

Godfrey, T., 204

Goldbach, C., 236; 249; Theorem, 249, 439

Golden section, 28, 142

Gonella, 486

Gopel, A., 418

Gordan, P., 346; 313, 345, 347, 348, 361,
431, 446; His theorem, 347

Gossard, H. C., 298

Gould, B. A., 383; 451

Gourne'rie, J. de la, 313, 296

Goursat, E. J. B., 385; 279, 325, 360, 372,

386, 413, 428

Gow, J., 10, 22, 53, 55, 59, 60, 62, 129
Grace, J. H., 348
Graffe, C. H., 364; 365, 366
Grammateus, 140
Grand!, G., 238; 250
Graphic statics, 294, 296
Grassmann, H., 289, 407, 408
Grassmann, H. G., 335~337; 289, 306, 322,

332, 338, 407, 408, 455
Graunt, J., 171, 380

Grave", D. A., 454 "V-x

Graves, J., 330
Gravitation, law of, 183, 199, 200, 232,

259, 260, 262
Gravity, center of, 183
Gray, P., 155

Greatest common divisor, 32, 58, 148, 180
Grebe, E. W., 299; 471
Green, G., 472; 281, 342, 419, 422, 460-462,

466, 468, 473
Green, G. M., 322
Greenhill, A. G., 344, 417, 458, 461
Greenwood, S. M., 383
Gregory, D., 201, 217
Gregory, D. F., 273; 330
Gregory, J., 143; 156, 189, 190, 206, 212, 216,

225, 228, 238
Griffith, F. L., 10
Gronwall,T.H.,46s
Grossmann, M., 480
Grote, 24
Grotefend, 4
Groups, 253, 282, 283, 285, 335, 346, 347,

349-366, 388, 438, 480; Continuous, 282,

306, 339, 355. 417, 357; Isometric, 325;

Use of word, 351; Abstract, 352, 353, 360;

Of regular solids, 353; Primitive, 354, 359;

Solvable, 357; Simple, 358, 360; Linear,

358, 359

Grunert, J. A., 320; 248, 336
Griison, J. P., 268
Gua, dej. P., 175, 248
Gubar-numerals, 68, 100
Guccia, G. B., 296
Gudermann, C., 424; 417, 484; Gudennan-

nian, 424

Guerry, A. M., 381

Guichard, C., 315, 392, 321, 322, 325, 426
Guimaraes, R., 142

Guldin, P., 159; 49, 161; His theorem, 159
Gundelfinger, S., 484; 366
Gunter, E., 151

Giinther, S., i, HI, 115, 204, 250, 341, 365
Guthri, F., 323
Giitzlaff, C. E., 417

4Q8

INDEX

Gylden, H., 453; 454

Haan, D. B. de, 372

Haas, A., 321

Habenicht, B., 250

Hachette, J. N. P , 276; 275, 296

Hadamard, J., 372; 324, 340, 375, 395, 402,

427, 439; Quoted, 392
Hadley, J., 204
Hagen, G. H. L., 382
Hahn, H., 372, 400, 406, 429
Hahn, Ph. M., 485
Halifax, J., 127
Hallam, 139
Halley, E., 156; 38, 41, 171, 190, 199, 200,

203, 219, 229, 251, 343, 380, 381, 448
Halphen, G. H., 313; 293, 321, 322, 345, 355,

375, 388, 390, 417
Halsted, G. B., 130, 304, 327, 389, 390, 425;

Quoted, 446
Hamburger, A., 387
Hamburger, M., 383
Hamel, G., 405

Hamilton, W., 172, 173, 273, 278, 331
Hamilton, W. R., 332; 255, 280, 314, 322,

330, 331, 333, 335, 337, 338, 340, 342, 347,

349, 384, 352, 353, 455, 456, 471, 473,

477; Conical refraction, 332, 471; Hamil-

tonian group, 357
Hammond, J., 345, 348
Hancock, H., 371, 372
Hankel, H., 423; 10, 25, 52, 57, 62, 91, oW-

96, 102, 105, 115, 1 20, 129, 135, 141, 220.

273, 337, 341, 367, 376, 400, 4°4, 423\

Principle of permanence, 337; Quoted,

290, 367

Hansen, P. A., 449; 382, 451, 452, 486
Hanus, P. H., 341
Ilann, J., 463, 464
Hardy, A. S., 265
Hardy, C., 164
Hardy, G. H., 439
Harkness, J., 433
Harnack, A., 404
Harmonics, theory of, 40, 46
Harmuth, Th., 366
Harpedonaptae, 10, 25
Harrington, M. W., 455
Harriot, T., 156, 157; 137, 141, 149, 158,

178, 179, 184
Hart, A. S., 298; 291, 318
Hart, H., 301
Hartogs, F. M., 401, 429
Harzer, P., 452
Haskell, M. W., 310
Haskins, C. N., 356
Hatzidakis, N., 321
Hawkes, H. E., 339

Hayashi, T., 78, 82

Hazlett, O.C., 339

Hearn, 155

Heat, theory of, 270, 391, 470-479

Heath, R. S., 308

Heath, T. L., 32, 41, 60, 62, 168, 302

Heaviside, O., 334, 474, 475

Heawood, P. J., 323

Hebrews, 7, 17

Hecke, E., 433

Hecker, J., 349 ,

Hedrick, E. R., 328, 372, 430

Heffter, L., 324, 480

Hegel, 447

Heiberg, J. H., 34, 35, 44

Heine, E., 377; 397, 398, 400, 470

Hellinger, E., 406

Helmholtz, H., 474-477; 459, 460, 461, 463,

464, 471, 477
Henderson, A., 317, 318
Henrici, O., 422, 423
Henry, J., 473
Hensel, K. VV. S., 431, 445
Herigone, P., 205
Hermann, J. M., 486
Hermes, O., 436
Hermite, C., 415, 416; 4, 7, 279, 345, 346,

348, 35°, 375, 388, 391, 412, 413, 418, 420,

432, 433, 444-446, 465, 470
Hermotimus, 28
Hero. See Heron
Herodianic signs, 52
Herodianus, 52
Herodotus, quoted, 9, n
Heron, 43-45; 42, 54, 61, 66, 84, 86, 101, 114,

131

Heron the Younger, 43
Herschel, J. F. W., 464; 126, 272, 405
Hertz, H. R., 281, 474
Hertzian waves, 393
Hess, W., 458
Hesse, L. 0., 311, 312; 313, 320, 341, 346,

35i, 361, 369, 384, 452
Hessel, L. 0., 291
"Hessian," 312, 345
Hettner, G., 425
Heuraet, H. van, 181
Hexagon, 6, 18, 166, 228, 290, 291, 318,

327

Hexagrammum mysticum. See Hexagon
Heywood, H. B., 383
Hicks, W. M., 460, 462
Hieratic writing, n
Hieroglyphics, n
Hilbert, D., 279, 309, 325, 326, 328, 341, 348,

372, 301, 393-395, 4°4, 430, 433, 443, 445,

446; Quoted, 430

INDEX

499

Hildebrandt, T. H., 406

Hill, G. F., 121

Hill, G. W., 450; 304, 341, 451, 453

Hill, J. E., 3ig

Hill, J. M., 384

Hill, Th., 324

Hilprecht, H. V., 7

Hilton, H., 360

Hindenburg, C. F., 373; 272

Hindu- Arabic numerals, 2, 52, 68, 88-90,

98, 100, 101, 107, 120, 121, 128, 147
Hindus, 83-98; 2, 8; Geometry, 83-86
Hipparchus, 43; 5, 45-47, 141, 481
Hippasus, 19
Hippias of Elis, 21
Hippocrates of Chios, 21; 22, 23, 25, 26, 30,

5i, 57, °i
Hippolytos, 59, 91
Hippopede, 42
Hirn, G. A., 476

History of Math's, why studied, 1-3
Hobson, E. W., 22, 144, 268, 446, 470;

Quoted, 398
Hodgkinson, E., 467
Hodograph, 332
Hoernly, R., 85
Holder, 0., 357, 358, 427
Holmboe, B., 414; 374, 411
Holzmann, W., 141
Homography, 293
Homological figures, 166
Hooke, R., 199, 200, 295
Hopital, G. F. A. See Hospital
Hoppe, R., 309; 453
Horn, J., 387
Horner, J., 366
Horner, W. G., 75

Horner's method, 72, 74, 75, 271, 365
Horsburgh, E. M., 329, 482, 485, 486
Hospital, G. F. A. 1', 224; 177, 213, 217,

222, 244

Hoiiel, J., 315; 304, 334, 369, 484

Hsu, 77

Hudde, J., 180; 178, 181, 193

Hudson, R. W. H. T., 3ig

Hugel, Th., 367

Hughes, T. M. P., 74

Humbert, M. G., 317

Hunain ibn Ishak, 101

Huntingdon, E. V., 357, 395, 409

Hunyadi, E., 341

Hurwitz, A., 376, 423, 426, 432, 444, 445,

446

Hurwitz, W. A., 395
Hussey, W. J., 455
Hutchinson, J. I., 319
Hutton, C., 247; 154, 155

Huxley, 344

Huygens, C., 182, 183; 143, 166, 160-171,
173, 179, 188, 190, 199, 200, 205, 206,
217, 221, 225, 230, 244, 470; Cycloidal
pendulum, 166

Hyde, E. W., 250, 337

Hydrodynamics, 223, 240, 242

Hydrostatics, 37

Hypatia, 50; 31

Hyperbola, 185, 188, 206, 247; Equilateral,
1 88. See Conies

Hyperbolic functions, 246, 424, 484

Hyperelliptic functions, 281, 411, 418

Hypergeometric equation, 282

Hypergeometric series, 185, 238, 373, 385,
386

Hyperspace. See Geometry of n dimensions

Hypsicles, 58; 5, 32, 42, 43, 101

lamblichus, 59; 6, 9, 19, 45, in

Ibbetson, W. J., 469

Ibn Albanna, 1 10

Ibn Al-Haitam, 104, 107, 109

Ibn Junos, 109

Ideal numbers, 442, 443

Ikeda, 80

Imaginaries, 225, 226, 229, 236, 237, 275,
279, 284, 288, 292-294, 315, 332, 390, 420,
434

Imaginary roots, 123, 135, 156, 179, 202,
248, 249, 363-365; Graphic representa-
tion of, 184, 237, 264, 265

Imamura Chisho, 79

Impact, 179

Imshenets ki, W. G., 385; 386

Incommensurables, 22, 28, 31, 32, 57, 126;
Indeterminate coefficients, 176, 204

Indeterminate equations, 60, 73, 74, 94-97,
106, 124, 167, 441; Of second degree, 62

Indeterminate form %, 222. 224

Indian notation. See Hindu-Arabic

Indicatrix, 275

Indivisibles, 126, 161, 162, 165, 172, 175,
184

Induction (math'l), 169, 331

Infinite products, 186, 187, 417

Infinite series, 75, 77, 80, 81, 106, 127, 172,
181, 187, 188, 192, 193, 196, 206, 212,
227, 232, 238, 246, 248, 257, 258, 361,
367, 373, 425, 434, 411; Convergence of,
227, 249, 270, 284, 367, 373-375, 417

Infinitely small, 160, 165, 194-198, 207,
210, 218, 220, 237

Infinitesimals, 24, 35, 48, 49, 51, 181, 189,

195, 196, 198, 210, 257, 258, 399
Infinitesimal calculus. See Differential

calculus
Infinity, 23, 24, 66, 126, 160, 166, 177, 184,

Soo

INDEX

185,- 219, 237, 241, 243, 257, 283, 287,

367, 402, 446, 447

Ingold, L., 328, 333

Ingram, J. R., 292

Insurance, 171, 223

Integral calculus, 3, 79, 81, 161, 209, 210,
222, 270, 393, 398, 367

Integral equations, 392-394, 405, 406, 413

Integrals, 36, 316, 376, 386, 388, 424;
Algebraic, 283; Definite, 189, 237, 263,
369, 385, 414, 416, 421, 466; Double, 420;
Elliptic, 239, 258, 266, 267, 414; Eulerian,
267, 272; Hyperelliptic, 413, 415; Lebes-
gue, 406, 407; Multiple, 284, 392; Pier-
pont, 406; Radon, 406; Riemann, 406, 407

Integraphs, 486

Integrations ante-dating the calculus, 189

Integra-differential equations, 392, 395,
405

International commission on teaching, 356

International congresses, 280

Interpolation, 186, 187, 192

Invariants, 282, 312, 316, 319, 321, 342-
348, 349, 351, 355, 356, 388, 417, 444;
Name 345

Inverse method of tangents, 180, 207-209

Inversion, Hindu method of, 92

Inversion (in geometry), 292

Involute, 183

Involution of points, 50, 166

Ionic school, 15-17

Irrationals, 2, 19, 22, 32, 43, 57, 61, 86, 93,
94, 103, 133, 140, 330, 396-400, 483;
First use of word, 68

Irrational roots. See Roots

Ishak ibn Hunain, 101

Isidorus, 51, 113, 121

Isochronous curve, 217, 221

Isomura Kittoku, 79

Isoperimetrical figures, 42, 221, 222, 234,

- 251

Isothermic surfaces, 314

Itelson, G., 286

Ivory, J., 273; 469; His theorem, 273

Jabir ibn Aflah, 109; 119

Jacob!, C. G. J., 414, 415; 266, 278, 289, 290,
311, 321, 332, 34°, 341, 351, 362, 365, 369-
37i, 377, 384, 385, 388, 4"-4i3, 417, 4i8,
421, 424, 425, 429, 436, 437, 441, 448,
455, 458, 469, 470; Jacobian, 345; Theory
of ultimate multiplier, 456

Jacobi, K. F. A., 298; 299

Jacobo de Billy, 158

Jacobs, J., 380

Jahnke, E., 335

Jahrbuch fiber die Fortschritte der Malhe-
malik, 278

Janni, G., 341

Japanese, 78-82

Jastremski, 381

Jeans, J. H., 450

Jellett, J. H., 370; 457, 468

Jerard of Cremona, 119; 123

Jerrard, G. B., 349

Jevons, W. S., 378; 281, 407

Joachim, G., See Rhaeticus

Joachimsthal, F., 424

Jochmann, E., 478

Johanson, A. M., 433

John of Palermo, 124

John of Seville, 118; 119, 147

Johnson, W. W., 30*

Joly, C. J., 333

Jones, G. W., 484

Jones, W., 155, 158, 235

Jordan, C., 318, 325, 326, 347, 348, 353, 354,
357-36o, 379, 300, 428; Jordan curve,
325, 326; Jordan's problem, 359

Jordanus Nemorarius, 118

Josephus, F., 79

Josephus problem, 79, 459

Joubert P., 417 •

Joule, J. P., 475; 477-479

Jourdain, P. E. B., 205, 212, 215, 257, 399,
400, 401, 402; Quoted, 407, 408, 409

Journal dcs Savans founded, 209

Journal of the Indian Malh. Society, 98

Journals (math'l), 82, 98, 209, 273, 278, 288,
289, 343, 346, 355, 4", 427

Jupiter's satellites, 252

Jurgensen, C., 414

Jurin, J., 219; 369

Kant, I., 261, 449

Karpinski, L. C., 68, 88, 89, 102, 103, 121

Karsten, W. J. G., 237; 265

Kasner, E., 318, 320, 322; Quoted, 324

Kastner, A. G., 434, 435; 204, 248

Kaye, G. R., 84-87, 91, 94, 96-98

Keill, J., 215; 216, 218

Kelland, P., 461; 474

Kelvin, Lord (Sir William Thomson), 472,
473; 271, 272, 279, 292, 323, 329, 334, 422,
457, 460, 461, 463, 466, 469-471, 475, 476,
479, 486; Thomson's principle, 284, 422,
428-430, 433, 473; Tide-calculating ma-
chine, 329; Vortex atoms, 323

Kempe, A. B., 286, 302, 323, 324, 409

Kempner, A. J., 444, 446

Kepler, J., 150-161; 131, 145, 146, 148, 154,
163, 170, 178, 184, 192, 199

Kepler's problem, 252

Ketteler, C., 471

Keyser, C. J., 66, 174, 285

Khayyam, Omar, 103, 108

INDEX

Killing, W., 308, 347, 358

Kimura, S., 335

Kinckhuysen's algebra, 193

Kirchhoff, G. R., 474; 281, 311, 457, 458,

461, 462, 466, 471

Kirkman, T. P., 323; 351; K. point, 290
Klein, F., 356; 177, 278, 293, 306-309, 311,

314, 318, 319, 328, 346, 347, 354. 355, 359,

360, 388, 390, 391, 417, 418, 431-433, 442,

457. 458, 480; Ikosaeder, 347, 359, 4*7
Klttgel, G. S., 234
Knapp, G. F., 381
Kneser, A., 327, 371, 372, 387, 393
Knoblauch, J., 425
Knots, theory of, 323
Knott, C. G., 333
Kobel, 122

Koch, N. F. Helge v., 326, 341, 439
Kochansky, A. A., 367
Koebe, P., 433
Kohn, G., 375, 293, 294
Konig, J., 401; 402, 409
Konigs, G. P. X., 302, 315; Quoted, 294
Konigsberger, L., 413, 417, 418, 348
Kopcke, H. A., 461
Korkine, A. N., 444; 384
Korn, A., 469
Korndorfer, G. H. L., 314
Kortum, L. H., 292
Kossak, H., 397
Kotjelnikofl, A. P., 335
Kotter, E., 287, 288
Kotter, F., 425
Kotteritzsch, T., 341

Kovalevski, Madame, 456; 415, 455, 456
Kowalewski, G., 324, 341, 393
Kramp, C., 341
Krause, K. C. F., 324
Krause, M., 418
Krazer, A., 418
Kries, J. v., 378, 379, 381
Kroman, 378 %

Kronecker, L., 362; 340, 342, 346, 347, 348,

350, 353, 354, 359, 398, 423, 424, 431, 436,

444; Quoted, 362
Kronig, A. K., 477, 478
Kuhn, H. W., 360
Kuhn, J., 205
Kiihnen, F.. 318
Kummer, E. E., 442-445; 168, 314, 319, 362,

375, 385, 424, 436, 438; K. surface, 314,

318, 319, 328, 418
Kurushima Gita, 81
Kustermann, W. W., 376
Lacroix, S. F., 324; 255, 272, 274, 336, 383,

419
Lagny, T. F. de, 203; 143

Lagrange, J., 250-259; 3, 62, 164, 169, 172,
100, 191, 203, 219, 227, 229-232, 234, 237,
240-243, 246, 247, 263, 264, 266, 267,
276, 292, 295, 306, 311, 314, 320, 336,
342, 349, 351-353, 356, 357, 365, 368,
369, 371, 372, 382-384, 405, 407, 426, 434,
435, 449, 452, t55-458, 460, 461, 464-467,
477, 480; Calcul des fonciions, 256;
Generalized coordinates, 255; Theorem on
groups, 253; Mecanique analytique, 252,
257; Method of derivatives, 258; Resolu-
tion dcs equations num., 227, 253, 256;
Theorie des fonctions, 256-258

Laguerre, E., 361; 248, 293, 308, 313, 375

Lahire, P. de., 166; 141, 167, 170, 222, 273,
288

Laisant, C. A., 334

Lalande, F. de, 250

Lalanne, L., 481

Lalesco, T., 393

Lallemand, E. A., 482

La Louveie, A., 165

Lamb, H., 26, 455, 461, 462; Quoted, 284,

474

Lambert, J. H., 245-247; 152, 153, 184, 287,
302, 305, 314, 407

Lambert, P. A., 365

Lam6, G., 467-470; 297, 388, 469, 470;
L. equation, 416; L. functions, 467

Lampe, E., 345; 278

Landau, E., 22, 439

Landen, J., 247; 257

Landry, F., 446

Laplace, P. S., 259-264; 164, 191, 223, 230-
232, 245, 252, 254, 256, 258, 266, 271, 273,
302, 336, 368, 369, 374, 377, 379, 380, 382,
384, 386, 405, 408, 420, 434, 438, 447-449,
451, 452, 457, 450, 462, 464, 466, 469, 470,
472, 473, 475; Great inequality, 261; L.
coefficients, 263; L. equation, 264; L.
theorem, 264; Laws of Laplace, 261;
Mcchanique Celeste, 261-263, 332, 338,
374; Systcme du monde, 261; Throrie
analytique des probabililes, 260, 262, 378

Laqui6re, E. M., 366

Larmor, J., 332

Lasswitz, K., 49

Last Theorem of Fermat, 106, 168, 239,
254, 442, 443

Latham, M., 48

Latin square, 239, 367

Laurent, P. M. H., 470; 420

Lauricella, G., 469

Lavoisier, A. L., 256

Lawrence, F. W. P., 446

Laws of motion, 171, 179, TQQ

Least action, 240, 255, 284, 429, 477

502

INDEX

Least resistance (solid of), 201, 234, 284

Least squares, 263, 266, 267, 273, 382, 434

Lebesgue, V. A., 405; 341, 360, 401, 402,
406, 421, 436

Lebon, E., 389

Legendre, A. M., 266-268; 231, 232, 246,
247, 256, 263, 293, 303, 306, 349, 351, 370,
371, 382, 383, 385, 405, 411-413, 4i7, 435,
438, 439, 442, 466, 469, 473, 482; Geome-
Irie, 268, 302; L. coefficients, 414; Num-
•ber theory, 170, 239

Lehmann-Filh6's, R., 383, 453

Lehmer, D. N., 440, 446

Leibniz, G. W., 205-219; 3, 51, 80, 146, 158,
161, 165, 169, 173, 175, 179, 182, 183, 188,
191, 193, 196-198, 220, 222, 224, 226,
236-239, 246, 248, 257, 323, 369, 373, 407,
408, 410, 485; De arte combinatoria, 205;
Notation of calculus, 207, 208, 210;
Other notations, 211, 157; Controversy
with Newton, 212-218

Leitzmann, H., 340

Lemniscate, 188, 221, 245

Lemoine, E., 209; 300, 379; L. point, 299;
L. circle, 300

Lemonnier, P. C., 256

Leodamas, 28

Leon, 28; 30

Leonardo de Vinci, 128

Leonardo of Pisa, 120-125; 13, 104, no,
120-125, I27, I28, 141

Leonelli, G. Z., 484; 155, 366

Lerch, M., 428

Leslie, J., 218, 145

Leuschner, A. O., 452

Le Vavasseur. See Vavasseur, Le

Lever, 37

Leverrier, U. J. J., 449; 332, 451

Levi ben Gerson, 128

Levi-Civita, T., 333, 356, 433, 453, 454

Levy, M., 469; 296

Lewis, C. I., 408

Lewis, G. N., 335, 481

Lewis, T. C., 460

Lexis, W., 381; 380; Dispersion theory, 381

Lezius, J., 98

L'Hospital. See Hospital

Li Ch' unftag, 71

Lie, S., 354; 279, 306, 307, 319, 320, 321,
346, 347, 354, 355, 356, 357, 384, 388, 414;
Quoted, 355

Light, corpuscular theory of, 204; Wave
theory of, 183

Ligowski, W., 484

Liguine, V., 302

Lilavati, 85, 87, 92

Lilienthal, R. v., 321

Limits, 24, 182, 184, 198, 220, 243, 257, 283,
367, 369, 373, 396, 398, 399, 421, 446

Lindeberg, J. W., 372

Lindelof, L. L., 370

Lindemann, F., 308, 362, 419, 446

Linear transformations, 282, 289, 340, 342

Ling, G. H., 357, 358

Linkages, 300-302, 344

Linteria, 221

Liouville, J., 440; 292, 321, 340, 351, 352,
364, 388, 393, 413, 418, 420, 436, 441,
446, 456, 473

Lipkin, 301

Lipschitz, R., 308; 376, 431, 449, 461 .

Listing, J. B., 323; 359

Little, C. N., 323

Littlewood, J. E., 439

Lituus, 226

Liu Hui, 71, 73

Lloyd, H., 471

Lobachevski, N. I., 303; 278, 304, 306, 307,
425

Local probability, 244

Local value, principle of, 5, 69, 78, 88, 94,
100, 102

Loewy, A., 360

Logarithmic curve, 156, 183, 236

Logarithmic series, 188

Logarithmic spiral, 156, 221

Logarithms, 140^ 149-156, 189, 235; Com-
mon, 151; Computation of, 153-156, 188;
radix method, 153, 155; Characteristic,
152; L. of cross-ratio, 293; L. of imagina-
ries; 225, 235-237, 243, 330, "Gaussian,"
484, Natural 1, 150, 152, 153; 247; Man-
tissa, 152; In China, 77; Logarithmic
curve, 156, 183, 236; Logarithmic spiral,
156, 221; Logarithmic tables, 482, 483

Logic, 22, 205, 246

Lommel, E., 449, 471

London, F., 292

Long, J., 155 -

Longomontanus, C., 169

Loomis, E., 463

Lorentz, H. A., 479, 481

Lorenz, J. F., 302

Lorenz, L., 471

Loria, G., 7, 22, 42, 156, 176, 177, 183, 245,
250, 275, 293, 301, 308, 320

Lottner, E., 458

Lotze, R. H., 309

Loud, F. H., 294

Love, A. E. H., 470

Lovett, E. O.; Quoted, 452, 453

Loxodromic curve, 142

Loxodromic spiral, 221

Lucas, E., 446; 366

INDEX

503

Lucas dc Burgo, 125, 187

Lucretius, T., 66

Ludlam, W., 302

Ludolph Van Ceulen. See Van Ceulen

Lunar theory. See Moon

Lune, squaring of, 22, 57

Lupton, S., 155

Liiroth, J., 422; 377

Mac CuLagh, J., 312; 461, 471

Mac Coll, H., 408; 379, 407

Macdonald, W. R., 150

Macfarlane, A., 335; 323, 334, 407, 441

Mach E., 37; 219

Machin, J., 206; 227

Mackay, J. S., 299; M. circle, 300

Maclaurin, C., 228, 229; 177, 202, 220, 226,

244, 267, 273, 277, 293, 369, 377; M

theorem, 226, 228, 365
Mac Mahon, P. A., 240, 343, 345, 348, 366,

430

Mac Millan, W. D., 433, 453
Magic circles, 76, 77, 79, 80
Magic cubes, 79, 81
Magic squares, 76, 77, 79, 80, 92, 93, 104

128, 141, 145, 167, 366, 367
Magic wheels, 79
Magnus, H. G., 459
Magnus, L. I., 295
Mahavira, 85; 86, 88, 96, 97
Maillet, E., 354; 357, 359, 366, 442, 444,

446

Main, R., 455
Mainardi, G., 370
M' Laren, J., 366
Malebranche, N., 222
Malfatti, G. F., 291; 349
Malfatti's problem, 81, 291, 314
Mangoldt, H. von, 314, 439
Mannheim, A., 324
Manning, Th., 155
Manning, W. A., 360
Mannoury, G., 403
Mansion, P., 384
Map coloring, 323, 324
Map construction, 48, 167, 295, 314
Marchi, L. de, 464
Marcolongo, R., 335, 469
Margetts, 481
Margules, M., 464
Marie, Abb6, 266; 252
Marie, M., 294; 43, 162
Mascheroni, L., 268; 47, 269
Maschke, H., 359; 318, 333, 347, 356
Maslama al-Majrltl, 104
Mason, M., 372, 391
Massau, J., 482
Mathematical induction, 142

Mathematical periodicals. See Journals
Mathematical physics, 392-394
Mathematical seminar, 424
Mathematical societies. See Societies
Mathematical Tables, 482-484
Mathematics, definition of, 285, 286
Mathews, G. B., 395, 414, 445
Mathieu, E., 353; 417, 358, 469, 475
Mathieu, P., 470

Matrices, 335, 337, 338, 339, 340, 344, 408
Matthiessen, L., 107, no, 253
Maudith, J., 128, 132
Maupertius, P., 244; 240, 477
Maurolycus, 141; 142, 145
Maxima and minima, 32, 40, 81, 142, 160,
163, 164, 180, 103, 196, 210, 229, 267,
291, 394, 370, 376, 384
Maxwell, C., 474-479; 271, 279, 281, 296,
322, 333, 334, 337, 451, 458, 460, 461, 468,
471, 472, 477, 479, 486
Maya, 69, 70
Mayer, A., 425; 353
Mayer, M., '363
Mayer, R., 475; 449
McClintock, E., 365; 360
McColl, H. See MacColl, H.
McCowan, J., 462
Mean Value, theorem of, 420
Measurement, 41; In Projective Geometry,
294; In theory of irrationals, 397; Of
areas, 455

Mechanics, 19, 37, 171, 172, 179, 183, 211,
229, 231, 240, 242, 255, 260, 276, 288, 296,
307, 308, 338, 384, 391, 447-464, 477, 481;
Theory of top, 458; Fluid motion, 460-
464; Least action, 240, 255 See Statics,
Dynamics
Mehler, G., 470

Mehmke, R., 266, 366, 483, 484
Mei Ku-ch'6ng, 77
Meissel, E., 416
Meissner, W., 443
Melanchthon, 140
Menaechmus, 27; 29, 39, 40, 107
Mendizdbel-Tamborrel, M. J. de, 483
Menelaus, 46, 47; 119; Lemma, 46
Mengoli, P., 173
Mensuration in Euclid, 33; in Boethius, 67;

in China, 71

Me*ray, C., 397; 400, 426
Mercator, G., 189, 295, 314
Mercator, N., 156; 188, 206
Meridian measured, 200, 244; Zero merid-
ian, 259
Mersenne, P., 156, 163, 168, 174, 177, i8r,

183, i'88; M. numbers, 167
Mertens, F., 373, 438, 439, 446

INDEX

Meteorology, 462-464
Method of exhaustion. See Exhaustion
Method of fluxions, 194, 196. See Fluxions
Method of tangents, 51, 163, 164, 177, 180,

189, 193, 207, 209, 212; Inverse method

of, 1 80, 207-209
Metius, A., 73

Metric system, 256, 259, 265, 266
Meusnier, J. B. M., 320
Meyer, A., 384
Meyer, G. F., 372
Meyer, O. E., 461; 469, 478
Meyer, R., 190
Meyer, W. F., 280, 300, 346
Meziriac. See Bachet de Meziriac
Michell, J., 230

Michelson, A. A., 465, 472, 479
Mikami, Y., 71, 78-81, 88
Mill, J. S., 379; 173
Miller, G. A., 82, 279, 341, 350, 351-355,

357-361
Milner, I., 248
Minchin, G. M., 457
Minding, F. A., 414; 321
Minimal surfaces. See Surfaces, minimal
Minkowski, H., 480; 335, 371, 404, 441, 445,

481

Miran Chelebi, no
Mirimanoff, D., 442, 443
Mises, R. M. E., 391
Mitchell, H. H., 360

Mittag-Leffler, G. M., 427; 279, 388, 426
Mitzsrherling, A., 437
Miyai Vnta', 81
Mobius, A. F., 289; 287, 297, 310, 323, 336,

337, 4°8, 423, 437, 449, 455
Models (geometric), 328, 329
Modular equations, 352, 416, 417
Modulus, first use of term, 265
Mohr, O., 296
Moigno, F., 370; 241, 364
Moivre, A. de. See De Moivre
Molenbroek, P., 335
Molien, Th., 339
Molk, J., 280
Moller, M., 464
Mollemp, P. J., 327
Mollweide, K. B., 235; 437
Moments of fluxions, 194
Moments of quantities, 195, 196
Momentum, 172
Monge, G., 274, 275; 41, 231, 232, 246, 266,

270, 276, 286, 287, 200, 296, 309, 315, 319,

320, 322, 371, 384, 385, 386
Montmort, P. R. de, 224; 222, 230, 383
Montucla, J. F., 250; 162, 185
Moon, theory of, 105, 204, 240, 245, 253,

260, 262, 449, 450, 451, 453, 462; Libra-

tion of, 252; Variation of, 106
Moore, C. L. E., 322
Moore, E. H., 318, 325, 357, 358, 394, 395,

404, 405; Quoted, 403
Moore, H. L., 380
Moore, R. L., 325, 328
Morera, G., 428
Mori Kambei Shigeyoshi, 78
Morin, 486

Moritz, R., 152, 345, 446, 447
Morley, E. W., 479
Morley, F., 319, 320, 433
Morley, S. G., 69
Mortality, 171
Moschopulus, M., 128
Moser, L., 381

Motion, laws of, 171, 179, 199
Moulton, F. R., 327, 450, 453, 459, 460;

Quoted, 389, 454
Mourraille, J. R., 247, 269, 364
Mouton, G., 206; 215
Muir, Th., 340, 341, 484
Miiller, F., 279

M tiller, J. See Regiomontanus
Miiller, J. H., 485
Miiller, R., 301
Multiple points, 295
Mu Niko, 77
Muramatsu, 79
Musa Sakir, 104
Musical proportion, 6
Mydorge, C., 166; 164
Nachreiner, V., 341
Nagelsbach, H., 341, 365
Napier, J., 149-155; 127, 145, 146, 148;

Analogies, 152; Rule of circular parts, 152
Napier, M., 147

Napoleon I, 31, 268, 270, 275, 276
Nascent quantities, 192
Naslr-Eddin, 108
Nau, M. F., 89

Navier, M. H., 465; 379, 383, 460, 461, 468
Nebular hypothesis, 260, 450
Negative numbers, 61, 71, 75, 93, 94, 107,

123, 138, 141, 276, 289
Neikirk, L. I., 360
Neil, W., 181; 188
Nekrasoff, P., 365
Nemorarius, J., 127; 118
Neocleides, 28

Nesselmann, G. H. F., 62, in
Netto, E., 341, 348, 354, 425
Neuberg, J., 300; his circle, 300
Neumann, C., 470; 311, 321, 393, 430, 449,

47i, 472
Neumann, F., 311, 313

INDEX

5°5

Neumann, F. E., 470; 468, 473, 474, 475, 477

Neville, E. H., 305

Newcomb, S., 451; 308, 309, 383

Newman, F. VV., 484

Newton, I., 101-205; 3, 42, 50, 80, 143, 161,
182, 183, 189, 190, 219, 220, 222-^24, 226,
228, 229, 231, 232, 234, 239, 242-245,
251, 257, 160, 262, 264, 273, 287, 303, 332,
342, 361, 366, 369, 373, 458, 464, 481,
Anagram, 213; Approximation to roots,
202, 227, 247, 269, 271, 364; Binomial
theorem, 186; Controversy with Leibniz,
212-218; De analysi per aequationes, 192,
196, 202, 212, 214; Enumeratio linearum
tertii ordinis, 204, 320; Fractional and
negative exponents, 178; Gravitation
(law of), 199; Least resistance (solid of),
201, 234, 284; Method of Fluxions, 193-
106, 202, 203; Notation of fluxions, 212,
^220; Parallelogram (Newton's), 203;
Portsmouth collection of MSS., 200,
204; Principia, 196, 199, 200, 210, 213,
220, 226, 234; Problem of Newton, 201;
Problem of Pappus, 176; Quadrature of
Curves, 196, 197, 2 14; Reflecting telescope,
204; Rule of imaginary roots, 202, 344;
Scholium (Prin. II, 7), 213, 214, 216;
Sextant, 204; Similitude, mechanical,
457; Universal Arithmetic, 201, 235

Newton, J., 152

Nicolai, F., 437

Nicole, F., 224; 227

Nicolo of Brescia. See Tartaglia

Nicomachus, 58; 48, 59, 67

Nicomedes, 42

Nieuwentijt, B., 218

Nievenglowski, B. A., 241

Nine-point-circle, 298

Nippur, library at, 4, 7

Niven, C., 473

Niven, W. D., 470

Noble, C. A., 430

Noether, M., 136-138, 295, 296, 313, 314,
316, 319, 340, 354, 419, 430, 431

pornography, 481, 482

Non-euclidean geometry, 32, 302-309; and
relativity, 481

Nonius. See Nunez

Norlund, N. E., 391

Norwood, R., 158

Notation of Algebra and Analysis: Abridged
in analytics, 310; Arabs, early, 100;
Arabs, late, in; Algebraic forms, 345,
346; Algebraic equations, 247; Calculus,
206, 207, 249, 272; Chinese, 76; Con-
tinued fractions, 239; Determinant-;,
340, 341; "Function of," 234; Greek, 125;

Infinity, 185; Hindu, 93, 125; Logic, 410;
Multiplication, 157, 158; Ratio, 157;
Renaissance, 125-127, 139, 156, 157;
Trigonometry, 158, 234, 272; Vector
analysis, 334; Symbols used by Diophan-
tus, 61, by Oughtred, 157; by Leibniz,
211; Symbols -j- and — , 139, 140; > and
<, 157; nl, 341; identity =E, 342; (), 158;
summation 2, 235; i for V~J, 235; e =
2.718..., 234; =, 140; -T, 140; V, 140;
if = 3.14159.. 158; X, 157; ",157; ^,
157; a3, 178; a*, 178; an, 192

Notation of arithmetic: Fractions, 12, 65;
Decimal fractions, 148; Proportion, 211

Notation of Geometry: Similarity, 211;
Congruence, 211

Notation of Numbers: Babylonian, 4-6;
Egyptian, n; Chinese, 72, 77; Hindu, 88-
oo; Maya, 69, 70; Greek, 52, 53; Roman,
63

Notation, principles of, 4, n

Nother, M. See Noether, M.

Nozawa, 79

Number-fields, 444-446

Number-mysticism, 7, 55, 56, 144, 145. See
Magic squares

Number-systems, 70. See Notation of
numbers

Numbers: Algebraic, 446, 447; Amicable, 56,
104, 109, 239; Complex, 442; Cardinal,
403, 447; Cubic residues, 414, 437; Bi-
quadratic residues, 437; Quadratic res-
idues, 435; Concept of 22; Defective, 56;
Excessive, 56; Heteromecic, 56; Ideal,
442; Negative, 61, 71, 75, 93, 94, 107, 123,
138, 141, 276, 289; Ordinal, 403, 447;
Partition of 239, 344, 367, 444; Perfect,
56, 104, 114, 167; Prime, 58, 167, 169,
239, 249, 254, 344, 438, 439; Pentagonal,
168; Polygonal, 168; Irrational, 2, 19,
22, 32, 43, 57, 61, 86, 93, 94, 103, 133, 140,
33°, 396-400, 483; Transcendental, 143,
440, 446, 447; Transfinite, 426; Trian-
gular, 56, 1 68, 173

Numbers, theory of, 48, 59, 106, 114, 124,
285, 344, 348, 362, 434-446; Euler, 168-
170, 239; Fermat, 167-169; Fields, 444,
445; Lagrange, 254; Legend re, 266, 267;
Law of quadratic red procity, 435; Law
of large numbers, 222, 380. See Last
theorem of Fermat, Magic squares.

Numerals. See Hindu-Arabic numerals,
Notation of numbers

Numerical equations, solution of, 74, 75-
77, no, 202, 203, 227, 247; Chinese, 74,
76, 77, ii i ; Continual fractions, 254;
Dandelin, 364, 365; Grilffe, 364-366;

5o6

INDEX

Homer, 74, 75, 271, 365; Infinite deter-
minants, 365; Infinite series, 227, 365;
Japanese, 80, 81; Leonardo of Pisa, 124;
Newton, 203; Raphson, 203; Recent re-
searches, 363-366. See Algebra, Equa-
tions, Roots

Nunez, P., 142; 143

Obenrauch, F. J., 297

Oberbeck, A., 464

Ocagne. See D'Ocagne

Odhner, W. T., 485

Oenopides, 17; 15

Ohm, G. S., 281

Ohm, M., 329; 330, 424

Ohrtmann, C., 278

Olbers, H. W. M., 452; 435, 437, 447,
448

Oldenburg, H., 178, 187, 212-214, 215

Olivier, T., 296

Omar Khayyam, 103, 107

Oppel, F. W., 235

Oppert, 8

Oppikoffer, J., 486

Oppolzer, T. v., 452; 455

Orchard, 155

Ordinate, 175

Oresme, N., 127; 148

Ongen, 67, 126

Orontius, 116

Orthocenter, 297

OsciUation, center of, 183

Osculating curves, 211

Osgood, W. F., 372, 405, 433

Ostrogradski, M., 369, 371, 456

Otho, V., 73, 132

Otto, V. See Otho, V

Oughtred, W., 157-159; «i, 137, 148, 152,
153, 155, 174, IQ2, 481

Ovals of Descartes, 176

Ovidio, E. d', 308

Ozanam, J., 170, 155

if, approximations to, 7, 10, 35, 71, 73, 77,
70-81, 87, 186, 206, 238, 104, 143, 483,
486; Determination of, 17, 185, 225, 238;
Notation of, 158; Proved irrational, 246,
268; Proved transcendental, 2, 143, 362,
440,446

Parioii, L., 128-130; 125, 133, 141, 144, 146

Pade, H., 375

Padmanabha, 85

Padoa, A., quoted, 410

Pagani, G. M., 330

Painleve, P., 279, 389, 453, 454

Pajot, L. L., 170

Palatine anthology, 59, 60

Paolis, R. de, 308

Papperitz, E., 286

Pappus, 49, 50; 21, 30, 33, 41, 42, 45, 54, 55,
142, 166; Problem of 50

Parabola, 162, 177, 185, 188, 206, 224;
Cubical, 182, 188, 208; Divergent, 204,
244; Semi-cubical, 181; Focus of, 50

Paradoxes, 400, 409. See Zeno

Parallel lines, 166, 302, 303, 327; Defined,
48. See Parallel postulate, Euclid, Non-
euclidean geometry

Parallel motion, 300

Parallel postulate, 32, 48, 108, 184, 302, 303,
305, 308; "proofs" of, 48, 108

Parallelogram of forces, 172

Parent, A., 167

Paris academy of sciences, 168, 182, 246

Parmenides, 24

Parseval, M. A., 376

Partial differential equations, 196, 242, 251,
255. 263, 264, 270, 275, 281, 392, 313,
355, 384-388, 422, 456

Partition of numbers, 239, 344, 367, 444

Pascal, B., 164-166; 76, 142, 146, 147, 163,
162, 167, 180, 183, 184, 187, 190, 206, 207,
246, 272, 273, 287, 311, 485; Pascal line,
290; On chance, 170; Theorem on hexa-
gon, 228, 166, 318, 327; Calculating ma-
chine, 165, 485

Pascal, Ernesto, 318, 319, 340, 371, 486;
Quoted, 371

Pasch, M., 309, 399, 400, 409

Pavanini, G., 453

Pascal, Etienne, 164

Peacock, G., 273; 122, 125, 272, 145, 330,
332; Principle of permanence, 273, 337

Peano, G., 285, 289, 309, 325, 326, 348, 387,
400, 401, 407-409; Formulaire, 408

Pearson, K., 380, 381, 383, 465, 468

Peaucellier, A., 301

Pedal curves, 228

Peirce, B., 338; 278, 285, 286, 332, 383, 451,
457; Linear associative algebra, 338, 339

Peirce, C. S., 407; 31, 285, 309, 337, 338, 339

Peletier, J., 137, 156

Pell, A. J., 396

Pell, J., 169; 206

Pell's equation, 96, 169

Pemberton, H., 219; 192, 199, 220

Pendulum, 183, 205, 266, 460, 478

Pendulum clocks, 183

Pentagram, 19

Pepin, Th., 436

Perfect numbers, 56, 104, 114, 167

Periodicals (mathematical). See Journals

Permutations, 221. See Probability

Pernter,J.M.,463

Perrault, C., 182

Perron, O., 348, 370

INDEX

5«>7

Perseus, 42

Persons, W. M., 380

Perspective, 166, 227. See Projective
geometry, Descriptive geometry

Perturbations of planets, 240 252, 261;
See Astronomy

Pesloiian, C. L de, 412

Peters A., 324

Peters, J., 483

Petersburg problem, 223, 243

Peterson, J., 436; 323

Petrus Hartsingius, 81

Peurbach, G., 127; 131, 132

Pezzo, Del, 307

Pfaff, J. F., 384; 434, 435

Philalethes Cantabrigiensis, 219. See
Jurin, J.

Phihppus of Mende, 29

Philolaus, rg; 25, 55

Philonides, 39

Phragme'n, E., 426, 427, 453

Piazzi, G , 447

Picard, C. E., 433; 279, 281, 315-31?, 35i,
387, 388, 391, 413, 415, 416, 432; Quoted,
258, 264, 353, 429, 447, 454, 477; Inte-
grals, 317

Picard, J., 200

Picone, M., 391

Picquet, H., 341

Piddington, H., 463

Pieri, M., 327; 309, 328, 400, 409

Pierpont, J., 406, 351; Quoted, 425, 430, 431

Pincherle, S , 394, 405

Piola, G., 467

Pitcher, A. D., 395

Pitigianis, F. de, 126

Pitiscus, B., 132; 137, 148

Pizzetti, P., 382

Plana, G. A. A., 449; 465, 473

Planetesimal hypothesis, 450

Planimeters, 486

Planisphere, 48

Planudes, M., 128

Plateau, J., 371; 461

Plato, 25-29; 7, 15, 19, 21, 23, 30, 59; In-
scription at his academy, 2, 26; Quoted,

9, 15 N

Plato Tiburtinus (Plato of Tivoli), 105, 1 18,

123

Platonic figures, 33
Platonic number, 7
Platonic school, 25-29
Playfair, J , 145, 192, 218, 302
Pulcker, J ., 309-312; 278, 288, 297, .500,

,U3, 3M, 335, 354; P- equations, 310; P.

lines, 290
J'hitarch, 15, 16, 34

Pohlke, K., 296

Poignard, 170

Poincare', H., 388-391; 327, 339, 341, 353,
355, 36i, 375, 378, 386, 387, 393, 4Oi, 402,
415, 429, 432, 433, 438, 450, 452, 453,

454, 462, 477, 479
Poinsot, L., 455; 293, 379, 458
Point, 26

Point sets, 325, 326, 394, 395, 404; Denu-
merable, 403; Non-measurable, 403. See
Aggregates

Poisson, S. D , 465-467; 164, 223, 293, 349,
369, 377, 379, 38o, 383, 413, 438, 449,

455, 456, 458, 460, 461, 464, 468, 470,
472, 473

Polar coordinates, 221, 224

Polars, theory of, 167

Pole, W., 383

Polenus, 485

Polya, G., 362

Polyhedra, 240

Poncelet, J. V., 287, 288; 166, 100, 268, 275,

276, 286, 290, 297, 298, 301, 308, 311,

354, 395, 407; P- paradox, 310
Poor, V. C., 335
Porism, 33
Porphyrius, 7, 45
Posidonius, 48

Position, principle of. See Local value
Postulates, 35; complete independence of,

395; Of Euclid, 31; Of geometry, 326-

328. See Parallel postulate
Potential, 256, 263, 264; 284, 314, 389, 393,

394, 419, 422, 472, 477
Potron, M., 360
Pouchet, 481
Powell, B., 260
Power series, 185, 387, 420, 431, 445. See

Series

Poynting, J. H., 474, 475
Prandl, L., 334
Precession of equinoxes, 242
Prestet, J., 248; 170
Preston, T., 476
Prihonski, F., 367
Prime numbers, 58, 167, 239, 249, 254, 344,

438, 439; Fermat, 169; Prime number

theorem, 439

Prime and ultimate ratios, 189, 257
Principia of Newton. See Newton
Principle of duality. See Duality
Principle of position. Sre Local value
Pringsheim, A., 374, 400, 475, 428
Probability, 165, 170, 171, 183, 221-224,

229, 230, 240, 243, 244, 258, 262, 263, 27.,.

,<44. 367, 377-383, 47^; Inverse pro-
bability, 230, 378; Local probability, 230,

5o8

INDEX

378, 379; Moral expectation, 223, 378;

Problem of points, 170, 224, 263; Law of

large numbers, 222, 380
Problems for quickening the mind, 114
Problem of Pappus, 176
Problem of three bodies. See Three bodies
Proclus, 51; 15, 17, 21, 28, 30, 31, 33, 42,

44, 48, 49, 142, 302
Probleme des rencontres, 366
Progression. See Arithmetical; Geometrical
Projection: Stereographic, 48, 167; Ortho-
graphic, 48; Globular, 167
Projective geometry, 276, 285, 202-294,

297, 308, 327, 328
Prony, F. M. de, 300, 301
Prony, G. Riche de, 482
Proportion, 6, 10, 16, 19, 20, 21, 31, 32, 56,

58, 73, 75 ; Euclid's theory, 32; Arith-
metical, 56; Geometrical, 56; Harmonic,

56; Musical, 56
Prym, F., 418
Pseudo-sphere, 305
Ptolemy, 46-48; 5, 7, 45, 87, 96, 101, 102,

105, 109, 127, 129-131, 160, 184, 306, 314;

Almagest, 5, 46, 49, 50, 54, 88, 100, 101,

104, 119, 120; Ptolemaic system, 46;

Tables of chords, 47
Puchta, A., 340
Puiseux, V. A., 420
Pulverizer, 95
Purbach, G., 127; 131, 132
Pyramids of Egypt, 9, 10, 14, 16
Pythagoras, 17-20; 2, 15, 55, 57, 68, 80,

104

Pythagorean school, 17-19
Pythagorean theorem, 2, 18, 30, 86-88, 97;

Nicknames of, 1 29
Pythagoreans, 7, 31, 239
Quadrant, centesimal division of, 259. See

Degree
Quadratic equations, 13, 57, 72, 74, 94;

Hindu method of solving, 94
Quadratic reciprocity, 239, 267
Quadratrix, 21, 28, 49
Quadratura curvarum (Newton's) 196, 197,

198
Quadrature of curves, 181, 184, 192, 206,

207
Quadrature of the circle, i, 2, 17, 74, 79, 133,

143, 169, 181, 182, 185, 212, 236, 246, 446;

Impossibility, i, 2, 143, 362, 440, 446
Quadrivium, 113
Quantity, 285, 396, 308
Quaternions, 307, 323, 330, 332~335, 337,

353, 473; Quaternion- Ass'n, 335
Quercu, a, 143
Querret, J. J., 273

Que'telet, A., 380; 144, 148, 378; Average

man, 380

Qu&elet, L. A. J., 289
Raabe, J. L., 374; 397
Radau, R., 452

Radians, 483; Origin of word, 484
Radius of curvature, 196, 221
Radon, J., 406
Raffy, L., 325
Rahn, J. H., 140; 169
Rallier des Ourmes, 170
Ramanujan, S., 367
Ramus, P., 142
Rangacarya, 85
Rankine, W. J. M., 476; 468
Ranum, A., 322

Raphson, J., 203; 227, 247, 271, 364
Rational, origin of word, 68
Rawlinson, R., 234

Rayleigh, Lord, 464; 448, 461, 462, 465, 474
Reciprocal polars, 288, 296
Reciprocal radii, 392
Recorde, R., 140; 146
Recurring series, 227, 230
Redfield, VV. C., 462
Reductio ad absurdum, 25
Reech, F., 457
Regiomontanus, 131; in, 132, 139, 141,

143, 145, 146, 147
Regnault, H. V., 473, 476
Regula falsa, 12, 13, 91, 93, 103, 137, 366;

Double, 44, 103, 1 10, 123
Regula sex quantitatum, 46, 47, 109
Regular solids, 18, 27, 29, 33, 43, 106, 159,

347

Reid, A., 155
Reid, W., 463
Reiff, R., 238
Reiss, M., 341

Relativity, principle of, 335, 470-481
Resal, H., 455
Residual analysis, 247
Resolvents, 253
Resultant, 249
Revue semestrielle, 278
Reye, T., 294, 464
Reymer, 178
Reynolds, O., 462
Rhaetius, 131, 132; 483
Rhind collection, 9
Riccati, J. F., 224; 223, 247; Riccati's

differential equation, 223, 225
Riccati, V., 247
Ricci, G., 333, 356
Ricci, M., 77

Richard, J. A., 401, 402, 409
Richard of Wallingford, 128

INDEX

5°9

Richardson, L. J., 68

Richelot, F. J., 417; 311, 313, 436

Richmond, H. W., 318, 319

Riemann, G. F. B., 421-423; 258, 279, 284,
306, 307, 313, 314, 316, 324, 342, 375, 376
385, 386, 387, 398, 400, 405, 418, 419, 428,
429, 430, 431, 432, 433, 438, 439, 445, 462;
Riemann's surface, 307, 347, 417, 422,
426, 470, 474; Zeta-function, 439

Riesz, F., 376, 396, 406

Riesz, M., 427

Rietz, H. L., 357, 360

Right triangle, 169. See Triangle

Rithmomachia, 116

Ritter, E., 433

Robb, A. A., 480

Robert of Chester, 119

Roberts S., 301, 302

Roberts W., 314

Roberval, G. P., 162; 42, 146, 163, 164, 165,
176, 177, 178, 180, 181, 183, 190, 191

Robins, B., 219; 220, 369, 458

Roch, G., 431

Rodriques, O., 469

Roe, N.f 152

Rogers, R. A. P., 313

Rohn, K., 319

Rolle, M., 224; 220; Rolle's theorem, 224;
Method of cascades, 224

Roman notation and numerals, 63, 178

Romanus, A., 143; 133, 138, 144

Romer, O., 190

Roots, 80, 106, in, 115, 123, 141, 440;
Chauchy's theorem, 363; Cube root, 71,
74, 123; Equal roots, 180; Every equa-
tion has a root, 349; Imaginary, 123, 135,
156, 179, 202, 248, 249, 363-365; In
Galois theory, 351; Irrational, 43, 61, 103;
Negative roots, 61, 107, 135, 141, 156;
Square root, 54, 71, 94, in; Sturm's
theorem, 363; Upper and lower limits,
180, 202, 225, 269, 361. See Equation,
Irrationals, Negative numbers

Rosenhain, J. G., 418; 414, 417

Ross, R., 383

Rossi, C., 365

Rothe, R., 322

Rothenberg, S., 384

Rouche1, E., 393

Rougier, L., 481

Roulettes, 81, 167, 224

Routh, E. J., 457, 458, 474

Rover, W. H., 459; Quoted, 460

Rowland, H. A., 472; 461, 474, 475

Royal Society of Ixindon, founded, 184

Royec, J., 410

Ru.lio, F., 17, 51

RudolS, Ch., 140

Ruffini, P., 349; 75, 271, 350, 352, 353;

Ruffini's theorem, 350
Riihlmann, R., 477

Rule of false position. See False position
Rule of three, 93, 103
Ruled surfaces, 295

Ruler and compasses. See Constructions
Runge, C., 426, 427
Runkle, J. D., 451
Russell, B., 285, 328, 399, 400, 402, 407,

409

Saccheri, G., 304; 184, 302, 305
Sachse, A., 375
Sacro Bosco, 129
Safford, T. H., 451
Saint-Venant, B. de, 468; 337, 350, 461, 466,

467, 471

Saladini, G., 221
Salmon, G. (1819-1904), 290, 297, 312, 313,

317, 320, 342, 345, 346, 361; Salmon

point, 291
Salvis, A. de, 172
Sand-counter, 54, 78, 90
Sang, E., 482, 486
Sangi pieces, 76, 78, 80
Sarasa, A. A. de, 181, 188
Sarrau, E., 471
Sarrus, P. F., 369; 301, 370
Sarton, G., 389
Sato Seiko, 79
Saturn's rings, 451
Saurin, J., 224; 143
Sauveur, J., 170, 367
Savart, F., 465
Savasorda, A., 121, 123
Sawaguchi, 79
Sawano Chuan, 81
Scaliger, J., 143
Scarpio, U., 357
Scheffers, G. W., 339, 355
Scheffler, H., 367
Schellbach, C. H., 291
Schepp, A., 458; 371, 377
Schering, E., 436; 308, 421
Scherk, H. F.( 340
Scheutz, 485
Schiaparelli, G. V., 382
Schilling, M., 328
Schimmack, R., 405
Schlafli, L., 417; 308, 318, 376, 470
Schlegel, V., 337; 309, 336
Schlesinger, J., 296
Schlesinger, L., 387
Schlick, O., 458
Sclilomilch, O., 449; 365
Schmidt, C., 383

INDEX

Schmidt, E., 341, 393, 394

Schmidt, F., 303

Schmidt, W., 44

Schone, H., 44

Schonemann, P., 348

Schonflies, A., 326, 400, 401

Schottenfels, I. M., 359

Schottky, F., 425, 431

Schreiber, G. 296; 276

Schreiber, H. See Grammateus

Schroder, E., 407; 285, 365, 405, 408

Schroter, H., 466; 314, 417

Schubert, F. T. v., 348

Schubert, H., 293

Schulze, J. K., 247

Schumacher, H. C., 411, 437, 449

Schur, F., 327

Schuster, A., 470

Schiitte, F., 176, 177, 183, 245, 275

Schwarz, H. A., 431-432; 293, 319, 347,
359, 368, 370, 372, 376, 390, 391, 428;
Schwarzian derivative, 432

Schweikart, F. K., 305

Schweins, F., 340

Schweitzer, A. R., 395

Scott, R. F., 341

Scotus, Duns, 126

Sebokht, S., 89

Section, golden, 28

Seeber, L., 444

Seelhoff, P., 167, 445

Segner, J. A., 248; 458

Segre, C., 289, 307, 318, 322, 431

Seguier, J. A. de, 360

Seidel, P. L. v., 377

Seitz, E. B., 379

Seki Kswa, 80; 81

Selling, E., 444

Sellmeyer, W., 471

Serenus, 45

Series, 75, 77, 80, 81, 106, 127, 172, 181, 187,
188, 192, 196, 206, 212, 227, 232, 238, 246,
248, 257, 258, 361, 367, 373, 425, 434, 4";
Alternating, 373; Asymptotic, 375; Con-
ditionally convergent, 374; Convergence
of, 227, 249, 270, 284, 367, 373-375, 417;
Divergent, 375, 454; Hypergeometric,
185, 387, 432; Product of two series, 373,
374; Of reciprocal powers, 238; Power-
series, 185, 387, 420, 431, 445; Trigo-
nometric series, 419, 431; Uniform con-
vergence, 84, 377; Recurrent, 127. See
Arithmetical progression, Geometrical
progression

Seiret, J. A., 314; 319, 352, 384, 385, 436, 456

Serson, 458

Servant, M. G., 325, 375 •

Servois, F., 273, 275, 288

Sets of curves, 405

Sets of lines, 405

Sets of planes, 405

Sets of points, 325, 326, 394, 395, 404

Seven, F., 293, 317, 319

Sexagesimal numbers, 4, 5; 43, 47, 88, TOO,

483; Fractions, 5, 54, 483; Invention of,

5,6

Sextant, 204
Sextus Empiricus, 48
Sextus Julius Africanus, 48
Shades and shadows, 297. See Descriptive

geometry
Shakespeare, 190
Shanks, W., 206
Sharp, A., 206; 24
Sharpe, F. R., 319
Shaw, H. S. Hele, 486
Shaw, J. B., 333, 339, 410
Sheldon, E. W., 372
Shstoku Taishi, 78
Siemens, W., 463
Silberstein, L., 479
Simart, G., 316

Similar polygons, 19, 22, 32, 184
Similitude, mechanical, 457
Simony, O., 323
Simplicius, 51; 22, 23, 48, 184
Simpson, T., 235; 227, 234, 244, 365, 382
Simson, R., 277; 31, 33
Sindhind, 99
Sine function, 104, no; Origin of name,

105, 119

Singhalesian signs, 89
Singular solutions, 211, 224, 227, 239, 245,

254, 255, 264, 383
Sinigallia, L., 395
Sisam, C. H., 322
Slide rule, 158, 159
Slobin, H. L., 446

Sluse, R. F. de, 180; 42, 188, 208, 209
Sluze. See Sluse
Smith, A., 457
Smith, D. E., 7, 68, 71, 78, 86, 88, 89, 116,

I2i, 128, 177, 184, 291, 332
Smith, H. J. S., 441; 342, 416, 438, 442, 444
Smith, R., 226
Smith, St., 292
Snellius, W., 143
Sniadecki, J. B., 258
Snyder, V., 320

Societies (mathematical), 279, 296
Socrates, 25
Sohncke, L. A., 417
Solar system (stability of), 284
Solids, regular, 19, 33, 106, 159

INDEX

Solidus, for writing fractions, 332

Sommer, J., 445

Sommerfeld, A., 458

Sommerville, D. M. Y., 305, 306, 329

Somoff, J. I., 457

Sophists, 20-25

Soroban, 78

Sosigenes, 66

Sound, 240, 251, 460, 464-470; Velocity of,
264

Space of » dimensions. See geometry, «
dimensions

Sparre, Comte de, 459

Specific gravity, 37

Speidell, J., 152; 158

Sperry, E. A., 458

Sphere, ig, 27, 33, 36, 42, 45, 50, 79, 106,
107, 3H

Spherical harmonics, 232, 263, 469

Spherical trigonometry. See Trigonometry

Sphero-circle (imaginary), 293

Spheroid, 36; Attraction of, 200, 267

Spirals of Archimedes, 36, 50; Fermatian,
224; Spherical, 50

Spitzer, S., 369; 365

Spottiswoode, 341; 337; Quoted, 281

Square root. See Root

Squaring the circle. See Quadrature of the
circle

S'ridhara, 85, 94

Stability of solar system, 260, 262

Stackel, P.', 184, 238, 426

"Stade," 24

Stager, H. W., 353

Stahl, H., 308

Star-polygons, 127

Statics, 171, 172, 181, 255, 282, 289; Theory
of couples, 455. See Graphic statics,
Mechanics

Statistics, 377-383; Arithmetic mean, 381,
382; Averages, theory of, 381; Normal
curve, 382; Median, 381; Frequency
curve, 383; Mode, 381; Mortality, 381;
Population, 381; In biology, 381; Stand-
ard deviation, 382

Staudt, von, 294-308; 280, 287, 297, 309,
3io

St. Augustine, 67

Steele, W. J., 457, 459

Stefano, A. B., 461

Steiger, O., 485

Stciner, J., 290-292; 287, 289, 297, 309,
310, 311, 312, 313, 317, 318, 320, 323, 336,
346, 362, 370, 411, 421, 423, 424; Steiner
point, 290; Steiner surface, 319

Stekloff, W. A., 393

Stephanos, C., 289; 348

Stereographic projection, 48

Stern, M. A., 364; 365, 421, 436

Sterneck, R. v., 439

Stevin, S., 147; 127, 137, 148, 171, 178, 187

Stewart, M., 277

Stieltjes, T. J., 375; 406, 446, 470

Stifel, M., 140; 139, 141, 144-146, 149, 183,
187

Stirling, J., 229; 204, 229, 377

St. Laurent, T. de, 273

Stokes, G. G., 460, 461; 281, 284, 332, 377,
461, 462, 465, 466, 468, 471, 475

Stolz, O., 425; 35, 368, 399, 400

Stone, E. J., 382

Story, W., 308, 323, 348

Stouffer, E. B., 322

Strassmaier, P. J. R., 8

Stratton, S. W., 465

Strauch, G. W., 370

Stringham, W. I., 308; 309

Stroh, E., 348

Stromgren, F. E., 453

Strutt, J. W. See Rayleigh, Lord

Struve, G. W., 437

Stubbs, J. W., 292

Stiibner, F. W., 248

Study, E., 289, 293, 308, 335, 339, 348, 356

Sturm, C., 363; 1 66, 269, 273, 310, 344, 420,
456, 457, 472, 473; Sturm's theorem, 363,
364, 366

Sturm, R., 317; 295, 318, 319

St. Venant, B. de, 297

St. Vincent, Gregory, 181; 182, 188, 190,
206

Suan-pan, 52, 76, 78

Substitutions, theory of, 281, 417; Orthog-
onal, 342

Sulvasutras, 84-86

Sumerians, 4

Sundman, K. F., 452, 454

Sun-Tsu, 72; 73, 75, 78

Surfaces, 49, 235, 275, 295, 296, 314-318,
321, 322; Anchor-ring, 323; Confocal,
293; Cubic, 295, 313, 314, 317, 318, 329,
343; Deformation, 321; Isothermic, 314;
Kummer, 418, 328; Minimal, 315, 325,
355, 37', 385, 432; Of negative curvature,
307; Plectoidal. 50; Polar, 307; Pseudo-
spherical, 321; Quintic, 319; Quartic, 329;
Ruled, 295, 319, 320; Third order, 291,
318; Fourth order, 314, 318, 319; Second
degree, 275, 290; Second order, 235, 295,
311; Universal, 296; Wave-surface, 311,

314, 319, 333
Surveying, 44, 66, 77
Surya siddhanta, 84
SUssmilch, J. P., 380

INDEX

Suter, H., 104, 109, 181

Swan-pan. See Suan-pan

Swedenborg, E., 260

Sylow, L., 354; 414, 352, 357, 361; Sylow's

theorem, 354
Sylvester, J. J., 343-349; 202, 249, 278, 282,

285, 297, 312, 313, 317, 323, 324, 333, 334,

340, 341, 342, 3Si, 361, 363, 379, 432, 418,

438, 441, 444, 455, 472; Link-motion, 301;

Partitions, 344; Reciprocants, 344
Symbolic logic, 205, 246
Symmedian point, 209
Symmetric functions, 235, 414
Synthesis, 27
Synthetic geometry, 166, 167, 286-309.

See geometry, Projective geometry
Syrianus, 51
Syzygies, 348

Tabit ibn Korra, 104; 101, no
Tables, mathematical, 482-484
Taber, H., 339, 340
Tait, P. G., 459; 272, 279, 322, 323, 333,

334, 335, 337, 382, 457, 466, 470, 473, 4?6;

Golf-ball, 459
Takebe, 80; 81
Talbot, H. F., 413
Tanaka Kisshin, 79; 81
Tangents, method of. See Method of tan-
gents

Tannery, J., 385; 314, 401, 351, 387, 433
Tannery, P., 401; 7, 24, 32, 43, 45, 46, 53,

60, 96, 177, 400; Quoted, 175
Tartaglia, 133, 134; 139, 141, 142, 146, 158,

170, 183

Taurinus, F. A., 305; 184
Tautochronous curve, 183, 413
Taylor, B., 226; 155, 218, 229, 242, 245, 251;

His theorem, 226, 251, 257, 258, 394, 369,

385, 422

Taylor, H. M., 300; His circle, 300
Tedenat, 273
Teixeira, G., 320
Telescope, 183; Reflecting, 204
Tenzan method, 80
Terquem, O., 228
Teubner, B. G. (firm of), 328
Thales, 15-18
Theaetetus, 28; 30, 31, 57
Theodoras of Cyrene. 57
Theodosius, 44, 45, 104, 118-120
Theon of Alexandria, 50; 31, 42, 43, 45, 54,

Theon of Smyrna, 59; 45, 48, 142
Theorem of Pythagoras. See Pythagorean

theorem

Theory of numbers. See Numbers, theory of
Theudius, 28, 30

Thiele, T. N., 378

Thomae, H., 400

Thomae, J, 416

Thoman, 155

Thomas, A., 127, 182

Thomas, Ch. X., 485

Thomas Aquinas, 126

Thom£, L. W., 387; 390

Thompson, S. P., 271, 334, 473

Thomson, J., 463; 472, 484, 486

Thomson, J. J., 461, 474

Thomson, W. See Kelvin, Lord

Three bodies, problem of, 240, 243, 252, 452-
455; Reduced problem of, 452

Thybaut, A. L., 325

Thymaridas, 59, in

Tichy, A., 481

Tides, 240, 264, 393, 336, 378, 449

Timaeus of Locri, 25

Time a fourth dimension, 306, 480

Timerding, H. E., 480

Tisserand, F. F., 388, 455

Todhunter, I., 370; 170, 171, 223, 230, 243-
245, 33i, 371, 437, 449, 465

Tohoku Mathematical Journal, 82

Tonelli, A., 423

Tonstall, C., 146

Top, theory of, 458

Torricelli, E., 162, 163; 146, 156, 190

Torsor, 335

Tortolini, B., 346

Tractrix, 182, 328

Trajectories, orthogonal, 217, 222, 322

Transcendental numbers, 2, 143, 362, 440,
446, 447

Transfinite numbers, 426

Transformation, birational, 295, 314, 316,
317, 319; Linear, 295; 297

Transon, A., 321

Treviso arithmetic, 128

Triangle, 16, 18, 19, 71, 116, 297-300;
Arithmetical, 76, 183, 187; Right, 10,
49^56, 66, 71, 86, 104, 160, 165, 166;
Similar, 16, 73; Spherical, 46, 48, 50;
Isosceles, 86, 104; Heron's formula for
area, 43, 66, 86, 123

Triangular numbers, 56, 168, 173

Triangulum characteristicum, 189, 207

Trigonometry, 108, 109, 127, 131, 138, 149-
iS7, 169. 222, 226, 229, 234-236, 265,
483, 484; Arabic, 104-106; First use of
word, 132; Greek, 43, 47; Hindu, 83, 96,
97; Notation for trig, functions, 158;
Notation for inverse functions, 223;
Spherical, 47, 76, 97, 105, 109, no, 132,
144, 267, 437, 481; Trigonometric func-
tions, cosecant, 106; Cosine, no, 151;

INDEX

Cotangent, 105, 151; Tangent, 104, 106,
132; Secant, 106, 132; Sine, 06, 99, TOJ,

105, no, 131, 132; Versed sine, 96
Trisection of an angle, 2, 20, 36, 42, 104,

106, 138, 142, 177, 202, 246; Proved im-
possible, 350

Trisectrix, 229

Tropfke, j., 211

Trouton, F. T., 471

Trudi, N., 341

Truel, H. D., 265

Tschebytschew. Sec Chebichev

Tschirnhausen, E. W., 225; 209, 210, 226,
253, 349; His transformation, 225, 349

Tsu Ch' ung-chih, 73

Tucker, R., 299; 414; His circle, 300

Two mean proportionals, 19

Tycho Brahe, 105, 159

Tzitzeica, G., 315

Ubaldo, G., 172

Uleg Beg, 108

Ultimate ratios. See Prime and ultimate r.

Underbill, A. L., 372

Undetermined coefficients. See Indetermi-
nate c.

Unger, F., 128

Uniformization, 433

Universities, math's in, 129

Unknown quantity, symbol for, 75

Vacca, G., 142

Vahlen, T., 327, 446

Vailati, G., 328

Valentin, G., 425

Valentiner, H., 359, 360

Vallee Poussin, C. J. de la, 376, 439

Valson, C. A., 368

Van Ceulen, L., 143

Vandermonde, C. A., 266; 253, 264

Vandiver, H. S., 443

Van Orstrand, C. E., 484

Van Schooten. See Schooten

Van Velzer, C. A., 341

Van Vleck, E. B., 405, 406; Quoted, 396,
405, 422

Varaha, Mihira, 84

Variable parameters, 211

Variables, complex, 420

Variations, calculus of. See Calculus of
variations

Variation of arbitrary constants, 240

Varignon, P., 224; 156, 185, 220, 222

Varicik, V., 481

Vasiliev, A., 425

Vavasseur, R. P. Le, 357, 360

Vcblen, ()., .soy, 324, 326, 328, 409

Vector analysis, 308, 322, 334, 335, 452. 476

Vega, G. F., 156, 482

Velaria, 221

V6ne, A. A., 270

Venn, J., 379; 407, 408

Venturi, 44

Venturi, A., 452

Verbiest, F., 77

Vernier, P., 142

Veronese, G., 291, 307, 309, 327

Vibrating strings, 226, 242, 251, 252, 258,
464

Vicat, L. J., 467:468

Victorius, 65

Vieta, F., 137-139; 41, in, 133, 141, 142,
143, 144, 146, 156, 174, 177, 178, 187, 192,
203, 253

Vigesimal system, 69, 70

Vija-ganita, 85

Villarceau, A. Y., 452; 482

Vincent, A. J. H., 330

Vincent, Gregory St., 181

Vinci, Leonardo da, 273

Virtual velocities, 29, 172, 255

Vitali, G., 405

Vivanti, G., 218, 427

Viviani, G., 409

Viviani, V., 162

Vlack, A., 151; 77, 152, 154, 482

Vogt, H., 29

Vogt, W., 308

Voigt, W., 471, 474

Volpi, R., 405

Volterra, V., 346, 372, 387, 393, 394, 395,

405, 406

Von Staudt, 280, 395, 409, 436
Voss, A., 297, 308, 325, 374, 394; Quoted,

410

Xenocrates, 26
Xy lander, 141
Wachter, F. L., 305
Wada, Nei, 81
Wagner, U., 128
Waldo, F., 463
Wallenberg, G. F., 387
Wallingford, R., 128

Wallis, J., 183-188; 88, 108, 137, 146, 148,
156, 157, 158, 165, 168, 169, 178, 179, 181,
190, 191, 192, 196, 202, 213, 215, 235, 265,
302, 306, 331, 343, 373
Wallner, C. R., 126
Waltershausen, W. S., 232
Walton, J., 219
Walton, W., 324
Wand, Th., 476
Wang Hs' lao-t'ung, 74
Wantzel, P. L., 350; 416
Ward, Seth, 157
Waring, E., 248-249; 143, 202, 241, 248,

INDEX

2S4» 320, 361, 364; Miscellanea analytica,
241; Waring's theorem, 248, 442, 443

Warren, J., 330

Watson, J. C., 455

Watson, S., 379

Watt, J., 300; Watt's curve, 300

Wave theory of light, 183

Weaver, J. H., 50

Weber, H., 361; 318, 353, 357, 390, 400, 418

Weber, W., 6, 421, 434, 467, 474

Weddle, Th., 365; 155, 319; Surface, 319

Weierstrass, K., 423-426; 32, 258, 279, 285;
326, 345, 346, 347, 362, 368, 370, 371, 372,
376, 385, 388, 397, 398, 399, 400, 415,
417, 418, 422, 428, 429, 430, 431, 432,
446, 453, 456; Weierstrass' Construction,
372

Weigel, E., 205

Weiler, A., 384

Weingarten, J., 314; 325

Weir, P., 482

Weissenborn, H., 115

Weldon, W. F. R., 381

Wendt, E., 357.

Werner, J., 141

Werner, H., 347 '

Wertheim, G., 60

Wertheim, W., 468

Wessel C., 265; 420

Westergaard, H., 380, 381

Wetli, 486

Weyl, H., 391, 465, 469

Wheatstone, C., 465

Whewell, W., 324; 37, 160, 240

Whipple, J. W., 485

Whist, 383

Whiston, W., 201

White, H. S., 278, 300; Quoted 3, 250, 395

Whitehead, A. N., 407, 409; Quoted, 294, 328

Whitley, J., 298

Whitney, W. D., 85

Whittaker, E. T., 386

Widmann, J., 139; 125

Wieferkh, A., 443

Wieleitner, H., 127, 174, 182, 235, 250

Wiener, A., 366

Wiener, C., 297; 274, 276, 317, 326

Wiener, H., 289, 329

Wiener, N., 409

Wikzynski, E. J., 322

Williams, K. P., 392

Williams, T., 265

Wilson, E. B., 335, 401, 481; Quoted, 327

Wilson, J., 248; 254; Wilson's theorem, 248,
254

Winckler, A., 469

Wing, V., 157

Wingate, E., 481

Winlock, J., 383

Winter, M., 410

Witt, F.de. See De Witt

Witting, A., 318

Wittstein, A., 291

Wittstein, T., 381

Woepcke, 68, 100

Wolf, C., 158, 175, 226

Wolf, R., 259, 379

WSlffing, E., 324

Wolfram, 247

Wolfshekl, F. P., 443

Wolstenholme, J., 379

Woodhouse, R., 272; 219, 370

Woodward, R. S., 459

Woolhouse, W. S. B., 365, 379

Wren, C., 166; 179, 181, 188, 199, 275

Wright, E., 153, 155, 189

Wright, J. E., 356

Wright, T., 451

Wronski, H., 340; 258; Wronskians, 340

Yan Hui, 75

Yendan method, 80

Yenri, 80, 81

Yoshida Shichibei Kflyu, 78; 79

Young, A., 348

Young, G. C., 325, 326

Young, J. R., 271

Young, J. W., 328

Young, Th., 470; n, 183, 464, 465

Young, W. H., 325, 326, 406

Yii, emperor, 76

Yule, G. V., 381

Zach, 484

Zehfuss, G., 341

Zeller, C., 436

Zeno of Elea, 23; 24, 29, 51; On motion,

48, 67, 126, 182, 219, 400
Zenodarus, 42; 370
Zermelo, E., 372, 401, 402, 403; Principle of,

401
Zero, invention and use of, 119, 121, 147;

2, 5, 116; by Maya, 69; Symbols for, 5,

S3, 69, 75, 78, 88, 89, 94, 100; division by,

94, 284

Zero-denominator, 185
Zero, first use of term, 128
Zerr, G. B. M., 379
Zeuner, G., 381

Zeuthen, H. G., 32, 190, 293, 314, 316, 320
Zeuxippus, 34
Zizek, F., 380
Zollner, F., 309
Zolotarev, G., 442, 444
Zorawski, K., 356
Zyklographic, 2Q7

Printed in the United States of America

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QA Cajori, Florian

A history of mathematics
C15 2d ed., rev. and enl.
1919

Phytkal 3e
Applied So,

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