m*
REESE LIBRARY
OF :
UNIVERSITY OF CALIFORNIA.
C/i
PREFACE.
THE subject-matter of this book is a historical sum
mary of the development of mathematics, illustrated by the
lives and discoveries of those to whom the progress of the
science is mainly due.
The first edition was substantially a transcript of some
lectures which I delivered in the year 1888 with the
object of giving a sketch of the subject that should be
intelligible to any one acquainted with the elements of
mathematics. In this edition I have revised the whole
and have made some changes in detail, but the general
character of the work as a popular account of the leading
facts in the history of mathematics remains unaltered.
The scheme of arrangement will be gathered from the
table of contents at the end of this preface. Shortly it is
as follows. The first chapter contains a brief statement
of what is known concerning the mathematics of the
Egyptians and Phoenicians: this is introductory to the
history of mathematics under Greek influence, but as the
Ionian Greeks were considerably indebted to the Egyptians
VI PREFACE.
and Phoenicians it is convenient to commence with a
concise account of the attainments of the latter. The
subsequent history is divided into three periods: first,
that under Greek influence, chapters II. to VII.; second,
that of the middle ages and renaissance, chapters VIII. to
xiii.; and lastly that of modern times, chapters XIV. to
XIX.
In discussing the mathematics of these periods I have
confined myself to giving the leading events in the history,
and frequently have passed in silence over men or works
whose influence was comparatively unimportant; doubtless
an exaggerated view of the discoveries of those mathe
maticians mentioned may be caused by the non-allusion
to minor writers who preceded and prepared the way for
them, but in all historical sketches this is to some extent
inevitable, and I have done my best to guard against it
by interpolating remarks on the progress of the science at
different times. Perhaps also I should here state that
generally I have omitted all reference to practical astro
nomers unless there was some mathematical interest in
the theories they proposed. In quoting results I have
commonly made use of modern notation ; the reader must
therefore recollect that, while the matter is the same as
that of any writer to whom allusion is made, his proof
is sometimes translated into a more convenient and
familiar language.
I am of opinion that it is undesirable to overload a
popular account with a mass of detailed references.
Usually therefore I have collected in a single footnote for
each school or mathematician references to the chief
PREFACE. Vll
authorities on which I have based my account or with
which I am acquainted, and I have not given the
authority for every particular fact mentioned unless I
regard it as difficult to verify without a definite reference.
I hope that these footnotes will supply the means of
studying in detail the history of mathematics at any
specified period should the reader desire to do so.
The greater part of my account is a compilation
from existing histories or memoirs, as indeed must be
necessarily the case where the works discussed are so
numerous and cover so much ground ; when authorities
disagree I have generally stated only that view which
seems to me to be the most probable, but if the question
be one of importance I believe that I have always indi
cated that there is a difference of opinion about it.
I have struck out the long list of standard histories
which I published in the first edition. Most of the facts and
opinions for the first and second periods into which I have
divided the history are quoted or criticized in the closely
printed pages of M. Cantor s elaborate Vorlesungen ilber
die Geschichte der Mathematik, to which the reader who
desires further information on any particular point would
naturally turn. To that work, to H. HankeFs brilliant
but fragmentary Geschichte der Mathematik, Leipzig, 1874;
and in a less degree to F. Hoefer s Histoire des mathe-
matiques, Paris, third edition, 1886, and to M. Marie s
Histoire des sciences mathematiques et physiques, 12
volumes, Paris, 1883 1888, I am usually indebted when
no specific reference is given : I frequently refer to these
works by the names of the authors only. For the last
viii PREFACE.
two or three centuries the general histories give but little
assistance, and the student must rely mainly on special
monographs.
My thanks are due to; various friends and corre
spondents who have eaUteS my atterition to points in the
first edition. No one who has not been engaged in such
work can realize how difficult it is to settle many a small
detail or how persistently mistakes which have once got
into print are reproduced in every subsequent account.
I shall be grateful for notices of additions or corrections
which may occur to any of my readers.
W. W. ROUSE BALL.
TRINITY COLLEGE, CAMBRIDGE.
April 21, 1893.
IX
TABLE OF CONTENTS.
PAGE
Preface . , . . v
Table of contents . ix
CHAPTER I. EGYPTIAN AND PHOENICIAN MATHEMATICS.
The history of mathematics begins with that of the Ionian Greeks . 1
Greek indebtedness to Egyptians and Phoenicians .... 2
Knowledge of the science of numbers possessed by the Phoenicians*. 2
Knowledge of the science of numbers possessed by the Egyptians . 3
Knowledge of the science of geometry possessed by the Egyptians . 5
Note on ignorance of mathematics shewn by the Chinese . . 9
^ntotr. JWartjcmattcs unter eSmfe Influence.
This period begins ivith the teaching of Thales, circ. 600 B. c. , and ends
with tJie capture of Alexandria by the Mohammedans in or about 641 A.D.
The characteristic feature of this period is the development of geometry.
CHAPTER II. THE IONIAN AND PYTHAGOREAN SCHOOLS.
CIRC. 600 B.C. 400 B.C.
Authorities 13
The Ionian School .......... 14
THALES, 640550 B.C 14
His geometrical discoveries 15
His astronomical teaching . . . . . . .17
Mamercus. Mandryatus. Anaximander, 611 545 B.C. . . 17
B. b
TABLE OF CONTENTS.
The Pythagorean School . . ... ,19
PYTHAGORAS, 569500 B.C ..... . . . .19
The Pythagorean geometry ...... 24
The Pythagorean theory of numbers ..... 27
Epicharmus. Hippasus. Philolaus. Archippus. Lysis . . 29
ARCHYTAS, circ. 400 B.C .......... 29
His solution of the duplication of a cube .... 30
Theodorus. Timaeus. Bryso ....... 31
Other Greek Mathematical Schools in the fifth century B.C. . . 31
(Enopides of Chios. Zeno of Elea. Democritus of Abdera . . 32
CHAPTER III. THE SCHOOLS OF ATHENS AND CYZICUS.
CIRC. 420300 B.C.
Authorities 34
Mathematical teachers at Athens prior to 420 B.C. . . ^ . 35
Anaxagoras. Hippias (The quadratrix). Antipho . . >35
The three problems in which these schools were specially interested aB"
HIPPOCRATES of Chios, circ. 420 B. c 39
Letters used to describe geometrical diagrams . . .39
Introduction in geometry of the method of reduction . 40
The quadrature of certain lunes . . . .. ,\ . 40
The Delian problem of the duplication of the cube . . 42
PlaJto, 429348 B.C . 43"""
Introduction in geometry of the method of analysis . . 44
Theorem on the duplication of the cube . ... . 45
EUDOXUS, 408 355 B.C 45
Theorems on the golden section 46
Invention of the method of exhaustions .... 46
Pupils of Plato and Eudoxus . . / . . . - . . . 47
Mi-iNAECHjuia, circ. 340 B.C . . 48
Discussion of the conic sections 48
His two solutions of the duplication of the cube . . 49
Aristaeus. Theaetetus ... . . . . . 49
Aristotle, 384 322 B.C. . . ; .-.- . . . .49
Questions on mechanics. Letters used to indicate magnitudes . 50
TABLE OF CONTENTS.
CHAPTER IV. THE FIRST ALEXANDRIAN SCHOOL.
CIRC. 30030 B.C.
Authorities . . . .
Foundation of Alexandria .
The third century before Christ _ . . . . .
EUCLID, circ. 330 275 B.C. . . * .
Euclid s Elements ,
The Elements as a text -book of geometry ....
The Elements as a text-book of the theory of numbers
Euclid s other works * . * . .
Aristarchus, circ. 310 250 B.C. * . . .
Method of determining the distance of the sun .
Conon. Dositheus. Zeuxippus. Nicoteles
AECHIMEDEJ&, 287 212 B.C
His works on plane geometry
His works on geometry of three dimensions
His two papers on arithmetic, and the "cattle problem" .
His works on the statics of solids and fluids
His astronomy -> w .
The principles of geometry assumed by Archimedes .
APOLLONIUS, circ. 260 200 B.C. .......
His conic sections ........
His other works .........
His solution of the duplication of the cube
Contrast between his geometry and that of Archimedes
Eratosthenes, 275194 B.C. (The sieve)
The second century before Christ .......
Hypsicles (Euclid, bk. xiv). Nicomedes (The conchoid) .
cissoid). Perseus. Zenodorus
, circ. 130 B.C. ... ....
Foundation of scientific astronomy and of trigonometry .
of Alexandria, circ. 125 B. c
Foundation of scientific engineering and of land-surveying
Area of a triangle determined in terms of its sides
62
xii TABLE OF CONTENTS.
PAGE
The first century before Christ . . . . . . .92
Theodosius. Dionysodorus . . . ; . . .92
End of the First Alexandrian Sclwol . . ... . . .93
Egypt constituted a Roman province . . . . . . 93
CHAPTER V. THE SECOND ALEXANDRIAN SCHOOL.
30 B.C. 641 A.D.
Authorities 94
The first century after Christ ........ 95
v Serenus. Menelaus. ......... 95
^v Nicomachus 95
Introduction of the arithmetic current in mediaeval Europe 96
The second century after Christ . . . . . v^ m 96
Theon of Smyrna. Thymaridas 96
PTOLEMY, died in 168 97
The Almagest . . . ... . . .97
Ptolemy s geometry ........ 99
The third century after Christ . . . . . . . . 100
Pappus, circ. 280 . . * . . . . . . . f . 100
The Swcrywy^j a synopsis of Greek mathematics . . 100
The fourth century after Christ . . * . . . . 102
Metrodorus. Elementary problems in algebra . 103
Three stages in the development of algebra . . . . . 104
4fcDioPHANTUS, circ. 320 (?) . . . . . . . .* . 105
Introduction of syncopated algebra in his Arithmetic . 106
The notation, methods, and subject-matter of the work . 106
His Porisms . . Ill
Subsequent neglect of his discoveries . . Ill
Theon of Alexandria. Hypatia . . . . .... 112
Hostility of the Eastern Church to Greek science .... 112
The Athenian School (in the fifth century) . . . . . 112
Proclus, 412485. Damascius (Euclid, bk. xv). Eutocius . . 113
TABLE OF CONTENTS. Xlll
PAGE
Roman Mathematics . . ... . . . . 114
Kind and extent of the mathematics read at Eome .... 114
Contrast between the conditions for study at Rome and at Alexandria 115
End of the Second Alexandrian School ...... 116
The capture of Alexandria, and end of the Alexandrian Schools . 116
CHAPTER VI. THE BYZANTINE SCHOOL. 641 1453.
Preservation of works of the great Greek mathematicians . . 118
Hero of Constantinople. Psellus. Planudes. Barlaam . . 119
Argyr.ua,. Nicholas Bhabdas of Smyrna. Pachymeres . . . 120
Moschopulus (Magic squares) . . - 120
Capture of Constantinople, and dispersal of Greek mathematicians 122
CHAPTER "VII. SYSTEMS OF NUMERATION AND PRIMITIVE
ARITHMETIC.
Authorities . . . . . . T^ .... 123
Methods of counting and indicating numbers among primitive races 123
Use of the abacus or swan-pan for practical calculation . . . 125
Methods of representing numbers in writing 128
The Roman and Attic symbols for numbers 129
The Alexandrian (or later Greek) symbols for numbers . . . 129^
Greek arithmetic .......... 130
Adoption of the Arabic system of notation among civilized races . 131
XIV TABLE OF CONTENTS.
^ertotr. Jttat&emattcs of tfje JWt&trte
antr of tfje
This period begins about the sixth century, and may be said to end
with the invention of analytical geometry and of the infinitesimal calculus.
The characteristic feature of this period is the creation of modern arith-
metic, algebra, and trigonometry.
CHAPTER VIII. THE RISE OF LEARNING IN WESTERN EUROPE.
CIRC. 6001200.
PAGE
Authorities 134
Education in the sixth, seventh, and eighth centuries . . . 134
The Monastic Schools . . . . . . . . . . 134
Boethius, circ. 475526 135
Mediaeval text-books in geometry and arithmetic . . 136
Cassiodorus, 480566. Isidorus of Seville, 570636 . . .136
The Cathedral and Conventual Schools ...... 137
The Schools of Charles the Great . . . . . . .137
Alcuin, 735804 . . . . . . . . . . 137
Education in the ninth and tenth centuries . . . . . 139
Gerbert (Sylvester II.), died in 1003. Bernelinus . .- .* ; . 140
The Early Mediaeval Universities . . . . . . . 142
The earliest universities arose during the twelfth century . . 142
The three stages through which the mediaeval universities passed . 143
Footnote on the early history of Paris, Oxford, and Cambridge . 144
Outline of the course of studies in a mediaeval university . . 148
VCHAPTER IX. THE MATHEMATICS OF THE ARABS.
Authorities . . . 150
Extent of mathematics obtained from Greek sources .... 150
The College of Scribes . . . . ... . .151
TABLE OF CONTENTS. XV
PAGE
Extent of mathematics obtained from the (Aryan) Hindoos . . 152
ARYA-BHATA, circ. 530 153
The chapters on algebra & trigonometry of his Aryabhathiya 153
BRAHMAGUPTA, circ. 640 . ., . j . . . . . 154
The chapters on algebra and geometry of his Siddhanta . 154
BHASKARA, circ. 1140 . . . / . .. .. . . .156
The Lilivati or arithmetic ; decimal numeration used . 157
The Bija Gani ta ox Algebra . .. . . . 159
The development of mathematics in Arabia . . . . . 161
ALKARISMI or AL-KHWARIZMI, circ. 830 ....... 162
His Al-gebr we I mukabala .. .. .. . . 163
His solution of a quadratic equation 163
Introduction of Arabic or Indian system of numeration . 164
TABIT IBN KORRA, 836 901 ; solution of a cubic equation . . 164
Alkayami ; solutions of various cubic equations . . . . 165
Alkarki, Development of algebra . -. . . . . . 166
Albategni. Albuzjani or Abul-Wafa. Development of trigonometry 166
Alhazen. Abd-al-gehl. Development of geometry .... 167
Characteristics of the Arabian school . . . . . . 168
CHAPTER X. INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
CIRC. 11501450.
The eleventh century 170
Geber ibn Aphla. Arzachel 170
The twelfth century 170
Adelhard of Bath. Ben-Ezra. Gerard. John Hispalensis . . 170
The thirteenth century 172
LEONARDO OF PISA, circ. 11751230 172
The Liber Abaci, 1202 173
The introduction of the Arabic numerals into commerce . 173
The introduction of the Arabic numerals into science . 173
The mathematical tournament ...... 174
Frederick II., 1194 1250 175
xvi TABLE OF CONTENTS.
PAGE
JORDANUS, circ. 1220 176
His geometry and algorism . . . ; . . . . 177
His De Numeris Datis, a syncopated algebra . . .177
Holywood. . 179
EOGER BACON, 12141294 . . . . . . . .180
Campanus ........... .182
The fourteenth century ......... 183
Bradwardine. Oresmus 183
The reform of the university curriculum 184
The fifteenth century 185
Beldomandi ........... 186
CHAPTER XI. THE DEVELOPMENT OF ARITHMETIC.
CIRC. 13001637.
Authorities . . - 187
The Boethian arithmetic 187
Algorism or modern arithmetic 188
The Arabic (or Indian) symbols : history of . . . . . 189
Introduction into Europe by science, commerce, and calendars . 191
Improvements introduced in algoristic arithmetic .... 193
(i) Simplification of the fundamental processes . . . 193
(ii) Introduction of signs for addition and subtraction, circ. 1489 200
(iii) Invention of logarithms, 1614 200
(iv) Use of decimals, 1619 202
CHAPTER XII. THE MATHEMATICS OF THE RENAISSANCE.
CIRC. 14501637.
Authorities . . . . . . . . . . 203
Effect of invention of printing. The renaissance .... 203
The development of syncopated algebra and trigonometry . . . 205
BEGIOMONTANUS, 14361476 . ... \ . . . .205
His De Triangulis (not printed till 1496) .... 206
Purbach, 1423 1461. xCusa, 1401 1464 . , . . . 209
TABLE OF CONTENTS. XV11
PAGE
Chuquet, circ. 1484 . . . . . . . . ; .210
Introduction of symbols + and - into German algorism v. . 210
Widman, circ. 1489. . . . 210
Pacioli or Lucas di Burgo, circ. 1500 . . ... 212
His arithmetic and geometry, 1494 . . . . . 213
Leonardo da Vinci, 1452 1519 . . . . . . . 216
Diirer, 14711528. Copernicus, 14731543 . ... .217
Eecord, 1510 1588 ; introduction of symbol for equality . . 218
Eudolff, circ. 1525. Eiese, 14891559 218
Stifel, 1486 1567. His Arithmetica Integra . .... 219
TARTAGLIA, 15001559 ~T "V . . 220
His solution of a cubic equation, 1535 .... 221
His arithmetic, 155660 ... A ... 222
CARDAN, 15011576 . . . . . . . .224
His Ars magna (1545), the third work printed on algebra . 226
His solution of a cubic equation . . . . . 228
Ferrari, 1522 1565 ; solution of a biquadratic equation . . 228
Eheticus, 15141576. Maurolycus, 14941575 .... 229
Borrel. Xylander. Cornmandino. Peletier. Eomanus. Pitiscus 230
Eamus, 15151572 230
Bombelli, circ. 1570 . . . * , 231
The development of symbolic algebra 232
VIETA, 15401603 233
Introduction of symbolic algebra, 1591 .... 234
Vieta s other works ........ 236
Girard, 1590 1633. Development of trigonometry and algebra . 238
NAPIER, 15501617. Introduction of logarithms, 1614 . . .239
Briggs, 15561631. Calculations of tables of logarithms . . 240
HARRIOT, 1560 1621. Development of analysis in algebra . . 241
Oughtred, 15741660 241
The origin of the more common symbols in algebra .... 243
CHAPTER XIII. THE CLOSE OF THE EENAISSANCE.
CIRC; 15861637.
The development of mechanics and experimental methods . . . 247
STEVINUS, 15481603 247
Commencement of the modern treatment of statics, 1586 . 248
XV111 TABLE OF CONTENTS.
PAGE
GAIOLEO, 1564 1642 . . . . ... . .249
Commencement of the science of dynamics . . . 250
Galileo s astronomy ........ 251
Francj.^ Bacon, 15611626 253
Guldinus, 15771643 254
Wright, 1560 1615. Construction of scientific maps . . . 255
Snell, 1591 1626. Discovery of law of refraction in optics . . 256
Revival of interest in pure geometry ....... 256
KEPLER, 15711630 256
His Paralipomena, 1604 ; principle of continuity . . 258
His Stereometria, 1615 ; use of infinitesimals . . . 258
Kepler s laws of planetary motion, 1609 and 1619 . . 258
, 15931662 259
His Brouillon project ; use of pro jective geometry . . 259
Mathematical knowledge at the close of the renaissance . . . 261
This period begins with the invention of analytical geometry and the
infinitesimal calculus. The mathematics is far more complex than that
produced in either of the preceding periods ; but it may be generally de
scribed as characterized by the development of analysis^ and its application
to the phenomena of nature.
CHAPTER XIV. FEATURES OF MODERN MATHEMATICS.
Invention of analytical geometry and the method of indivisibles . 265
Invention of the calculus . . . . . . . . 265
Development of mechanics ........ 266
Application of mathematics to physics . . . . 267
Recent development of pure mathematics . . . . 268
TABLE OF CONTENTS. XIX
CHAPTER XV. HISTORY OF MATHEMATICS FROM DESCARTES
TO HUYGENS. CIRC. 16351675.
PAGE
1596 1650 . . . . . . . . . 270
His views on philosophy . . . . * . 273
His invention of analytical geometry, 1637. . . . 273
His algebra, optics, and theory of vortices .... 277
Cavalieri, 15981647 . . . 279
The method of indivisibles . . ^ 280
PASCAL. 1623 1662 . . . . 282
His geometrical conies ~. 284
The arithmetical triangle . 285
Foundation of the theory of probabilities, 1654 . . . 286
His discussion of the cycloid . . . . 288
WALLIS, 1616 1703 . .. .... . . . . . .288
The Arithmetica Infinitorum, 1656 . . . . . 289
Law of indices in algebra . . . . . - . . . 290
Use of series in quadratures - . \ .... 290
Earliest rectification of curves, 1657 291
Wallis s analytical conies, algebra, and other works . . 292
FERMAT, 16011665 293
His investigations on the theory of numbers . . . 295
His use in geometry of analysis and of infinitesimals . 299
Foundation of the theory of probabilities, 1654 . . . 300
HUTGENS, 16291695 302
The Horologium Oscillator ium, 1673 303
The undulatory theory of light 304
Other mathematicians of this time ....... 306
Bachet de M6ziriae . ... 306
Mydorge. Mersenne ; theorem on primes and perfect numbers . 307
De Beaune. Koberval. Van Schooten 308
Saint-Vincent. Torricelli. Hudde . . . . u . . ... . .309
Frenicle. Laloubere. Kinckhuysen. Courcier. Eicci. Mercator 310
Barrow; the tangent to a curve determined by the angular coefficient 311
Brouncker 314
James Gregory; distinction between convergent and divergent series. 315
Sir Christopher Wren 315
Hooke 316
Collins. Fell. Sluze 317
Tschirnhausen. Eoemer. , 318
XX TABLE OF CONTENTS.
CHAPTER XVI. THE LIFE AND WORKS OP NEWTON.
PAGE
Newton s school and undergraduate life ...... 320
Investigations in 1665 1666 on fluxions, optics, and gravitation . 321
His views on gravitation ....... 322
Work in 16671669 323
Elected Lucasian professor, 1669 324
Optical lectures and discoveries, 16691671 324
Emission theory of light, 1675 326
Letters to Leibnitz, 1676 327
Discoveries on gravitation, 1679 330
Discoveries and lectures on algebra, 16731683 . . . .331
Discoveries and lectures on gravitation, 1684 333
The Principia, 16851686 334
Footnote on the contents of the Principia .... 336
Publication of the Principia 343
Investigations and work from 1686 to 1696 344
Appointment at the mint, and removal to London, 1696 . . . 345
Publication of the Optics, 1704 345
Appendix on classification of cubic curves .... 346
Appendix on quadrature by means of infinite series . . 348
Appendix on method of fluxions 349
The invention of fluxions and the infinitesimal calculus . . . 352
The dispute as to the origin of the differential calculus . . . 352
Newton s death, 1727 . . . 353
List of his works . . . 353
Newton s character . . . . . . . . . . 354
Newton s discoveries . , 356
CHAPTER XVII. LEIBNITZ AND THE MATHEMATICIANS
OF THE FIRST HALF OF THE EIGHTEENTH CENTURY.
Leibnitz and the Bernoullis . . . . ... .359
LEIBNITZ, 16461716 . . . . . . ... . 359
His system of philosophy, and services to literature . . 361
The controversy as to the origin of the calculus ; . 362
His memoirs on the infinitesimal calculus . . . . . 368
His papers on various mechanical problems . . . 369
Characteristics of his work . . . . . , 371
TABLE OF CONTENTS. XXI
PAGE
JAMES BERNOUILLI, 16541705 , . . . . . . 372
JOHN BEBNOUILLI, 1667 1748. ... . . . . . 373
The younger Bernouillis . f f , , . . . . 374
The development of analysis on the continent ..... 375
L Hospital, 16611704 . . . , . . . . .375
Varignon, 16541722 . . . . ... . .376
De Montmort. Nicole. Parent. Saurin. De Gua . . . 377
Cramer, 17041752. Kiccati, 16761754. Fagnano, 16821766 378
Viviani, 16221703. De la Hire, 16401719 . . . .379
Eolle, 16521719 . . . . . . . , . . . .380
CLAIBAUT, 17131765 . . . . . ^ . . . .380
D ALEMBERT, 1717 1783 . . *-".. 382
Solution of a partial differential equation of the second order 383
Daniel Bernoulli, 17001782 . . ... . . .385
The English mathematicians of the eighteenth century . . . 386
David Gregory, 16611708. Halley, 16561742 .... 387
Ditton, 16751715 . - . . . . 388
BROOK TAYLOB, 16851731 . 388
Taylor s theorem . 388
Taylor s physical researches 389
Cotes, 1682 1716 . . . . 390
Demoivre, 16671754 .......... 391
MACLAUBIN, 1698 1746 . . . . V . . .392
His geometrical discoveries . . , ; . . . 392
The Treatise of fluxions, and propositions on attractions . 394
Thomas Simpson, 17101761 396
CHAPTER XVIII. LAGRANGE, LAPLACE, AND THEIR CON
TEMPORARIES. CIRC. 1740 1830.
Characteristics of the mathematics of the period .... 398
The development of analysis and mechanics 399
EULEB, 17071783 399
The Introductio in Analysin Infinitorum, 1748 . . . 400
The Institutiones Calculi Differ entialis, 1755 . . . 402
The Institutiones Calculi Integralis, 17681770 . . 402
The Anleitung zur Algebra, 1770 403
His works on mechanics and astronomy .... 404
XX11 TABLE OF CONTENTS.
Lambert, 17281777 . . , .
Bfeout, 17301783. Trembley, 17491811. Arbogast, 17591803
LAGRANGE, 17361813
Memoirs on various subjects
The Mecanique analytique, 1788
The Theorie des f auctions and Calcul des fonctions
The Resolution des equations numeriques, 1798 .
Characteristics of his work
LAPLACE, 17491827
Use of the potential and spherical harmonics .
Memoirs on problems in astronomy .....
The Mecanique celeste and Exposition du systeme du monde
The Theorie analytique des probabilites, 1812 .
Laplace s physical researches
Character of Laplace
LEGENDRE, 17521833
His memoirs on attractions ......
The Theorie des nombres, 1798 . . . . .
The Calcul integral and the Fonctions elliptiques
Pfaff, 17651825 . . - . .
The creation of modern geometry
Monge, 17481818
Lazare Carnot, 17531823 , s
Poncelet, 1788 1867
The development of mathematical physics . .....
Cavendish. Eumford. Young. Wollaston. Dalton .
FOURIER, 17681830 . . . . . . . . ..
Sadi Carnot; foundation of thermodynamics . . .
POISSON, 1781 1840
Ampere. Fresnel. Biot. Arago . . . . . .
The introduction of analysis into England . . . .
Ivory, 17651845 . . . . . . . . ...
The Cambridge Analytical School . . . ...
Woodhouse, 1773 1827 . . ... . . . .
Peacock, 17911858 . . . . . . . . .
Babbage, 17921871. Sir John Herschel, 17921871 . . .
TABLE OF CONTENTS. xxill
CHAPTER XIX. MATHEMATICS OF RECENT TIMES.
Difficulty in discussing the mathematics of this century . . . 449
Account of contemporary work not intended to be exhaustive . . 449
Authorities . . . . . . 450
GAUSS, 17771855 . . . . . . ... .451
Investigations in astronomy, electricity, &c. . . . 452
The Disquisitiones Arithmeticae, 1801 .... 454
His other discoveries ......... 455
Comparison of Lagrange, Laplace, and Gauss . . . 456
Development of the Theory of Numbers . . . . . 457
Dirichlet, 18051859 . . . . . . . . .457
Eisenstein, 1823 1852 ." . . ... . . .457
Henry Smith, 1826 1883 . . . . .^ . , . . 458
Notes on other writers on the Theory of Numbers .... 461
Development of the Theory of Functions of Multiple Periodicity . 463
ABEL, 18021829 463
JACOBI, 1804 1851 . . . . V . . . . . 464
BIEMANN, 18261866 ......... 465
"7 Memoir on functions of a complex variable, 1850 . . 465
Memoir on hypergeometry, 1854 466
Investigations on functions of multiple periodicity, 1857 . 468
Paper on the theory of numbers . . . . . 468
Notes on other writers on Elliptic and Abelian Functions . . 468
The Theory of Functions . . . . > - . . . ~" . 470
Development of Higher Algebra ....... 471
CAUCHY, 1759 1857 .471
Development of analysis and higher algebra . . . 473
Argand, born 1825 ; geometrical interpretation of complex numbers 474
SIB WILLIAM HAMILTON, 18051865 . . . . . . 474
Introduction of quaternions, 1852 . ... . 475
Hamilton s other researches 475
GRASSHANN, 18091877 . . 476
The introduction of non-commutative algebra, 1844 . . 476
DE MORGAN, 18061871 476
Notes on other writers on Algebra, Forms, and Equations . . 477
Notes on modern writers on Analytical Geometry .... 480
XXIV TABLE OF CONTENTS.
PAGE
Notes on other writers on Analysis . . . . . .481
Development of Synthetic Geometry ...... 482
Steiner, 17961863 482
Von Staudt, 1798 1867 . . . . . . . .483
Other writers on modern Synthetic Geometry . .... 484
Development of the Theory of jCcraphics . ":.:. . . . . 484
Clifford, 1845 1879 . .^V >% ,. . . " . . . . 485
Development of Theoretical Mechanics and Attractions . . . 486
Green, 1793 1841 . . f *.^-:- <.. 486
Notes on other writers on Mechanics . . . ,, . 487
Development of Theoretical Astronomy . . . . . . 488
Bessel, 17841846 . . - 489
Leverrier, 18111877 . . . -. . . . . .489
Adams, 18191892 490
Notes on other writers on Theoretical Astronomy . . . . 491
Development of Mathematical Physics . . . . . . 493
INDEX . . . . . . . . . - . .499
PRESS NOTICES . . . ... . . ..- . 521
ERRATA.
Page 22, line 26. For 410 read 409356.
Page 238, line 18. For Vieta read Snell.
Page 338. Dele lines 610 of footnote.
Page 339, line 15 of note. For second and third editions read third
edition.
Page 339, line 18 of note. For Cotes read Pemberton.
Page 390, line 11. For should have learnt read might have known.
CHAPTEE I.
EGYPTIAN AND PHOENICIAN MATHEMATICS.
THE history of mathematics cannot with certainty be
traced back to any school or period before that of the Ionian
Greeks, but the subsequent history may be divided into three
periods, the distinctions between which are tolerably well
marked. The first period is that of the history of mathematics
under Greek influence, this is discussed in chapters n. to vn. :
the second is that of the mathematics of the middle ages and
the renaissance, this is discussed in chapters VIH. to xiu. : the
third is that of modern mathematics, and this is discussed in
chapters xiv. to xix.
Although the history commences with that of the Ionian
schools, there is no doubt that those Greeks who first paid
attention to mathematics were largely indebted to the previous
investigations of the Egyptians and Phoenicians. This chapter
is accordingly devoted to a statement of what is known con
cerning the mathematical attainments of those races, but our
acquaintance with the subject is so imperfect that the following
notes must be regarded merely as a brief summary of the
conclusions which seem to me most probable. The actual
history of mathematics begins with the next chapter.
On the subject of pre-historic mathematics, we may observe
le first place that, though all early races which have left
B. 1
2 EGYPTIAN AND PHOENICIAN MATHEMATICS.
records behind them knew something of numeration and
mechanics, and though the majority were also acquainted with
the elements of land-surveying, yet the rules which they
possessed were in general founded only on the results of
observation and experiment, and were neither deduced from
nor did they form part of any science. The fact then that
various nations in the vicinity of Greece had reached a high
state of civilization does not justify us in assuming that they
had studied mathematics.
The only races with whom the Greeks of Asia Minor
(amongst whom our history begins) were likely to have come
into frequent contact were those inhabiting the eastern littoral
of the Mediterranean: and Greek tradition uniformly assigned
the special development of geometry to the Egyptians, and that
of the science of numbers either to the Egyptians or to the
Phoenicians. I will consider these subjects separately.
First, as to the science of numbers. So far as the acquire
ments of the Phoenicians on this subject are concerned it is
impossible to speak with any certainty. The magnitude of the
commercial transactions of Tyre and Sidon must have neces
sitated a considerable development of arithmetic, to which
it is probable the name of science might be properly applied*
According to Strabo the Tyrians paid particular attention to
the sciences of numbers, navigation, and astronomy; they had
we know considerable commerce with their neighbours and
kinsmen the Chaldaeans ; and Bb ckh says that they regularly
supplied the weights and measures used in Babylon. Now
the Chaldaeans had certainly paid some attention to arithmetic
and geometry, as is shewn by their astronomical calculations ;
and, whatever was the extent of their attainments in arithmetic,
it is almost certain that the Phoenicians were equally proficient,
while it is likely that the knowledge of the latter, such as it
was, was communicated to the Greeks. On the whole I am
inclined to think that the early Greeks were largely indebted
to the Phoenicians for their knowledge of practical arithmetic
or the art of calculation. It is perhaps worthy of note that
EARLY EGYPTIAN ARITHMETIC. 3
Pythagoras was a Phoenician ; and according to Herodotus, but
this is more doubtful, Thales was also of that race.
Next, as to the arithmetic of the Egyptians. Their civili
zation, and in particular their astronomical calculations, have
been generally accepted as implying that they were fairly
proficient in the science of numbers. But about twenty-five
years ago a hieratic papyrus* forming part of the R-hind
collection in the British Museum was deciphered, and this has
thrown considerable light on the mathematical attainments
of the Egyptians. The manuscript was written by a priest
named Ahmes somewhere between the years 1700 B.C. and
1100 B.C., and is believed to be itself a copy, with emenda
tions, of an older treatise of about 3400 B.C. The work is
called "directions for knowing all dark things," and consists
of a collection of problems in arithmetic and geometry ; the
answers are given, but in general not the processes by which
they are obtained.
The first part deals with the reduction of fractions of the
form 2/(2n+ 1) to a sum of fractions , whose numerators are
each unity : for example, Ahmes states that -^ is the sum of
"* irV* TTT> and %&$ > and T is the sum f 5lf> TTTTTJ TT6--
In all the examples n is less than 50. Probably he had no
rule for forming the component fractions, and the answers
given represent the accumulated experiences of many previous
writers : in one solitary case however he has indicated his
method, for, after having asserted that -| is the sum of | and i,
he adds that therefore two-thirds of one-fifth is equal to the
sum of a half of a fifth and a sixth of a fifth, that is, to
To + uV r ^ ne nex t part of the book is devoted to examples in
* See Ein mathematisches Handbuch der alien Aegypter by A. Eisen-
lohr, second edition, Leipzig, 1891 ; see also Cantor, chap. i. ; and
Gow s History of Greek Mathematics, Cambridge, 1884, arts. 12 14.
Beside- itiese authorities the papyrus has been discussed in memoirs by
lei, A. Favaro, V. Bobynin, and E. Weyr. I may add that there
is in the British Museum another and older roll on a mathematical
subject which has not been yet deciphered.
12
4 EGYPTIAN AND PHOENICIAN MATHEMATICS.
division and subtraction. Ahmes then proceeds to the solution
of some simple numerical equations. For example, he says
" heap, its seventh, its whole, it makes nineteen," which means
find a number such that the sum of it and one-seventh of it
shall be together equal to 19; and he gives as the answer
16 + J + |, which is correct. The latter part of the book
contains various geometrical problems to which I allude later.
He concludes the work with some arithmetico-algebraical
questions, two of which deal with arithmetical progressions
and seem to indicate that he knew how to sum such series.
This appears to represent the most advanced arithmetic with
which the Egyptians became acquainted at any rate it is all
that they communicated to the Greeks.
Throughout the work Ahmes rarely explains the process
by which he arrives at a result, but in one numerical example,
where he requires to multiply a certain number, say &, by 13,
he points out the method he has used. In this instance he
first multiplied by 2 and got 2, then he doubled the result
and got 4a, then he again doubled the result and got &a, and
lastly he added together a, 4&, and Sa a process strictly
analogous to what is now called "practice."
The arithmetical part of the papyrus indicates that Ahmes
had some idea of algebraic symbols. The unknown quantity
is always represented by the symbol which means a heap ;
addition is represented by a pair of legs walking forwards,
subtraction by a pair of legs walking backwards or by a flight
of arrows ; and equality by the sign /_. As we shall see in
the next chapters the Greeks shewed no aptitude for algebra,
and it was not until the development of mathematics passed
again into the hands of members of a Semitic race that any
considerable progress was made in the subject.
A large part of Ahmes s arithmetic is devoted to fractions.
It may be noticed in passing that the treatment of fractions
presented great difficulty to all early races. The Egyptians
and Greeks reduced a fraction to the sum of several fractions,
in each of which the numerator was unity, so that they had
EARLY EGYPTIAN MATHEMATICS. 5
to consider only the various denominators : the sole excep
tions to this rule being the fractions and f . This remained
the Greek practice nmtil the sixth century of our era. The
Romans, on the other hand, generally kept the denominator
constant and equal to twelve, expressing the fraction (approxi
mately) as so many twelfths. The Babylonians did the same
in astronomy, except that they used sixty as the constant
denominator; and from them through the Greeks the modern
division of a degree into sixty equal parts is derived. Thus
in one way or the other the difficulty of having to consider
changes in both numerator and denominator was evaded.
Before leaving the question of early arithmetic I should
mention that for practical purposes the almost universal use
of the abacus or swan-pan rendered it easy to add and
subtract, or even to multiply and divide, without any know
ledge of theoretical arithmetic. These instruments will be
described later in chapter vn. ; it will be sufficient here to say
that they afford a concrete way of representing a number
in the decimal scale, and enable the results of addition and
subtraction to be obtained by a merely mechanical process.
This, coupled with a means of representing the result in writing,
all that was required in primitive times.
Second, LS to the science of geometry. Geometry is supposed
to have had its origin in land-surveying \ but while it is difficult
to say when the study of numbers and calculation some know
ledge of which is essential in any civilized state became a
;ience, it is comparatively easy to distinguish between the
abstract reasonings of geometry and the practical rules of land-
The principles of land-surveying must have been
understood from very early times, but the universal tradition
of antiquity Asserted that the origin of geometry must be
sought In Egypt. That it was not indigenous to Greece and
that it arose from the necessity of surveying is rendered the
more probable by the derivation of the word from yjj the earth
and /tcrpcco I measure. Now the Greek geometricians, as far
as WH can judge by their extant works, always dealt with the
6 EGYPTIAN AND PHOENICIAN MATHEMATICS.
science as an abstract one : they sought for theorems which
should be absolutely true, and would have argued that to
measure quantities in terms of a unit "which might have
been incommensurable with some of the magnitudes considered
would have made their results mere approximations to the
truth. The name does not therefore refer to their practice.
It is not however unlikely that it indicates the use which
was made of geometry among the Egyptians from whom the
Greeks learned it. This also agrees with the Greek traditions,
which in themselves appear probable ; for Herodotus states
that the periodical inundations of the Nile (which swept away
the land-marks in the valley of the river, and by altering
its bed increased or decreased the taxable value of the adjoin
ing lands) rendered a tolerably accurate system of surveying
ground indispensable, and thus led to a systematic study of
the subject by the priests. The Egyptians certainly studied
geometry. A small piece of evidence which tends to shew that
the Phoenicians and Jews had not paid much attention to it
is to be found in the mistake made in /. Kings , ch. 7, v. 23,
and //. Chronicles, ch. 4, v. 2, where it is stated that the
circumference of a circle is three times its diameter : the
Babylonians* also assumed that TT was equal to 3.
Assuming then that a knowledge of geometry was first
derived by the Greeks from Egypt, we must next discuss the
range and nature of Egyptian geometry f . For any accurate
account of this we have to rely on the Rhind papyrus men
tioned above : this, as I have already stated, was probably a
summary of the information which was familiar to the priests,
and was not a book of research. At any rate we have reason
to believe that some time before the year 2000 B.C. (that
is some centuries before it was written) the following method
of obtaining a right angle was used in laying out the ground-
plan of certain buildings. The Egyptians were we know very
* See J. Oppert, Journal Asiatique, August, 1872, and October, 1874.
t See Eisenlohr ; Cantor, chap. n. ; Grow, arts. 75, 76 ; and Die
Geometric der alten Aegypter by E. Weyr, Vienna, 1884.
EARLY EGYPTIAN GEOMETRY. / 7
particular about the exact orientation of their temples ; and they
had therefore to obtain with accuracy a north and south line,
and also an east and west line. By observing the points on the
horizon where a star rose and set, and taking a plane midway
between them, they could obtain a north and south line. To
get an east and west line, which had to be drawn at right
angles to this, certain professional " rope-fasteners " were
employed, who stretched a rope round three pegs (the two of
them which were nearest together being fixed along the north
and south line) so that the sides of the triangle formed were
in the ratio of 3 : 4 : 5 ; the angle opposite the longest side
would then be a right angle. A similar method is constantly
used at the present time by practical engineers. This property
can be deduced as a particular case of Euc. i. 48 : and there is
reason to think that the Egyptians were acquainted with the
results of this proposition and of Euc. i. 47 for triangles whose
sides are in the ratio mentioned above. They must also, there
is little doubt, have known that the latter proposition was true
for an isosceles right-angled triangle, as that is obvious if a
floor be paved with tiles of that shape. But though these are
interesting facts in the history of the Egyptian arts we must
not press them too far as shewing that geometry was then
studied as a science.
Our real knowledge of the nature of Egyptian geometry
depends almost entirely on the Rhirid papyrus, and therefore
at the earliest does not go further back than the year 1700 B.C.
Ahmes commences that part of the papyrus which deals with
geometry by giving several numerical instances of the contents
of barns. Unluckily we do not know what was the usual
shape of an Egyptian barn, but where it is defined by three
linear measurements, say a, 6, and c, the answer is always
given as if he had formed the expression a x b x (c + Jc). He
next proceeds to find the areas of certain rectilineal figures
(in some of which he is certainly wrong) and then to find
the area of a circular 6eld of diameter 12 no unit of length
. mentioned. In the latter case he gives the area as
8 EGYPTIAN AND PHOENICIAN MATHEMATICS.
(d ^) 2 , where d is the diameter of the circle : this is equi
valent to taking 3-1604 as the value of ?r, the actual value
being very approximately 3*1416. Lastly Ahmes gives some
problems on pyramids. These long proved incapable of inter
pretation, but Cantor and Eisenlohr have shewn that Ahmes
was attempting to find, by means of data obtained from the
measurement of some of the external dimensions of a building,
the ratio of certain other dimensions which could not be
directly measured : his process is equivalent to determining
the trigonometrical ratios of certain angles. The data and
the results given agree closely with the dimensions of some of
the existing pyramids.
It is noticeable that all the specimens of Egyptian geo
metry which we possess deal only with particular numerical
problems and not with general theorems ; and even if a result
be stated as universally true, it was probably proved to be
so only by a wide induction. We shall see later that Greek
geometry was from its commencement deductive. There are
reasons for thinking that Egyptian geometry and arithmetic
made little or no progress subsequent to the date of Ahmes s
work: and though for nearly two hundred years after the time
of Thales Egypt was recognized by the Greeks as an important
school of mathematics, it would seem that, almost from the
foundation of the Ionian school, the Greeks outstripped their
former teachers.
It may be added that Ahmes s book gives us much that
idea of Egyptian mathematics which we should have gathered
from statements about it by various Greek and Latin authors,
some of whom lived nearly fifteen centuries later. Previous
to its translation it was commonly thought that these state
ments exaggerated the acquirements of the Egyptians, and its
discovery must increase the weight to be attached to the
testimony of these authorities.
We know nothing of the applied mathematics (if there
were any) of the Egyptians or Phoenicians. The astronomical
attainments of the Egyptians and Chaldaeans were no doubt
EARLY CHINESE MATHEMATICS. 9
considerable, though they were chiefly the results of obser
vation : the Phoenicians are said to have confined themselves
to studying what was required for navigation. Astronomy
however lies outside the range of this book.
I do not like to conclude the chapter without a brief
J mention of the Chinese, since at one time it was asserted that
I they were familiar with the sciences of arithmetic, geometry,
* mechanics, optics, navigation, and astronomy nearly three
thousand years ago, and a few writers were inclined to suspect
(for no evidence was forthcoming) that some knowledge of
this learning had filtered across Asia to the West. It is
indeed almost certain that the Chinese were then acquainted
with several geometrical or rather architectural implements,
such as the rule, square, compasses, and level ; with a few
mechanical machines, such as the wheel and axle ; that they
knew of the characteristic property of the magnetic needle ;
and were aware that astronomical events occurred_Jn cycles^-
But the careful investigations of L. A. Sedillot* have shewn
that the Chinese of that time had made no serious attempt to
classify or extend the few rules of arithmetic or geometry
which they knew, or to explain the causes of the phenomena
with which they were acquainted. The idea that the Chinese
had made considerable progress in theoretical mathematics
seems to have been due to a misapprehension of the Jesuit
missionaries who went to China in the sixteenth century. In
the first place they failed to distinguish between the original
science of the Chinese and the views which they found preva
lent on their arrival ; the latter being founded on the work
and teaching of Arab missionaries who had come to China in
the course of the thirteenth century, and while there introduced
a knowledge of spherical trigonometry. In the second place,
finding that one of the most important government depart
ments was known as the Board of Mathematics, they supposed
* See Boncompagni s Bullettino di bibliografia e di storia delle scienze
matematiche e fisiche for May, 1868, vol. i., pp. 161 166. On Chinese
mathematics, mostly of a later date, see Cantor, chap. xxxi.
10 EGYPTIAN AND PHOENICIAN MATHEMATICS.
that its function was to promote and superintend mathematical
studies in the empire. Its duties were really confined to the
annual preparation of an almanadk, the dates and predictions
in which regulated many affairs both in public and domestic
life. All extant specimens of this almanadk are extraordinarily
inaccurate arid defective. The only geometrical theorem with
which, as far as I am aware, the ancient Chinese were ac
quainted was that in certain cases (najnely when the ratio of
the sides was 3 : 4 : 5 or 1 : 1 : ^/2) the area of the square
described on the hypotenuse of a right-angled triangle is equal
to the sum of the areas of the squares described on the sicles.
It is barely possible that a few geometrical theorems which can
be demonstrated in the quasi-experimental way of superposi
tion were also known to them. Their arithmetic was decimal
in notation, but their knowledge seems to have been con
fined to the art of calculation by means of the swan-pan,
and the power of expressing the results in writing. Our
acquaintance with the early attainments of the Chinese, slight
though it is, is more complete than in the case of most of
their contemporaries. It is thus specially instructive, and
serves to illustrate the fact that a nation may possess consider
able skill in the applied arts while they are almost entirely
ignorant of the sciences on which those arts are founded.
From the foregoing summary it will be seen that our
knowledge of the mathematical attainments of those who
o
preceded the Greeks is very limited ; but we may reasonably
infer that from one source or another the early Greeks learned
as much mathematics as is contained or implied in the Rhind
papyrus, and it is probable that they were not acquainted with
much more. In the next six chapters I shall trace the de
velopment of mathematics under Greek influence.
11
FIRST PERIOD.
JWatfjemattcs unfcn (5mfe influence.
This period begins with the teaching of Thales, circ. 600 B.C.,
and ends with the capture of Alexandria by the Mohammedans
in or about 641 A.D. The characteristic feature of this period
is the development of geometry.
12
It will be remembered that I commenced the last chapter
by saying that the history of mathematics might be divided
into three periods, namely, that of mathematics under Greek
influence, that of the mathematics of the middle ages and of
the renaissance, and lastly that of modern mathematics. The
next four chapters (chapters n., in., iv. and v.) deal with the
history of mathematics under Greek influence : to these it will
be convenient to add one (chapter vi.) on the Byzantine school,
since through it the results of Greek mathematics were trans
mitted to western Europe; and another, (chapter vn.) on the
systems of numeration which were ultimately displaced by
the system introduced by the Arabs. I should add that many
of the dates mentioned in these chapters are not known with
certainty and must be regarded as only approximately correct.
13
CHAPTER II.
THE IONIAN AND PYTHAGOKEAN SCHOOLS*.
CIRC. 600 B.C. 400 B.C.
WITH the foundation of the Ionian and Pythagorean
schools we emerge from the region of antiquarian research and
conjecture into the light of history. The materials at our dis
posal for estimating the knowledge of the philosophers of these
schools previous to about the year 430 B.C. are however very
scanty. Not only have all but fragments of the different
mathematical treatises then written been lost, but we possess
no copies of the elaborate histories of mathematics written
about 325 B.C. by Eudemus (who was a pupil of Aristotle)
and Theophrastus respectively. Luckily Proclus, who about
450 A. D. wrote a commentary on Euclid s Elements, was familiar
with the history of Eudemus and gives a summary of that
part of it which dealt with geometry. We have also a frag
ment of the General View of Mathematics written by Geminus
about 50 B.C., in which the methods of proof used by the
early Greek geometricians are compared* with those current
at a later date. In addition to these general statements we
* The history of these schools is discussed by Cantor, chaps, v. vm. ;
by G. J. Allman in his Greek Geometry from Thales to Euclid, Dublin,
1889 ; by C. A. Bretschneider in his Die Geometrie mid die Geometer
vor Eukleides, Leipzig, 1870 ; and partially by H. Hankel in his post
humous Geschichte der Mathematik, Leipzig, 1874.
14 THE IONIAN AND PYTHAGOREAN SCHOOLS.
have biographies of a few of the leading mathematicians, and
some scattered notes in various writers in which allusions are
made to the lives and works of others. The original authorities
are criticized and discussed at length in the works mentioned
in the footnote to the heading of the chapter.
The Ionian School.
Thales*. The founder of the earliest Greek school of
mathematics and philosophy was Thales, one of the seven sages
of Greece, who was born about 640 B.C. at Miletus and died in
the same town about 550 B.C. The materials for an account of
his life consist of little more than a few anecdotes which have
been handed down by tradition. During the early part of his
life he was engaged partly in commerce and partly in public
affairs ; and to judge by two stories that have been preserved,
he was then as distinguished for shrewdness in business and
readiness in resource as he was subsequently celebrated in
science. It is said that, once when transporting some salt
which was loaded on mules, one of the animals slipping in
a stream got its load wet and so caused some of the salt
to be dissolved, finding its burden thus lightened it rolled
over at the next ford to which it came; to break it of
this trick Thales loaded it with rags and sponges which, by
absorbing the water, made the load heavier and soon effectually
cured it of its troublesome habit. At another time, according
to Aristotle, when there was a prospect of an unusually
abundant crop of olives Thales got possession of all the olive-
presses of the district ; and, having thus " cornered " them, he
was able to make his own terms for lending them out, and thus
realized a large sum. These tales may be apocryphal but it is
certain that he must have had considerable reputation as a man
of affairs and as a good engineer since he was employed to
construct an embankment so as to divert the river Halys in
such a way as to permit of the construction of a ford.
* See Cantor, chap. v. ; Allman, chap. i.
THALES. 15
It was probably as a merchant that Thales first went to
Egypt, but during his leisure there he studied astronomy and
geometry. He was middle-aged when he returned to Miletus ;
he seems then to have abandoned business and public life,
and to have devoted himself to the study of philosophy and
science subjects which in the Ionian, Pythagorean, and
perhaps also the Athenian schools, were inextricably involved :
he continued to live at Miletus till his death circ. 550 B.C.
His views on philosophy do not here concern us.
We cannot form any exact idea as to how Thales presented
his geometrical teaching : we infer however from Proclus that
it consisted of a number of isolated propositions which were
not arranged in a logical sequence, but that the proofs were
deductive, so that the theorems were not a mere statement
of an induction from a large number of special instances,
as probably was the case with the Egyptian geometricians.
The deductive character which he thus gave to the science
is his chief claim to distinction.
The following comprise all the propositions that we can
now with reasonable probability refer back to him.
(i) The angles at the base of an isosceles triangle are
equal (Euc. I. 5). Proclus seems to imply that this was
proved by taking another exactly equal isosceles triangle,
turning it over, and then superposing it on the first; a sort
of experimental demonstration.
(ii) If two straight lines cut one another the vertically
opposite angles are equal (Euc. i. 15). Thales may have
regarded this as obvious, for Proclus adds that Euclid was the
first to give a strict proof of it.
(iii) A triangle is determined if its base and base angles
be given (cf. Euc. I. 26). Apparently this was applied to find
tl : ;i ship at sea; the base being a tower, and the
base angles beini obtained by observation.
The si i^s of equiangular triangles are proportionals
(1 vi. 4, or perhaps rather Euc. vi. 2). This is said to
ha been used by Thales when in Egypt to find the height of
16 THE IONIAN AND PYTHAGOREAN SCHOOLS.
a pyramid. In a dialogue given by Plutarch, the speaker
addressing Thales says " placing your stick at the end of
the shadow of the pyramid, you made by the sun s rays two
triangles, and so proved that the [height of the] pyramid was
to the [length of the] stick as the shadow of the pyramid to
the shadow of the stick." The king Amasis, who was present,
is said to have been amazed at this application of abstract
science, and the Egyptians seem to have been previously unac
quainted with the theorem.
(v) A circle is bisected by any diameter. This may have
been enunciated by Thales, but it must have been recognized
as an obvious fact from the earliest times.
(vi) The angle in a semicircle is a right angle (Euc. in.
31). This appears to have been regarded as the most re
markable of the geometrical achievements of Thales, and it is
stated that on inscribing a right-angled triangle in a circle he
sacrificed an ox to the immortal gods. It is supposed that he
proved the proposition by joining the centre of the circle to
the apex of the right angle, thus splitting the triangle into two
isosceles triangles, and then applied the proposition (i) above:
if this be the correct account of his proof, he must have been
aware that the sum of the angles of a right-angled triangle
is equal to two right angles.
It has been ingeniously suggested that the shape of the
tiles used in paving floors may have afforded an experimental
demonstration of the latter result, namely, that the sum of
the angles of a triangle is equal to two right angles. We
know from Eudemus that the first geometers proved the
general property separately for three species of triangles, and
it is not unlikely that they proved it thus. The area about a
point can be filled by the angles of six equilateral triangles or
tiles, hence the proposition is true for an equilateral triangle.
Again a rectangle (the sum of whose angles is four right
angles) can be divided into two equal right-angled triangles,
hence the proposition is true for a right-angled triangle :
and it will be noticed that tiles of such a shape would give an
THALES. 17
ocular demonstration of this case it would appear that this
proof was given at first only in the case of isosceles right-
angled triangles, but probably it was extended later so as
to cover any right-angled triangle. Lastly any triangle can be
split into the sum of two right-angled triangles by drawing
a perpendicular from the biggest angle on the opposite side,
and therefore again the proposition is true. The first of these
proofs is evidently included in the last, but the early Greek
geometers were timid about generalizing their proofs, and
were afraid that any additional condition imposed on the
triangle might vitiate the general result.
Thales wrote an astronomy, and among his contemporaries
was more famous as an astronomer than as a geometrician. It
is said that, one night, when walking out, he was looking so
intently at the stars that he tumbled into a ditch, on which an
old woman exclaimed " How can you tell what is going on
in the sky when you can t see what is lying at your own feet ?"
an anecdote which was often quoted to illustrate the un
practical character of philosophers.
Without going into astronomical details it may be mentioned
that he taught that a year contained 365 days, and not (as
was previously reckoned) twelve months of thirty days each.
According to some recent critics he believed the earth to be a
disc, but it seems to be more probable that he was aware that
it was spherical. He explained the causes of the eclipses both
of the sun and moon, and it is well known that he predicted a
solar eclipse which took place at or about the time he foretold :
the actual date was May 28, 585 B.C. But though this pro
phecy and its fulfilment gave extraordinary prestige to his
teaching, and secured him the name of one of the seven sages
of Greece, it is most likely that he only made use of one of the
Egyptian or Chaldaean registers which stated that solar eclipses
recur at intervals of 18 years and 11 H^.s
Among the pupils of Thale; were Anaximander, Mamercus,
and Mandryatus. Of tho two mentioned last we know next
to nothing. Anax winter is better known; he was born in
B. 2
18 THE IONIAN AND PYTHAGOREAN SCHOOLS.
611 B.C. and died in 545 B.C., and succeeded Thales as head of
the school at Miletus. According to Suidas he wrote a treatise
on geometry in which tradition says he paid particular attention
to the properties of spheres, and dwelt at length on the philo
sophical ideas involved in the conception of infinity in space
and time. He constructed terrestrial and celestial globes. He
is alleged to have introduced the use of the style or gnomon into
Greece. This, in principle, consisted only of a stick stuck
upright in a horizontal piece of ground. It was originally used
as a sun-dial, in which case it was placed at the centre of three
concentric circles so that every two hours the end of its shadow
passed from one circle to another. Such sun-dials have been
found at Pompeii and Tusculum. It is said that he employed
these styles to determine his meridian (presumably by marking
the lines of shadow cast by the style at sunrise and sunset on
the same day, and taking the plane bisecting the angle so
formed) ; and thence, by observing the time of year when the
noon-altitude of the sun was greatest and least, he got the
solstices ; thence, by taking half the sum of the noon-altitudes
of the sun at the two solstices, he found the inclination of the
equator to the horizon (which determined the latitude of the
place), and, by taking half their difference, he found the incli
nation of the ecliptic to the equator. There seems good reason
to think that he did actually determine the latitude of Sparta,
but it is more doubtful whether he really made the rest of
these astronomical deductions.
We need not here concern ourselves further with the
successors of Thales. The school he established continued to
flourish till about 400 B.C., but, as time went on, its members
occupied themselves more and more with philosophy and less
with mathematics. We know very little of the mathematicians
comprised in it, but they would seem to have devoted most of
their attention to astrcroray. They exercised but slight in
fluence on the further advance of Greek mathematics, which
was made almost entirely under the influence of the Pythago
reans, who not only immensely developed the science of
PYTHAGORAS. 1 9
geom ; i v l)ii e of numbers. If Thales was
the in -hi iu uirect general attention to geometry, it was Pytha
goras, says Proclus, quoting from Eudemus, who "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... intellectual manner" : and it is accordingly
to Pythagoras that we must now direct attention.
The Pythagorean School.
Pythagoras*. Pythagoras was born at Samos about
569 B.C., perhaps of Tyrian parents, and died in 500 B.C. He
was thus a contemporary of Thales. The details of his life are
somewhat doubtful, but the following account is I think sub
stantially correct. He studied first under Pherecydes of Syros,
and then under Ariaximander; by the latter he was recom
mended to go to Thebes, and there or at Memphis he spent
some years. After leaving Egypt he travelled in Asia Minor,
and then settled, at Samos, where he gave lectures but without
much success. /* About 529 B.C. he migrated to Sicily with his
mother, and with a single disciple who seems to have been the
sole fruit of his labours at Samos. Thence he went to Tarentum,
but very shortly moved to Croton, a Dorian colony in the south
of Italy. Here the schools that he opened were crowded with
an enthusiastic audience; citizens of all ranks, especially those
of the upper classes, attended, and even the women broke a law
which forbade their going to public meetings and nocked to
hear him. Amongst his most attentive auditors was Theano,
the young and beautiful daughter of his host Milo, whom, in
spite of the disparity of their ages, he married : she wrote a
biography of her husband but unfortunately it is lost.
* See Cantor, chaps, vi., vii. ; Allman, chap. n. ; Hankel, pp. 92 111 ;
Hoefer, pp. 87 130 ; and various papers by P. Tannery. For an account
of Pythagoras s life, embodying the Pythagorean traditions, see the bio
graphy by lamblichus, of which there are two or three English trans
lations.
2 .3
20 THE IONIAN AND PYTHAGOREAN SCHOOLS.
Pythagoras was really a philosopher and moralist, but his
philosophy and ethics, as we shall shortly see, rested on a
mathematical basis. He divided those who attended his lectures
into two classes, the listeners or TrvOayoptioi and the mathe
maticians or TrvOay opt/cot. In general, a " listener" after passing
three years as such could be initiated into the second class,
to whom alone were confided the chief discoveries of the
school. Following the modern usage I confine the use of the
word Pythagoreans to the latter class.
The Pythagoreans formed a brotherhood with all things in
common, holding the same philosophical beliefs, engaged in the
same pursuits, and bound by oath not to reveal the teaching or
secrets of the school. Their food was simple ; their discipline
severe ; and their mode of life arranged to encourage self-
command, temperance, purity, and obedience. They rose
before the sun, and began by recalling the events of the pre
ceding day, next they made a plan for the day then com
mencing, and finally on retiring to rest they were expected to
compare their performances with this plan.
One of the symbols which they used for purposes of re-
cognftion was the pentagram, sometimes also called the triple-
triangle TO rpLirXovv r/otywi/ov. The pentagram is merely a
regular re-entrant pentagon; it was considered symbolical of
health, and the angles were denoted by the letters of the word
PYTHAGORAS. 21
(see below p. 39), the diphthong ct being replaced by a 0]
it will be noticed that it consists of a single broken line*, a
feature to which a certain mystical importance was attached,
lamblichus, to whom we owe the disclosure of this symbol,
tells us how a certain Pythagorean, when travelling, fell ill at
a roadside inn where he had put up for the night; he was poor
and sick, but the landlord who was a kindhearted fellow
nursed him carefully and spared no trouble or expense to
relieve his pains. However, in spite of all efforts, the student
got worse; feeling that he was dying and unable to make the
landlord any pecuniary recompense, he asked for a board on
which he inscribed the pentagram-star; this he gave to his host,
begging him to hang it up outside so that all the passers-by
might see it, and assuring him that he would not then regret
his kindness as the symbol on it would ultimately shew. The
scholar died and was honourably buriedj and the board was
duly exposed. After a considerable time had elapsed a traveller
one day riding by saw the sacred symbol; dismounting, he
entered the inn, and after hearing the story, handsomely re
munerated the landlord, Such is the anecdote, which if not
true is at least ben trovato.
t- f ^ The majority of those who attended the lectures of Pytha-
s goras were only "listeners"; but his philosophy was intended to
colour the whole life, political and social, of all his followers.
In advocating self-control, government by the best men in the
state, strict obedience to legally constituted authorities, and an
appeal to eternal principles of right and wrong, he represented a
view of society totally opposed to that of the democratic party of
that time, and thus naturally most of the brotherhood were aris
tocrats. It had affiliated members in many of the neighbouring
cities, and its method of organization and strict discipline gave
it great political power; but like all secret societies it was an
object of suspicion to those who did not belong to it. For a
short time the Pythagoreans triumphed, but a popular revolt
* On the theory of such figures, see my Mathematical Recreations
and Prnhlfims T.nnHon, 1892, chap. vi.
22 THE IONIAN AND PYTHAGOREAN SCHOOLS.
in 501 B.C. overturned the civil government, and in the riots
that accompanied the insurrection the mob burnt the house
of Milo (where the students lived) and killed many of the
most prominent members of the school. Pythagoras himself
escaped to Tarentum, and thence fled to Metapontum, where
he was murdered in another popular outbreak in 500 B.C.
Though the Pythagoreans as a political society were thus
rudely broken up and deprived of their head, they seem to
have re-established themselves at once as a philosophical and
mathematical society, having Tarentum as their head-quarters.
They continued to flourish for a hundred or a hundred and
fifty years after the death of their founder, but they remained
to the end a secret society, and we are therefore ignorant of
the details of their history.
Pythagoras himself did not allow the use of text-books, and
the assumption of his school . was not only that all their
knowledge was held in common and secret from the outside
world, but that the glory of any fresh discovery must be
referred back to their founder: thus Hippasus (circ. 470 B.C.)
is said to have been drowned for violating his oath by publicly
boasting that he had added the dodecahedron to the number of
regular solids enumerated by Pythagoras. Gradually, as the
society became more scattered, it was found convenient to alter
this rule, and treatises containing the substance of their teach
ing and doctrines were written. The first book of the kind
was composed by Philolaus (circ. 410 B.C.), and we are told
that Plato contrived to buy a copy of it. We may say that
during the early part of the fifth century before Christ the
Pythagoreans were considerably in advance of their contem
poraries, but by the end of that time their more prominent
discoveries and doctrines had become known to the outside
world, and the centre of intellectual activity was transferred to
Athens.
Though it is impossible to separate precisely the discoveries
of Pythagoras himself from those of his school of a later date,
we know from Proclus that it was Pythagoras who gave
PYTHAGORAS. 23
geometry that rigorous character of deduction which it still
bears, and made it the foundation of a liberal education; and
there is good reason to believe that he was the first to arrange
the leading propositions of the subject in a logical order. It
was also, according to Aristoxenus, the glory of his school that
they raised arithmetic above the needs of merchants. It was
their boast that they sought knowledge and not wealth, or in
the language of one of their maxims, "a figure and a step
forwards, not a figure to gain three oboli."
Pythagoras was primarily a moral reformer and practical
philosopher, but his system of morality and philosophy was
built on a mathematical foundation. In geometry he himself
probably knew and taught the substance of what is contained
in the first two books of Euclid, and was acquainted with a
few other isolated theorems including some elementary pro
positions on irratiooal magnitudes (while his successors added
several of the propositions in the sixth and eleventh books of
Euclid); but it is thought that many of his proofs were not
rigorous, and in particular that the converse of a theorem was
frequently assumed without a proof. What philosophical
doctrines were based on these geometrical results is now only a
matter of conjecture. In the theory of numbers he was con
cerned with four different kinds of problems which dealt re
spectively with polygonal numbers, ratio and proportion, the
factors of numbers, and numbers in series; but many of his
arithmetical inquiries, and in particular the questions on poly
gonal numbers and proportion, were treated by geometrical
methods. Knowing that measurement was essential to the
accurate definition of form Pythagoras thought that it was also
to some extent the cause of form, and he therefore taught that
the foundation of the theory of the universe was to be found in
the science of numbers. He was confirmed in this opinion by
discovering that the note sounded by a vibrating string de
pended (other things being the same) only on the length of the
string, and in particular that the lengths which gave a note,
its fifth, and its octave were in the ratio 1 : : J. This may
24 THE IONIAN AND PYTHAGOREAN SCHOOLS.
have been the reason why music occupied so prominent a
position in the exercises of his school. He also believed that
the distances of the heavenly bodies from the earth formed a
musical progression: hence the phrase "the harmony of the
spheres." Taking the science of numbers as the foundation of
his philosophy he went on to attribute properties to numbers
and geometrical figures : for example the cause of colour was
the number five; the origin of fire was to be found in the
pyramid; a solid body was analogous to the tetrad, which
represented matter as composed of the four primary elements,
fire, air, earth, and water; and so on. The tetrad like the
pentagram was a sacred symbol, and the initiate s oath ran
vat //.a TOV dfjiTpa i/^x TrapaSovra TtTpaKrvv
Trayav devvaov <i;crea>s
The philosophical views of Pythagoras do not further con
cern us, arid I now proceed to discuss his work on mathematics
in rather greater detail. The Pythagoreans began by dividing
the subjects with which they dealt into four divisions: numbers
absolute or arithmetic, numbers applied or music, magnitudes
at rest or geometry, and magnitudes in motion or astronomy.
This " quadrivium " was long considered as constituting the
necessary and sufficient course of study for a liberal education.
Here I confine myself to describing the Pythagorean treatment
of geometry and arithmetic.
First, as to their geometry. We are of course unable to
reproduce the whole body of Pythagorean teaching on this
subject, but we gather from the notes of Proclus on Euclid and
from a few stray remarks in other writers that it included the
following propositions, most of which are on the geometry of
areas.
(i) It commenced with a number of definitions, which
probably were rather statements connecting mathematical ideas
with philosophy than explanations of the terms used. One
has been preserved in the definition of a point as unity having
position.
PYTHAGORAS.
25
(ii) The sum of the angles of a triangle was shewn to
be equal to two right angles (Euc. I. 32); and in the proof,
which has been preserved, the results of the propositions Euc.
I. 13 and the first part of Euc. I. 29 are quoted. The demon
stration is substantially the same as that in Euclid, and it
is most likely that the proofs there given of the two propo
sitions last mentioned are also due to Pythagoras himself.
(iii) Pythagoras certainly proved the properties of right-
angled triangles which are given in Euc. I. 47 and i. 48. We
know that the proofs of these propositions which are found
in Euclid were of Euclid s own invention ; and a good deal of
curiosity has been excited to discover what was the demon
stration which was originally offered by Pythagoras of the first
of these theorems*. It would seem most likely to have been
one of the two following.
(a) Any square ABC D can be split up as in Euc. u. 4
into two squares BK and DK and two equal rectangles AK
and CK : that is, it is equal to the square on FK, the square
on JEK, and four times the triangle AEF. But, if points be
taken, G on BC, H on CD, and E on DA, so that BG, CH,
* A collection of over thirty proofs of Euc. i. 47 was published in Der
Pythagorische Lehrsatz by Joh. Jos. Ign. Hoffmann, second edition,
Mainz, 1821.
26
THE IONIAN AND PYTHAGOREAN SCHOOLS.
and DE are each equal to AF, it can be easily shewn
that EFGH is a square, and that the triangles AEF, BFG,
CGH. and DUE are equal : thus the square A BCD is also
equal to the square on EF and four times the triangle AEF.
Hence the square on EF is equal to the sum of the squares on
FK and EK.
(/3) Let ABC be a right-angled triangle, A being the right
angle. Draw AD perpendicular to BC. The triangles ABC
and DBA are similar,
.-. BC :AB=AB: BD.
Similarly BC : AC = AC : DC.
Hence AB 2 + AC 2 = BC (BD + DC) - BC 2 .
This proof requires a knowledge of the results of Euc. 11. 2,
vi. 4, and vi. 17, with all of which Pythagoras was acquainted.
(iv) Pythagoras is also credited with the discovery of the
theorems Euc. i. 44 and i. 45, and with giving a solution of
the problem Euc. u. 14. It is said that on the discovery of
the necessary construction for the problem last mentioned he
sacrificed an ox, but as his school had all things in common
the liberality was less striking than it seems at first. The
Pythagoreans of a later date were aware of the extension
given in Euc. vi. 25, and Dr Allman thinks that Pythagoras
himself was acquainted with it, but this must be regarded as
doubtful. It will be noticed that Euc. n. 14 is a geometrical
solution of the equation x 2 = ab.
(v) Pythagoras shewed that the plane about a point could
be completely filled by equilateral triangles, by squares, or by
regular hexagons results that must have been familiar where-
ever tiles of these shapes were in common use.
PYTHAGORAS. 27
(vi) The Pythagoreans were said to have solved the quad
rature of the circle : they stated that the circle was the most
beautiful of all plane figures. -^V\XA/^
(vii) They knew that there were five regular solids inscri-
bable in a sphere, which was itself, they said, the most beautiful
of all solids.
(viii) From their phraseology in the science of numbers
and from other occasional remarks it would seem that they
were acquainted with the methods used in the second and
fifth books of Euclid, and knew something of irrational
magnitudes. In particular, there is reason to believe that
Pythagoras proved that the side and the diagonal of a square,
were incommensurable; and that it was this discovery which led;
the Greeks to banish the conceptions of number and measure
ment from their geometry. A proof of this proposition which
is not unlikely to be that due to Pythagoras is given below
(see p. 61).
Next, as to their theory of numbers*. I have already re
marked that in this the Pythagoreans were chiefly concerned
>vith (i) polygonal numbers, (ii) the factors of numbers,
iii) numbers which form a proportion, and (iv) numbers in
> series.
Pythagoras commenced his theory of arithmetic by dividing
11 numbers into even or odd : the odd numbers being termed
nomons. An odd number such as 2n + 1 was regarded as the
inference of two square numbers (n+ I) 2 and n 2 , and the sum
the gnomons from 1 to 2n + 1 was stated to be a square
lumber, viz. (n + I) 2 , its square root was termed a side. Pro-
lucts of two numbers were called plane, and, if a product had no
>xact square root, it was termed an oblong. A product of three
lumbers was called a solid number, and, if the three numbers
vvere equal, a cube. All this has obvious reference to geometry,
ind the opinion is confirmed by Aristotle s remark that when
i gnomon is put round a square the figure remains a square
* See the appendix Sur Varithmetique pythagoriennt to Tannery s La
n ,iencc r r . "iris. 1887.
28
THE IONIAN AND PYTHAGOREAN SCHOOLS.
though it is increased in dimensions. Thus, in the annexed
figure in which n is taken equal to 5, the gnomon AKC (con
taining 11 small squares) when put round the square AC
(containing 5 2 small squares) makes a square HL (containing
6 2 small squares). The numbers (2n 2 42/1 + 1), (2n 2 + 2n), and
H K
(2n +1) possessed special importance as representing the hypo
tenuse and two sides of a right-angled triangle : Cantor thinks
that Pythagoras knew this fact before discovering the geo
metrical proposition Euc. I. 47. A more general expression
for such numbers is (m 2 + n*\ 2mn, and (m 2 n 2 ) : it will be
noticed that the result obtained by Pythagoras can be deduced
from these expressions by assuming m = n + 1 ; at a later time
Archytas and Plato gave rules which are equivalent to
taking n 1 ; Diophantus knew the general rule.
After this preliminary discussion the Pythagoreans pro
ceeded to the four special problems already alluded to. Pytha
goras was himself acquainted with triangular numbers, but
probably not with any other polygonal numbers : the latter
were discussed by later members of the school. A triangular
number represents the sum of a number of counters laid in
rows on a plane ; the bottom row containing n y and each
succeeding row one less ; it is therefore equal to the sum of
the series
n + (n- 1)+ (r&-2) + ... + 2 + 1,
PYTHAGORAS. 29
that is, to %n(n+l). Thus the triangular number corre
sponding to 4 is 10. This is the explanation of the language
of Pythagoras in the well-known passage in Lucian where the
merchant asks Pythagoras what he can teach him. Pythagoras
replies, "I will teach you how to count." Merchant, "I know
that already." Pythagoras, " How do you count ] " Merchant,
"One, two, three, four " Pythagoras, "Stop! what you take
to be four is ten, a perfect triangle, and our symbol."
As to the work of the Pythagoreans on the factors of
numbers we know very little : they classified numbers by com
paring them with the sum of their integral factors, calling
a number excessive, perfect, or defective according as it was
greater than, equal to, or less than the sum of these factors.
These investigations led to no useful result.
--The third class of problems which they considered dealt
with numbers which formed a proportion ; presumably these
were discussed with the aid of geometry as is done in the fifth
book of Euclid.
Lastly the Pythagoreans were concerned with series of
numbers in arithmetical, geometrical, harmonica!, and musical
progressions. The three progressions first-mentioned are well
known ; four integers are said to be in musical progression
when they are iu the ratio a : 2ab/ (a + b) : J (a + 6) : b, for
example, 6, 8, 9, and 12 are in musical progression.
After the death of Pythagoras, his teaching seems to have
been carried on by Epicharmus, and Hippasus; and sub
sequently by Philolaus, Archippus, and Lysis. About a century
after the murder of Pythagoras we find Archytas recognized
as the head of the school.
Archytas*. Archytas, circ. 400 B.C., was one of the most
* See Allman, chap. iv. A catalogue of the works of Archytas is
given by Fabricius in his Bibliotlieca Graeca, vol. i., p. 833: most of
the fragments on philosophy were published by Thomas Gale in his
Opuscula Mythologia, Cambridge, 1670 ; and by Thomas Taylor as an
appendix to his translation of lambiichus s Life of Pythagoras, London,
1818. See also the references given by Cantor, vol. i., p. 203.
30 THE IONIAN AND PYTHAGOREAN SCHOOLS.
influential citizens of Tarentum, and was made governor of
the city no less than seven times. His influence among his
contemporaries was very great, and he used it with Dionysius
on one occasion to save the life of Plato. He was noted
for the attention he paid to the comfort and education of his
slaves and of children in the city. He was drowned in a
shipwreck near Tarentum, and his body washed on shore: a
fit punishment, in the eyes of the more rigid Pythagoreans,
for his having departed from the lines of study laid down by
their founder. Several of the leaders of the Athenian school
were among his pupils and friends, and it is believed that
much of their work was due to his inspiration.
The Pythagoreans at first made no attempt to apply their
knowledge to mechanics, but Archytas is said to have treated
it with the aid of geometry : he is alleged to have invented
and worked out the theory of the pulley, and is credited with
the construction of a flying bird and some other ingenious
mechanical toys. He introduced various mechanical devices
for constructing curves and solving problems : these were
objected to by Plato, who thought that they destroyed the
value of geometry as an intellectual exercise, and later Greek
geometricians confined themselves to the use of two species
of instruments, namely, rulers and compasses. Archytas was
also interested in astronomy ; he taught that the earth was
a sphere rotating round its axis in 24 hours, and round which
the heavenly bodies moved.
Archytas was one of the first to give a solution of the
problem to duplicate a cube, that is, to find the side of a cube
whose volume is double that of a given cube. This was
one of the most famous problems of antiquity (see below,
pp. 38, 42). The construction given by Archytas is equivalent
to the following. On the diameter OA of the base of a right
circular cylinder describe a semicircle whose plane is perpen
dicular to the base of the cylinder. Let the plane containing
this semicircle rotate round the generator through 0, then the
surface traced out by the semicircle will cut the cylinder in a
ARCHYTAS. THEODORUS. 31
tortuous curve. This curve will be cut by a right cone whose
axis is OA and semi- vertical angle is (say) 60 in a point ^such
that the projection of OP on the base of the cylinder will be to
the radius of the cylinder in the ratio of the side of the required
cube to that of the given cube. The proof given by Archytas
is of course geometrical*; it will be enough here to remark
that in the course of it he shews himself acquainted with the
results of the propositions Euc. in. 18, in. 35, and xi. 19.
To shew analytically that the construction is correct, take OA
as the axis of a?, and the generator through as axis of 2, then,
with the usual notation in polar coordinates, and if a be the
radius of the cylinder, we have for the equation of the surface
described by the semicircle, r = 2a sin \ for that of the cylinder,
r sin = 2a cos < ; and for that of the cone, sin 6 cos $ = J.
These three surfaces cut in a point such that sin 3 = |, and
therefore, if p be the projection of OP on the base of the
cylinder, then p 3 = (r sin O) 3 = 2a 3 . Hence the volume of the
cube whose side is p is twice that of a cube whose side is a.
I mention the problem and give the construction used by Ar
chytas to illustrate how considerable was the knowledge of the
Pythagorean school at that time.
Theodoras. Another Pythagorean of about the same date
as Archytas was Tkeodorus of ^Gyrene who is said to have
proved geometrically that the numbers represented by ^/3, ^/5,
x/6, /T, V 8 > V10, v/ n > x/12, V13, V14, V15 and ^17 are
incommensurable with unity. Theaetetus was one of his pupils.
Perhaps Timaeus of Locri and Bryso of Heraclea should be
mentioned as other distinguished Pvthagoreans of this time.
Other Greek Mathematical Schools in the fifth century B.C.
Id be a mistake to suppose that Miletus and Tarentum
were the only places where, in the fifth century, Greeks were
i laying a scientific foundation for the study of
* It is printed by Allman, pp. Ill 113.
32 THE IONIAN AND PYTHAGOREAN SCHOOLS.
mathematics. These towns represented the centres of chief
activity, but there were few cities or colonies of any importance
where lectures on philosophy and geometry were not given.
Among these smaller schools I may mention those at Chios,
Elea, and Thrace.
The best known philosopher of the School of Chios was
(Enopides, who was born about 500 B.C. and died about 430 B.C.
He devoted, himself chiefly to astronomy, but he had studied
geometry in Egypt, and is credited with the solution of the
two problems, namely, (i) to draw a straight line from a given
external point perpendicular to a given straight line (Euc. i. 12),
and (ii) at a given point to construct an angle equal to a given
angle (Euc. i. 23).
Another important centre was at Elea in Italy. This
was founded in Sicily by Xenophanes. He was followed by
Parmenides, Zeno, and Melissus. The members of the Eleatic
School were famous for the difficulties they raised in con
nection with questions that required the use of infinite series,
such for example as the well-known paradox of Achilles and
the tortoise, enunciated by Zeno, one of their most prominent
members, who was born in 495 B.C., and was executed at Elea
in 435 B.C. in consequence of some conspiracy against tfte
state. He was a pupil of Parmenides, with whom he visited
Athens, circ. 455 450 BC.
Zeno argued that if Achilles ran ten times as fast as a
tortoise, yet if the tortoise had (say) 1000 yards start it could
never be overtaken : for, when Achilles had gone the 1000
yards, the tortoise would still be 100 yards in front of him;
by the time he had covered these 100 yards, it would still be
10 yards in front of him ; and so on for ever : thus Achilles
would get nearer and nearer to the tortoise but never overtake
it. The fallacy is usually explained by the argument that the
time required to overtake the tortoise can be divided into
an infinite number of parts, as stated in the question, but
these get smaller and smaller in geometrical progression, and
the sum of them all is a finite time : after the lapse of that
THE ELEATIC AND ATOMISTIC SCHOOLS. 33
time Achilles would be in front of the tortoise. Probably
Zeno would have replied that this argument rests on the
assumption that space is infinitely divisible, which is the
question under discussion ; he himself asserted that magnitudes
were not infinitely divisible.
These paradoxes made the Greeks look with suspicion on
the use of infinite series, and ultimately led to the invention
of the method of exhaustions.
The Atomistic School, having its head-quarters in Thrace,
was another important centre. This was founded by Leucippus,
who was a pupil of Zeno. He was succeeded by Democritus
and Epicurus. Its most famous mathematician was Democritus,
born at Abdera in 460 B.C. and said to have died in 370 B.C.,
who besides his philosophical works wrote on plane and solid
geometry, incommensurable lines, perspective, and numbers.
These works are all lost.
But though several distinguished individual philosophers
may be mentioned who during the fifth century lectured at
different cities, they mostly seem to have drawn their inspi
ration from Tarentum, and towards the end of the century to
have looked to Athens as the intellectual capital of the Greek
world : and it is to the Athenian schools that we owe the next
great advance in mathematics.
B.
CHAPTER III.
THE SCHOOLS OF ATHENS AND CYZICUS*.
CIRC. 420 B.C. 300 B.C.
IT was towards the close of the fifth century before Christ
that Athens first became the chief centre of mathematical
studies. Several causes conspired to bring this about. During
that century she had become, partly by commerce, partly by
appropriating for her own purposes the contributions of her
allies, the most wealthy city in Greece; and the genius of her
statesmen had made her the centre on which the politics of the
peninsula turned. Moreover whatever states disputed her
claim to political supremacy her intellectual pre-eminence was
admitted by all. There was no school of thought which had
not at some time in that century been represented at Athens
by one or more of its leading thinkers ; and the ideas of the
new science, which was being so eagerly studied in Asia Minor
and Graecia Magna, had been brought before the Athenians
on various occasions.
* The history of these schools is discussed at length in G. J. Allman s
Greek Geometry from Thales to Euclid, Dublin, 1889 ; it is also
treated by Cantor, chaps, ix., x., and xi. ; by Hankel, pp. Ill 156;
and by C. A. Bretschneider in his Die Geometrie und die Geometer
vor Eukleides, Leipzig, 1870; a critical account of the original autho
rities is given by P. Tannery in his Geometrie Grecque, Paris, 1887, and
other papers.
ANAXAGORAS. THE SOPHISTS. HIPPIAS. 35
Anaxagoras. Amongst the most important of the philo
sophers who resided at Athens and prepared the way for the
Athenian school I may mention Anaxagoras of Clazomenae,
who was almost the last philosopher of the Ionian school. He
was born in 500 B.C. and died in 428 B.C. He seems to have
settled at Athens about 440 B.C., and there taught the results
of the Ionian philosophy. Like all members of that school
he was much interested in astronomy. He asserted that the
sun was larger than the Peloponnesus: this opinion, together
with some attempts he had made to explain various physical (
phenomena which had been previously supposed to be due to
the direct action of the gods led to a prosecution for impiety,
and he was convicted. While in prison he is said to have
written a treatise on the quadrature of the circle. ,/ ^
The Sophists. The sophists can hardly be considered as
belonging to the Athenian school, any more than Anaxagoras
can; but like him they immediately preceded and prepared the
way for it, so that it is desirable to devote a few words to
them. One condition for success in public life at Athens was
the power of speaking well, and as the wealth and power of
the city increased a considerable number of "sophists" settled
there who undertook amongst other things to teach the art of
oratory. Many of them also directed the general education of ]
their pupils, of which geometry usually formed a part. We
are told that two of those who are usually termed sophists
made a special study of geometry these were Hippias of Elis
and Antipho and one made a special study of astronomy
this was Meton, after whom the metonic cycle is named.
Hippias. The first of these geometricians, Hippias of Elis
(circ. 420 B.C.), is described as an expert arithmetician; but he
is best known to us through his invention of a curve called the
quadratrix, by means of which an angle could be trisected, or
indeed divided in any given ratio. If the radius of a circle
rotate uniformly round the centre from the position OA
through a right angle to OB, and in the same time a straight
line drawn perpendicular to OB move uniformly parallel to
32
36
THE SCHOOLS OF ATHENS AND CYZICUS.
itself from the position OA to BC, the locus of their inter
section will be the quadra trix.
Let OR and MQ be the positions of these lines at any
time; and let them cut in P, a point on the curve. Then
angle AOP : angle AOB=OM : OB.
Similarly, if OR be another position of the radius,
angle AOP : angle AOB = OM : OB.
:. angle AOP : angle AOP ^OM : M ;
. . angle AOP : angle POP = OM : M M.
Hence, if the angle AOP be given, and it be required to
divide it in any given ratio, it is sufficient to divide OM
in that ratio at M , and draw the line M P \ then OP will
divide AOP in the required ratio.
If OA be taken as the initial line, OP=r, the angle AOP=0,
and OA = a, we have 6 : \tr = r sin : a, and the equation of
the curve is 7rr = 2aO cosec 0.
Hippias devised an instrument to construct the curve
mechanically; but constructions which involved the use of any
mathematical instruments except a ruler and a pair of com
passes were objected to by Plato, and rejected by most
geometricians of a subsequent date.
ANTIPHO. BRYSO. 37
Antipho. The second sophist whom I mentioned was
Antipho (circ. 420 B.C.). He is one of the very few writers
among the ancients who attempted to find the area of a circle
by considering it as the limit of an inscribed regular polygon
with an infinite number of sides. He began by inscribing an
equilateral triangle; on each side in the smaller segment he
inscribed an isosceles triangle, and so on ad infinitum.
Bryso. Another mathematician, probably of about the
same time, who attacked the quadrature problem in a similar
way to that used by Antipho was Bryso of Heraclea, who
seems to have been a Pythagorean (see above, p. 31). It is
said that he began by inscribing and circumscribing squares,
and finally obtained polygons between whose areas the area of
the circle lay. It is possible but not probable that for some
time he taught at Athens.
No doubt there were other cities in Greece where similar
and equally meritorious work was being done, though the
record of it has now been lost; I have mentioned the investi
gations of these three writers, partly in order to give an idea
of the kind of work which was then going on all over Greece,
but chiefly because they were the immediate predecessors of
those who created the Athenian school.
The history of the Athenian school begins with the teaching \
of Hippocrates about 420 B.C. ; the school was established on
a permanent basis by the labours of Plato and Eudoxus; and,
together with the neighbouring school of Oyzicus, continued
to extend on the lines laid down by these three geometricians
until the foundation (about 300 B.C.) of the new university
at Alexandria drew most of the talent of Greece thither. ,
Eudoxus, who was among the most distinguished of the
Athenian mathematicians, is also reckoned as the founder of
the school at Cyzicus. The connection between this school
and that of Athens was very close, and it is now impossible
to disentangle their histories. It is said that Hippocrates,
Plato, and Theaetetus belonged to the Athenian school ; while
Eudoxus, Menaechmus, and Aristaeus belonged to that of
38 THE SCHOOLS OF ATHENS AND CYZICUS.
Cyzicus. There was always a constant intercourse between
the two schools, the earliest members of both had been under
the influence either of Archytas or of his pupil Theodorus of
Gyrene, and there was no difference in their treatment of the
subject, so that they may be conveniently treated together.
Before discussing the work of the geometricians of these
schools in detail I may note that they were especially interested
in three problems*: namely, (i) the duplication of a cube,
that is, the determination of the side of a cube whose volume
is double that of a given cube; (ii) the trisection of an angle;
and (iii) the squaring of a circle, that is, the determination
of a square whose area is equal to that of a given circle.
Now the first two of these problems (considered analytically)
require the solution of a cubic equation : and, since a con
struction by means of circles (whose equations are of the form
^ 2 + 2/ 2 + ax + by + c = 0) and straight lines (whose equations are
of the form ax + /3y+y = Q) cannot be equivalent to the
solution of a cubic equation, the problems are insoluble if in
our constructions we restrict ourselves to the use of circles and
straight lines, i.e. to Euclidean geometry. If the use of the
conic sections be permitted, both of these questions can be
solved in many ways. The third problem is equivalent to
finding a rectangle whose sides are equal respectively to the
radius and to the semiperimeter of the circle. These lines
have been long known to be incommensurable, but it is only
recently that it has been shewn by Lindemann that their ratio
cannot be the root of a rational algebraical equation. Hence
this problem also is insoluble by Euclidean geometry. The
Athenians and Cyzicians were thus destined to fail in all three
problems, but the attempts to solve them led to the discovery
of many new theorems and processes. Besides attacking these
problems the later Platonic school collected all the geometrical
theorems then known and arranged them systematically. These
* On these problems, solutions of them, and the authorities for their
history, see my Mathematical Recreations and Problems, London, 1892,
chap. vin.
HIPPOCRATES. 39
collections comprised the bulk of the propositions in Euclid s
Elements, books I. ix., XL, and xn., together with some of
the more elementary theorems in conic sections.
Hippocrates. Hippocrates of Chios (who must be carefully
distinguished from his contemporary, Hippocrates of Cos, the
celebrated physician) was one of the greatest of the Greek
geometricians. He was born about 470 B.C. at Chios, and
began life as a merchant. The accounts differ as to whether
he was swindled by the Athenian custom-house officials who
were stationed at the Chersonese, or whether one of his
vessels was captured by an Athenian pirate near Byzantium ;
but at any rate somewhere about 430 B.C. he came to Athens
to try to recover his property in the law courts. A foreigner
was not likely to succeed in such a case, and the Athenians
seem only to have laughed at him for his simplicity, first in
allowing himself to be cheated, and then in hoping to recover
his money. While prosecuting his cause he attended the
lectures of various philosophers, and finally (in all probability
to earn a livelihood) opened a school of geometry himself. He
seems to have been well acquainted with the Pythagorean
philosophy, though there is no sufficient authority for the
statement that he was ever initiated as a Pythagorean.
He wrote the first elementary text-book of geometry, a
text-book on which Euclid s Elements was probably founded;
and therefore he may be said to have sketched out the lines
on which geometry is still taught in English schools. It is
supposed that the use of letters in diagrams to describe a
figure was made by him or introduced about his time, as he i
employs expressions such as "the point on which the letter
A stands" and "the line on which AB is marked." Cantor
however thinks that the Pythagoreans had previously been
accustomed to represent the five vertices of the pentagram-
star by the letters v y i a (see above, p. 21); and though this
was a single instance, they may perhaps have used the method
generally. The Indian geometers never employed letters to aid
them in the description of their figures. Hippocrates also
40 THE SCHOOLS OF ATHENS AND CYZICUS.
denoted the square on a line by the word SiW/u?, and thus
gave the technical meaning to the word power which it still
retains in algebra: there is reason to think that this use of the
word was derived from the Pythagoreans, who are said to have
enunciated the result of the proposition as Euc. i. 47, in the
form that "the total power of the sides of a right-angled
triangle is the same as that of the hypothenuse."
In this text-book Hippocrates introduced the method of
"reducing" one theorem to another, which being proved, the
thing proposed necessarily follows; of which plan the reductio
ad absurdum is a particular case. No doubt the principle had
been used occasionally before, but he drew attention to it as
a legitimate mode of proof which was capable of numerous
applications. He may be said to- have introduced the geometry
of the circle. He discovered that similar segments of a circle
contain equal angles; that the angle subtended by the chord of
a circle is greater than, equal to, or less than a right angle
as the segment of the circle containing it is less than, equal
to, or greater than a semicircle (Euc. in. 31); and probably
several other of the propositions in the third book of Euclid.
It is most likely that he also enunciated the propositions that
[similar] circles are to one another as the squares of their
diameters (Euc. xn. 2), and that similar segments are as the
squares of their chords. The proof given in Euclid of the first
of these theorems is believed to be due to Hippocrates, but the
latter mathematician does not seem to have realized that all
circles are similar.
The most celebrated discoveries of Hippocrates were how
ever in connection with the quadrature of the circle and the
duplication of the cube, and it was owing to his influence that
these problems played such a prominent part in the history of
the Athenian school.
The following propositions will sufficiently illustrate the
method by which he attacked the quadrature problem.
(a) He commenced by finding the area of a lune contained
between a semicircle and a quadrantal arc standing on the
AREA OF A LUNE.
same chord. This he did as follows. Let ABC be an isosceles
right-angled triangle inscribed in the semicircle ABOC whose
B O C
centre is 0. On AB and AC as diameters describe semicircles
as in the figure. Then, since BC 2 = AC 2 + AB 2 (Euc. I. 47),
therefore, by Euc. xn. 2,
area \ on BC = sum of areas of -| Q s on A C and AB.
Take away the common parts
.-. area A ABC = sum of areas of lunes AECD and AFBG.
Hence the area of the lime AECD is equal to half that of the
triangle ABC.
(/3) He next inscribed half a regular hexagon A BCD in P
}
semicircle whose centre was 0, and on OA, AB, BC, and CD
as diameters described semicircles of which those on OA and
42 THE SCHOOLS OF ATHENS AND CYZICUS.
AB are drawn in the figure. Then AD is double any of the
lines OA, AB, BC and CD,
. -. area ABOD=sum of areas of s on OA, AB, BC, and CD.
Take away the common parts
. . area trapezium A BCD = 3 lime AEBF + -J-O on OA.
If therefore the area of this latter lune be known, so is that of
the semicircle described on OA as diameter. According to
Simplicius, Hippocrates assumed that the area of this lune was
the same as the area of the lune found in proposition (a); if this
" be so, he was of course mistaken, as in this case he is dealing
with a lune contained between a semicircle and a sextantal
arc standing on the same chord; but it seems probable that
Simplicius misunderstood Hippocrates.
Hippocrates also enunciated various other theorems con
nected with lunes (which have been collected by Bretsch-
neider and by Allman) of which the theorem last given is a
typical example. I believe that they are the earliest instances
in which areas bounded by curves were determined by geometry.
The other problem to which Hippocrates turned his atten
tion was the duplication of the cube, that is, the determination
of the side of a cube whose volume is double that of a given
cube.
Th - Mem was known in ancient times as the Delian
proble } - jirn consequence of a legend that the Delians had
consul teu Jr lato on the subject. In one form of the story,
which is related by Philoponus, it is asserted that the
Athenians in 430 B.C., wnen suffering from the plague of
eruptive typhoid fever, consulted the oracle at Delos as to
how they could stop it. Apollo replied that they must
double the size of his altar which was in the form of a cube.
To the unlearned suppliants nothing seemed more easy, and
a new altar was constructed either having each of its edges
double that of the old one (from which it followed that the
HIPPOCRATES. PLATO. 43
volume was increased eight-fold) or by placing a similar cubic
altar next to the old one. Whereupon, according to the
legend, the indignant god made the pestilence worse than before,
arid informed a fresh deputation that it was useless to trifle with
him, as his new altar must be a cube and have a volume exactly
double that of his old one. Suspecting a mystery the Athenians /
applied to Plato, who referred them to the geometricians,
and especially to Euclid, who had made a special study of the
problem. The introduction of the names of Plato and Euclid
is an obvious anachronism. Eratosthenes gives a somewhat
similar account of its origin, but with king Minos as the pro-
pounder of the problem.
Hippocrates reduced the problem of duplicating the cube
to that of finding two means between one straight line (a),
and another twice as long (2a). If these means be x and
?/, we have a : x = x : y = y : 2a. from which it follows that
x 3 = 2a 3 . It is in this form that the problem is always pre
sented now. Formerly any process of solution by finding
these means was called a mesolabum. Hippocrates did not
succeed in finding a construction for these means.
Plato. The next philosopher of the Athenian school who
requires mention here was Plato, who was born at Athens in
429 B.C. He was, as is well known, a pupil for eight yean
of Socrates, and much of the teaching of the latter is inferred
from Plato s dialogues. After the execution of his master in
399 B.C. Plato left Athens, and being possessed of - " .< Table
wealth he spent some years in travelling : it was : ^ this
time that he studied mathematics. He visited Egypt with
Eudoxus, and Strabo says that in his time the apartments they
occupied at Heliopolis were still shewn. Thence Plato went
to Gyrene, where he studied under Theodorus. Next he moved
to Italy, where he became intimate with Archytas the then
head of the Pythagorean school, Eurytas of Metapontum, and
Timaeus of Locri. He returned to Athens about the year
380 B.C., and formed a school of students in a suburban gym
nasium called the "Academy." He died in 348 B.C.
44 THE SCHOOLS OF ATHENS AND CYZICUS.
Plato, like Pythagoras, was primarily a philosopher-, and
perhaps his philosophy should be regarded as founded on the
Pythagorean rather than on the Socratic teaching. At any
rate it, like that of the Pythagoreans, was coloured with the
idea that the secret of the universe was to be found in
number and in form ; hence, as Eudemus says, " he exhibited
on every occasion the remarkable connection between mathe
matics and philosophy." All the authorities agree that, unlike
many later philosophers, he made a study of geometry or
some exact science an indispensable preliminary to that of
philosophy. The inscription over the entrance to his school
ran " Let none ignorant of geometry enter my door," and on
one occasion an applicant who knew no geometry is said to
have been refused admission as a student.
Plato s position as one of the masters of the Athenian
mathematical school rests not so much on his individual dis
coveries and writings as on the extraordinary influence he
exerted on his contemporaries and successors. Thus the ob
jection that he expressed to the use in the construction of
curves of any instruments other than rulers and compasses
was at once accepted as a canon which must be observed in
such problems. It is probably due to Plato that subsequent
geometricians began the subject with a carefully compiled series
of definitions, postulates, and axioms. He also systematized
the methods which could be used in attacking mathematical
questions, and in particular directed attention to the value of
analysis. The analytical method of proof begins by assuming
that the theorem or problem is solved, and thence deducing
some result : if the result be false, the theorem is not true or
the problem is incapable of solution : if the result be known to
be true, and if the steps be reversible, we get (by reversing
them) a synthetic proof; but if the steps be not reversible,
no conclusion can be drawn. Numerous illustrations of the
method will be found in any modern text-book on geometry.
If the classification of the methods of legitimate induction
given by Mill in his work on logic had been universally ac-
PLATO. EUDOXUS. 45
cepted and every new discovery in science had been justified
by a reference to the rules there laid down, he would, I
imagine, have occupied a position in reference to modern
science somewhat analogous to that which Plato occupied in
regard to the mathematics of his time.
Almost the only extant instance of a mathematical theorem
attributable to Plato is the following, which is traditionally
assigned to him. If CAB and DAB be two right-angled
triangles, having one side, AB, common, their other sides,
AD and BC, parallel, and their hypothenuses, AC and BD,
at right angles, then, if these hypothenuses cut in P, we have
PC : PB = PB : PA = PA : PD. This theorem was used in
duplicating the cube, for, if such triangles can be constructed
having PD = 2P(7, the problem will be solved. It is easy to
make an instrument by which the figure can be drawn.
Eudoxus*. Of Eudoxus, the third great mathematician of
the Athenian school and the founder of that at Cyzicus, we
know very little. He was born in Cnidus in 408 B.C. Like
Plato, he went to Tarentum and studied under Archytas the
then head of the Pythagoreans. Subsequently he travelled
with Plato to Egypt, and then settled at Cyzicus where he
founded the school of that name. Finally he and his pupils
moved to Athens. There he seems to have taken some part
in public affairs, and to have practised medicine ; but the
hostility of Plato and his own unpopularity as a foreigner
made his position uncomfortable, and he returned to Cyzicus
or Cnidus shortly before his death. He died while on a journey
to Egypt in 355 B.C.
His mathematical work seems to have been of a high order
of excellence. He discovered most of what we now know as
the fifth book of Euclid, and proved it in much the same
form as that in which it is there given.
* The discoveries of Eudoxus have been discussed in considerable
detail by H. Kiinssberg of Dinkelsbiibl, in addition to the authors
mentioned above in the footnote on p. 34.
46 THE SCHOOLS OF ATHENS AND CYZTCUS.
He discovered some theorems on what was called " the
golden section." The problem to
cut a line AB in the golden section, A H ./ B
that is, to divide it, say at H, in
extreme and mean ratio (i.e. so that AB : AH = AH : HE] is
solved in Euc. n. 11, and probably was known to the Pythago
reans at an early date. If we denote AB by I, AH by a, and
HB by 6, the theorems that Eudoxus proved are equivalent
to the following algebraical identities, (i) (a + ^l) 2 - 5 (J) 2 .
(Euc. xni. 1.) (ii) Conversely, if (i) be true, and AH be
taken equal to a, then AB will be divided at H in a golden
section. (Euc. xni. 2.) (iii) (b + %a) 2 = 5 (Ja) 2 . (Euc. xm. 3.)
(iv) l 2 + b 2 = 3a*. (Euc. xni. 4.) (v) .1 + a : 1 = 1 :a, which
gives another golden section. (Euc. xm. 5.) These propo
sitions were subsequently put by Euclid at the commence
ment of his thirteenth book, but they might have been
equally well placed towards the end of the second book. All
of them are obvious algebraically, since l=a + b and a 2 = bl.
Eudoxus further established the "method of exhaustions ; "
which depends on the proposition that "if from the greater
of two unequal magnitudes there be taken more than its half,
and from the remainder more than its half, and so on, there
will at length remain a magnitude less than the least of the
proposed magnitudes." This proposition was placed by Euclid
as the first proposition of the tenth book of his Elements,
but in most modern school editions it is printed at the
beginning of the twelfth book. By the aid of this theorem
the ancient geometers were able to avoid the use of infini
tesimals : the method is rigorous, but awkward of application.
A good illustration of its use is to be found in the demon
stration of Euc. xn. 2, namely, that the square of the radius
of one circle is to the square of the radius of another circle
as the area of the first circle is to an area which is neither
less nor greater than the area of the second circle, and
which therefore must be exactly equal to it : the proof given
by Euclid (as was usual) is completed by a rediictio ad
EUDOXUS. 47
absurdum. Eudoxus applied the principle to shew that the
volume of a pyramid (or a cone) is one-third that of the prism
(or cylinder) on the same base and of the same altitude (Euc.
xn. 7 and 10). Some writers attribute the proposition Euc.
xn. 2 to him, and not to Hippocrates.
Eudoxus also considered certain curves other than the
circle, but there seems to be no authority for the statement,
which is found in some old books, that he studied the
properties of the conic sections. He discussed some of the
plane sections of the anchor ring, that is, of the solid gene
rated by the revolution of a circle round a straight line lying
in its plane ; but he assumed that the line did not cut the
circle. A section by a plane through this line consists of
two circles ; if the plane be moved parallel to itself the sec
tions are lemniscates ; when the plane first touches the surface
the section is a " figure of eight," generally called Bernoulli s
lenmiscate, whose equation is r 2 = a 2 cos 20. All this is ex
plained at length in books on solid geometry. Eudoxus
applied these curves to explain the apparent progressive and
retrograde motions of the planets, but we do not know the
method he used.
Eudoxus constructed an orrery, and wrote a treatise on
practical astronomy, in which he adopted a hypothesis pre
viously propounded by Philolaus (409 B.C. 356 B.C.), and
supposed a number of moving spheres to which the sun,
moon, and stars were attached, and which by their rotation
produced the effects observed. Jn all he required twenty-
seven spheres. As observations became more accurate, sub
sequent astronomers who accepted his theory had continually
to introduce fresh spheres to make the theory agree with
the facts. The work of Aratus on astronomy, which was
written about 300 B.C. and is still extant, is founded on that
of Eudoxus.
Plato and Eudoxus were contemporaries. Among Plato s
pupils were the mathematicians Leodamas, Neocleides, Amyclas,
and to their school also belonged Leon, Theudius (both of whom
48 THE SCHOOLS OF ATHENS AND CYZICUS.
wrote text-books on plane geometry), Cyzicenus, Thasus
Hermotimus, Philippus, and Theaetetus. Among the pupils
of Eudoxus are reckoned Menaechmus, his brother Dino-
stratus (who applied the quadratrix to the duplication and
trisection problems), and Aristaeus.
Menaechmus. Of the above-mentioned mathematicians
Menaechmus requires special mention. He was born about
375 B.C. and died about 325 B.C. He was a pupil of Eudoxus,
and probably succeeded him as head of the school at Cyzicus.
Menaechmus acquired great reputation as a teacher of geo-
1 metry, and was for that reason appointed one of the tutors
to Alexander the Great. In answer to Alexander s request
to make his proofs shorter, he made the well-known reply, " In
the country, sire, there are private and even royal roads, but
in geometry there is only one road for all."
Menaechmus was the first to discuss the conic sections,
which were long called the Menaechmian triads. He divided
them into three classes, and investigated their properties, not
by taking different plane sections of a fixed cone, but by
keeping his plane fixed and cutting it by different cones. He
shewed that the section of a right cone by a plane perpen
dicular to a generator is an ellipse, if the cone be acute-
angled ; a parabola, if it be right-angled ; and a hyperbola, if
it be obtuse-angled ; and he gave a mechanical construction
for curves of each class. It seems almost certain that he was
acquainted with the fundamental properties of these curves;
but some writers think that he failed to connect them with
the sections of the cone which he had discovered, and there
is no doubt that he regarded the latter not as plane loci but
as curves drawn on the surface of a cone.
He also shewed how these curves could be used in either-
of the two following ways to give a solution of the problem
to duplicate a cube. In the first of these, he pointed out that
two parabolas having a common vertex, axes at right angles,
and such that the latus rectum of the one is double that of
the other will intersect in another point whose abscissa (or
MENAECHMUS. ARISTAEUS. THEAETETUS. 49
ordinate) will give a solution : for (using analysis) if the equa
tions of the parabolas be y* = 2ax and x 2 = ay, they intersect in
a point whose abscissa is given by x 3 = 2a*. It is probable
that this method was suggested by the form in which Hip
pocrates had cast the problem : namely, to find x and y so
that a : x x : y y : 2a, whence we have y? ay and if 2ax.
The second solution given by Menaechmus was as follows.
Describe a parabola of latus rectum I. Next describe a rect
angular hyperbola, the length of whose real axis is 4, and
having for its asymptotes the tangent at the vertex of the
parabola and the axis of the parabola. Then the ordinate and
the abscissa of the point of intersection of these curves are
the mean proportionals between I and 21. This is at once
obvious by analysis. The curves are x 2 = ly and xy 2l 2 .
These cut in a point determined by x s = 2F and ?/ 3 = 4 3 .
Hence I : x x : y = y : 21.
Aristaeus and Theaetetus. Of the other members of
these schools the only mathematicians of first-rate power were
Aristaeus and Theaetetus, whose works are entirely lost. We
know however that Aristaeus wrote on the five regular solids
and on conic sections, and that Theaetetus developed the
theory of incommensurable magnitudes. The only theorem
we can now definitely ascribe to the latter is that given by
Euclid in the ninth proposition of the tenth book of the
Elements, namely, that the squares on two commensurable
right lines have one to the other a ratio which a square
number has to a square number (and conversely); but the
squares on two incommensurable right lines have one to the
j other a ratio which cannot be expressed as that of a square
! number to a square number (and conversely). This theorem
includes the results given by Theodorus (see above, p. 31).
The contemporaries or successors of these mathematicians
wrote some fresh text-books on the elements of geometry and
ithe conic sections, introduced problems concerned with finding
lloci, and efficiently carried out the work commenced by Plato
of systematizing the knowledge already acquired.
Aristotle. An account of the Athenian school would be
B.
50 THE SCHOOLS OF ATHENS AND CYZICUS.
incomplete if there were no mention of Aristotle, who was born
at Stagira in Macedonia in 384 B.C. and died at Chalcis in
Euboea in 322 B.C. Aristotle however, deeply interested
though he was in natural philosophy, was chiefly concerned
with mathematics and mathematical physics as supplying illus
trations of correct reasoning. A small book containing a few
questions on mechanics which is sometimes attributed to him
is of doubtful authority; but, though in all probability it is due
to another writer, it is interesting, partly as shewing that the
principles of mechanics were beginning to excite attention, and
partly as containing the earliest known employment of letters
to indicate magnitudes.
The most instructive parts of the book are the dynamical
proof of the parallelogram of forces for the direction of the
resultant, and the statement that "if a be a force, ft the mass to
which it is applied, y the distance through which it is moved,
and 8 the time of the motion, then a will move ^/3 through
2y in the time 8, or through y in the time J8": but the author
goes on to say that "it does not follow that ^a will move /3
through |y in the time 8, because Ja may not be able to move
/3 at all; for 100 men may drag a ship 100 yards, but it does not
follow that one man can drag it one yard." The first part of
this statement is correct and is equivalent to the statement
that an impulse is proportional to the momentum produced,
but the second part is wrong.
The author also states the fact that what is gained in
power is lost in speed, and therefore that two weights which
keep a [weightless] lever in equilibrium are inversely pro
portional to the arms of the lever ; this, he says, is the
explanation why it is easier to extract teeth with a pair of
pincers than with the fingers.
Among other questions raised, but not answered, are why
a projectile should ever stop, and why carriages with large
wheels are easier to move than those with small. I ought to
add that the book contains some gross blunders, and as a whole
is not as able or suggestive as might be inferred from the
above extracts.
51
CHAPTER IV.
THE FIRST ALEXANDRIAN SCHOOL*.
CIRC. 300 B.C. 30 B.C.
THE earliest attempt to found a university, as we understand
the word, was made at Alexandria. Hichly endowed, supplied
with lecture rooms, libraries, museums, laboratories, gardens,
and all the plant and machinery that ingenuity could suggest,
it became at once the intellectual metropolis of the Greek race,
and remained so for a thousand years. It was particularly
fortunate in producing within the first century of its existence
three of the greatest mathematicians of antiquity Euclid,
Archimedes, and Apollonius. They laid down the lines on
which mathematics were subsequently studied; and, largely
owing to their influence, the history of mathematics centres
more or less round that of Alexandria until the destruction
of the city by the Arabs in 641 A.D.
* The history of the Alexandrian schools is discussed by Cantor,
chaps, xii. xxin. ; and by J. Gow in his interesting History of Greek
Mathematics, Cambridge, 1884. The subject of Greek algebra is treated
by E. H. F. Nesselmann in his Die Algebra der Griechen, Berlin, 1842;
see also L. Matthiessen, Grundziige der antiken und modernen Algebra
der litteralen Gleichungen, Leipzig, 1878. The Greek treatment of the
conic sections forms the subject of a recent work by H-G. Zeuthen
entitled Die Lehre von den Kegelschnitten in Altertum, Copenhagen,
1886. The materials for the history of these schools have been subjected
to a searching criticism by P. Tannery, and most of his papers are
collected in his Geometrie Grecque, Paris, 1887.
42
52 THE FIRST ALEXANDRIAN SCHOOL.
The city and university of Alexandria were created under
the following circumstances. Alexander the Great had as
cended the throne of Macedonia in 336 B.C. at the early age of
20, and by 332 B.C. he had conquered or subdued Greece, Asia
Minor, and Egypt. Following the plan he adopted whenever
a commanding site had been left unoccupied, he founded a new
city on the Mediterranean near one mouth of the Nile ;
and he himself sketched out the ground-plan, and arranged
for drafts of Greeks, Egyptians, and Jews to be sent to occupy
it. The city was intended to be the most magnificent in the
world, and, the better to secure this, its erection was left in the
hands of Dinocrates, the architect of the temple of Diana at
Ephesus.
After Alexander s death in 323 B.C. his empire was divided,
and Egypt fell to the lot of Ptolemy, who chose Alexandria
as the capital of his kingdom. A short period of confusion
followed, but as soon as Ptolemy was settled on the throne, say
about 306 B.C., he determined to attract, as far as he was able,
learned men of all sorts to his new city; and he at once began
the erection of the university buildings on a piece of ground
immediately adjoining his palace. The university was ready to
be opened somewhere about 300 B.C., and Ptolemy, who wished
to secure for its staff the most eminent philosophers of the time,
naturally turned to Athens to find them. The great library
which was the central feature of the scheme was placed under
Demetrius Phalereus, a distinguished Athenian ; and so rapidly
did it grow that within 40 years it (together with the Egyptian
annexe) possessed about 600,000 rolls. The mathematical de
partment was placed under Euclid, who was thus the first, as
he was one of the most famous, of the mathematicians of the
Alexandrian school.
It happens that contemporaneously with the foundation of
this school the information on which our history is based be
comes more ample and certain. Many of the works of the
Alexandrian mathematicians are still extant; and we have
besides an invaluable treatise by Pappus, described below, in
EUCLID. 53
which their best known treatises are collated, discussed, and
criticized. It curiously turns out that just as we begin to be
able to speak with certainty on the subject-matter which was
taught, we find that our information as to the personality of
the teachers becomes uncertain; and we know very little of
the lives of the mathematicians mentioned in this and the next
chapter, even the dates at which they lived being frequently
uncertain.
The third century before Christ.
Euclid*. This century produced three of the greatest
mathematicians of antiquity, namely Euclid, Archimedes, and
Apollonius. The earliest of these was Euclid. Of his life we
know next to nothing, save that he was of Greek descent,
and was born about 330 B.C.; he died about 275 B.C. It would
appear that he was well acquainted with the Platonic geometry,
but he does not seem to have read Aristotle s works ; and these
facts are supposed to strengthen the tradition that he was
educated at Athens. Whatever may have been his previous
training and career, he proved a most successful teacher when
settled at Alexandria. He impressed his own individuality on
the teaching of the new university to such an extent that to
his successors and almost to his contemporaries the name Euclid
* Besides Cantor, chaps, xn. xin., and Gow, pp. 72 86, 195 221,
see the article Eucleides by A. De Morgan in Smith s Dictionary of Greek
and Roman Biography, London, 1849 ; the article on Irrational Quantity
by A. De Morgan in the Penny Cyclopaedia, London, 1839 ; and Litterar-
geschichtliche Studien fiber Euklid, by J. L. Heiberg, Leipzig, 1882.
The latest complete edition of all Euclid s works is that by J. L. Heiberg
and H. Menge in Teubner s library at Leipzig, 18831887. An English
translation of the thirteen books of the Elements was published by
J. Williamson in 2 volumes, Oxford, 1781, and London, 1788, but the
notes are not always reliable : there is another translation by Isaac
Barrow, London and Cambridge, 16f>0.
54 THE FIRST ALEXANDRIAN SCHOOL.
meant (as it does to us) the book or books he wrote, and not
the man himself. Some of the mediaeval writers went so far
as to deny his existence, and with the ingenuity of philologists
they explained that the term was only a corruption of VK\L a
key, and Sis geometry. The former word was presumably
derived from K\LS. I can only explain the meaning assigned
to Sis by the conjecture that as the Pythagoreans said that
the number two symbolized a line possibly a schoolman
may have thought that it could be taken as indicative of
geometry.
From the meagre notices of Euclid which have come down
to us we find that the saying that there is no royal road to
geometry was attributed to Euclid as well as to Menaechmus;
but it is an epigrammatic remark which has had many imi
tators. Euclid is also said to have insisted that knowledge
was worth acquiring for its own sake, and Stobaeus (who is a
somewhat doubtful authority) tells us that when a lad who
had just begun geometry asked "What do I gain by learning
all this stuff? 7 Euclid made his slave give the boy some
coppers, " since," said he, " he must make a profit out of what
he learns."
According to Pappus^ Euclid, in making use of the work
of his predecessors when writing the Elements, dealt most
gently with those who had in any way advanced the science:
and the Arabian writers, who may perhaps convey to us the
traditions of Alexandria, uniformly represent him as a gentle
and kindly old man.
Euclid was the author of several works, but his reputation
has rested mainly on his Elements. This treatise contains a
systematic exposition of the leading propositions of elementary
geometry (exclusive of conic sections) and of the theory of
numbers. It was at once adopted by the Greeks as the
standard text-book for the elements of pure mathematics, and
it is probable that it was written for that purpose and not as a
philosophical attempt to shew that the results of geometry
and arithmetic are necessary truths.
EUCLID. 55
The modern text* is founded on an edition prepared by
Theon, the father of Hypatia, and is practically a transcript of
Theon s lectures at Alexandria (circ. 380 A.D.). There is at
the Vatican a copy of an older text, and we have besides
quotations from the work and references to it by numerous
writers of various dates. From these sources we gather that
the definitions, axioms, and postulates were re-arranged and
slightly altered by subsequent editors, but that the propositions
themselves are substantially as Euclid wrote them.
As to the matter of the work. The geometrical part is to
a large extent a compilation from the works of previous writers.
Thus the substance of books I. and n. is probably due to
Pythagoras; that of book in. to Hippocrates; that of book v.
to Eudoxus; and the bulk of books iv., vi., XL, and xu. to
the later Pythagorean or Athenian schools. But this material
was re-arranged, obvious deductions were omitted (e.g. the
proposition that the perpendiculars from the angular points of
a triangle on the opposite sides meet in a point was cut out),
and in some cases new proofs substituted. (The part con^
cerned with the theory of numbers would seem to have been
taken from the works of Eudoxus and Pythagoras, except that
portion (book x.) which deals with irrational magnitudes.
This latter may be founded on the lost book of Theaetetus ;
but much of it is probably original, for Proclus says that while
Euclid arranged the propositions of Eudoxus he completed many
of those of Theaetetus.
The way in which the propositions are proved, consisting of
enunciation, statement, construction, proof, and conclusion, is
due to Euclid: so also is the synthetical character of the work,
each proof being written out as a logically correct train of
reasoning but without any clue to the method by which it was
obtained.
* Most of the modern text-books in English are founded on Simson s
edition, issued in 1758. Robert Simson, who was born in 1687 and died
in 1768, was professor of mathematics at the university of Glasgow, and
left several valuable works on ancient geometry.
56 THE FIRST ALEXANDRIAN SCHOOL.
The defects of Euclid s Elements as a text-book of geometry
have been often stated ; the most prominent are these, (i) The
definitions and axioms contain many assumptions which are
not obvious, and in particular the so-called axiom about parallel
lines is not self-evident*, (ii) No explanation is given as
to the reason why the proofs take the form in which they are
presented, that is, the synthetical proof is given but not the
analysis by which it was obtained, (iii) There is no attempt
made to generalize the results arrived at, for instance, the idea
of an angle is never extended so as to cover the case where it
is equal to or greater than two right angles : the second half
of the 33rd proposition in the sixth book, as now printed,
appears to be an exception ; but it is due to Theon and not to
Euclid, (iv) The principle of superposition as a method of
proof might be used more frequently with advantage, (v) The
classification is imperfect. And (vi) the work is unnecessarily
long and verbose.
On the other hand, the propositions in Euclid are arranged
so as to form a chain of geometrical reasoning, proceeding from
certain almost obvious assumptions by easy steps to results of
considerable complexity. The demonstrations are rigorous,
often elegant, and not too difficult for a beginner. Lastly,
nearly, all the elementary metrical (as opposed to the graphical)
properties of space are investigated. The fact that for two
thousand years it has been the recognized text-book on the
subject raises further a strong presumption that it is not
unsuitable for the purpose. During the last few years some
determined efforts have been made to displace it in our schools,
but the majority of teachers still appear to regard it as the
best foundation for geometrical teaching that has been yet pub
lished. The book has been however generally abandoned on
the continent, though apparently with doubtful advantage to the
teaching of geometry. To these arguments in its favour may
* It would seem from the recent researches of Grassmann, Riemann,
and Lobatschewsky that it is incapable of proof : see passim my Mathe
matical Recreations and Problems, London, 1892, chap. x.
EUCLID. 57
be added the fact that some of the greatest mathematicians of
modern times, such as Descartes, Pascal, Newton, and Lagrange,
have advocated its retention as a text-book: and Lagrange
said that he who did not study geometry in Euclid would be as
one who should learn Latin and Greek from modern works
written in those tongues. It must be also remembered that
there is an immense advantage in having a single text-book in
universal use in a subject like geometry. The unsatisfactory
condition of the teaching of geometrical conies in schools is a
standard illustration of the evils likely to arise from using
different text-books in such ,a subject. Some of the objections
urged against Euclid do not apply to certain of the recent
school editions of his Elements.
I do not think that all the objections above stated can
fairly be urged against Euclid himself. He published a
collection of problems generally known as the AeSo/xeVa or
Data. This contains 95 illustrations of the kind of deductions
which frequently have to be made in analysis ; such as that, if
one of the data of the problem under consideration be that one
angle of some triangle in the figure is constant, then it is
legitimate to conclude that the ratio of the area of the rectangle
under the sides containing the angle to the area of the triangle
is known (prop. 66). Pappus says that the work was written for
those " who wish to acquire the power of solving problems."
It is in fact a graduated series of exercises in analysis ; and
this seems a sufficient answer to the second objection.
Euclid also wrote a work called Ilept AtaipeVecoi or De
Divisionibus, which is known to us only through an Arabic
translation which may be itself imperfect. This is a collection
of 36 problems on the division of areas into parts which bear
to one another a given ratio. It is not unlikely that this was
only one of several such collections of examples possibly
including the Fallacies and the Porisms but even by itself it
shews that the value of exercises and riders was fully recognized
by Euclid.
I may here add a suggestion thrown out by De Morgan,
58 THE FIRST ALEXANDRIAN SCHOOL.
who is perhaps the most acute of all the modern critics of
Euclid. He thinks it likely that the Elements were written
towards the close of Euclid s life, and that their present form
represents the first draft of the proposed work, which, with the
exception of the tenth book, Euclid did not live to revise. If
this opinion be correct, it is probable that Euclid would in
his revision have removed the fifth objection.
The geometrical* parts of the Elements are so well known
that I need do no more than allude to them. The first four
books and book vi. deal with plane geometry; the theory of
proportion (of any magnitudes) is discussed in book v. ; and
books xi. and XH. treat of solid geometry. On the hypothesis
that the Elements are the first draft of Euclid s proposed
work, it is possible that book xiu. is a sort of appendix
containing some additional propositions which would have
been put ultimately in one or other of the earlier books.
Thus, as mentioned above (see p. 46), the first five propositions
which deal with a line cut in golden section might be added to
the second book. The next seven propositions are concerned
with the relations between certain incommensurable lines in
plane figures (such as the radius of a circle and the sides of an
inscribed regular triangle, pentagon, hexagon, and decagon)
which are treated by the methods of the tenth book and as an
illustration of them. The five regular solids are discussed in
the last six propositions. Bretschn eider is inclined to think
that the thirteenth book is a summary of part of the lost work
of Aristaeus : but the illustrations of the methods of the tenth
book are due most probably to Theaetetus.
* Euclid supposed that his readers had the use of a ruler and a pair
of compasses. Lorenzo Mascheroni (who was born at Castagneta on
May 14, 1750, and died at Paris on July 30, 1800) set himself the task to
obtain by means of constructions made only with a pair of compasses
the same results as Euclid had given. Mascheroni s treatise on the
geometry of the compass which was published at Pavia in 1795 is
so curious a tour de force that it is worth chronicling. He was pro
fessor first at Bergamo and afterwards at Pavia, and left numerous minor
works.
EUCLID. 59
Books vii. , viii., ix., and x. of the Elements are given up
to the theory of numbers. The mere art of calculation or
AoyioriK^ was taught to boys when quite young, it was stig
matized by Plato as childish, and never received much attention
from Greek mathematicians ; nor was it regarded as forming
part of a course of mathematics. We do not know how it was
taught, but the abacus certainly played a prominent part in it.
The scientific treatment of numbers was called apitf/x^riK?/,
which I have here generally translated as the science of num
bers. It had special reference to ratio, proportion, and the
theory of numbers. It is with this alone that most of the
extant Greek works deal.
- HJI discussing Euclid s arrangement of the subject, we must
therefore bear in mind that those who attended his lectures
were already familiar with the art of calculation. The system
of numeration adopted by the Greeks is described later (see
below, chap, vii.), but it was so clumsy that it rendered the
scientific treatment of numbers much more difficult than that
of geometry; hence Euclid commenced his mathematical course
with plane geometry. At the same time it must be observed
that the results of the second book though geometrical in form
are capable of expression in algebraical language, and the fact
that numbers could be represented by lines was probably
insisted on at an early stage, and illustrated by concrete
examples. This graphical method of using lines to represent
numbers possesses the obvious advantage of leading to proofs
which are true for all numbers, rational or irrational. It will
be noticed that among other propositions in the second book
we get geometrical proofs of the distributive and commutative
laws, of rules for multiplication, and finally geometrical solu
tions of the equations a (a x) = x 2 , that is, x 2 + ax a 2 =
(Euc. II. 11), and x 2 ab = Q (Euc. n. 14): the solution of
the first of these equations is given in the form \J a 2 + (%a) 8 - \a.
The solutions of the equations ax 2 bx + c = and ax*+ bx-c=0
are given later in Euc. vi. 28 and vi. 29; the cases when
a 1 can be deduced from the identities proved in Euc. IT.
60 THE FIRST ALEXANDRIAN SCHOOL.
5 and 6, but it is doubtful if Euclid would have detected
this.
The results of the fifth book in which the theory of propor
tion is considered apply to any magnitudes, and therefore are
true of numbers as well as of geometrical magnitudes. In the
opinion of many writers this is the easiest way of treating
the theory of proportion on a scientific basis; and it was used
by Euclid as the foundation on which he built the theory of
numbers. The theory of proportion given in this book is
believed to be due to Eudoxus. The treatment of the same
subject in the seventh book is less elegant, and is supposed
to be a reproduction of the Pythagorean teaching. This
double discussion of proportion is, as far as it goes, in
favour of the conjecture that Euclid did not live to revise
the work.
In books vii., viii., and ix. Euclid discusses the theory of
rational numbers. He commences the seventh book with some
definitions founded on the Pythagorean notation. In propo
sitions 1 to 3 he shews that if, in the usual process for finding
the greatest common, measure of two numbers, the last divisor
be unity, the numbers must be prime; and he thence deduces
the rule for finding their G.C.M. Propositions 4 to 22 include
the theoiy of fractions, which he bases on the theory of pro
portion; among other results he shews that ab = ba (prop. 16).
In propositions 23 to 34 he treats of prime numbers, giving
many of the theorems in modern text-books on algebra. In
propositions 35 to 41 he discusses the least common multiple
of numbers, and some miscellaneous problems.
The eighth book is chiefly devoted to numbers in continued
proportion, i.e. in a geometrical progression ; and the cases
where one or more is a product, square, or cube are specially
considered.
In the ninth book Euclid continues the discussion of geo
metrical progressions, and in proposition 35 he enunciates the
rule for the summation of a series of n terms, though the
proof is given only for the case where n is equal to 4. He
EUCLID. 61
also develops the theory of primes, shews that the number of
primes is infinite (prop. 20), and discusses the properties of
odd and even numbers. He concludes by shewing how to
construct a "perfect" number (prop. 36).
In the tenth book Euclid treats of irrational magnitudes ;
and, since the Greeks possessed no symbolism for surds, he was
forced to adopt a geometrical representation. Propositions 1
to 21 deal generally with incommensurable magnitudes. The
rest of the book, namely, propositions 22 to 117, is devoted to
the discussion of every possible variety of lines which can be
represented by J( >J.a ^6), where a and b denote commensur
able lines. There are twenty-five species of such lines, and
that Euclid could detect and classify them all is in the opinion
of so competent an authority as Nesselmann the most striking
illustration of his genius. It seems at first almost impossible
that this could have been done without the aid of algebra, but
it is tolerably certain that it was actually effected by abstract
reasoning. No further advance in the theory of incom
mensurable magnitudes was made until the subject was taken
up by Leonardo and Cardan after an interval of more than a
thousand years.
In the last proposition of the tenth book (x. 117) the side
and diagonal of a square are proved to be incommensurable.
The proof is so short and easy that I may quote it. If
possible let the side be to the diagonal in a commensurable
ratio, namely, that of the two integers a and b. Suppose this
ratio reduced to its lowest terms so that a and b have no
common divisor other than unity, that is, they are prime to
one another. Then (by Euc. i. 47) b 2 = 2a 2 ; therefore b 2 is an
even number; therefore b is an even number; hence, since a is
prime to 6, a must be an odd number. Again, since it has
been shewn that b is an even number, b may be represented
by 2?i; therefore (2n) 2 = 2a 2 j therefore a 2 = 2?i 2 ; therefore a 2
is an even number; therefore a is an even number. Thus the
same number a must be both odd and even, which is absurd;
therefore the side and diagonal are incommensurable. Hankel
62 THE FIRST ALEXANDRIAN SCHOOL.
believes that this proof was due to Pythagoras, and was
inserted on account of its historical interest. This proposition
is also proved in another way in Euc. x. 9.
In addition to the Elements and the two collections of
riders above mentioned (which are extant) Euclid wrote the
following books on geometry : (i) an elementary treatise on
conic sections in four books; (ii) a book on curved surfaces
(probably chiefly the cone and cylinder); (iii) a collection of
geometrical fallacies, which were to be used as exercises in the
detection of errors; and (iv) a treatise on porisms arranged in
three books. All of these are lost, but the work on porisms was
discussed at such length by Pappus, that some writers have
thought it possible to restore it. In particular Chasles in 1860
published what purports to be a reproduction of it, in which
will be found the conceptions of cross ratios and projection
in fact those ideas of modern geometry which Chasles and other
writers of this century have used so largely. This is brilliant
and ingenious, and of course no one can prove that it is not
exactly what Euclid wrote, but the statements of Pappus con
cerning this book have come to us only in a mutilated form,
and De Morgan frankly says that he found them unintelligible,
an opinion in which most of those who read them will, I think,
concur.
Euclid published two books on optics, namely the Optics
and the Catoptrica. Of these the former is extant. A work
which purports to be the latter exists in the form of an Arabic
translation, but there is some doubt as to whether it repre
sents the original work written by Euclid ; in any case, the
text is extraordinarily corrupt. The Optics commences with
the assumption that objects are seen by rays emitted from the
eye in straight lines, "for if light proceeded from the object
we should not, as we often do, fail to perceive a needle on the
floor." It contains 61 propositions founded on 12 assumptions.
The Catoptrica consists of 31 propositions dealing with reflex
ions in plane, convex, and concave mirrors. The geometry of
both books is ingenious.
EUCLID. ARISTARCHUS. 63
Euclid also wrote the Phaenomena, a treatise on geometrical
astronomy. It contains references to the work of Autolycus*
and to some book on spherical geometry by an unknown
writer. Pappus asserts that Euclid also composed a book on
the elements of music : this may refer to the Sectio Canonis
which is by Euclid, and deals with musical intervals.
To these works I may add the following little problem,
which occurs in the Palatine Anthology and is attributed by
tradition to Euclid. "A mule and a donkey were going to
market laden with wheat. The mule said If you gave me
one measure I should carry twice as much as you, but if I
gave you one we should bear equal burdens. Tell me, learned
geometrician, what were their burdens." It is impossible to
say whether the question is genuine, but it is the kind of
question he might have asked.
It will be noticed that Euclid dealt only with magnitudes,
and did not concern himself with their numerical measures,
but it would seem from the works of Aristarchus and Archi
medes that this was not the case with all the Greek mathe
maticians of that time. As one of the works of the former
is extant it will serve as another illustration of Greek mathe
matics of this period.
Aristarchus. Aristarchus of Samos, born, in 310 B.C. and
died in 250 B.C., was an astronomer rather than a mathema
tician. He asserted, at any rate as a working hypothesis, that
the sun was the centre of the universe, and that the earth
revolved round the sun. This view, in spite of the simple
explanation it afforded of various phenomena, was generally
rejected by his contemporaries. But his propositions t on the
* Autolycus lived at Pitane in Aeolis and flourished about 330 B.C.
His two works on astronomy, containing 43 propositions, are the oldest
extant Greek mathematical treatises. They exist in manuscript at
Oxford. A Latin translation has been edited by F. Hultsch, Leipzig, 1885.
t Ilept fjieytOw /ecu aTrocrr^/xdra;* HXtou /ecu ZeA?Jj/?7s, edited by E. Nizze,
Stralsund, 1856. Latin translations were issued by F. Commandino in
1572 and by J. Wallis in 1688 ; and a French translation was published
by F. d Urban in 1810 and 1823.
04 THE FIRST ALEXANDRIAN SCHOOL.
measurement of the sizes and distances of the sun and moon
were accurate in principle, and his results were generally ac
cepted (for example, by Archimedes in his ^a/x/xn-^s, see below,
p. 73) as approximately correct. There are 19 theorems,
of which I select the seventh as a typical illustration, because
it shews the way in which the Greeks evaded the difficulty
of finding the numerical value of surds.
Aristarchus observed the angular distance between the
moon, when dichotomized and the sun, and found it to be
twenty-nine thirtieths of a right angle. It is actually about
89 21 , but of course his instruments were of the roughest
description. He then proceeded to shew that the distance of
the sun is greater than eighteen and less than twenty times
the distance of the moon in the following manner.
Let S be the sun, E the earth, and M the moon. Then
when the moon is dichotomized, that is, when the bright part
which we see is exactly a half -circle, the angle between MS
and ME is a right angle. With E as centre, and radii ES
and EM describe circles, as in the figure above. Draw EA
perpendicular to ES. Draw EF bisecting the angle AES, and
AUISTAKCHUS. ARCHIMEDES. G5
EG bisecting the angle AEF, as in the figure. Let EM (pro
duced) cut AF in //. The angle AEM is by hypothesis ^th
of a right angle. Hence we have
angle AEG : angle AEH = ^ rt. L\ ^rt. L = 15 : 2,
.-. AG :AH[=t*nAEG:t&nAEII]>l5 : 2 (a)
Again FG* : AG 2 = ,B7 72 : EA* (Euc. vi. 3)
= 2:1 (Euc. i. 47),
.*. J^6? a :AG*>3 : 25,
.-. 7 T :AG >1 : 5,
.-. AF : AG>12 : 5,
.-. AE :^^>12:5 ().
Compounding the ratios (a) and (/?), we have
4A 1 : AU> 18 : 1.
But the triangles EMS and ^^17/ are similar,
.-. ES : EM>IS : 1.
I will leave the second half of the proposition to amuse any
reader who may care to prove it: the analysis is straightfor
ward. In a somewhat similar way Aristarchus found the ratio
of the radii of the sun, earth, and moon.
We know very little of Conon and Dositheus, the imme
diate successors of Euclid at Alexandria, or of their contem
poraries Zeuxippus and Nicoteles, who most likely also lectured
there, except that Archimedes, who was a student at Alexandria
probably shortly after Euclid s death, had a high opinion of
their ability and corresponded with the three first mentioned.
Their work and reputation has been overshadowed completely
by that of Archimedes whose marvellous mathematical powers
have been surpassed only by those of Newton.
Archimedes*. Archimedes^ who probably was related to
* Besides Cantor, chaps, xiv. , xv., and Gow, pp. 221 244, see
Quacslhmi t Archimedeae, by J. L. Heibcrg, Copenhagen, 1879 ; and Marie,
vol. i., pp. 81 134. The latest and best edition of the extant works of
Archimedes is that by J. L. lleiberg, in 3 vols., Leipzig, 18801881.
B. 5
66 THE FIRST ALEXANDRIAN SCHOOL.
the royal family at Syracuse, was born there in 287 B.C. and
died in 212 B.C. He went to the university of Alexandria
and attended the lectures of Conon but, as soon as he had
finished his studies, returned to Sicily where he passed the
remainder of his life. He took no part in public affairs, but
his mechanical ingenuity was astonishing, and, on any diffi
culties which could be overcome by material means arising, his
advice was generally asked by the government.
Archimedes, like Plato, held that it, was undesirable for
a philosopher to seek to apply the results of science to any
practical use ; but, whatever might have been his view of what
ought to be the case, he did actually introduce a large number
of new inventions. The stories of the detection of the fraudu
lent goldsmith and of the use of burning glasses to destroy the
ships of the Roman blockading squadron will recur to most
readers. " Perhaps it is not as well known that Hiero, who had
built a ship so large that he could not launch it off the
slips, applied to Archimedes. The difficulty was overcome
by means of an apparatus of cogwheels worked by an endless
screw, but we are not told exactly how the machine was used.
It is said that it was on this occasion, in acknowledging the
compliments of Hiero, that Archimedes made the well-known
remark that had he but a fixed fulcrum he could move the
earth. Most mathematicians are aware that the Archimedean
screw was another of his inventions. It consists of a tube,
open at both ends, and bent into the form of a spiral like a
cork-screw. If one end be immersed in water, and the axis of
the instrument (i.e. the axis of the cylinder on the surface of
which the tube lies) be inclined to the vertical at a sufficiently
big angle, and the instrument turned round it, the water will
flow along the tube and out at the other end. In order that
it may work, the inclination of the axis of the instrument to
the vertical must be greater than the pitch of the screw. It
was used in Egypt to drain the fields after an inundation of
the Nile ; and was also frequently applied to pump water out
of the hold of a ship. The story that Archimedes set fire to
ARCHIMEDES. G7
the Roman ships by means of burning glasses and concave
mirrors is not mentioned till some centuries after his death,
and is generally rejected : but it is not so incredible as is com
monly supposed. The mirror of Archimedes is said to have
been made of a hexagon surrounded by several polygons, each
of 24 sides; and Buffon* in 1747 contrived, with the aid of
a single composite mirror made on this model with 168 small
mirrors, to set fire to wood at a distance of 150 feet, and to
melt lead at a distance of 140 feet. This was in April and in
Paris, so in a Sicilian summer and with several mirrors the
deed would be possible, and if the ships were anchored near
the town would not be difficult. It is perhaps worth mention
ing that a similar device is said to have been used in the
defence of Constantinople in 514 A.D., and is alluded to by
writers who either were present at the siege or obtained their
information from those who were engaged in it. But what
ever be the truth as to this story, there is no doubt that
Archimedes devised the catapults which kept the Romans,
who were then besieging Syracuse, at bay for a considerable
time. These were constructed so that the range could be made
either short or long at pleasure, and so that they could be
discharged through a small loophole without exposing the
artillerymen to the fire of the enemy. So effective did they
prove that the siege was turned into a blockade, and three
years elapsed before the town was taken (212 B.C.).
Archimedes was killed during the sack of the city which
followed its capture, in spite of the orders, given by the consul
Marcellus who was in command of the Romans, that his house
and life should be spared. It is said that a soldier entered his
study while he was regarding a geometrical diagram drawn in
sand on the floor, which was the usual way of drawing figures
in classical times. Archimedes told him to get off the diagram,
and not spoil it. The soldier, feeling insulted at having orders
given to him and ignorant of who the old man was, killed him.
* See Memoires de Vacaddmie royale des sciences for 1747, Paris,
1752, pp. 82101.
52
68 THE FIRST ALEXANDRIAN SCHOOL.
According to another and more probable account, the cupidity
of the troops was excited by seeing his instruments, constructed
of polished brass which they supposed to be made of gold.
The Romans erected a splendid tomb to Archimedes on
which was engraved (in accordance with a wish he had ex
pressed) the figure of a sphere inscribed in a cylinder, in com
memoration of the proof he had given that the volume of a
sphere was equal to two-thirds that of the circumscribing
right cylinder, and its surface to four times the area of a^great
circle. Cicero* gives a charming account of his efforts (which
were successful) to re-discover the tomb in 75 B.C.
It is difficult to explain in a concise form the works or
discoveries of Archimedes, partly because he wrote on nearly
all the mathematical subjects then known, and partly because
his writings are contained in a series of disconnected mono
graphs. Thus, while Euclid aimed at producing systematic
treatises which could be understood by all students who had
attained a certain level of education, Archimedes wrote a
number of brilliant essays addressed chiefly to the most educated
mathematicians of the day. The work for which he is perhaps
now best known is his treatment of the mechanics of solids
and fluids; but he and his contemporaries esteemed his geo
metrical discoveries of the quadrature of a parabolic area and
of a spherical surface, and his rule for finding the volume of a
sphere as more remarkable; while at a somewhat later time his
numerous mechanical inventions excited most attention.
(i) On plane geometry the extant works of Archimedes are
three in number, namely, (a) the Measure of the Circle, (b)
the Quadrature of the Parabola, and (c) one on Spirals.
(a) The Measure of the Circle contains three propositions.
In the first proposition Archimedes proves that the area is the
same as that of a right-angled triangle whose sides are equal
respectively to the radius a and the circumference of the circle,
i.e., the area is equal to \a (2?ra). In the second proposition
* See his Tusc. Disput., v. 23.
ARCHIMEDES. 69
he shews that ira* : (laf - 11 : 14 very nearly; and next, in
the third proposition, that TT is less than 3| and greater than
3yy. These theorems are of course proved geometrically. To
demonstrate the two latter propositions, he inscribes in and
circumscribes about a circle regular polygons of ninety-six
sides, calculates their perimeters, and then assumes the cir
cumference of the circle to lie between them. It would seem
from the proof that he had some (at present unknown) method
of extracting the square roots of numbers approximately.
(6) The Quadrature of the Parabola contains twenty-four
propositions. Archimedes begins this work, which was sent
to Dositheus, by establishing some properties of conies (props.
1 5). He then states correctly the area cut off from a para
bola by any chord, and gives a proof which rests on a pre
liminary mechanical experiment of the ratio of areas which
balance when suspended from the arms of a lever (props. 6
17) ; and lastly he gives a geometrical demonstration of this
result (props. 18 24). The latter is of course based on the
method of exhaustions,- but for brevity I will, in quoting it,
use the method of limits.
Let the area of the parabola (see figure on next page) be
bounded by the chord PQ. Draw VM the diameter to the
chord PQ, then (by a previous proposition), V is more remote
from PQ than any other point in the arc PVQ. Let the area
of the triangle PVQ be denoted by A. In the segments
bounded by VP and VQ inscribe triangles in the same way as
the triangle PVQ was inscribed in the given segment. Each of
these triangles is (by a previous proposition of his) equal to ^A,
and their sum is therefore ^A. Similarly in the four segments
left inscribe triangles ; their sum will be yV-^- Proceeding in
this way the area of the given segment is shewn to be equal to
the limit of
A A A
+ 4 + 16 + + 4~" + -
when n is indefinitely large.
The problem is therefore reduced to finding the sum
70
THE FIRST ALEXANDRIAN SCHOOL.
of a geometrical series. This he effects as follows. Let
A, B, (7, ..., Jj K be a series of magnitudes such that each
is one fourth of that which precedes it. Take magnitudes
6, c, ..., k equal respectively to B, i(7, ..., K. Then
... + J);
Hence (5+ (7 + ... +JT) + (6 + c + ... + &) - %(A +
but, by hypothesis, (6 + c + ... +j + k) = ^(fi+C + .
Hence the sum of these magnitudes exceeds four times the
third of the largest of them by one-third of the smallest of
them.
Returning now to the problem of the quadrature of the
parabola A stands for A, and ultimately K is indefinitely
small ; therefore the area of the parabolic segment is four-
thirds that of the triangle PVQ, or two-thirds that of a rect
angle whose base is PQ and altitude the distance of V from PQ.
While discussing the question of quadratures it may be
ARCHIMEDES. 71
added that in the fifth and sixth propositions of his work on
conoids and spheroids he determined the area of an ellipse/^
(c) The work on Spirals contains twenty-eight proposi
tions on the properties of the curve now known as the spiral
of Archimedes. It was sent toDositheus at Alexandria accom
panied by a letter, from which it appears that Archimedes had
previously sent a note of his results to Conon, who had died
before he had been able to prove them. The spiral is defined
by saying that the vectorial angle and radius vector both in
crease uniformly, hence its equation is r = cO. Archimedes
finds most of its properties, and determines the area inclosed
between the curve and two radii vectores. This he does (in
effect) by saying, in the language of the infinitesimal cal
culus, that an element of area is > J r 2 dO and < J (r + drf dO :
to effect the sum of the elementary areas he gives two lemmas
in which he sums (geometrically) the series a 2 + (2a) 2 + (3a) 2 +
... 4- (no) 2 (prop. 10), and a -f 2a + 3& + ... + na (prop. 11).
(d) In addition to these he wrote a small treatise on
geometrical methods, and works on parallel lines, triangles, the
properties of right-angled triangles, data, the heptagon inscribed
in a circle, and systems of circles touching one another ; possibly
he wrote others too. These are all lost, but it is probable that
fragments of four of the propositions in the last mentioned
work are preserved in a Latin translation from an Arabic
manuscript entitled Lemmas of Archimedes.
(ii) On geometry of three dimensions the extant works
of Archimedes are two in number, namely, (a) the Sphere and
Cylinder, and (b) Conoids and Spheroids.
(a) The Sphere and Cylinder contains sixty propositions
arranged in two books. Archimedes sent this like so many
j of his works to Dositheus at Alexandria ; but he seems to
; have played a practical joke on his friends there, for he pur-
; posely misstated some of his results " to deceive those vain
geometricians who say they have found everything but never
give their proofs, and sometimes claim that they have discovered
| what is impossible." He regarded this work as his master-
72 THE FIRST ALEXANDRIAN SCHOOL.
piece. It is too long for me to give an analysis of its contents,
but I remark in passing that in it he finds expressions for the
surface and volume of a pyramid, of a cone, and of a sphere,
as well as of the figures produced by the revolution of polygons
inscribed in a circle about a diameter of the circle. There are
several other propositions on areas and volumes of which perhaps
the most striking is the tenth proposition of the second book,
namely that "of all spherical segments whose surfaces are
equal the hemisphere has the greatest volume." In the second
proposition of the second book he enunciates the remarkable
theorem that a line of length a can be divided so that
a x : b = 4a 2 : 9# 2 (where b is a given length), only if b be
less than i; that is to say, the cubic equation x 3 -ax 2 + a*b = Q
can have a real and positive root only if a be greater than 35.
This proposition was required to complete his solution of the
problem to divide a given sphere by a plane so that the volumes
of the segments should be in a given ratio. One very simple
cubic equation occurs in the Arithmetic of Diophantus, but
with that exception no such equation appears again in the
history of European mathematics for more than a thousand
years.
(6) The Conoids and Spheroids contains forty propositions
on quadrics of revolution (sent to Dositheus in Alexandria)
mostly concerned with an investigation of their volumes.
(c) Archimedes also wrote a treatise on the thirteen semi-
regular polyhedrons, that is, solids contained by regular but
dissimilar polygons. This is lost.
(iii) On arithmetic, Archimedes wrote two papers. One
. (addressed to Zeuxippus) was on the principles of numeration ;
this is now lost. The other (addressed to Gelon) was called
^a/x/xt-n/s (the sand-reckoner), and in this he meets an objection
which had been urged against his first paper.
The object of the first paper had been to suggest a con
venient system by which numbers of any magnitude could be
represented ; and it would seem that some philosophers at Syra
cuse had doubted whether the system was practicable. Archime-
ARCHIMEDES, 73
des says people talk of the sand on the Sicilian shore as some
thing beyond the power of calculation, but he can estimate it, and
further he will illustrate the power of his method by finding a
superior limit to the number of grains of sand which would fill
the whole universe, i.e. a sphere whose centre is the earth, and
radius the distance of the sun. He begins by saying that in
ordinary Greek nomenclature it was only possible to express
numbers from 1 up to 10 8 : these are expressed in what he
says he may call units of the first order. If 10 8 be termed a
unit of the second order, any number from 10 8 to 10 18 can be
expressed as so many units of the second order plus so many
units of the first order. If 10 16 be a unit of the third order
any number up to 10 24 can be then expressed ; and so on.
Assuming that 10000 grains of sand occupy a sphere whose
radius is not less than -g^th of a finger breadth, and that the
diameter of the universe is not greater than 10 10 stadia, he finds
that the number of grains of sand required to fill the universe
is less than 10 63 .
Probably this system of numeration was suggested merely as
a scientific curiosity. The Greek system of numeration with
which we are acquainted had been only recently introduced,
most likely at Alexandria, and was sufficient for all the purposes
for which the Greeks then required numbers ; and Archimedes
used that system in all his papers. On the other hand it has been
conjectured that Archimedes and Apollonius had some symbolism
based on the decimal system for their own investigations, and it
is possible that it was the one here sketched out. The units
suggested by Archimedes form a geometrical progression,
having 10 8 for the radix. He incidentally adds that it will
be convenient to remember that the product of the mth and ?ith
terms of a geometrical progression, whose first term is unity, is
equal to the (ra + n)fh term of the series, i.e. that r m x r n - r m+n .
To these two arithmetical papers, I may add the following
celebrated problem which he sent to the Alexandrian mathe
maticians. The sun had a herd of bulls and cows, all of
which were either white, grey, dun, or piebald : the number
74 THE FIRST ALEXANDRIAN SCHOOL.
of piebald bulls was less than the number of white bulls by
5/6ths of the number of grey bulls, it was less than the
number of grey bulls by 9/20ths of the number of dun bulls,
and it was less than the number of dun bulls by 13/42nds
of the number of white bulls : the number of white cows was
7/12ths of the number of grey cattle (bulls and cows), the
number of grey cows was 9/20ths of the number of dun
cattle, the number of dun cows was ll/30ths of the number
of piebald cattle, and the number of piebald cows was 13/42nds
of the number of white cattle. The problem was to find the
composition of the herd. The problem is indeterminate, but
the solution in lowest integers is
white bulls, ....... 10,366,482; white cows, 7,206,360;
grey bulls, 7,460,514; grey cows, 4,893,246;
dun bulls, 7,358,060; dun cows, 3,515,820;
piebald bulls, 4,149,387; piebald cows, 5,439,213.
In the classical solution, attributed to Archimedes, these num
bers are multiplied by 80.
Nesselmann believes, from internal evidence, that the pro
blem has been falsely attributed to Archimedes. It certainly
is unlike his extant work, but it was attributed to him among
the ancients, and is generally thought to be genuine though
possibly it has come down to us in a modified form. It is
in verse, and a later copyist has added the additional con
ditions that the sum of the white and grey bulls shall be a
square number, and the sum of the piebald and dun bulls a
triangular number.
It is perhaps worthy of note that in the enunciation the
fractions are represented as a sum of fractions whose numera
tors are unity : thus Archimedes wrote 1/7 + 1/6 instead of
13/42, in the same way as Ahmes would have done (see above,
p. 4).
(iv) On mechanics the extant works of Archimedes are
two in number, namely, (a) his Mechanics, and (c) his Hydro
statics,
ARCHIMEDES. 75
a) The Mechanics is a work on statics with special refer
ence to the equilibrium of plane laminas and to properties of
their centres of gravity ; it consists of twenty-five propositions
in two books. In the first part of book I. most of the ele
mentary properties of the centre of gravity are proved (props.
1 8); and in the remainder of book I. (props. 9 15) and in
book II. the centres of gravity of a variety of plane areas, such
as parallelograms, triangles, trapeziums, and parabolic areas,
are determined.
(b) Archimedes also wrote a treatise on levers and perhaps
on all the mechanical machines. The book is lost, but we
know from Pappus that it contained a discussion of how a
given weight could be moved with a given power. It was in
this work probably that Archimedes discussed the theory of
a certain compound pulley consisting of three or more simple
pulleys which he had invented and which was used in some
public works in Syracuse. It is well known that he boasted
that, if he had but a fixed fulcrum, he could move the whole
^arth (see above, p. 66); and a commentator of later date
ays that he added he would do it by using a compound pulley.
(c) His work vn floating bodies contains nineteen proposi-
ions in two books, and was the first attempt to apply mathe
matical reasoning to hydrostatics. The story of the manner in
hich his attention was directed to the subject is told by
Vitruvius. Hiero, the king of Syracuse, had given some gold
o a goldsmith to make into a crown. The crown was delivered,
made up, and of the proper weight, but it was suspected that
he workman had appropriated a considerable proportion of the
*old, replacing it by an equal weight of silver. Archimedes was
hereupon consulted. Shortly afterwards, when in the public
>aths, he noticed that his body was pressed upwards by a force
which increased the more completely he was immersed in the
water. Recognizing the value of the observation, he rushed
)ut, just as he was, and ran home through the streets, shouting
tvprjKa, "I have found it, I have found it." There (to
ollow a later account) on making accurate experiments he
76 THE FIRST ALEXANDRIAN SCHOOL.
found that when equal weights of gold and silver were weighed
in water they no longer appeared equal : each seemed lighter
than before by the weight of the water it displaced, and as the
silver was more bulky than the gold its weight was more
diminished. Hence, if on a balance hfc weighed the crown
against an equal weight of gold and then immersed the whole
in water, the gold would outweigh the crown if any silver had
been used in its construction. Tradition says that the gold
smith was found to be fraudulent-^/
Archimedes began the work by proving that the surface of
a fluid at rest is spherical, the centre of the sphere being at the
centre of the earth. He then proved that the pressure of the
fluid on a body, wholly or partially immersed, is equal to the
weight of the fluid displaced ; and thence found the position
of equilibrium of a floating body, which he illustrated by
spherical segments and paraboloids of revolution floating on a
fluid. Some of the latter problems involve geometrical reason
ing of great complexity.
The following is a fair specimen of the questions considered.
A solid in the shape of a paraboloid of revolution of height h
and latus rectum 4a floats in water, with its vertex immersed
and its base wholly above the surface. If equilibrium be
possible when the axis is not vertical, then the density of the
body must be less than (h - 3a) 2 /ti* (book n. prop. 4). When
it is recollected that Archimedes was unacquainted with
trigonometry or analytical geometry, the fact that he could
discover and prove a proposition such as that just quoted will
serve as an illustration of his powers of analysis.
As an illustration of the influence of Archimedes on the
history of mathematics I may mention that the science of
statics rested on his theory of the lever until 1586 when
Stevinus published his treatise on statics; and no distinct
advance was made in the theory of hydrostatics until Stevinus
in the same work investigated the laws which regulate the
pressure of fluids (see below, p. 248).
(v) We know, both from occasional references in his works
ARCHIMEDES. APOLLONIUS. 77
and from remarks by other writers, that Archimedes was largely
occupied in astronomical observations. He wrote a book, Ilepi
o"<^)tpo7rotta9, on the construction of a celestial sphere, which is
lost ; and he constructed a sphere of the stars, and an orrery.
These after the capture of Syracuse were taken by Marcellus
to Rome, and were preserved as curiosities for at least two or
three hundred years.
This mere catalogue of his works will shew how wonderful
were his achievements ; but no one who has not actually read
some of his writings can form a just appreciation of his extra
ordinary ability. This will be still further increased if we
recollect that the only principles used by Archimedes, in
addition to those contained in Euclid s Elements and Conic
sections, are that of all lines like
AGE, ADB, ... connecting two
points A and B, the straight line
is the shortest, and of the curved
lines, the inner one ALE is A B
shorter than the outer one AGE\ together with two similar
statements for space of three dimensions.
In the old and mediaeval world Archimedes was unanimously
reckoned as the first of mathematicians : and in the modern world
there is no one but Newton who can be compared with him.
Perhaps the best tribute to his fame is the fact that those
writers who have spoken most highly of his work and ability
are those who have been themselves the most distinguished men
of their own generation.
Apollonius * . The third great mathematician of this century
was Apollonius of Perga, who is chiefly celebrated for having
produced a systematic treatise on the conic sections which not
* In addition to Zeuthen s work and the other authorities mentioned
in the footnote on p. 51, see Litterargeschichtliche Studien iibcr Euklid,
by J. L. Heiberg, Leipzig, 1882. A collection of the extant works of
Apollonius was issued by E. Halley, Oxford, 1706 and 1710: a new
edition of the conies with a critical commentary is now being issued by
J. L. Heiberg.
78 THE FIRST ALEXANDRIAN SCHOOL.
only included all that was previously known about them but
immensely extended the knowledge of these curves. This work
was accepted at once as the standard text-book on the subject,
and completely superseded the previous treatises of Menaech-
mus, Aristaeus, and Euclid which until that time had been in
general use.
We know very little of Apollonius himself. He was born
about 260 B.C. and died about 200 B.C. He studied in
Alexandria for many years, and probably lectured there ; he
is represented by Pappus as "vain, jealous of the reputation
of others, and ready to seize every opportunity to depreciate
them." It is curious that while we know next to nothing
of his life, or of that of his contemporary Eratosthenes, yet
their nicknames, which were respectively epsilon and beta,
have come down to us. Dr Gow has ingeniously suggested
that the lecture rooms at Alexandria were numbered, and
that they always used the rooms numbered 5 and 2 respec
tively.
Apollonius spent some years at Pergamum in Pamphylia,
where a university had been recently established and endowed
in imitation of that at Alexandria. There he met Eudemus
and Attains to whom he subsequently sent each book of his
conies as it came out with an explanatory note. He returned
to Alexandria, and lived there till his death, which was nearly
contemporaneous with that of Archimedes.
In his great work on conic sections he so thoroughly
investigated the properties of these curves that he left but
little for his successors to add. But his proofs are long and
involved, and I think most readers will be content to accept
a short analysis of his work, and the assurance that his
demonstrations are valid. Dr Zeuthen believes that many of
the properties enunciated were obtained in the first instance
by the use of coordinate geometry, and that the demonstrations
were translated subsequently into a geometrical form. If this
be so, we must suppose that the classical writers were familiar
with some branches of analytical geometry Dr Zeuthen says
APOLLONIUS. 79
the use of orthogonal and oblique coordinates, and of transfor
mations depending on abridged notation that this knowledge
was confined to a limited school, and was finally lost. This
is a mere conjecture and is unsupported by any direct evidence,
but it has been accepted by many critics as affording an ex
planation of the extent and arrangement of the work.
The treatise contained about four hundred propositions
and was divided into eight books ; we have the Greek text of
the first four of these, and we also possess copies of the
commentaries by Pappus and Eutocius on the whole work.
In the ninth century an Arabic translation was made of the
first seven books, which were the only ones then extant;
we have two manuscripts of this version. The eighth book
is lost.
In the letter to Eudemus which accompanied the first book
Apollonius says that he undertook the work at the request of
Naucrates, a geometrician who had been staying with him
t Alexandria, and, though he had given some of his friends a
ough draft of it, he had preferred to revise it carefully before
snding it to Pergamum. In the note which accompanied the
ext book, he asks Eudemus to read it and communicate it to
thers who can understand it, and in particular to Philonides
certain geometrician whom the author had met at Ephesus.
The first four books deal with the elements of the subject,
nd of these the first three are founded on Euclid s previous
pork (which was itself based on the earlier treatises by
^Tenaechmus and Aristaeus). Heracleides asserts that much
f the matter in these books was stolen from an unpublished
r ork of Archimedes, but a critical examination by Heiberg
as shewn that this is improbable.
Apollonius begins by defining a cone on a circular base.
le then investigates the different plane sections of it, and
lews that they are divisible into three kinds of curves which
e calls ellipses, parabolas, and hyperbolas. He proves the
reposition that, if A, A be the vertices of a conic and if P be
ny point on it and PM the perpendicular drawn from P on
80
THE FIRST ALEXANDRIAN SCHOOL.
AA , then (in the usual notation) the ratio MP 2 : AM . MA is
constant in an ellipse or hyperbola,
and the ratio MP 2 : AM is constant
in a parabola. These are the charac
teristic properties on which almost
all the rest of the work is based.
He next shews that, if A be the
vertex, I the latus rectum, and if
AM and MP be the abscissa and
ordinate of any point on a conic,
then MP 2 is less than, equal to, or
greater than I . AM according as
the conic is an ellipse, parabola, or
hyperbola ; hence the names which he gave to the curves and
by which they are still known.
^He had no idea of the directrix, and was not aware that
the parabola had a focus, but, with the exception of the propo
sitions which involve these, his first three books contain most
of the propositions which are found in modern text-books.
In the fourth book he develops the theory of lines cut
harmonically, and treats of the points of intersection of systems
of conies. In the fifth book he commences with the theory of
maxima and minima; applies it to find the centre of curva
ture at any point of a conic, and the evolute of the curve;
and discusses the number of normals which can be drawn
from a point to a conic. In the sixth book he treats of
similar conies. The seventh and eighth books were given up
to a discussion of conj ugate diameters, the latter of these was
conjecturally restored by E. Halley in 1710.
The verbose and tedious explanations make the book re
pulsive to most modern readers ; but the logical arrangement
and reasoning are unexceptionable, and it has been not unfitly
described as the crown of Greek geometry. It is the work on
which the reputation of Apollonius rests, and it earned for him
the name of " the great geometrician."
Besides this immense treatise he wrote numerous shorter
APOLLONIUS. 81
works ; of course the books were written in Greek, but they
are usually referred to by their Latin titles : those about which
we now know anything are enumerated below. He was
the author of a work on the problem " given two co-planar
straight lines Aa and Bb, drawn through fixed points A and B;
to draw a line Gab from a given point outside them cutting
them in a and 6, so that A a shall be to Bb in a given ratio " :
he reduced the question to seventy-seven separate cases and
gave an appropriate solution, with the aid of conies, for each
case; this was published by E. Halley (translated from an Arabic
copy) in 1706. He also wrote a treatise De Sectione Spatii
(restored by E. Halley in 1706) on the same problem under
the condition that the rectangle Aa . Bb was given. He
wrote another entitled De Sectione Determinates (restored by
R. Simson, Glasgow, 1749), dealing with problems such as to
find a point P in a given straight line AB so that PA 2 shall
be to PB in a given ratio. He wrote another De Tactionibus
(restored by Yieta in 1600 ; see below, p. 238) on the construc
tion of a circle which shall touch three given circles. Another
work was his De Inclinationibus (restored by M. Ghetaldi,
Venice, 1607) on the problem to draw a line so that the
intercept between two given lines, or the circumferences of two
given circles, shall be of a given length. He was also the
author of a treatise in three books on plane loci, De Locis Planis,
(restored by Fermat in 1637, and by R. Simson in 1746), and
of another on the regular solids. And lastly he wrote a treatise
on unclassed incommensurableSj being a commentary on the
tenth book of Euclid. It is believed that in one or more of
the lost books he used the method of conical projections.
Besides these geometrical works he wrote on the methods of
arithmetical calcidation. All that we know of this is derived
from some remarks of Pappus. Friedlein thinks that it was
merely a sort of ready-reckoner. It would however seem that
Apollonius here suggested a system of numeration similar to
that proposed by Archimedes (see above, p. 73), but proceeding
by tetrads instead of octads, and described a notation for it.
B. 6
82
THE FIRST ALEXANDRIAN SCHOOL.
It will be noticed that our modern notation goes by hexads,
a million = 10 6 , a billion = 10 12 , a trillion = 10 18 , &c. It is not
impossible that Apollonius also pointed out that a decimal
system of notation, involving only nine symbols, would facilitate
numerical multiplications.
Apollonius was interested in astronomy, and wrote a book
on the stations and regressions of the planets of which Ptolemy
made some use in writing the Almagest. He also wrote a
treatise on the use and theory of the screw in statics.
This is a long list, but I should suppose that most of these
works were short tracts on special points.
Like so many of his predecessors he too gave a construction
for finding two mean proportionals between two given lines, and
thereby duplicating the cube. It was as follows. Let OA and
OB be the given lines. Construct a rectangle OADB, of which
they are adjacent sides. Bisect AB in C. Then, if with C as
centre we can describe a circle cutting OA produced in a and
cutting OB produced in 6, so that aDb shall be a straight line,
the problem is effected. For it is easily shewn that
Similarly
Hence
That is,
Ob . Bb + CB* = Cb 2 .
Oa . Aa=0b.b.
Oa : Ob = Bb : Aa,
APOLLONIUS. ERATOSTHENES. 83
But, by similar triangles,
BD : Eb = Oa : Ob = Aa : AD.
Therefore OA : Bb = Bb : Aa = Aa : OB,
that is, Bb and Oa are the two mean proportionals between
OA and OB. It is impossible to construct the circle whose
centre is C by Euclidean geometry, but Apollonius gave a
mechanical way of describing it. This construction is quoted
by several Arabic writers.
In one of the most brilliant passages of his Apergu histo-
rique Chasles remarks that, while Archimedes and Apollonius
were the most able geometricians of the old world, their
works are distinguished by a contrast which runs through
the whole subsequent history of geometry. Archimedes, in
attacking the problem of the quadrature of curvilinear areas,
laid the foundation of the geometry which rests on measure
ments; this naturally gave rise to the infinitesimal calculus,
and in fact the method of exhaustions as used by Archi
medes does not differ in principle from the method of limits
as used by Newton. Apollonius, on the other hand, in
investigating the properties of conic sections by means of
transversals involving the ratio of rectilineal distances and of
perspective, laid the foundations of the geometry of form and
position.
Eratosthenes*. Among the contemporaries of Archimedes
and Apollonius I may mention Eratosthenes. Born at Gyrene
in 275 B.C., he was educated at Alexandria perhaps at the
same time as Archimedes of whom he was a personal friend
and Athens, and was at an early age entrusted with the care
of the university library at Alexandria, a post which probably
he occupied till his death. He was the Admirable Crichton
of his age, and distinguished for his athletic achievements not
less than for his literary and scientific attainments: he was
* The works of Eratosthenes exist only in fragments. A collection
of these was published by G. Bernhardy at Berlin in 1822 : some
additional fragments were printed by E. Hiller, Leipzig, 1872.
62
84 THE FIRST ALEXANDRIAN SCHOOL.
also something of a poet. He lost his sight by ophthalmia,
then as now a curse of the valley of the Nile, and, refusing
to live when he was no longer able to read, he committed
suicide by starvation in 194 B.C.
In science he was chiefly interested in astronomy and geodesy,
and he constructed various astronomical instruments which
were used for some centuries at the university. He introduced
the calendar (now known as Julian), in which every fourth year
contains 366 days; and he determined the obliquity of the
ecliptic as 23 5 1 20". He measured the length of a degree on
the earth s surface, making it to be about 79 miles, which is too
long by nearly 10 miles, and thence calculated the circum
ference of the earth to be 252000 stadia, which, if we take the
Olympic stadium of 202 \ yards, is equivalent to saying that
the radius is about 4600 miles. The principle used in the
determination is correct.
Of Eratosthenes s work in mathematics we have two extant
illustrations : one in a description of an instrument to dupli
cate a cube, and the other in the rule he gave for constructing
a table of prime numbers. The former is given in many
books. The latter, called the " sieve of Eratosthenes," was as
follows: write down all the numbers from 1 upwards; then
every second number from 2 is a multiple of 2 and may be
cancelled; every third number from 3 is a multiple of 3 and
may be cancelled; every fifth number from 5 is a multiple of 5
and may be cancelled; and so on. It has been estimated
that it would involve workiDg for about 300 hours to thus
find the primes in the numbers from 1 to 1,OOQOOO. The
labour of determining whether any particular ryimber is a
prime may be however much shortened by observing that if a
number can be expressed as the product of two factors one
must be less and the other greater than the square root of the
number, unless the number is the square of a prime in which
case the two factors are equal. Hence every composite number
must be divisible by a prime which is not greater than its
square root.
HYPSICLES. NICOMEDES. 85
>C
The second century before Christ.
The third century before Christ, which opens with the
career of Euclid and closes with the death of Apollonius, is the
most brilliant era in the history of Greek mathematics. But
the great mathematicians of that century were geometricians,
and under their influence attention was directed almost solely
to that branch of mathematics. With the methods they used,
and to which their successors were by tradition confined, it
was hardly possible to make any further great advance : to
fill up a few details in a work that was completed in its
essential parts was all that could be effected. It was not till
after the lapse of nearly 1800 years that the genius of Descartes
opened the way to any further progress in geometry, and I
therefore pass over the numerous writers who followed Apollo
nius with but slight mention. Indeed it may be said roughly
that during the next thousand years Pappus was the sole
geometrician of great ability; and during this long period
almost the only other pure mathematicians of exceptional
genius were Hipparchus and Ptolemy who laid the foundations
of trigonometry, and Diopharitus who laid those of algebra.
Early in the second century, circ. 180 B.C., we find the
names of three mathematicians Hypsicles, Nicomedes, and
Diocles who in their own day were famous.
Hypsicles. The first of these was Hypsicles who added a
fourteenth book to Euclid s Elements in which the regular
solids were discussed. In another small work, entitled Risings,
Hypsicles^ developed the theory of arithmetical progressions
which had been so strangely neglected by the earlier mathe
maticians, and here for the first time in Greek mathematics
we find a right angle divided in the Babylonian manner into
90 degrees ; possibly Eratosthenes may have previously esti
mated angles by the number of degrees they contain, but this
is only a matter of conjecture.
Nicomedes. The second was Nicomedes who invented the
curve known as the conchoid or the shell-shaped curve. If
86 THE FIRST ALEXANDRIAN SCHOOL.
from a fixed point S a line be drawn cutting .a given fixed
straight line in Q and if P be taken on SQ so that the length
QP is constant (say d), then the locus of P is the conchoid.
Its equation may be put in the form r = a sec =t d. It is easy
with its aid to trisect a given angle or to duplicate a cube ; and
this no doubt was the cause of its invention.
Diocles. The third of these mathematicians was Diodes
the inventor of the curve known as the cissoid or the ivy-
shaped curve which, like the conchoid, was used to give a
solution of the duplication problem. He defined it thus: let
AOA and BOB be two fixed diameters of a circle at right angles
to one another. Draw two chords QQ and RR parallel to
BOB and equidistant from it. Then the locus of the inter
section of AR and QQ will be the cissoid. Its equation can be
expressed in the form y 2 (2a x) =x 3 . Diocles also solved (by
the aid of conic sections) a problem which had been proposed
by Archimedes, namely, to draw a plane which will divide a
sphere into two parts whose volumes shall bear to one another
a given ratio.
Perseus. Zenodorus. About a quarter of a century later,
say about 150 B.C., Perseus investigated the various plane
sections of the anchor-ring (see above, p. 47), and Zenodorus
wrote a treatise on isoperimetrical figures. Part of the latter
work has been preserved ; one proposition which will serve to
shew the nature of the problems discussed is that "of segments
of circles, having equal arcs, the semicircle is the greatest."
Towards the close of this century we find two mathema
ticians who, by turning their attention to new subjects, gave a
fresh stimulus to the study of mathematics. These were
Hipparchus and Hero.
Hipparchus*. Hipparchus was the most eminent of Greek
astronomers his chief predecessors being Eudoxus, Aristarchus,
Archimedes, and Eratosthenes. Hipparchus is said to have been
born about 160B.C. at Nicaea in Bithynia; it is probable that
* See Delambre, Histoire de V astronomic ancienne, Paris, 1817, vol. i.
pp. 106189.
HIPPAUCHUS. 87
he spent some years at Alexandria, but finally he took up his
abode at Rhodes where he made most of his observations.
Delambre has obtained an ingenious confirmation of the tradi
tion which asserted that Hipparchns lived in the second
century before Christ. Hipparchus in one place says that
the longitude of a certain star rj Canis observed by him was
exactly 90, and it should be noted that he was an extremely
careful observer. Now in 1750 it was 116 4 10", and, as
the first point of Aries regredes at the rate of 50 2" a year,
the observation was made about 120 B.C.
Except for a short commentary on a poem of Aratus
dealing with astronomy all his works are lost, but Ptolemy s
great treatise, the Almagest (see below, pp. 97, 98), was founded
on the observations and writings of Hipparchus, and from
the notes there given we infer that the chief discoveries of
Hipparchus were as follows. He determined the duration of
the year to within six minutes of its true value. He calculated
the inclination of the ecliptic and equator as 23 51 ; it was
actually at that time 23 46 . He estimated the annual pre
cession of the equinoxes as 59" ; it is 50 -2". He stated the
lunar parallax as 57 , which is nearly correct. He worked
out the eccentricity of the solar orbit as 1/24 ; it is very
approximately 1/30. He determined the perigee and mean
motion of the sun and of the moon, and he calculated the
extent of the shifting of the plane of the moon s motion.
Finally he obtained the synodic periods of the five planets
then known. I leave the details of his observations and
calculations to writers who deal specially with astronomy such
as Delambre ; but it may be fairly said that this work placed
the subject for the first time on a scientific basis.
To account for the lunar motion Hipparchus supposed the
moon to move with uniform velocity in a circle, the earth
occupying a position near (but not at) the centre of this circle.
This is equivalent to saying that the orbit is an epicycle of the
first order. The longitude of the moon obtained on this
hypothesis is correct to the first order of small quantities for a
88 THE FIRST ALEXANDRIAN SCHOOL.
few revolutions. To make it correct for any length of time
Hipparchus further supposed that the apse line moved forward
about 3 a month, thus giving a correction for evection. He
explained the motion of the sun in a similar manner. This
theory accounted for all the facts which could be determined
with the instruments then in use, and in particular enabled him
to calculate the details of eclipses with considerable accuracy.
He commenced a series of planetary observations to enable
his successors to frame a theory to account for their motions ;
and with great perspicacity he predicted that to do this it
would be necessary to introduce epicycles of a higher order,
that is, to introduce three or more circles the centre of each
successive one moving uniformly on the circumference of the
preceding one.
He also formed a list of the fixed stars. It is said that the
sudden appearance in the heavens of a new and brilliant star
called his attention to the need of such a catalogue; and the
appearance of such a star during his lifetime is confirmed
by Chinese records.
No further advance in the theory of astronomy was made
until the time of Copernicus, though the principles laid down
by Hipparchus were extended and worked out in detail by
Ptolemy.
Investigations such as these naturally led to trigono
metry, and Hipparchus must be credited with the invention
of that subject. It is known that in plane trigonometry he
constructed a table of chords of arcs, which is practically the
same as one of natural sines; and that in spherical trigonometry
he had some method of solving triangles : but his works are
lost, and we can give no details. It is believed however that
the elegant theorem, printed as Euc. vi. D and generally
known as Ptolemy s Theorem, is due to Hipparchus arid was
copied from him by Ptolemy. It contains implicitly the
addition formulae for sin (A B) and cos (AB); and Carnot
shewed how the whole of elementary plane trigonometry could
be deduced from it.
HERO OF ALEXANDRIA. 89
I ought also to add that Hipparchus was the first to in
dicate the position of a place on the earth by means of its
latitude and longitude.
Hero*. The second of these mathematicians was Hero of
Alexandria (circ. 125 B.C.) who placed engineering and land-
surveying on a scientific basis. He was a pupil of Ctesibus
who invented several ingenious machines and is alluded to as
if he were a mathematician of note.
In pure mathematics Hero s principal and most character
istic work consists of (i) some elementary geometry, with
applications to the determination of the areas of fields of given
shapes; (ii) propositions on finding the volumes of certain
solids, with applications to theatres, baths, banquet-halls, and
so on; (iii) a rule to find the height of an inaccessible object;
and (iv) tables of weights and measures. He invented a
solution of the duplication problem which is practically the
same as that which Apollonius had already discovered (see
above, p. 82). Some commentators think that he knew how
to solve a quadratic equation even when the coefficients were
not numerical ; but this is doubtful. He proved the formula
that the area of a triangle is equal to {s(s a) (s b) (s - c)}^,
where s is the semiperimeter, and a, 6, c, the lengths of the
sides, and gave as an illustration a triangle whose sides were
13, 14, and 15. He was evidently acquainted with the trigono
metry of Hipparchus, but he nowhere quotes a formula or
expressly uses the value of the sine, and it is probable that
like the later Greeks he regarded trigonometry as forming an
introduction to, and being an integral part of, astronomy.
* See Eecherches sur la vie et les ouvrages d 1 Heron d Altxandrie by
T. H. Martin in vol. iv. of Memoires presentes . . .a Vacademie d j inscriptio)is,
Paris, 1854; see also Cantor, chaps, xvui, xix. On the work entitled
Definitions which is attributed to Hero, see Tannery, chaps, xiu, xiv,
and an article by G. Friedlein in Boncompagni s Bullctino di bibliografia,
March, 1871, vol. iv, pp. 93 126. An edition of the extant works of
Hero was published by F. Hultsch, Berlin, 1864. An English translation
of the lIvev/jLariKd was published by B. Woodcroft and J. G. Greenwood
at London in 1851.
90.
THE FIRST ALEXANDRIAN SCHOOL.
The following is the manner * in which he solved the problem
to find the area of a triangle ABC the lengths of whose sides
are a, 6, c. Let s be the semiperimeter of the triangle. Let
the inscribed circle touch the sides in D, E, F, and let be
its centre. On BC produced take H so that CH = AF, therefore
Bff=s. Draw OK at right angles to OB, and CK at right
angles to BC ; let them meet in K. The area ABC or A is equal
to the sum of the areas OBC, OCA, OAB ar + br + cr = sr,
that is, is % equal to Ell . OD. He then shews that the angle
OAF= angle CBK-, hence the triangles OAF and CBK are
similar ;
.-. BC : CK^AF: OF=Cff: OD,
.-. BC : CH = CK : OD = CL : LD,
.-. BH: Cff=CD : LD.
.-. BH* : CH . BH = CD . ED : LD . BD = CD . BD : OD 2
* In his Dioptra, Hultsch, pp. 235237.
HERO 0V ALEXANDRIA. 91
Hence
A - nil . OD = {CH . EH . CD . ED $ = {(* - a) * (* - c) (s - 6)}*.
In applied mathematics Hero discussed the centre of gravity,
the live simple machines, and the problem of moving a given
weight with a given power; and in one place he suggested
a way in which the power of a catapult could be tripled.
He also wrote on the theory of hydraulic machines. He
described a theodolite and cyclometer, and pointed out various
problems in surveying for which they would be useful. But
the most interesting of his smaller works are his nvcv/xart/ca
and AvTo/jLOLTa, containing descriptions of about 100 small
machines and mechanical toys, many of which are very in
genious. In the former there is an account of a small
stationary steam-engine which is of the form now known
as Avery s patent : it was in common use in Scotland at the
beginning of this century, but is not so economical as the form
introduced by Watt. There is also an account of a double
forcing pump to be used as a fire-engine. It is probable that
in the hands of Hero these instruments never got beyond
models. It is only recently that general attention has been
directed to his discoveries, though Arago had alluded to them
. in his eloge on Watt.
All this is very different from the classical geometry and
arithmetic of Euclid, or the mechanics of Archimedes. Hero
did nothing to extend a knowledge of abstract mathematics ;
he learnt all that the text- books of the day could teach him,
but he was interested in science only on account of its prac
tical applications, and so long as his results were true he
cared nothing for the logical accuracy of the process by which
he arrived at them. Thus in finding the area of a triangle
he took the square root of the product of four lines. The
classical Greek geometricians permitted the use of the square
and the cube of a line because these could be represented
geometrically, but a figure of four dimensions is inconceivable,
and certainly they would have rejected a proof which involved
such a conception.
92 THE FIRST ALEXANDRIAN SCHOOL.
It is questionable if Hero or his contemporaries were aware
of the existence of the Rhind papyrus, but it would seem that
treatises founded on it and of a similar character were then
current in Egypt, and while I am passing these sheets through
the press the manuscript of a text-book of this kind though
most likely some eight centuries or so later in date has been
discovered and reproduced.* Doubtless it was from some such
source that Hero drew his inspiration. Two or three reasons
have led modern commentators to think that Hero, who was
born in Alexandria, was a native Egyptian. If this be so, it
affords an interesting illustration of the permanence of racial
characteristics and traditions. Hero spoke and wrote Greek,
and it is believed that he was brought up under Greek
influence ; yet the rules he gives, his methods of proof, the
figures he draws, the questions he attacks, and even the
phrases of which he makes use, recall the earlier w*brk of
Ahmes.
The first century before Christ.
The successors of Hipparchus and Hero did not avail them
selves of the opportunity thus opened of investigating new
subjects, but fell back on the well-worn subject of geometry.
Amongst the more eminent of these later geometricians were
Theodosius and Dionysodorus, both of whom flourished about
50 B.C.
Theodosius. Theodosius was the author of a complete
treatise on the geometry of the sphere, which was edited by
Barrow, Cambridge, 1675, and by Nizze, Berlin, 1852. He
also wrote two works on astronomy which were published by
Dasypodius in 1572.
Dionysodorus. Dionysodorus is known to us only by his
solution of the problem to divide a hemisphere by a plane
* The Akhmim papyrus by J. Baillet in the Memoires de la mission
archeologique frangaise au Caire, vol. ix, pp. 1 88, Paris, 1892.
THE FIRST ALEXANDRIAN SCHOOL. 93
parallel to its base into two parts, whose volumes shall be in
a given ratio. Like the solution by Diocles of the similar
problem for a sphere above alluded to, it was effected by the
aid of conic sections : it is reproduced in Suter s Geschichte
der mathematischen Wissenschaften (p. 101). Pliny says that
Dionysodorus determined the length of the radius of the earth
approximately as 42000 stadia, which, if we take the Olympic
stadium of 202^ yards, is a little less than 5000 miles ; we do
not know how it was obtained. This may be compared with the
result given by Eratosthenes (see above, p. 84).
End of the first Alexandrian School.
The administration of Egypt was definitely undertaken
by Rome in 30 B.C. The closing years of the dynasty of the
Ptolemies and the earlier years of the Roman occupation of
the country were marked by much disorder, civil and political.
The studies of the university were naturally interrupted, and
it is customary to take this time as the close of the first
Alexandrian school.
CHAPTER V.
THE SECOND ALEXANDRIAN SCHOOL*.
30 B.C. 641 A.D.
I CONCLUDED the last chapter by stating that the first
school of Alexandria may be said to have come to an end at
about the same time as the country lost its nominal inde
pendence. But, although the schools at Alexandria suffered
from the disturbances which affected the whole Roman world
in the transition, in fact if not in name, from a republic to
the empire, there was no break of continuity; the teaching in
the university was never abandoned ; and as soon as order
was again established students began once more to flock to
Alexandria. This time of confusion was however contempo
raneous with a change in the prevalent views of philosophy
which thenceforward were mostly neo-platonic or neo-pytha-
gorean, and it therefore fitly marks the commencement of a
new period. These mystical opinions reacted on the mathe
matical school, and this may partially account for the paucity
of good work.
* For authorities, see footnote above on p. 51. All dates given
hereafter are to be taken as anno domini, unless the contrary is expressly
stated.
SERENUS. MENELAUS. NICOMACHUS. 95
Though Greek influence was still predominant and the
Greek language always used, Alexandria now became the in
tellectual centre for most of the Mediterranean nations which
were subject to Rome. It should be added however that
the direct connection with it of many of the mathematicians
of this time is at least doubtful, but their knowledge was
ultimately obtained from the Alexandrian teachers, and they
are usually described as of the second Alexandrian school.
Such mathematics as were taught at Rome were derived from
Greek sources, and we may therefore conveniently consider
their extent in connection with this chapter.
The first century after Christ.
There is no doubt that throughout the first century after
Christ geometry continued to be that subject in science to
which most attention was devoted. But by this time it was
evident that the geometry of Archimedes and Apollonius was
not capable of much further extension; and such geometrical
treatises as were produced consisted mostly of commentaries
on the writings of the great mathematicians of a preceding age.
In this century the only original works of any ability were
two by Serenus and one by Menelaus.
Serenus. Menelaus. Those by Serenus of Antissa, circ. 70,
were on the plane sections of the cone and cylinder ; these were
edited by E. Halley, Oxford, 1710. That by Menelaus of
Alexandria, circ. 98, was on spherical trigonometry, investigated
in the Euclidean method ; this was translated by E. Halley,
Oxford, 1758. The fundamental theorem on which the sub
ject is based is the relation between the six segments of the
sides of a spherical triangle, formed by the arc of a great circle
which cuts them (book in. prop. 1). Menelaus also wrote on
the calculation of chords, i.e. on plane trigonometry ; this is
lost,
Nicomachus. Towards the close of this century, circ. 100,
96 THE SECOND ALEXANDRIAN SCHOOL.
Nicomachus, a Jew, who was born at Gerasa in 50 and died
circ. 110, published an Arithmetic, which (or rather the Latin
translation, of it) remained for a thousand years a standard
authority on the subject. The work has been edited by
R. Hoche, Leipzig, 1866. Geometrical demonstrations are
here abandoned, and the work is a mere classification of the
results then known, with numerical illustrations : the evidence
for the truth of the propositions enunciated, for I cannot call
them proofs, being in general an induction from numerical
instances. The object of the book is the study of the
properties of numbers, and particularly of their ratios. Nico-
machus commences with the usual distinctions between even,
odd, prime, and perfect numbers; he next discusses fractions
in a somewhat clumsy manner; he then turns to polygonal and
to solid numbers; and finally treats of ratio, proportion, and
the progressions. Arithmetic of this kind is usually termed
Boethian, and the work of Boethius on it was a recognized
text-book in the middle ages.
The second century after Christ.
Theon. Another arithmetic on much the same lines as
that of Nicomachus was produced by Theon of Smyrna , circ.
130; but it was even less scientific than that of Nicomachus.
It was edited by J. J. de Gelder, Leyden, 1827; and by E.
Hiller, Leipzig, 1878. Theon also wrote a work on astronomy
which was edited by T. H. Martin, Paris, 1849.
Thymaridas. Another mathematician of about the same
date was Thymaridas, who is worthy of notice from the fact
that he is the earliest known writer who explicitly enunciated
an algebraical theorem. He stated that, if the sum of any
number of quantities be given, and also the sum of every pair
which contains one of them, then this quantity is equal to
one (n - 2)th part of the difference between the sum of these
pairs and the first given sum. Thus, if
x l + x 2 + . . . + x n = S,
I TOLEMY. 97
and if x l -\- x, 2 =s. 2 , X 1 + x 3 = s 3 , . . . , arid x 1 + x n = s n ,
then x L = (s z + s 3 + ...+s n - 8)1 (n - 2).
He does not seem to have used a symbol to denote the unknown
quantity, but he always represented it by the same word, which
is an approximation to symbolism.
Ptolemy*. About the same time as these writers Ptolemy
of Alexandria, who died in 168, produced his great work on
astronomy, which will preserve his name as long as the history
of science endures. This treatise is usually known as the
Almagest: the name is derived from the Arabic title al mid-
scliisti, which is said to be a corruption of peyLO-Tr) [fjLaOrjfjiaTLKrj]
o"vvrais. The work is founded on the writings of Hipparchus,
and, though it did not sensibly advance the theory of the
subject, it presents the views of the older writer with a com
pleteness and elegance which will always make it a standard
treatise. We gather from it that Ptolemy made observations
at Alexandria from the years 125 to 150; he however was
but an indifferent practical astronomer, and the observations
of Hipparchus are generally more accurate than those of his
expounder.
The work is divided into thirteen books. In the first
book Ptolemy discusses various preliminary matters; treats of
trigonometry, plane and spherical ; gives a table of chords, i.e.
of natural sines (which is substantially correct and is probably
taken from the lost work of Hipparchus) ; and explains the
obliquity of the ecliptic ; in this book he uses degrees, minutes,
and seconds as measures of angles. The second book is devoted
chiefly to phenomena depending on the spherical form of the
earth: he remarks that the explanations would be much
simplified if the earth were supposed to rotate on its axis once
* See the article Ptolemaeus Claudius by A. De Morgan in Smith s
Dictionary of Greek and Roman Biography, London, 1849 ; and
Delambre, Histoire de V astronomic ancienne, Paris, 1817, vol. 11. An
edition of all the works of Ptolemy which are now extant was published
at Bale in 1551. The Almagest with various minor works was edited by
M. Halma, 12 vols, Paris, 181328, and this is the standard edition.
B
98 THE SECOND ALEXANDRIAN SCHOOL.
a day, but points out that this hypothesis is inconsistent with
known facts. In the third book he explains the motion of the
sun round the earth by means of excentrics and epicycles: and
in the fourth and fifth books he treats the motion of the moon
in a similar way. The sixth book is devoted to the theory of
eclipses; and in it he gives 3 8 30", that is 3 T y^-, as the
approximate value of TT, which is equivalent to taking it equal
to 3 14l6. The seventh and eighth books contain a catalogue
of 1022 fixed stars determined by indicating those, three or
more, that are in the same straight line (this was probably
copied from Hipparchus) : and in another work Ptolemy added
a list of annual sidereal phenomena. The remaining books
are given up to the theory of the planets.
This work is a splendid testimony to the ability of its
author. It became at once the standard authority on as
tronomy, and remained so till Copernicus and Kepler shewed
that the sun and not the earth must be regarded as the centre
of the solar system.
The idea of excentrics and epicycles on which the theories
of Hipparchus and Ptolemy are based has been often ridiculed
in modern times. No doubt at a later time, when more accu
rate observations had been made, the necessity of introducing
epicycle on epicycle in order to bring the theory into accord
ance with the facts made it very complicated. But De Morgan
has acutely observed that in so far as the ancient astronomers
supposed that it was necessary to resolve every celestial motion
into a series of uniform circular motions they erred greatly,
but that, if the hypothesis be regarded as a convenient way
of expressing known facts, it is not only legitimate but con
venient. It was as good a theory as with their instruments
and knowledge it was possible to frame, and in fact it exactly
corresponds to the expression of a given function as a sum of
sines or cosines, a method which is of frequent use in modern
analysis.
In spite of the trouble taken by Delambre it is almost
impossible to separate the results due to Hipparchus from
PTOLEMY. 99
those due to Ptolemy. But Delambre and De Morgan agree
in thinking that the observations quoted, the fundamental
ideas, and the explanation of the apparent solar motion are
due to Hipparchus; while all the detailed explanations and
calculations of the lunar and planetary motions are wholly
due to Ptolemy.
The Almagest shews that Ptolemy was a geometrician of
the first rank, though it is with the application of geometry to
astronomy that he is chiefly concerned. He was however the
author of numerous other treatises, most of which were on
pure mathematics.
Amongst these treatises is one on pure geometry in which
he proposed to cancel the twelfth axiom of Euclid on parallel
lines and to prove it in the following manner. Let the
straight line EFGH meet the two straight lines AB and CD
so as to make the sum of the angles BFG and FGD equal
to two right angles. It is required to prove that AB and CD
are parallel. If possible let them not be parallel, then they
will meet when produced say at M (or N). But the angle
AFG is the supplement of BFG, and is therefore equal to
FGD: similarly the angle FGC is equal to the angle BFG.
Hence the sum of the angles AFG and FGC is equal to two
H
right angles, and the lines BA and DC will therefore if pro
duced meet at N (or J/). But two straight lines cannot enclose
a space, therefore AB and CD cannot meet when produced,
that is, they are parallel. Conversely, if AB and CD be
parallel, then AF and CG are not less parallel than FB and
72
100 THE SECOND ALEXANDRIAN SCHOOL.
GD j and therefore whatever be the sum of the angles AFG
and FGC such also must be the sum of the angles FGD and
BFG. But the sum of the four angles is equal to four right
angles, and therefore the sum of the angles BFG and FGD
must be equal to two right angles.
Ptolemy wrote another work to shew that there could not
be more than three dimensions in space: he also discussed
orthographic and stereographic projections with special reference
to the construction of sun-dials. He wrote on geography, and
stated that the length of one degree of latitude is 500 stadia.
A book on optics and another on sound are sometimes attributed
to him, but their authenticity is doubtful.
The third century after Christ.
, Pappus. Ptolemy had shewn not only that geometry could
be applied to astronomy, but had indicated how new methods
of analysis like trigonometry might be thence developed. He
found however no successors to take up the work he had com
menced so brilliantly, and we must look forward 150 years
before we find another geometrician of any eminence. That !
geometrician was Pappus who lived and taught at Alexandria
about the end of the third century. We know that he had
numerous pupils, and it is probable that he temporarily revived
an interest in the study of geometry.
Pappus wrote several books, but the only one which has
come down to us is his Swaywy^, a collection of mathematical
papers arranged in eight books of which the first and part of
the second have been lost; it has been published by F. Hultsch,
Berlin, 1876 8. This collection was intended to be a syn
opsis of Greek mathematics together with comments and
additional propositions by the editor. A careful comparison of
various extant works with the account given of them in this
book shews that it is trustworthy, and we rely largely on it for
our knowledge of other works now lost. It is not arranged
chronologically, but all the treatises on the same subject
PAPPUS. 101
are grouped together, and it is most likely that it gives
roughly the order in which the classical authors were read at
Alexandria. Probably the first book, which is now lost, was
on arithmetic. The next four books deal with geometry ex
clusive of conic sections : the sixth with astronomy including,
as subsidiary subjects, optics and trigonometry : the seventh
with analysis, conies, and porisms: arid the eighth with
mechanics.
The last two books contain a good deal of original work by
Pappus; at the same time it should be remarked that in two
or three cases he has been detected in appropriating proofs
from earlier authors, and it is possible he may have done this
in other cases.
Subject to this suspicion we may say that he discovered
the focus in the parabola, and the directrix in the conic
sections, but in both cases he investigated only a few isolated
properties: the earliest comprehensive account of the foci was
given by Kepler, and of the directrix by Newton and Boscovich.
In mechanics, he shewed that the centre of mass of a
triangular lamina is the same as that of an inscribed triangular
lamina whose vertices divide each of the sides of the original
o
triangle in the same ratio. He also discovered the two theorems
on the surface and volume of a solid of revolution which are
still quoted in text-books under his name : these are that the
volume generated by the revolution of a curve about an axis is
equal to the product of the area of the curve and the length
of the path described by its centre of mass; and the surface
is equal to the product of the perimeter of the curve and the
length of the path described by its centre of mass.
Pappus s best work is in geometry. As an illustration of
his power I may mention that he solved (book vii., prop. 107)
the problem to inscribe in a given circle a triangle whose sides
produced shall pass through three collinear points. This
question was in the eighteenth century generalised by Cramer
by supposing the three given points to be anywhere; and
was considered a difficult problem. It was sent in 1742 as a
102 THE SECOND ALEXANDRIAN SCHOOL.
challenge to Castillon, and in 1776 he published a solution.
Lagrange, Euler, Lhulier, Fuss, and Lexell also gave solutions
in 1 780. A few years later the problem was set to a Nea
politan lad Oltaiano, who was only 16 but who had shewn
marked mathematical ability, and he extended it to the case
of a polygon of n sides which pass through n given points, and
gave a solution both simple and elegant. Poncelet extended
it to conies of any species and subject to other restrictions.
The problem just mentioned is but a sample of many
brilliant but isolated theorems which were enunciated by
Pappus. His work as a whole and his comments shew that he
was a geometrician of great power; but it was his misfortune
to live at a time when but little interest was taken in geometry,
and when the subject, as then treated, had been practically
exhausted.
Possibly a small tract on multiplication and division of
sexagesimal fractions, which would seem to have been written
about this time, is due to Pappus. It was edited by C. Henry,
Halle, 1879, and is valuable as an illustration of practical Greek
arithmetic.
The fourth century after Christ.
Throughout the second and third centuries, that is, from
the time of Nicomachus, interest in geometry had steadily
decreased, and more and more attention had been paid to the
theory of numbers though the results were in no way com
mensurate with the time devoted to the subject. It will
be remembered that Euclid used lines as symbols for any
magnitudes, and investigated a number of theorems about
numbers in a strictly scientific manner, but he confined him
self to cases where a geometrical representation was possible.
There are indications in the works of Archimedes that he was
prepared to carry the subject much further : he introduced
numbers into his geometrical discussions and divided lines by
lines, but he was fully occupied by other researches and had
METRODORUS. 103
no time to devote to arithmetic. Hero abandoned the geo
metrical representation of numbers but he, Nicomachus, and
other later writers on arithmetic did not succeed in creating
any other symbolism for numbers in general, and thus when
they enunciated a theorem they were content to verify it by
a large number of numerical examples. They doubtless knew
how to solve a quadratic equation with numerical coefficients
for, as pointed out above, geometrical solutions of the equa
tions ax 2 - bx + c = and ax 2 + bx c = Q are given in Euc. vi.
28 and 29 but probably this represented their highest attain
ment.
It would seem then that, in spite of the time given to its
study, arithmetic and algebra had not made any sensible advance
since the time of Archimedes. The problems of this kind
which excited most interest in the third century may be illus
trated from a collection of questions, printed in the Palatine
Anthology, which was made by Metrodorus at the beginning
of the next century, about 310. Some of them are due to
the editor, but some are of an anterior date, and they fairly
illustrate the way in which arithmetic was leading up to
algebraical methods. The following are typical examples.
" Four pipes discharge into a cistern : one fills it in one
day ; another in two days ; the third in three days ; the
fourth in four days : if all run together how soon will they
fill the cistern]" "Demochares has lived a fourth of his life
as a boy; a fifth as a youth ; a third as a man ; and has spent
thirteen years in his dotage : how old is he ? " " Make a
crown of gold, copper, tin, and iron weighing 60 minae : gold
and copper shall be two-thirds of it ; gold and tin three-
fourths of it ; and gold and iron three-fifths of it : find the
weights of the gold, copper, tin, and iron which are required."
The last is a numerical illustration of Thymaridas s theorem
quoted above.
The German commentators have pointed out that though
these problems lead to simple equations, they can be solved
by geometrical methods, the unknown quantity being repre-
104 THE SECOND ALEXANDRIAN SCHOOL.
sented by a line. Dean Peacock has also remarked that they
can be solved by the method used in similar cases by the
Arabians and many mediaeval writers. This method, usually
known as the ride of false assumption, consists in assuming any
number for the unknown quantity, and, if on trial the given
conditions be not satisfied, altering the number by a simple
proportion as in rule of three. For example, in the second
problem, suppose we assume that the age of Demochares is 40,
then, by the given conditions, he would have spent 8f (and not
13) years in his dotage, and therefore we have the ratio of
&% to 13 equal to the ratio of 40 to his actual age, hence his
o -I O "
actual age is 60.
But the most recent writers on the subject think that the
problems were solved by rhetorical algebra, that is, by a process
of algebraical reasoning expressed in words and without the
use of any symbols. This, according to Nesselmann, is the first
stage in the development of algebra, and we find it used both
by Ahmes and by the earliest Arabian, Persian, and Italian
algebraists : examples of its use in the solution of a geometrical
problem and in the rule for the solution of a quadratic equation
are given later (see below, pp. 207, 214). On this view then
a rhetorical algebra had been gradually evolved by the Greeks,
or was then in process of evolution. Its development was
however very imperfect.^ Hankel, who is no unfriendly critic,
says that the results attained as the net outcome of the work of
600 years onTTne theory of numbers are, whether we look at
the form or the substance, unimportant or even childish and
are not in any way the commencement of a science.
In the midst of this decaying interest in geometry and
these feeble attempts at algebraic arithmetic, a single algebraist
of marked originality suddenly appeared who created what
was practically a new science. This was Diophantus who
introduced a system of abbreviations for those operations and
quantities which constantly recur, though in using them he
observed all the rules of grammatical syntax. The resulting
science is called by Nesselmann syncopated algebra : it is a sort
DIOPHANTUS. 105
of shorthand. Broadly speaking, it may be said that European
algebra did not advance beyond this stage until the close of
the sixteenth century.
Modern algebra has progressed one stage further and is
entirely symbolic ; that is, it has a language of its own and a
system of notation which has no obvious connection with the
things represented, while the operations are performed accord
ing to certain rules which are distinct from the laws of gram
matical construction.
Diophantus*. All that we know of Diophantus is that he
lived at Alexandria, and that most likely he was not a Greek.
Even the date of his career is uncertain, but probably he
flourished in the early half of the fourth century, that is,
shortly after the death of Pappus. He was 84 when he died.
In the above sketch of the lines on which algebra has
developed I credited Diophantus with the invention of synco
pated algebra. This is a point on which opinions differ, and
some writers believe that he only systematized the knowledge
which was familiar to his contemporaries. In support of this
latter opinion it may be stated that Cantor thinks that there
are traces of the use of algebraic symbolism in Pappus, and
Friedlein mentions a Greek papyrus in which the signs / and 9
are used for addition and subtraction respectively ; but no other
direct evidence for the non-originality of Diophantus has been
produced, and no ancient author gives any sanction to this view.
Diophantus wrote a short essay on polygonal numbers ; a
treatise on algebra which has come down to us in a mutilated
condition ; and a work on porisms which is lost.
The Polygonal Numbers contains ten propositions, and was
probably his earliest work. In this he abandons the em
pirical method of Nicomachus, and reverts to the old and
classical system by which numbers are represented by lines, a
construction is (if necessary) made, and a strictly deductive
* See Diophantos of Alexandria by T. L. Heath, Cambridge, 1885;
also Die Arithmetic uml die Schrift Hbcr Polygonalznhlfn des Diophantus
by G. Wcrtheim, Leipzig, 1890.
106 THE SECOND ALEXANDRIAN SCHOOL.
proof follows : it may be noticed that in it he quotes propo
sitions, such as Euc. II. 3 and n. 8, as referring to numbers and
not to any magnitudes.
His chief work is his Arithmetic. This is really a treatise
on algebra ; algebraic symbols are used, and the problems are
treated analytically. Diophantus tacitly assumes, as is done in
nearly all modern algebra, that the steps are reversible. He
applies this algebra to find solutions (though frequently only
particular ones) of several problems involving numbers. I
propose to consider successively the notation, the methods of
analysis employed, and the subject-matter of this work.
First, as to the notation. Diophantus always employed a
symbol to represent the unknown quantity in his equations,
but as he had only one symbol he could never use more than
one unknown at a time (see, however, below, p. 109). The
unknown quantity is called o dpiOfAos, and is represented by
$* or 5" . It is usually printed as s. In the plural it is
denoted by 95 or ss l . This symbol may be a corruption of OP,
or possibly is an old hieratic symbol for the word heap (see
above, p. 4), or it may stand for the final sigma of the word.
The square of the unknown is called Suva/xis, and denoted
by &: the cube KU/:?OS, and denoted by K \ and so on up to
the sixth power.
The coefficients of the unknown quantity and its powers are
numbers, and a numerical coefficient is written immediately after
the quantity it multiplies : thus s d = x, and ss 01 ta = ssia = llx.
An absolute term is regarded as a certain number of units or
/xova Ses which are represented by ju: thus /x s d - 1, /x 5 ta = 11.
There is no sign for addition beyond mere juxtaposition.
Subtraction is represented by >/i, and this symbol affects all the
symbols that follow it. Equality is represented by i. Thus
represents (x* + Sx) - (5x 2 + !) = #.
Diophantus also introduced a somewhat similar notation
DIOPHANTUS. 107
for fractions involving the unknown quantity, but into the
details of this I need not here enter.
It will be noticed that all these symbols are mere abbre
viations for words, and Diophantus reasons out his proofs,
writing these abbreviations in the middle of his text. In
most manuscripts there is a marginal summary in which the
symbols alone are used and which is really symbolic algebra ;
but probably this is the addition of some scribe of later times.
This introduction of a contraction or a symbol instead of a
word to represent an unknown quantity marks a greater ad
vance than anyone not acquainted with the subject would
imagine, and those who have never had the aid of some such
abbreviated symbolism find it almost impossible to understand
complicated algebraical processes. It is likely enough that it
might have been introduce! earlier, but for the unlucky system
of numeration adopted by the Greeks by which they used all
the letters of the alphabet to denote particular numbers and
thus make it impossible to employ them to represent any
number.
Next, as to the knowledge of algebraic methods shewn in
the book. Diophantus commences with some definitions whichA
include an explanation of his notation, and in giving the)
symbol for minus he states that a subtraction multiplied by/
a subtraction gives an addition ; by this he means that the
product of 6 and d in the expansion of (a b)(c- d) is
+ bd, but in applying the rule he always takes care that the
numbers a, 5, c, d are so chosen that a is greater than b and
c is greater than d.
The whole of the work itself, or at least as much as is now ,
extant, is devoted to solving problems which lead to equa
tions. It contains the rules for solving a simple equation of the
first degree and a binomial quadratic. The rule for solving
any quadratic equation is probably in one of the lost books,
but where the equation is of the form ax 2 + bx + c = he
seems to have multiplied by a and then " completed the
in much the same way as is now done : when the roots
108 THE SECOND ALEXANDRIAN SCHOOL.
are negative or irrational* the equation is rejected as " im
possible," and even when both roots are positive he never
gives more than one, always taking the positive value of the
square root. Diophantus solves one cubic equation, namely,
x 3 + x = 4# 2 + 4 (book vi., prob. 19).
The greater part of the work is however given up to in
determinate equations between two or three variables. When
the equation is between two variables, then, if it be of the
first degree, he assumes a suitable value for one variable and
solves the equation for the other. Most of his equations are
of the form y 2 = Ax 2 + Bx + C. Whenever A or C is absent,
he is able to solve the equation completely. When this is not
the case, then, if A a 2 , he assumes y ax + m ; if C = c 2 , he
assumes y mx + c ; and lastly, if the equation can be put in
the form y 2 (ax b) 2 + c 2 , he assumes y = mx : where in each
case m has some particular numerical value suitable to the
problem under consideration. A few particular equations of
a higher order occur, but in these he generally alters the pro
blem so as to enable him to reduce the equation to one of the
above forms.
The simultaneous indeterminate equations involving three
variables, or " double equations " as he calls them, which he
considers are of the forms y 2 - Ax 2 + Bx -f C and z 2 = ax 2 + bx+c.
If A and a both vanish, he solves them in one of two ways.
It will be enough to give one of his methods which is as
follows : he subtracts and thus gets an equation of the form
y 2 z 2 = mx + n ; hence, if yz = \, then y =p z (mx + n)/\ ;
and solving he finds y and z. His treatment of " double
equations " of a higher order lacks generality and depends on
the particular numerical conditions of the problem.
Lastly, as to the matter of the book. The problems he
attacks and the analysis he uses are so various that they
cannot be described concisely and I have therefore selected five
typical problems to illustrate his methods. What seems to
strike his critics most is the ingenuity with which he selects
as his unknown some quantity which leads to equations such
DIOPHANTUS. 109
as he can solve, and the artifices by which he finds numerical
solutions of his equations.
I select the following as characteristic examples.
(i) Find four numbers, the sum of every arrangement three
at a time being given; say, 22, 24, 27, and 20 (book I., prob. 17).
Let oJ be the sum of all four numbers j hence the num
bers are x - 22, x - 24, x - 27, and x - 20.
.-. x = (x - 22) + (x - 24) + (x - 27) + (x- 20).
.-. a; = 31.
.-. the numbers are 9, 7, 4, and 11.
(ii) Divide a number, suck as 13 which is the sum of two
squares 4 and 9, into two other squares (book n., prob. 10).
He says that since the given squares are 2 2 and 3 2 he will
take (x + 2) 2 and (mx - 3) 2 as the required squares, and will
assume m = 2.
.-. (x + 2) 2 + (2x-3) 2 =l3.
.-. a = 8/5.
.-. the required squares are 324/25 and 1/25.
(iii) Find two squares such that the sum of the product
and either is a square (book II., prob. 29).
Let x 2 and y* be the numbers. Then x~y 2 + y~ and x*y* + x 2
are squares. The first will be a square if x 2 + I be a square,
which he assumes may be taken equal to (x 2) 2 , hence
#=3/4. He has now to make 9(?/ 2 +l)/16 a square, to do
this he assumes that 9?/ 2 + 9 = (3i/ 4) 2 , hence y = 7/24. There
fore the squares required are 9/16 and 49/576.
It will be recollected that Diophantus had only one symbol
for an unknown quantity : and in this example he begins by
calling the unknowns x 2 and 1, but as soon as he has found x
he then replaces the 1 by the symbol for the unknown quan
tity, and finds it in its turn.
110 THE SECOND ALEXANDRIAN SCHOOL.
(iv) To find a [rational] right angled triangle such that the
line bisecting an acute angle is rational (book vi., prob. 18).
His solution is as follows. Let ABC be the triangle of
which C is the right- angle. Let the bisector AD = 5x, and
A
B DC
let DC = 3x, hence AC = \x. Next let EG be a multiple of 3,
say 3, .-. JBJ) = 3-3x, hence AB=-kx (by Euc. vi. 3).
Hence (4 - x) 2 = 3 2 + (4x) 2 (Euc. i. 47), .-.a = 7/32. Multi
plying by 32 we get for the sides of the triangle 28, 96, and
100 ; and for the bisector 35.
(v) A man buys x measures of ivine, some at 8 drachmae
a measure, the rest at 5. He pays for them a square number of
drachmae, such that, if 60 be added to it, the resulting number
is x 2 . Find the number he bought at each price (book v.,
prob. 33).
The price paid was x 2 60, hence Sx > x 2 - 60 and
5x < x 2 - 60. From this it follows that x must be greater
than 11 and less than 12.
Again x 2 - 60 is to be a square ; suppose it is equal to
(x m) 2 then x= (m 2 + 60)/2m, we have therefore
.-. 19<ra<21.
Diophantus therefore assumes that m is equal to 20, which
gives him x= 11|- ; and makes the total cost, i.e. x 2 60, equal
to 72^ drachmae.
He has next to divide this cost into two parts which shall
give the cost of the 8 drachmae measures and the 5 drachmae
measures respectively. Let these parts be y and z.
DIOPHANTUS. Ill
Then ** + i(72i-z) = ll.
5 x 79 8 x 59
Therefore z = - 9 -, and y = . ^
Therefore the number of 5 drachmae measures was 79/12, and
of 8 drachmae measures was 59/12.
From the enunciation of this problem it would seem
that the wine was of a poor quality, and M. Tannery has
ingeniously suggested that the prices mentioned for such a
wine are higher than were usual until after the end of the
second century. He therefore rejects the view which was
formerly held that Diophantus lived in that century, but he
does not seem to be aware that De Morgan had previously
shewn that this opinion was untenable. M. Tannery inclines
to think that Diophantus lived half a century earlier than
I have supposed.
I mentioned that Diophantus wrote a third work entitled
Porisms. The book is lost, but we have the enunciations of
some of the propositions and though we cannot tell whether
they were rigorously proved by Diophantus they confirm our
opinion of his ability and sagacity. It has been suggested
that some of the theorems which he assumes in his arithmetic
were proved in the porisms. Among the more., striking of
these results are the statements that the difference of two
cubes can be always expressed as the sum of two cubes ; that
no number of the form 4?z 1 can be expressed as the sum
of two squares ; and that no number of the form Sn 1 (or
possibly 2in + 7) can be expressed as the sum of three squares :
to these we may perhaps add the proposition that any number
can be expressed as a square or as the sum of two or three or
four squares.
The writings of Diophantus exercised no perceptible influ
ence on Greek mathematics ; but his Arithnielicj when trans
lated into Arabic in the tenth century, influenced the Arabian
school, and so indirectly affected the progress of European
mathematics. A copy of the work was discovered in 1462;
112 THE SECOND ALEXANDRIAN SCHOOL.
it was translated into Latin and published by Xy lander in
1575 ; tho translation excited general interest, but by that
time the European algebraists had on the whole advanced
beyond the point at which Diophantus had left off.
The names of two commentators will practically conclude
the long roll of Alexandrian mathematicians.
Theon. The first of these is Tkeon of Alexandria who
flourished about 370. He was not a mathematician of
special " note, but we are indebted to him for an edition of
Euclid s Elements and a commentary on the Almagest] the
latter gives a great deal of miscellaneous information about
the numerical methods used by the Greeks, it was translated
with comments by M. Halma and published at Paris in 1821.
Hypatia. The other was Hypatia the daughter of Theon.
She .was more distinguished than her father, and was the last
Alexandrian mathematician of any general reputation : she
wrote a commentary on the Conies of Apollonius and possibly
some other works, but nothing of hers is now extant. She was
murdered at the instigation of the Christians in 415.
Tho fate of Hypatia may serve to remind us that the
Christians, as soon as they became the dominant party in
the state, shewed themselves bitterly hostile to all forms of
learning. That very singleness of purpose which had at first
so materially aided their progress developed into a one-
sidediiess which refused to see any good outside their own
body; those who did not actively assist them were persecuted,
and the manner in which they carried on their war against
the old schools of learning is pictured in the pages of Kingsley s
novel. The final establishment of Christianity in the East
marks the end of the Greek scientific schools, though nominally
they continued to exist for two hundred years more.
The Athenian school (in the fifth century).
The hostility of the Eastern church to Greek science is fur
ther illustrated by the fall of the later Athenian school. This
PROCLUS. DAMASCIUS. EUTOCIUS. 113
school occupies but a small space in our history. Ever since
Plato s time a certain number of professional mathematicians
had lived at Athens ; and about the year 420 this school again
ai-ij uired considerable reputation, largely in consequence of the
numerous students who after the murder of Hypatia migrated
tli ere from Alexandria. Its most celebrated members were
Proclus, Damascius, and Eutocius.
Proclus*. Proclus was born at Constantinople in February
1 1 1 and died at Athens on April 17, 485. He wrote a com
mentary on Euclid s Elements, of which that part which deals
with the first book is extant and contains a great deal of valu
able information on the history of Greek mathematics : he is
verbose and dull but luckily he has preserved for us quotations
from other and better authorities. His commentary has been
edited by G. Friedlein, Leipzig, 1873. Proclus was succeeded
as head of the school by Marinus, and Marinus by Isidorus.
Damascius. Eutocius. Two pupils of Isidorus, who in
their turn subsequently lectured at Athens, may be mentioned
in passing. Damascius of Damascus, circ. 490, added to Euclid s
Eli inmds a fifteenth book on the inscription of one regular
solid in another. Eutocius^ circ. 510, wrote commentaries on
the first four books of the Conies of Apollonius and on
various works of Archimedes ; he also published some examples
of practical Greek arithmetic. His works have never been
edited though they would seem to deserve it.
This later Athenian school was carried on under great
difficulties owing to the opposition of the Christians. Proclus,
for example, was repeatedly threatened with death because he
was "a philosopher." His remark "after all, my body does
not matter, it is the spirit that I shall take with me when
I die," which he made to some students who had offered to
def.-nd him, has been often quoted. The Christians, after seve
ral ineffectual attempts, at last got a decree from Justinian in
J!) that " heathen learning " should no longer be studied at
* Srt> rntsrxiH Inuiiii n nlt-r die ncu tinfiirj nmlcnt n Hrhulicn (It 1 *
by J. H. Knoche, Herford, 1865.
U.
THE SECOND ALEXANDRIAN SCHOOL.
Athens. That date therefore marks the end of the Athenian
school.
The church at Alexandria was less influential, and the
city was more remote from the centre of civil power. The
schools there were thus suffered to continue, though their
existence was of a precarious character. Under these con
ditions mathematics continued to be read there for another
hundred years but all interest in the study had gone.
Roman Mathematics*.
I ought not to conclude this part of the history without
any mention of Roman mathematics, for it was through Rome
that mathematics first passed into the curriculum of mediaeval
Europe, and in Rome all modern history has its origin. There
is however very little to say on the subject. The chief study of
the place was in fact the art of government, whether by law,
by persuasion, or by those material means on which all govern
ment ultimately rests. There were no doubt professors who
could teach the results of Greek science but there was no
demand for a school of mathematics. Italians who wished to
learn more than the elements of the science went to Alex
andria or to places which drew their inspiration from Alex
andria.
The subject as taught in the mathematical schools at Rome
seems to have been confined in arithmetic to the art of calcula
tion (no doubt by the aid of the abacus) and perhaps some of
the easier parts of the work of Nicomachus ; and in geometry
to a few practical rules ; though some of the arts founded on a
knowledge of mathematics (especially that of surveying) were
carried to a high pitch of excellence. It would seem also that
special attention was paid to the representation of numbers by
* The subject is discussed by Cantor, chaps, xxv., xxvi., and xxvu.;
also by Hankel, pp. 294304,
ROMAN MATHEMATICS. 115
signs. The manner of indicating numbers up to ten by the
use of fingers must have been in practice from quite early
times, but about the first century it had been developed by
the Romans into a finger-symbolism by which numbers up to
10000 or perhaps more could be represented : this would seem
to have been taught in the Roman schools. The system would
hardly be worth notice but that its use has still survived in
the Persian bazaars.
I am not aware of any Latin work on the principles of
mechanics, but there were numerous books on the practical
side of the subject which dealt elaborately with architectural
and engineering problems. We may judge what they were like
by the Matkematici Veteres, which is a collection of various
short treatises on catapults, engines of war, &c. (an edition
was published in Paris, in 1693): and by the Keo-rot, written
by Sextus Julius Africanus about the end of the second century,
which contains, amongst other things, rules for finding the
breadth of a river when the opposite bank is occupied by an
enemy, how to signal with a semaphore, &c.
In the sixth century Boethius published a geometry con
taining a few propositions from Euclid and an arithmetic
founded on that of Nicomachus ; and about the same time
Cassiodorus discussed the foundation of a liberal education
which, after the preliminary trivium of grammar, logic, and
rhetoric, meant the quadrivium of arithmetic, geometry, music,
and astronomy. These works were written at Rome in the
closing years of the Athenian and Alexandrian schools and
I therefore mention them here, but as their only value lies in
the fact that they became recognized text-books in mediaeval
education I postpone their consideration to chapter vin.
Theoretical mathematics was in fact an exotic study at
Rome ; not only was the genius of the people essentially prac
tical, but, alike during the building of their empire, while it
lasted, and under the Goths, all the conditions were unfavour
able to abstract science.
On the other hand, Alexandria was exceptionally well
82
116 THE SECOND ALEXANDRIAN SCHOOL.
placed to be a centre of science. From the foundation of the
city to its capture by the Mohammedans it was disturbed
neither by foreign nor by civil war, save only for a few years
when the rule of the Ptolemies gave way to that of Rome : it
was wealthy, and its rulers took a pride in endowing the uni
versity : and lastly, just as in commerce it became the meeting-
place of the east and the west, so it had the good fortune to be
the dwelling-place alike of Greeks and of various Semitic people;
the one race shewed a peculiar aptitude for geometry, the other
for all sciences which rest on measurement. Here too, how
ever, as time went on the conditions gradually became more
unfavourable, the endless discussions by the Christians on
theological dogmas and the increasing insecurity of the empire
tending to divert men s thoughts into other channels.
End of the second Alexandrian School.
The precarious existence and unfruitful history of the last
two centuries of the second Alexandrian School need no record.
In 632 Mohammed died, and within ten years his successors
had subdued Syria, Palestine, Mesopotamia, Persia, and Egypt.
The precise date on which Alexandria fell is doubtful but
the most reliable Arab historians give Dec. 10, 641 a date
which at any rate is correct within eighteen months.
With the fall of Alexandria the long history of Greek
mathematics came to a conclusion. It seems probable that the
greater part of the famous university library and museum had
been destroyed by the Christians a hundred or two hundred
years previously, and what remained was unvalued and neg
lected. Some two or three years after the first capture of
Alexandria a serious revolt occurred in Egypt, which was
ultimately put down with great severity. I see no reason to
doubt the truth of the account that after the capture of the
city the Mohammedans destroyed such university buildings and
FALL OF ALEXANDRIA. 117
collections as were still left. It is said that, when the Arab
commander ordered the library to be burnt, the Greeks made
such energetic protests that he consented to refer the matter to
the caliph Omar. The caliph returned the answer, " as to the
books you have mentioned, if they contain what is agreeable
with the book of God, the book of God is sufficient without
them ; and, if they contain what is contrary to the book of God,
there is no need for them; so give orders for their destruction."
The account goes on to say that they were burnt in the public
baths of the city, and that it took six months to consume
them all.
118
CHAPTER VI.
THE BYZANTINE SCHOOL.
6411453.
IT will be convenient to consider the Byzantine school in.
connection with the history of Greek mathematics. After the
capture of Alexandria by the Mahommedans the majority of
the philosophers, who previously had been teaching there,
migrated to Constantinople which then became the centre of
Greek learning in the East arid remained so for 800 years.
But though the history of the Byzantine school stretches over
so many years a period about as long as that from the
Norman Conquest to the present day it is utterly barren of
any scientific interest ; and its chief merit is that it preserved
for us the works of the different Greek schools. The revelation
of these works to the West in the fifteenth century was one
of the most important sources of the stream of modern European
thought, and the history of the Byzantine school may be
summed up by saying that it played the part of a conduit-pipe
in conveying to us the results of an earlier and brighter age.
The time was one of constant war, and men s minds during
the short intervals of peace were mainly occupied with theo
logical subtleties and pedantic scholarship. I should not have
mentioned any of the following writers had they lived in the
HERO. PSELLUS. PLANUDES. BARLAAM. 119
Alexandrian period, but in. default of any others they may be
noticed as illustrating the character of the school. I ought
also perhaps to call the attention of the reader explicitly to
the fact that I am here departing from chronological order,
and that the mathematicians mentioned in this chapter were
contemporaries of those discussed in the chapters devoted to
the mathematics of the middle ages. The Byzantine school
was so isolated that I deem this the best arrangement of the
subject.
Hero. One of the earliest members of the Byzantine
school was Hero of Constantinople, circ. 900, sometimes called
the younger to distinguish him from Hero of Alexandria.
There is some difficulty in separating the works of these two
writers. Hero would seem to have written on geodesy and
mechanics as applied to engines of war.
During the tenth century two emperors Leo VI. and Con-
stantine VII. shewed considerable interest in astronomy and
mathematics, but the stimulus thus given to the study of these
subjects was only temporary.
Psellus. In the eleventh century Michael Psellus, born
in 1020, wrote a pamphlet on the quadrivium. It is now in
the National Library at Paris; it was printed at Bale in 1556.
He also wrote a Compendium Mathematicum which was printed
at Ley den in 1647.
In the fourteenth century we find the names of three
monks who paid attention to mathematics.
Planudes. The first of the three was Maximus Planudes ;
he wrote a commentary on the first two books of the Arithmetic
of Diophantus; this was published by Xylander, Bale, 1575:
a work on Hindoo arithmetic in which he introduced the
use of the Arabic numerals into the eastern empire ; this was
published by C. J. Gerhardt, Halle, 1865 : and another on
proportions which is now in the National Library at Paris.
Barlaam. The next was a Calabrian monk named Barlaam,
who was born in 1290 and died in 1348. He was the author
of a work on the Greek methods of calculation from which we
120 THE BYZANTINE SCHOOL.
derive a good deal of our information as to the way in which
the Greeks practically treated fractions : this was published in
Paris in 1606. Barlaam seems to have been a man of great
intelligence. He was sent as an ambassador to the pope at
Avignon, and acquitted himself creditably of a difficult mission;
while there he taught Greek to Petrarch. He was famous at
Constantinople for the ridicule he threw on the preposterous
pretensions of the monks at Mount Athos x who taught that
those who joined them could, by standing naked resting their
beards on their breasts and steadily regarding their stomachs,
see a mystic light which was the essence of .God; Barlaam
advised them to substitute the light of reason for that of their
stomachs a piece of advice which nearly cost him his life.
Argyrus. The last of these monks was Isaac Argyrus,
who died in 1372. He wrote three astronomical tracts, the
manuscripts of which are in the libraries at the Vatican,
Leyden, and Vienna: one on geodesy, the manuscript of which
is at the Escurial : one on geometry, the manuscript of which is
in the National Library at Paris : one on the arithmetic of
Nicomachus, the manuscript of which is in the National Library
at Paris : and one on trigonometry, the manuscript of which
is in the Bodleian at Oxford.
Nicholas Rhabdas. In the fourteenth or perhaps the
fifteenth century Nicholas Rhabdas of Smyrna wrote two
papers on arithmetic which are now in the National Library
at Paris and have been edited by P. Tannery, Paris, 1886.
He gave an account of the finger-symbolism (see above, p. 115)
which the Romans had introduced into the East and was then
current there; this is described by Bede and therefore would
seem to have been known as far west as Britain ; Jerome
also alludes to it.
Pachymeres. Early in the fifteenth century Pachymeres
wrote tracts on arithmetic, geometry, and four mechanical
machines.
Moschopulus. A few years later Emmanuel Moschopulus,
who died in Italy circ. 1460, wrote a treatise on magic squares.
MOSCHOPULUS.
121
A magic square* consists of a number of integers arranged
in the form of a square so that the sum of the numbers in
every row, in every column, and in each diagonal is the same.
If the integers be the consecutive numbers from 1 to n 2 , the
square is said to be of the nth order, and it is easily seen
that in this case the sum of the numbers in any row, column,
or diagonal is equal to \n (n 2 +1). Thus the first 16 integers,
arranged in either of the forms given below, form a magic
1
15
14
4
12
6
10
7
11
9
5
8
13
3
2
16
square of the fourth order, the sum of the numbers in every
row, every column, and each diagonal being 34.
In the mystical philosophy then current certain metaphy
sical ideas were often associated with particular numbers, and
thus it was natural that such arrangements of numbers should
attract attention and be deemed to possess magical properties.
The theory of the formation of magic squares is elegant and
several distinguished mathematicians have written on it,
but I need hardly say it is not useful : it is largely due to
De la Hire who gave rules for the construction of a magic
square of any order higher than the second. Moschopulus
seems to have been the earliest European writer who attempted
to deal with the mathematical theory, but his rules apply only
to odd squares. The astrologers of the fifteenth and sixteenth
centuries were much impressed by such arrangements. In
particular the famous Cornelius Agrippa (1486 1535) con
structed magic squares of the orders 3, 4, 5, 6, 7, 8, 9 which
* On the formation and history of magic squares, see my Mathematical
Recreations and Problems, London, 1892, chap. v. On the work of Mos
chopulus, see chap. iv. of S. Giinther s Geschichte der mathcmutischen
Wissenschaften, Leipzig, 1876.
122 THE BYZANTINE SCHOOL.
were associated respectively with the seven astrological
"planets:" namely, Saturn, Jupiter, Mars, the Sun, Venus,
Mercury, and the Moon. He taught that a square of one
cell, in which unity was inserted, represented the unity and
eternity of God ; while the fact that a square of the second
order could not be constructed illustrated the imperfection of
the four elements, air, earth, fire, and water ; and later writers
added that it was symbolic of original sin. A magic square
engraved on a silver plate was often prescribed as a charm
against the plague, and one (namely, that in the first diagram
on the last page) is drawn in the picture of melancholy painted
about the year 1500 by Albrecht Diirer. Such charms are
worn still in the East.
Constantinople was captured by the Turks in 1453, and the
last semblance of a Greek school of mathematics then disap
peared. Numerous Greeks took refuge in Italy. In the West
the memory of Greek science had vanished and even the names
of all but a few Greek writers were unknown ; thus the books
brought by these refugees came as a revelation to Europe, and
as we shall see later gave an immense stimulus to the study of
science.
123
CHAPTER VII.
SYSTEMS OF NUMERATION AND PRIMITIVE ARITHMETIC*.
I HAVE in many places alluded to the Greek method of
expressing numbers in writing, and I have thought it best to
defer to this chapter the whole of what I wanted to say on the
various systems of numerical notation which were displaced
by the system introduced by the Arabs.
First, as to symbolism and language. The plan of indi
cating numbers by the digits of one or both hands is so natural
that we find it in universal use among early races, and the
members of all tribes now extant are able to indicate by signs
numbers at least as high as ten : it is stated that in some
languages the names for the first ten numbers are derived from
the fingers used to denote them. For larger numbers we soon
however reach a limit beyond which primitive man is unable
to count, while as far as language goes it is well known that
many tribes have no word for any number higher than ten, and
some have no word for any number beyond four, all higher
numbers being expressed by the words plenty or heap : in
connection with this it is worth remarking that the Egyptians
* The subject of this chapter is discussed by Cantor and by Hankel.
See also the Philosophy of Arithmetic by John Leslie, second edition,
Edinburgh, 1820. Besides these authorities the article on Arithmetic
by George Peacock in the Encyclopaedia Metropolitana, Pure Sciences,
London, 1845; E. B. Tylor s Primitive Culture, London, 1873; Lea
signes numeraux et Varithmetique chez les peuples de Vantiquite...\)y
T. H. Martin, Rome, 1864; and Die Zahlzeichen...by G. Friedlein,
Erlangen, 1869, should be consulted.
124 SYSTEMS OF NUMERATION
used the symbol for the word heap to denote an unknown
quantity in algebra (see above, p. 4).
The number five is generally represented by the open hand,
and it is said that in almost all languages the words five and
hand are derived from the same root. It is possible that in
early times men did not readily count beyond five, and things if
more numerous were counted by multiples of it. Thus the
Roman symbol X for ten probably represents two "V"s,
placed apex to apex and seems to point to a time when things
were counted by fives*. ID connection with this it is worth
noticing that both in Java and also among the Aztecs a week
consisted of five days*
The members of nearly all races of which we have now
any knowledge seem however to have used the digits of both
hands to represent numbers. They could thus count up to and
including ten, and therefore were led to take ten as their radix
of notation. In the English language for example all the
words for numbers higher than ten are expressed on the decimal
system: those for 11 and 12, which at first sight seem to be
exceptions, being derived from Anglo-Saxon words for one and
ten and two and ten respectively.
Some tribes seem to have gone further and by making use
of their toes were accustomed to count by multiples of twenty.
The Aztecs, for example, are said to have done so. It may be
noticed that we still count some things (e.g. sheep) by scores,
the word score signifying a notch or scratch made on the
completion of the twenty ; while the French also talk of
quatre-vingt, as though at one time they counted things by
multiples of twenty. I am not, however, sure whether the
latter argument is worth anything, for I have an impression
that I have seen the word octante in old French books ; and
there is no question t that septante and nonante were at one
* See also the Odyssey, iv. 413 415 in which apparently reference is
made to a similar custom.
t See for example, V. M. de Kempten s Practique . . .a ciffrer, Antwerp,
1556.
AM) PRIMITIVE ARITHMETIC. 125
time common words for seventy and ninety, and indeed they
are still retained in^^me dialects.
The only tribes of whom I have read who did not count in
terms either of five or of some multiple of five are the Bolans
of West Africa who are said to have counted by multiples of
seven, and the Maories who are said to have counted by
multiples of eleven.
Up to ten it is comparatively easy to count, but primitive
people found and still find great difficulty in counting higher
numbers; apparently at first this dilficulty was only overcome
by the method (still in use in South Africa) of getting two men,
one to count the units up to ten on his fingers, and the other
to count the number of groups of ten so formed. To us it is
obvious that it is equally effectual to make a mark of some
kind on the completion of each group of ten, but it is alleged
that the members of many tribes never succeeded in counting
numbers higher than ten unless by the aid of two men.
Most races who shewed any aptitude for civilization pro
ceeded further and invented a way of representing numbers by
\ means of pebbles or counters arranged in sets of ten ; and this
1 in its turn developed into the abacus or swan-pan. This in
strument was in use among nations so widely separated as the
Etruscans, Greeks, Egyptians, Hindoos, Chinese, and Mexicans;
, and was, it is believed, invented independently at several
different centres. It is still in common use in Russia, China,
and Japan.
In its simplest form (fig. i) the abacus consists of a wooden
board with a number of grooves cut in it, or of a table covered
with sand in which grooves are made with the fingers. To re
present a number, as many counters or pebbles (calculi) are put
on the first groove as there are units, as many on the second
as there are tens, and so on. When * by its aid a^ number of
objects are counted, for each object a pebble is put on the first
groove; and, as soon as there are ten pebbles there, they are
taken off and one pebble put on the second groove ; and so on.
It was sometimes, as in the Aztec quipus, made with a number
126
SYSTEMS OF NUMERATION
Fig. i.
Q
Fig. iii.
munm
i
AND PRIMITIVE ARITHMETIC. 127
of parallel wires or strings stuck in a piece of wood on which
beads could be threaded; and in that form is called a swan-pan.
In the number represented in each of the instruments drawn
on the opposite page there^ are seven thousands, three hun
dreds, no tens, and five units, that is, the number is, 7305.
Some races counted from left to right, others from right to left,
but this is a mere matter of convention.
The Roman abaci seem to have been rather more elaborate.
They contained two marginal grooves or wires, one with four
beads to facilitate the addition of fractions whose denominators
were four, and one with twelve beads for fractions whose de
nominators were twelve: but otherwise they do not differ in
principle from those described above. They were generally
made to represent numbers up to 100,000000. There are no
Greek abaci now in existence but there is no doubt that they
were similar to the Roman ones. The Greeks and Romans
used their abaci as boards on which they played a game
something like backgammon.
In the Russian tschotii (fig. ii) the instrument is improved
by having the wires set in a rectangular frame, and ten (or nine)
beads are permanently threaded on each of the wires, the wires
being considerably longer than is necessary to hold them. If
the frame be held horizontal, and all the beads be towards one
side, say the lower side of the frame, it is possible to represent
any number by pushing towards the other or upper side as
many beads on the first wire as there are units in the number,
as many beads on the second wire as there are tens in the
number, and so on. Calculations can be made somewhat more
rapidly if the five beads on each wire next to the upper side
be coloured differently to those next to the lower side, and they
can be still further facilitated if the first, second, ..., ninth
counters in each column be respectively marked with symbols
for the numbers 1, 2,..., 9. Gerbert is said to have introduced
the use of such marks, called apices, towards the close of the
tenth century (see below, p. 141).
Figure iii represents the form of swan-pan in common use in
128 SYSTEMS OF NUMERATION
China and Japan. There the development is carried one step
further, and five beads on each wire are replaced by a single
bead of a different form or on a different division, but apices
are not used. I am told that an expert Japanese can by the
aid of a swan- pan add numbers as rapidly as they can be read
out to him. It will be noticed that the instrument represented
in figure iii on p. 126 is made so that two numbers can be ex
pressed at the same time on it.
The use of the abacus in addition and subtraction is
evident. It can be used also in multiplication and division ;
rules for these processes, illustrated by examples, are given
in the arithmetic known as The Grounds of Artes* which was
published by Eecord at London in 1540.
The abacus is obviously only a concrete way of representing
a number in the decimal system of notation, that is, by means
of the local value of the digits. Unfortunately the method of
writing numbers developed on different lines, and it was not
until about the thirteenth century of our era when a symbol
zero used in conjunction, with nine other symbols was intro
duced that a corresponding notation in writing was adopted in
Europe.
Next, as to the means of representing numbers in writing. ,
In general we may say that in the earliest times a number
was (if represented by a sign and not a word) indicated by the
requisite number* of strokes. Thus in an inscription from
Tralles in Caria of the date 398 B.C. the phrase seventh year is
represented by creos | | | | | | | . These strokes may have been
mere marks; or perhaps they originally represented fingers,
since in the Egyptian hieroglyphics the symbols for the
numbers 1, 2, 3, are one, two, and three fingers respectively
though in the later hieratic writing these symbols had become
reduced to straight lines. Additional symbols for 10 and 100
were soon introduced : and the oldest extant Egyptian and
Phoenician writings repeat the symbol for unity as many times
(up to 9) as was necessary, and then repeat the symbol for ten as
* Edition of 1610, pp. 225262.
AND PRIMITIVE ARITHMETIC. 120
many times (up to 9) as was necessary, and so on. No speci
mens of Greek numeration of a similar kind are in existence,
but there is every reason to believe the testimony of lamblichus
who asserts that this was the method by which the Greeks first
expressed numbers in writing.
This way of representing numbers remained in current use
throughout Roman history; and for greater brevity they or the
Etruscans added separate signs for 5, 50, &c. The Roman
[symbols are generally merely the initial letters of the names
jof the numbers; thus c stood for centum or 100, M for mille
I or 1000. The symbol v for 5 seems to have originally repre-
I sen ted an open palm with the thumb extended. The symbols
JL for 50 and D for 500 are said to represent the upper
jhalves of the symbols used in early times for c and M. The
Isubtractive forms like iv for mi are probably of a later
origin.
Similarly in Attica five was denoted by II the first letter
of TreVre, or sometimes by F; ten by A the initial letter of
Se /ca; a hundred by H for e/cardv; a thousand by X for ;(i\ioi;
while 50 was represented by a A written inside a II; and so
;0n. These Attic symbols continued to be used for inscriptions
and formal documents until a late date.
This, if a clumsy, is a perfecly intelligible system; but the
iGreeks at some time in the third century before Christ aban
doned it for one which offers no special advantages in denoting
a given number, while it makes all the operations of arithmetic
(exceedingly difficult. In this, which is known from the place
(where it was introduced as the Alexandrian system, the
numbers from 1 to 9 are represented by the first nine letters
Nof the alphabet; the tens from 10 to 90 by the next nine
(fetters; and the hundreds from 100 to 900 by the next nine
letters. To do this the Greeks wanted 27 letters, and as
hoir alphabet contained only 24, they re-inserted two letters
(the digamma and koppa) which had formerly been in it but
had become obsolete, and introduced at the end another symbol
taken from the Phoenician alphabet. Thus the ten letters
B.
130 SYSTEMS OF NUMERATION
a to t stood respectively for the numbers from 1 to 10; the
next eight letters for the multiples of 10 from 20 to 90; and
the last nine letters for 100, 200, &c. up to 900. Intermediate
numbers like 11 were represented as the sum of 10 and 1, that
is, by the symbol ta . This afforded a notation for all numbers
up to 999 ; and by a system of suffixes and indices it was
extended so as to represent numbers up to 100,000000.
There is no doubt that these signs were at first only used
as a way of expressing a result attained by some concrete or
experimental method, and the idea of operating with the
symbols themselves in order to obtain the results is of a later
growth, and is one with which the Greeks never became
familiar. The non-progressive character of Greek arithmetic
may be partly due to their unlucky adoption of the Alex
andrian system which caused them for most practical purposes
to rely on the abacus, and to supplement it by a table of multi
plications which was learnt by heart. The results of the mul
tiplication or division of numbers other than those in the
multiplication table might have been obtained by the use of
the abacus, but in fact they were generally got by repeated
additions and subtractions. Thus, as late as 944, a certain
mathematician who in the course of his work wants to multiply ^
400 by 5 finds the result by addition. The same writer, when
he wants to divide 6152 by 15, tries all the multiples of 15
until he gets to 6000, .this gives him 400 and a remainder
152; he then begins again with all the multiples of 15 until
he gets to 150, and this gives him 10 and a remainder 2.
Hence the answer is 410 with a remainder 2.
A few mathematicians however such as Hero of Alex
andria, Theon, and Eutocius multiplied and divided in what
is essentially the same way as we do. Thus to multiply 18 by
13 they proceeded as follows.
7-x ty= (t + y) (c +*) 13 x 18 = (10 + 3) (10 + 8)
= i(i + ri) + y(i, + ri) = 10 (10 + 8) + 3 (10 + 8)
= p + TT 4- X + KS = 1 00 + 80 + 30 + 24
= crX8 -234,
AND PRIMITIVE ARITHMETIC. 131
I suspect that the last step, in which they had to add four
numbers together, was obtained by the aid of the abacus.
These however were men of exceptional genius, and we
must recollect that for all ordinary purposes the art of calcu
lation was performed only by the use of the abacus and the
multiplication table, while the term arithmetic was confined
to the theories of ratio, proportion, and of numbers (see above,
p. 59).
All the systems here described were more or less clumsy,
and they have been displaced among civilized races by the Arabic
system in which there are ten digits or symbols, namely, nine
for the first nine numbers and another for zero. In this
system an integral number is denoted by a succession of digits,
each digit representing the product of that digit and a power
of ten, and the number being equal to the sum of these pro
ducts. Thus, by means of the local value attached to nine
symbols and a symbol for zero, any number in the decimal
scale of notation can be expressed. The history of the develop
ment of the science of arithmetic with this notation will be
considered in a subsequent chapter (ch. XL).
92
132
SECOND PERIOD.
JWatfjemattcs of tfjc JJXtlfole &ges antr of tfte
This period begins about the sixth century, and may be said
to end with the invention of analytical geometry and of the
infinitesimal calcidus. The characteristic feature of this period
is the creation of modern arithmetic, algebra, and trigonometry.
133
I commenced this history by dividing it in three periods.
I have discussed the history of mathematics under Greek influ
ence, and I now come to that of the mathematics of the middle
ages and renaissance. The history of this period has not been
investigated with the same fulness as that of earlier or of later
times, and the relative importance of some mathematicians
who lived in this period has been estimated differently by
different writers.
I shall consider first, in chapter vin., the rise of learning
in western Europe, and the mathematics of the middle ages.
Next, in chapter ix., I shall discuss the nature and history of
Arabian mathematics, and in chapter x. their introduction into
Europe. I shall then, in chapter XL, trace the subsequent
progress of arithmetic to the year 1637. Next, in chapter XIL,
I shall treat of the general history of mathematics during the
renaissance, from the invention of printing to the beginning of
the seventeenth century, say, from 1450 to 1637; this contains
an account of the commencement of the scientific treatment of
arithmetic, algebra, and trigonometry. Lastly, in chapter xni.,
I shall consider the revival of interest in mechanics, experi
mental methods, and pure geometry which marks the last few
years of this period, and serves as a connecting link between
the mathematics of the renaissance and the mathematics of
modern times.
134
CHAPTER VIII.
THE RISE OF LEARNING IN WESTERN EUROPE.*
CIRC. 6001200.
Education in the sixth, seventh, and eighth centuries.
THE first few centuries of this second period of our history
are singularly barren of interest; and indeed it would be
strange if we found science or mathematics studied by those
who lived in a condition of perpetual war. Broadly speaking
we may say that from the sixth to the eighth centuries the
only places of study in western Europe were the Benedictine
monasteries. We may find there some slight attempts at a
study of literature ; but the science usually taught was con
fined to the use of the abacus, the method of keeping accounts,
and a knowledge of the rule by which the date of Easter could
be determined. Nor was this unreasonable, for the monk had
renounced the world, and there was no reason why he should
learn more science than was required for the services of the
church and his monastery. The traditions of Greek and Alexan
drian learning gradually died away. Possibly in Rome and
a few favoured places copies of the works of the great Greek
* The mathematics of this period has been discussed by Cantor ;
by M. S. Gtinther, Geschichte des mathematischen Unterrichtes im deut-
schen Mittelalter, Berlin, 1887 ; and by H. Weissenborn, Kenntnix* </rr
Mathematik des Mittelalter^ Berlin, 1888.
BOETHIUS. 135
mathematicians were obtainable, though with difficulty, but
there were no students, the books were unvalued, and in
time became very scarce.
Three authors of the sixth century Boethius, Cassiodorus,
and Isidorus may be named whose writings serve as a con
necting link between the mathematics of classical and of
mediaeval times. As their works remained standard text
books for some six or seven centuries it is necessary to mention
them, but it should be understood that this is the only reason
for doing so and they shew no special mathematical ability.
It will be noticed that these authors were contemporaries of the
later Athenian and Alexandrian schools (see above-, p. 115).
Boethius. Anicius Manlius Severinus Boethius , or as the
name is sometimes written Boetius, born at Rome about
475 and died in 526, belonged to a family which for the two
preceding centuries had been esteemed one of the most illus
trious in Rome. It was formerly believed that he was educated
at Athens : this is somewhat doubtful, but at any rate he was
exceptionally well read in Greek literature and science. He %
would seem to have wished to devote his life to literary
pursuits ; but recognizing " that the world would be happy
only when kings became philosophers or philosophers kings,"
he yielded to the pressure put on him and took an active
share in politics. He was celebrated for his extensive
charities, and, what in those days was very rare, the care that
he took to see that the recipients were worthy of them. He was
elected consul at an unusually early age, and took advantage of
his position to reform the coinage and to introduce the public
use of sun-dials, water-clocks, &c. He reached the height of
his prosperity in 522 when his two sons were inaugurated as
consuls. His integrity and attempts to protect the provincials
from the plunder of the public officials brought on him the
hatred of the Court. He was sentenced to death while absent
from Rome, seized at Ticinum, and in the baptistery of the
church there tortured by drawing a cord round his head till
the eyes were forced out of the sockets, and finally beaten to
136 THE RISE OF LEARNING IN WESTERN EUROPE.
death with clubs on Oct. 23, 526. Such at least is the account
that has come down to us. At a later time his merits were
recognized, and tombs and statues erected in his honour by
the state.
Boethius was the last Eoman of any note who studied the
language and literature of Greece, and his works afforded to
mediaeval Europe the means of entering into the intellectual
life of the old world. His importance in the history of litera
ture is thus very great, but it arises merely from the
accident of the time at which he lived. After the introduction
of Aristotle s works in the thirteenth century his fame died
away, and he has now sunk into an obscurity which is as great
as was once his reputation. He is best known by his Conso-
latio, which was translated by Alfred the Great into Anglo-
Saxon. For our purpose it is sufficient to note that the teaching
of early mediaeval mathematics was mainly founded on his
geometry and arithmetic.
His Geometry consists of the enunciations (only) of the first
book of Euclid, and of a few selected propositions in the third
and fourth books, but with numerous practical applications to
finding areas, <fec. He adds an appendix with proofs of the
first three propositions to shew that the enunciations may be
relied on. He also wrote an Arithmetic, founded on that of
Mcomachus. These works have been edited by G. Friedlein,
Leipzig, 1867. A text-book on music by him was in use at
Oxford within the present century.
Cassiodorus. A few years later another Roman, Magnus
Aurelius Cassiodorus, who was born about 480 and died in
566, published two works, De Institutione Divinarum Litte-
rarum and De Artibus ac Disciplinis, in which not only the
preliminary trivium of grammar, logic, and rhetoric were dis
cussed, but also the mathematical quadrivium of arithmetic,
geometry, music, and astronomy. These were considered
standard works during the middle ages : the former was
printed at Venice in 1729.
Isidorus. Isidorus, bishop of Seville, born in 570 and
ALCUIN. 137
died in 636, was the author of an encyclopaedic work in 20
volumes called Origines, of which the third volume is given
up to the quadrivium. It was published at Leipzig in 1833.
The Cathedral and Conventual Schools*.
When, in the latter half of the eighth century, Charles the
Great had established his empire, he determined to promote
learning so far as he was able; and he began by commanding
that schools should be opened in connection with every
cathedral and monastery in his kingdom; an order which was
approved and materially assisted by the popes. It is interest
ing to us to know that this was done at the instance and
under the direction of two Englishmen, Alcuin and Clement,
who had attached themselves to his court; a fact which may
serve to remind us that during the eighth century England
and Ireland were in advance of the rest of Europe as far as
learning went.
Alcuin f. Of these the more prominent was Alcuin who
was born in Yorkshire in 735 and died at Tours in 804. He
was educated at York under archbishop Egbert his "beloved
master" whom he succeeded as director of the school there.
Subsequently he became abbot of Canterbury, and was sent to
Rome by Offa to procure the pallium for archbishop Eanbald.
On his journey back he met Charles at Parma; the emperor
took a great liking to him, and finally induced him to take up
his residence at the imperial court, and there teach rhetoric,
logic, mathematics, and divinity. Alcuin remained for many
years one of the most intimate and influential friends of
Charles who constantly employed him as a confidential ambas-
* See The Schools of Charles the Great and the Restoration of Educa
tion in the ninth century by J. B. Mullinger, London, 1877.
t See the life of Alcuin by F. Lorentz, Halle, 1829, translated by
J. M. Slee, London, 1837; Alcuin nml w/ Jnhrliuntlert by K. Werner,
Paderborn, 1876 ; and Cantor, vol. i. pp. 712721.
138 THE RISE OF LEARNING IN WESTERN EUROPE.
sador: as such he spent the years 791 and 792 in England, and
while there reorganized the studies at his old school at York.
In 801 he begged permission to retire from the court so as to
be able to spend the last years of his life in quiet: with dif
ficulty he obtained leave, and went to the abbey of St. Martin
at Tours, of which he had been made head in 796. He estab
lished a school in connection with the abbey which became
very celebrated, and he remained and taught there till his
death on May 19, 804.
Most of the extant writings of Alcuin deal with theology
or history, but they include a collection of arithmetical proposi
tions suitable for the instruction of the young. The majority
of the propositions are easy problems, either determinate or
indeterminate, and are, I presume, founded on works with
which he had become acquainted when at Rome. The follow
ing is one of the most difficult, and will give an idea of the
character of the work. If one hundred bushels of corn be
distributed among one hundred people in such a manner
that each man receives three bushels, each woman two, and
each child half a bushel: how many men, women and chil
dren were there? The general solution is (20 3^) men, 6n
women, and (80 - 2n) children, ^where n may have any of
the values 1, 2, 3, 4, 5, 6. Alcuin only states the solution for
which n = 3 ; that is, he gives as the answer 1 1 men, 1 5 women,
and 74 children.
This collection however was the work of a man of excep
tional genius, and probably we shall be correct in saying that
mathematics, if taught at all in a school, was generally con
fined to the geometry of Boethius, the use of the abacus and mul
tiplication table, and possibly the arithmetic of Boethius ; while
except in one of these schools or in a Benedictine cloister it
was hardly possible to get either instruction or opportunities
for study. It was of course natural that the works used should
come from Roman sources, for Britain and all the countries
included in the empire of Charles had at one time formed part
of the western half of the Roman empire, and their inhabitants
EDUCATION IN THE NINTH CENTURY. 139
continued for a long time to regard Rome as the centre of
civilization, while the higher clergy kept up a tolerably constant
intercourse witji Rome.
After the death of Charles many of the schools confined
themselves to teaching Latin, music, and theology, that is,
those subjects some knowledge of which was essential to the
worldly success of the higher clBrgy. Hardly any science or
mathematics was taught, but the continued existence of the |
schools gave an opportunity to any teacher whose learning or
zeal exceeded the narrow limits fixed by tradition ; and though
there were but few who availed themselves of the oppor
tunity, yet the number of those desiring instruction was so
large that it would seem as if any one who could teach was
certain to attract a considerable audience. A few schools at
which this was the case became large and acquired a certain
degree of permanence, but even in them the teaching was still
usually confined to the trivium and quadrivium. The former
comprised the three arts of grammar, logic, and rhetoric,
but practically meant the art of reading and writing Latin ;
nominally the latter included arithmetic and geometry with
their applications, especially to music and astronomy, but in
fact it rarely meant more than arithmetic sufficient to enable
one to keep accovints, music for the church services, geometry
for the purpose of land surveying, and astronomy sufficient to
enable one to calculate the feasts and fasts of the church.
The seven liberal arts are enumerated in the line, Lingua,
tropuSj ratio; numerus, tonus, angulus, astra. Any student
who got beyond the trivium was looked on as a man of great
erudition, Qui tria, qui septein. qui totum scibile novit, as
a verse of the eleventh century runs. The special questions
which then and long afterwards attracted the best thinkers
were logic and certain portions of transcendental theology and
philosophy. We may sum the matter up by saying that during
the ninth and tenth centuries the mathematics taught was
still usually confined to that comprised in the two works of
Boethiua together with the practical use of the abacus and
140 THE RISE OF LEARNING IN WESTERN EUROPE.
the multiplication table, though during the latter part of the
time a wider range of reading was undoubtedly accessible.
Gerbert*. In the tenth century a man appeared who would
in any age have been remarkable and who gave a great stimulus
to learning. This was Gerbert, an Aquitanian by birth,
who died in 1003 at about the age of fifty. His abilities
attracted attention to him even when a boy, and procured his
removal from the abbey school at Aurillac to the Spanish
inarch where he received a good education. He was in Rome
in 971, and his proficiency in music and astronomy excited
considerable interest : at that time he was not much more than
twenty, but he had already mastered all the branches of the
trivium and - quadriviuin, as then taught, except logic ; and to
learn this he moved to Rheims which archbishop Adalbero
had made the most famous school in Europe. Here he was at
once invited to teach, and so great was his fame that to him
Hugh Capet entrusted the education of his son Robert who was
afterwards king of France. Gerbert was especially famous for
his construction of abaci and of terrestrial and celestial globes ;
he was accustomed to use the latter to illustrate his lectures.
These globes excited great admiration which he utilized by
offering to exchange them for copies of classical Latin works,
which seem already to have become very scarce ; and the
better to effect this he appointed agents in the chief towns
of Europe. To his efforts it is believed we owe the preserva
tion of several Latin works, but he made a rule to reject the
Christian fathers and Greek authors from his library. In 982
he received the abbey of Bobbio, and the rest of his life was
taken up with political intrigues; he became archbishop of
Rheims in 991, and of Ravenna in 998 ; in 999 he was elected
pope, when he took the title of Sylvester II. ; as head of the
Church, he at once commenced an appeal to Christendom to
* Weissenborn, in the work already mentioned, treats Gerbert very
fully ; see also La vie et les oeuvres de Gerbert, by A. Olleris, Clermont,
1867 ; and Gerbert von Aurillac, by K. Werner, 2nd Edition, Vienna,
1881.
GEKHERT. 141
arm and defend the Holy Land, thus forestalling Peter the
Hermit by a century, but he died on May 12, 1003 before he
had time to elaborate his plans. His library is I believe pre
served in the Vatican.
So remarkable a personality left a deep impress on his
generation, and all sorts of fables soon began to collect around
his memory. It seems certain that he made a clock which was
long preserved at Magdeburg, and an organ worked by steam
which was still at Rheims two centuries after his death. All
this only tended to confirm the suspicions of his contemporaries
that he had sold himself to the devil ; and the details of his
interviews with that gentleman, the powers he purchased, and
his effort to escape from his bargain when he was dying, may
be read in the pages of William of Malrnesbury, Orderic
Vitalis, and Platina. To these anecdotes the first named
writer adds the story of the statue inscribed with the words
" strike here," which having amused our ancestors in the Gesta
Romanorum has been recently told again in the Earthly
Paradise.
Extensive though his influence was, it must not be supposed
that Gerbert s writings shew any great originality. His mathe
matical works comprise a treatise on the use of the abacus, one
on arithmetic entitled De Numerorum Divisione, and one on
geometry. An improvement in the abacus, attributed by some
writers to Boethius but which is more probably due to Gerbert,
is the introduction in every column of beads marked by different
characters, called apices, for each of the numbers from 1 to 9
instead of nine exactly similar counters or beads. These apices
were probably of Indian or Arabic origin, and lead to a repre
sentation of numbers essentially the same as the Gobar numerals
reproduced below (see p. 190), there was however no symbol for
zero ; the step from this concrete system of denoting numbers
by a decimal system on an abacus to the system of denoting
them by similar symbols in writing seems to us to be a small
one, but it would appear that Gerbert did not make it. His
work on geometry is of unequal ability; it includes a few
142 THE RISE OF LEARNING IN WESTERN EUROPE.
applications to land-surveying and the determination of the
heights of inaccessible objects, but much of it seems to be
copied from some pythagorean text-book. In the course of it
he however solves one problem which was of remarkable diffi
culty for that time. The question is to find the sides of a
right-angled triangle whose hypothenuse and area are given.
He says, in effect, that if these latter be denoted respectively
by c and 7* 2 , then the lengths of the two sides will be
h* + Jc 2 -4;k 2 } and 1 {^/TTlA* - Jc 2 ~^W}.
Bernelinus. One of Gerbert s pupils Bernelinus published
a work on the abacus (reprinted in Olleris s edition of Gerbert s
works, pp. 311 326) which is, there is very little doubt, a
reproduction of the teaching of Gerberfc. It is valuable as
indicating that the Arabic system of writing numbers was still
unknown in Europe.
The rise of the early mediaeval universities*.
At the end of the eleventh century or the beginning of the
twelfth a great revival of learning took place at several of these
cathedral or monastic schools; or perhaps we should rather
say that in some cases teachers who were not members of
the school settled in its vicinity and with the sanction of the
authorities gave lectures which were in fact always on theo
logy, logic, or civil law. As the students at these centres
grew in numbers, it became possible and desirable to act to
gether whenever any interest common to all was concerned.
The association thus formed was a sort of guild or trades union,
or in the language of the time a universitas magistrorum et
scholarium. This was the first stage in the development of
* Nearly all the known facts 011 the subject of the mediaeval uni
versities are collected in Die Universitaten des Mittelalters Ms 1400 by
P. H. Denifle, Berlin, 1885; see also vol. i. of the University oj
Cambridge by J. B. Mullinger, Cambridge, 1873.
MKDLAEVAL UNIVERSITIES.
every early mediaeval university. I In some cases, as at Paris,
the governing body of the university was formed by the teachers
alone, in others, as at Bologna, by both teachers and students ;
but in all cases precise rules for the conduct of business and
the regulation of the internal economy of the guild were
formulated at an early stage in its history. 1 The municipalities
and numerous societies which existed in Italy supplied plenty
of models for the construction of such rules. We are, almost
inevitably, unable to fix the exact date of the commencement
of these voluntary associations, but they existed at Paris,
Bologna, Salerno, Oxford, and Cambridge before the end
of the twelfth century. Whether such a loosely associated
and self-constituted guild of students can be correctly de
scribed as a university is a doubtful point. | These societies
seem to have arisen in connection with schools established by
some church or monastery, and I believe that nearly all the
mediaeval universities grew up under the protection of some
bishop or abbot. They were not however ecclesiastical organi
zations, and, though the bulk of their members were ordained,
their connection with the church arose chiefly from the fact
that clerks were then the only class of the community who
were left free by the state to pursue their studies. The guild
was thus at first in some undefined manner subject to the
special authority of the bishop or his chancellor, from the latter
of whom the head of the university subsequently took his
title. The schools from which the universities sprang con
tinued for a long time to exist under the direct control of the
cathedral or monastic authorities, by the side of the guilds
formed by the teachers on the more advanced subjects.
The next stage in the development of the university was
iti recognition by the sovereign of the kingdom in which it
was situated. |A universitas scholarium, if successful in at
tracting students and acquiring permanency, always sought
special legal privileges, such as the right of fixing the price of
provisions and the power of trying legal actions in which its
members were concerned. These privileges generally led to a
144 THE RISE OF LEARNING IN WESTERN EUROPE.
recognition, explicit or implicit, of the guild by the crown as
a studium generate, that is, a body with power to grant degrees
which conferred a right of teaching anywhere within the
kingdonffr The university was frequently incorporated at or
about the same time.l I believe no university was thus ac
knowledged before the end of the twelfth century. Paris
received its charter in 1200, and probably was the earliest
university in Europe thus recognized. A medical school
existed at Salerno as early as the ninth century, and a legal
school at Bologna as early as 1138, but at these the education
was technical rather than general ; I therefore consider that the
universities to which these schools respectively gave rise should
be referred to a later date.
\ The last step in the evolution of a mediaeval university
was the acknowledgment of its corporate existence by the
pope (or emperor), and the recognition of its degrees as a title
to teach throughout Christendom : thenceforward it became
a recognized member of a body of closely connected corpora
tions. Paris was thus recognized in 1283.
A mediaeval university therefore passed through three
stages : first, it was a self-constituted guild of students ; second,
legal privileges were conferred on it by the state, and usually
it was incorporated; third, it was recognized by the pope
and its degrees declared current throughout the whole of
Christendom. In later times the title of university was con
fined to degree-granting bodies, and any other place of higher
education was termed a studium generale. I add in a foot
note a few additional particulars connected with the early
history of Paris, Oxford, and Cambridge*.
* Paris is probably the oldest European university, and as not only is
it usually taken as the typical mediaeval university, but as it also served
as the model on which Oxford and Cambridge were subsequently con
stituted, its history possesses special interest for English readers. The
first of these stages in its history perhaps may be dated as far back as
1109 when William of Champeaux began to teach logic, and certainly
may be said to have commenced when his pupil Abelard was lecturing
MEDIAEVAL UNIVERSITIES. 145
The standard of education in mathematics has been largely
fixed by the universities, and most of the mathematicians of
on logic and divinity. The faculty of arts and (probably) its form of
self-government existed in 11G9, for Henry II. proposed to refer his
quarrel with Thomas a Becket to it and two other bodies: it is also
alluded to in two decretals of the pope in 1180. By an ordinance of the
king of France in 1200 the university entered on the second of these
stages, and its members were granted exemption from all ordinary tri
bunals : in 1206 it was incorporated and thus put on a permanent basis,
which its mere recognition by the state did not effect. The first definite
body of statutes seems to have been formed in 1208. In 1215 the
cardinal legate Robert de Couron laid down a curriculum, and from
that time European universities have imposed a definite course of study
combined with certain periodical tests of proficiency on their junior mem
bers ; the modern system of university education dates from this order.
In 12G7 theology, and in 1281 law and medicine, were created separate
faculties. About the same time the pope Nicholas IV. decreed that
doctors of the university should enjoy the privileges and rank of doctors
throughout Christendom.
The collegiate system also originated in Paris. The religious orders
established hostels for their own students about the middle of the twelfth
century, but these are now considered to have been independent of the
university. It is possible that St Thomas s College and the Danish
College in the Rue de la Montagne were founded about 1200; but if we
reject these, the dates of their foundation being uncertain, the first
regular college was that founded by Robert de Sorbonne in 1250. The
college of Navarre which far surpassed all others in wealth and numbers
was founded in 1305. Two hundred years later there were 18 colleges
and 82 hostels, the latter being really mere boarding houses and gene
rally unendowed : by that time all the colleges had specialized their
higher teaching on some one subject, and all but one had thrown their
lectures open to the university, while the masters and tutors of the hostels
had abandoned teaching except in the case of Latin grammar. The want
of discipline among the non-collegiate students led to their suppression
at an early date.
It would take me beyond my limits if I were to trace the history of the
university of Paris further. Its decay is generally dated from the year
1719. Until that time a teacher or regent received from his college
board, lodging, and sufficient money to enable him to live, but he de
pended for his luxuries on the fees of those who attended his lectures ;
hence there was every encouragement to make the lectures efficient. The
stipends of the professors also depended to a large extent on their
B. 10
146 THE RISE OF LEARNING IN WESTERN EUROPE.
subsequent times have been closely connected with one or i
more of them; and therefore I may be pardoned for adding/
efficiency. This was altered in 1719, and professors whose lectures
were gratuitous were subsequently appointed for life at a fixed stipend.
Perhaps the eighteenth century was an unfavourable time for the ex
periment, but the result was disastrous ; those graduates of the colleges,
who continued to charge fees, soon found their lecture-rooms deserted ;
within forty years the number of hostels was reduced to less than 40,
and that of the colleges to 10, most of which were heavily in debt ; in
1764 the hostels were shut up ; finally, on Sept. 15, 1793, the Convention
suppressed the university and colleges, and appropriated their revenues.
The present centralized university of France is a creation of Napoleon I.
The first reliable mention of Oxford as a place of education refers to
the year 1133 when Robert Pullen came from Paris and lectured on
theology. A little later, in 1149, Vacarius came from Bologna and taught
civil law. It is not unlikely that the Benedictine monastery of St
Frideswyde was ruled by French monks, and that the lectures were given
under their influence and in their monastery : but the references seem
to imply that there was then no university there. In 1180 there is an
allusion to a scholar in the Acta Sanctorum (p. 579), and in 1184
Giraldus Cambrensis lectured to the masters and scholars. (Gir. Camb.
vol. i. p. 23.) Hence it is almost certain that the university had its
origin between 1150 and 1180. Mr Rashdall believes that it developed
out of a migration from Paris in 1167, but the available data do not
seem to justify a definite statement about it. In 1214 the university was
given legal jurisdiction whenever one party was a scholar or the servant
of a scholar. In 1244 it was incorporated by Henry III. The collegiate
system commenced with the foundation of Merton College in 1264:
though money for building University College was given in 1249, and for
building Balliol College in 1263. The university was recognized by
Innocent IV. in 1252, but it was not till 1296 that the masters received
from Boniface VIH. permission to teach anywhere in Christendom.
I wish I could be equally explicit about Cambridge, but unfortunately
its early records and charters were burnt. All the mediaeval universities
were divided into "nations" according to the place of birth of their
students. There was a constant feud at Cambridge between those born
north of the Trent and those born to the south of it. In 1261 a
desperate fight, lasting some days, took place between the two factions
in the course of which the university records were burnt. A similar
disturbance took place in 1322. Again in 1381, under cover of the
popular disturbances then prevalent throughout the kingdom, a mob of
townsmen broke into St Mary s Church, seized the university chest,
MEDIAEVAL UNIVERSITIES. 147
a few words on the general course of studies in a university
in mediaeval times, referring the reader who wishes for fuller
and burnt the charters and documents therein contained. The original
charters having been destroyed, we are compelled in their absence to rely
on allusions to them in trustworthy authorities. Now it was the custom
at both universities to solicit a renewal of their privileges at the be
ginning of each reign (an opportunity of which they often took advantage
to get them extended) and it is possible that the dates here given may be
those of the renewals of original charters which are now lost. At any
rate it would seem certain that the university existed in its first stage,
i.e. as a self -constituted and self-governing community, before 1209, since
several students from Oxford migrated in that year to the university of
Cambridge ; and it is clear it did not exist in 1112 when the canons of
St Giles s opened schools at their new priory at Barnwell. It was at
some time then between these two dates that the university entered on
its first stage of existence. In 1225 there is an allusion in some legal
proceedings (Record office, Coram Hege Bolls Hen. III. Nos. 20 and 21)
to the chancellor of the university. In 1229, after some disturbances in
Paris, Heury III. invited French students to come and settle at Oxford
or Cambridge, and some hundreds came to Cambridge. In 1231 Henry
III. gave the university jurisdiction over certain classes of townsmen,
in 1251 he extended it so as to give exclusive legal jurisdiction in all
matters concerning scholars, and finally confirmed all its rights in 1260.
[These privileges were given by letters and enactments, and the first
charter of which we now know anything was that given by Edward I. in
1291.] The collegiate system commenced with the foundation of what
was afterwards known as Peterhouse in or before 1280. The university
was recognized by letters from the pope in 1233, but in 1318 John XXII.
gave it all the rights which were or could be enjoyed by any university in
Christendom. Under these sweeping terms it obtained exemption from
the jurisdiction both of the bishop of Ely and the archbishop of Canter
bury (as settled in the Barnwell process, 1430). I may add that just as
the old monastic schools continued to exist by the side of the university of
Paris, so the grammar schools, which had originally attracted students to
Cambridge and from which therefore the university may be said to have
sprung, continued to exist until the middle of the sixteenth century.
We can express these results in a tabular form thus :
7 J m Oxford ( <nnf>ri<l<n:
In existence before the year 1169 li84 1209
Legal privileges conferred by the state 1200 1214 1231
Foundation of first college 1250 1264 1280
Degrees current throughout Christendom ...1283 1296 1318
102
148 THE RISE OF LEARNING IN WESTERN EUROPE.
details as to their organization of studies, their system of
instruction, and their constitution to my History of the Study
of Mathematics at Cambridge, 1889.
The students entered when quite young, sometimes not
being more than 11 or 12 years old when first coming into
residence. It is misleading to describe them as undergraduates,
for their age, their studies, the discipline to which they were
subjected, and their position in the university shew that
they should be regarded as schoolboys. The first four years
of their residence were supposed to be spent in the study
of the trivium, i.e. Latin grammar, logic, and rhetoric. The
majority of students in quite early times did not progress beyond
the study of Latin grammar they formed an inferior faculty
and were eligible only for the degree of master of grammar
but the more advanced students (and in later times all students)
spent these years in the study of the trivium.
The title of bachelor of arts was conferred at the end of
s this course, and x signified that the student was no longer a
schoolboy and therefore in pupilage. The average age of a
commencing bachelor may be taken as having been about 1 7 or
I 18. Thus at Cambridge in the presentation for a degree the
technical term still used for an undergraduate is juvenis, while
that for a bachelor is vir. A bachelor could not take pupils,
could teach only under special restrictions, and probably occupied
a position closely analogous to that of an undergraduate now-a-
days. y Some few bachelors proceeded to the study of civil or
canon law, but it was assumed in theory that they next studied
the quadrivium, the course for which took three years, and
which included about as much science as was to be found in
the pages of Boethius and Isidorus.
The degree of master of arts was given at the end of this
course.^ In the twelfth and thirteenth centuries it was merely
a license to teach : no one sought it who did not intend to use
it for that purpose and to reside, and only those who had a
natural aptitude for such work were likely to enter so ill paid
a profession as that of a teacher. I The degree was obtainable by
MEDIAEVAL UNIVERSITIES. 149
any student who had gone through the recognized course of
study and shewn that he was of good moral character. Out
siders were also admitted, but not as a matter of course. I
may here add that towards the end of the fourteenth century
students began to find that a degree had a pecuniary value,
and most universities subsequently conferred it only on con
dition that the new master should reside and teach for at least
a year. V A few years later the universities took a further step
and began to refuse degrees to those who were not intellectually
qualified. This power was assumed on the precedent of a case
which arose in Paris in 1426 when the university declined to
confer a degree on a student a Slavonian, one Paul Nicholas,
who thus has the distinction of being the first student ever
" plucked " who had performed the necessary exercises in a
very indifferent manner : he took legal proceedings to compel
the university to grant the degree, but their right to withhold
it was established.
Although science and mathematics were recognized as the
standard subjects of study for a bachelor, it is probable that
until the renaissance the majority of the students devoted most
of their time to logic, philosophy, and theology. The subtleties
of the scholastic theology and logic, which were the favourite
intellectual pursuit of these centuries, may seem to us dreary
and barren, but it is only just to say that they afforded an
intellectual exercise which fitted men at a later time to de-
velope science, and certainly were in advance of what had been
previously taught.
We have now arrived at a time when the results of Arab
and Greek science became known in Europe. The history of
Greek mathematics has been already discussed ; I must now
temporarily leave the subject of mediaeval mathematics, and
trace the development of the Arabian schools to the same date ;
and I must then explain how the schoolmen became acquainted
with the Arab and Greek text-books, and how their introduc
tion affected the progress of European mathematics.
150
CHAPTER IX.
THE MATHEMATICS OF THE ARABS*.
THE story of Arabian mathematics is known to us in its
general outlines, but we are as yet unable to speak with cer
tainty on many of its details. It is however quite clear that
while part of the early knowledge of the Arabs was derived
from Greek sources, part was obtained from Hindoo works;
and that it was on those foundations that Arab science was
built. I will begin by considering in turn the extent of mathe
matical knowledge derived from these sources.
Extent of mathematics obtained from Greek sources.
According to their traditions, in themselves very probable,
the scientific knowledge of the Arabs was at first derived from
* The subject is discussed atjlength by Cantor, chaps, xxxn. xxxv. ;
by Hankel, pp. 172 293 ; and by A. von Kremer in Kulturgescliiclite ties
Orientes unter den Chalifen, Vienna, 1877. See also Materiaux pour servir
a Vhistoire compares des sciences mathematiques chez les Grecs et les
Orientaux, by L. A. Sedillot, Paris, 18459 : and the following five
articles by Fr. Woepcke, Sur Vemploi des chiffres Indiens par les Arabes ;
Sur Vhistoire des sciences mathematiques chez les Orientaux (2 articles),
Paris, 1855 ; Sur V introduction de Varithmetique Indienne en Occident,
Eome, 1859 ; and Memoire sur la propagation des chiffres Indiens, Paris,
1863.
THE MATHEMATICS OF THE ARABS. 151
the Greek doctors who attended the caliphs at Bagdad. It is
said that when the Arabian conquerors settled in towns they
became subject to diseases which had been unknown to them
in their life in the desert. The study of medicine was then
confined almost entirely to Greeks, and many of these, en
couraged by the caliphs, settled at Bagdad, Damascus, and
other cities ; their knowledge of all branches of learning was
far more extensive and accurate than that of the Arabs, and
the teaching of the young, as has often happened in similar
cases, soon fell into their hands. The introduction of European
science was rendered the more easy as various small Greek
schools existed in the countries subject to the Arabs : there
had for many years been one at Edessa among the Nestorian
Christians, and there were others at Antioch, Emesa, and
even at Damascus which had preserved the traditions and
some of the results of Greek learning.
The Arabs soon remarked that the Greeks rested their
medical science on the works of Hippocrates, Aristotle, and
Galen ; and these books were translated into Arabic by order
of the caliph Haroun Al Raschid about the year 800. The
translations excited so much interest that his successor Al
Mamuii (813 833) sent a commission to Constantinople to
obtain copies of as many scientific works as was possible, while
an embassy for a similar purpose was also sent to India. At
the same time a large staff of Syrian clerks was engaged, whose
duty it was t* translate the works so obtained into Arabic and
Syriac. To disarm fanaticism these clerks were at first termed
the caliph s doctors, but in 851 they were formed into a college,
and their most celebrated member Honein ibn Ishak was
made its first president by the caliph Mutawakkil (847 861).
Hoiiein and his son Ishak ibn Honein revised the transla
tions before they were finally issued. Neither of them knew
much mathematics, and several blunders were made in the
works issued on that subject, but another member of the
college, Tabit ibn Korra, shortly published fresh editions which
thereafter became the standard texts.
152 THE MATHEMATICS OF THE AKABS.
In this way before the end of the ninth century the Arabs
obtained translations of the works of Euclid, Archimedes,
Apollonius, Ptolemy, and others ; and in some cases these
editions are the only copies of the books now extant. It is
curious as indicating how completely Diophantus had dropped
out of notice that as far as we know the Arabs got no manu
script of his great work till 150 years later, by which time
they were already acquainted with the idea of algebraic notation
and processes.
Extent of mathematics obtained from Hindoo sources.
The Arabs had considerable commerce with India, and a
knowledge of one or both of the two great original Hindoo
works on algebra had been thus obtained in the caliphate of
Al Mansur (754 775), though it was not until fifty or sixty
years later that they attracted much attention. The algebra
and arithmetic of the Arabs were largely founded on these
treatises, and I therefore devote this section to the considera
tion of Hindoo mathematics.
The Hindoos, like the Chinese, have pretended that they
are the most ancient people on the face of the earth, and
that to them all sciences owe their creation. But it would
appear from all recent investigations that these pretensions
have no foundation; and in fact no science or useful art
(except a rather fantastic architecture and sculpture) can be
traced back to the inhabitants of the Indian peninsula prior
to the Aryan invasion. This invasion seems to have taken place
at some time in the latter half of the fifth century or in the
sixth century after Christ, when a tribe of the Aryans entered
India by the north-west frontier and established themselves as
rulers over a large part of the country. Their descendants,
wherever they have kept their blood pure, may be still recog
nized by their superiority over the races they originally con
quered \ but as is the case with the modern Europeans they
AUYA-BHATA. 153
found the climate trying, and gradually degenerated. For
the first two or three centuries they however retained their
intellectual vigour, and produced one or two writers of great
ability.
Arya-Bhata. The first of these is Arya-Bhata, who was
born at Patna in the year 476. He is frequently quoted by
Brahmagupta, and in the opinion of many commentators he
created algebraic analysis though it has been suggested that
he may have seen Diophantus s Arithmetic. The chief work of
Arya-Bhata with which we are acquainted is his Aryabhathiya
which consists of the enunciations of various rules and pro
positions written in verse. There are no proofs, and the
language is so obscure and concise that it long defied all efforts
to translate it*.
The book is divided into four parts : of these three are
devoted to astronomy and the elements of spherical trigono
metry ; the remaining part contains the enunciations of thirty-
three rules in arithmetic, algebra, and plane trigonometry. It
is probable that Arya-Bhata, like Brahmagupta and Bhaskara
who are mentioned next, regarded himself as an astronomer,
and studied mathematics only so far as it was useful to him in
his astronomy.
In algebra Arya-Bhata gives the sum of the first, second,
and third powers of the first n natural numbers ; the general
solution of a quadratic equation ; and the solution in integers
of certain indeterminate equations of the first degree. His
solutions of numerical equations have been supposed to imply
that he was acquainted with the decimal system of numeration.
In trigonometry he gives a table of natural sines of the
angles in the first quadrant, proceeding by multiples of 3|,
* A Sanskrit text of the Aryabhathiya, edited by H. Kern, was
published at Leyden in 1874 ; there is also an article on it by the same
editor in the Journal of the Asiatic Society, London, 1863, vol. xx.,
pp. 371387 : a French translation by L. Rodet of that part which deals
with algebra and trigonometry is given in the Journal Axiatique, 1879,
Paris, series 7, vol. xui., pp. 393 434.
154 THE MATHEMATICS OF THE HINDOOS.
defining a sine as the semichord of double the angle. Assuming
that for the angle 3f the sine is equal to the circular measure,
he takes for its value 225, i.e. the number of minutes in the
angle. He then enunciates a rule which is nearly unintelligible
but probably is the equivalent of the statement
sin (n + 1) a sin na = sin na sin (n 1) a sin na cosec a,
where a stands for 3| ; and working with this formula he
constructs a table of sines, and finally finds the value of sin 90
to be 3438. This result is correct if we take 3 141 6 as the
value of TT, and it is interesting to note that this is the number
which in another place he gives for IT. The correct trigono
metrical formula is
sin (n + 1) a - sin na = sin na sin (n 1) a 4 sin na sin 2 |a.
Arya-Bhata therefore took 4 sin 2 |a as equal to cosec a, i.e. he
supposed that 2 sin a = 1 + sin 2a : using the approximate
values of sin a and sin 2a given in his table, this reduces to
2 (225) = 1 + 449, and hence to that degree of approximation
his formula is correct. A large proportion of the geometrical
propositions which he gives are wrong.
Brahmagupta. The next Hindoo writer of considerable
note is Brahmagupta, who is said to have been born in 598
and probably was alive about 660. He wrote a work in verse
entitled Brahma-Sphuta-Siddhanta, that is, the Riddhanta or
system of Brahma in astronomy. In this two chapters (chaps,
xii. and XVIH.) are devoted to arithmetic, algebra, and
geometry*.
The arithmetic is entirely rhetorical. Most of the problems
are worked out by the rule of three, and a large proportion of
them are on the subject of interest.
In his algebra, which is also rhetorical, he works out the
fundamental propositions connected with an arithmetical pro
gression, and solves a quadratic equation (but gives only the
* These two chapters (chaps, xii. and xvm.) were translated by H. T.
Colebrooke, and published at London in 1817.
BRAHMAGUPTA. 155
positive value to the radical). As an illustration of the pro
blems given I may quote the following, which was reproduced
in slightly different forms by various subsequent writers, but
I replace the numbers by letters. "Two apes lived at the
top of a cliff of height h, whose base was distant mh from a
neighbouring village. One descended the cliff and walked to
the village, the other flew up a height x and then flew in a
straight line to the village. The distance traversed by each
was the same. Find x." Brahmagupta gave the correct
answer, namely x = mh/(m + 2). In the question as enun
ciated originally li 100, m = 2.
Brahmagupta finds solutions in integers of several in
determinate equations of the first degree, using the same
method as that now practised. He states one indeterminate
equation of the second degree, namely, nx 2 + 1 = y 2 , and gives
as its solution x = 2t/(t 2 - n) and y = (t 2 + n)/(t 2 - n). To obtain
this general form he proved that, if one solution either of that
or of certain allied equations could be guessed, the general
solution could be written down ; but he did not explain how
one solution could be obtained. He added that the equation
y 2 = nx* 1 could not be satisfied by integral values of x and y
unless n could be expressed as the sum of the squares of two
integers. Curiously enough the former of these equations was
sent by Fermat as a challenge to Wallis and Lord Brouncker
in the seventeenth century, and the latter found the same
solutions as Brahmagupta had previously done. It is perhaps
worth noticing that the early algebraists, whether Greeks,
Hindoos, Arabs, or Italians, drew no distinction between the
problems which led to determinate and those which led to
indeterminate equations. It was only after the introduction
of syncopated algebra that attempts were made to give general
solutions of equations, and the .difficulty of giving such solu
tions of indeterminate equations other than those of the first
degree has led to their practical exclusion from elementary
algebra.
In geometry Brahmagupta proved the pythagorean property
156 THE MATHEMATICS OF THE HINDOOS.
of a right-angled triangle (Euc. i. 47). He gave expressions for
the area of a triangle and of a quadrilateral inscribable in a
circle in terms of their sides ; and shewed that the area of a
circle was equal to that of a rectangle whose sides were the
radius and semiperimeter. He was less successful in his
attempt to rectify a circle, and his result is equivalent to
taking \/10 for the value of TT. He also determined the sur
face and volume of a pyramid and cone ; problems over which
Arya-Bhata had blundered badly. The next part of his
geometry is almost unintelligible, but it seems to be an at
tempt to find expressions for several magnitudes connected
with a quadrilateral inscribed in a circle in terms of its sides :
most of this is wrong.
It must not be supposed that in the original work all the
propositions which deal with any one subject are collected
together, and it is only for convenience that I have tried to
arrange them in that way. It is impossible to say whether
the whole of Brahmagupta s results given above are original.
He knew of Arya-Bhata s work, for he reproduces the table
of sines there given ; and it is likely that some progress in
mathematics had been made by Arya-Bhata s immediate suc
cessors, and that Brahmagupta was acquainted with their
works ; but there seems no reason to doubt that the bulk of
Brahmagupta s algebra and arithmetic is original, although
perhaps influenced by Diophantus s writings : the origin of
the geometry is more doubtful, probably some of it is derived
from Hero s works.
Bhaskara. To make this account of Hindoo mathematics
complete, I may depart from the chronological arrangement
and say that the remaining great Indian mathematician was
Bhaskara who was born in 1114. He is said to have been
the lineal successor of Brahmagupta as head of an astronomical
observatory at Ujein or as it is sometimes written Ujjayini.
He wrote an astronomy of which only four chapters have been
translated. Of these one termed Lilavati is on arithmetic ; a
second termed Bija Ganita is on algebra; the third and fourth
BHASKARA. 157
are on astronomy and the sphere*. This work was I believe
known to the Arabs almost as soon as it was written and
influenced their subsequent writings, though they failed to
utilize or extend most of the discoveries contained in it. The
results thus became indirectly known in the West before the
end of the twelfth century, but the text itself was not intro
duced into Europe till within recent times.
The treatise is in verse but there are explanatory notes
in prose. It is not clear whether it is original or whether it
is merely an exposition of the results then known in India;
but in any case it is most probable that Bhaskara was ac
quainted with the Arab works which had been written in the
tenth and eleventh centuries, and with the results of Greek
mathematics as transmitted through Arabian sources. The
algebra is syncopated and almost symbolic, which marks a
great advance over that of Brahmagupta and of the Arabs.
The geometry is also superior to that of Brahmagupta, but
apparently this is due to the knowledge of various Greek works
obtained through the Arabs.
The first book or Lilavati commences with a salutation
to the god of wisdom. The general arrangement of the work
may be gathered from the following table of contents. Systems
of weights and* measures. Next decimal numeration, briefly
described. Then the eight operations of arithmetic, namely,
addition, subtraction, multiplication, division, square, cube,
square-root, and cube-root. Reduction of fractions to a common
denominator, fractions of fractions, mixed numbers, and the
eight rules applied to fractions. The " rules of cipher," namely,
a a, O 2 = 0, v = 0, a -4- = GO , The solution of some
simple equations which are treated as questions of arithmetic.
The rule of false assumption. Simultaneous equations of the
first degree with applications. Solution of a few quadratic
* See the article Viga Ganita in the Penny Cyclopaedia, London,
1843 ; and the translations of the Lilavati and the Bija Ganita issued
by H. T. Colebrooke, London, 1817. The two chapters on astronomy
and the sphere were edited by L. Wilkinson, Calcutta, 1842.
158 THE MATHEMATICS OF THE HINDOOS.
equations. Rule of three and compound rule of three, with
various cases. Interest, discount, and partnership. Time of
filling a cistern by several fountains. Barter. Arithmetical
progressions, and sums of squares and cubes. Geometrical pro
gressions. Problems on triangles and quadrilaterals. Approxi
mate value of TT. Some trigonometrical formulae. Contents
of solids. Indeterminate equations of the first degree. Lastly
the book ends with a few questions on combinations.
This is the earliest known work which contains a syste
matic exposition of the decimal system of numeration. It is
possible that Arya-Bhata was acquainted with it, and it is
most likely that Brahmagupta was so, but in Bhaskara s arith
metic we meet with the Arabic or Indian numerals and a sign
for zero as part of a well-recognized notation. It is impossible
at present to definitely trace these numerals further back than
the eighth century, but there is no reason to doubt the assertion
that they were in use at the beginning of the seventh century.
Their origin is a difficult and disputed question. I mention
below (see p. 189) the view which on the whole seems most
probable and perhaps is now generally accepted, and I reproduce
there some of the forms used in early times.
To sum the matter up briefly it may be said that the
Lilavati gives the rules now current for addition, subtraction,
multiplication, and division, as well as the more common pro
cesses in arithmetic; while the greater part of the work is
taken up with the discussion of the rule of three, which is
divided into direct and inverse, simple and compound, and
is used to solve numerous questions chiefly on interest and
exchange the numerical questions being expressed in the
decimal system of notation with which we are familiar.
Bhaskara was celebrated as an astrologer no less than as a
mathematician. He learnt by this art that the event of his
daughter Lilavati marrying would be fatal to himself. He
therefore declined to allow her to leave his presence, but by
way of consolation he not only called the first book of his
work by her name, but propounded many of his problems in
BHASKARA. 159
the form of questions addressed to her. For example, " Lovely
and dear Lilavati, whose eyes are like a fawn s, tell me what
are the numbers resulting from 135 multiplied by 12. If thou
be skilled in multiplication, whether by whole or by parts,
whether by division or by separation of digits, tell me, auspi
cious damsel, what is the quotient of the product when divided
by the same multiplier."
I may add here that the problems in the Indian works give
a great deal of interesting information about the social and
economic condition of the country in which they were written.
Thus Bhaskara discusses some questions on the price of slaves,
and incidentally remarks that a female slave was generally
supposed to be most valuable when 16 years old, and subse
quently to decrease in value in inverse proportion to the age;
for instance, if when 16 years old she were worth 32 nishkas,
her value when 20 would be represented by (16 x 32) H- 20
nishkas. It would appear that, as a rough average, a female
slave of 16 was worth about 8 oxen which had worked for
two years. The interest charged for money in India varied
from 3 to 5 per cent, per month. Amongst other data thus
given will be found the price of provisions and labour.
The chapter termed Bija Ganita commences with a sentence
so ingeniously framed that it can be read as the enunciation
of a religious, or a philosophical, or a mathematical truth.
Bhaskara after alluding to his Lilavati or arithmetic states that
he intends in this book to proceed to the general operations of
analysis. The idea of the notation is as follows. Abbrevia
tions and initials are used for symbols; subtraction is indicated
by a dot placed above the coefficient of the quantity to be
subtracted; addition by juxtaposition merely; but no symbols
are used for multiplication, equality, or inequality, these being
written at length. A product is denoted by the first syllable
of the word subjoined to the factors, between which a dot is
sometimes placed. In a quotient or fraction the divisor is
written under the dividend without a line of separation. The
two sides of an equation are written one under the other,
160 THE MATHEMATICS OF THE HINDOOS.
confusion being prevented by the recital in words of all the
steps which accompany the operation. Various symbols for
the unknown quantity are used, but most of them are the
initials of names of colours, and the word colour is often used
as synonymous with unknown quantity; its Sanscrit equivalent
also signifies a letter, and letters are sometimes used either
from the alphabet or from the initial syllables of subjects of
the problem. In one or two cases symbols are used for the
given as well as for the unknown quantities. The initials of
the words square and solid denote the second and third powers,
and the initial syllable of square root marks a surd. Poly
nomials are arranged in powers, the absolute quantity being
always placed last and distinguished by an initial syllable de
noting known quantity. Most of the equations have numerical
coefficients, and the coefficient is always written after the un
known quantity. Positive or negative terms are indiscrimi
nately allowed to come first ; and every power is repeated on
both sides of an equation, with a zero for the coefficient when
the term is absent. After explaining his notation, Bhaskara
goes on to give the rules for addition, subtraction, multiplica
tion, division, squaring, and extracting the square root of alge
braical expressions : he then gives the rules of cipher as in the
Lilavati \ solves a few equations ; and lastly concludes with
some operations on surds. Many of the problems are given in
a poetical setting with allusions to fair damsels and gallant
warriors.
Other chapters on algebra, trigonometry, and geometrical
applications exist, and fragments of them have been translated
by Colebrooke. Amongst the trigonometrical formulae is one
which is equivalent to the equation d (sin 0) = cos dO.
I have departed from the chronological order in treating
here of Bhaskara, but as he was the only remaining Hindoo
writer of exceptional eminence I thought it better to mention
him at the same time as I was discussing his compatriots. It
must be remembered however that he flourished subsequently
to all the Arab mathematicians considered in the next section.
miASKARA. 1G1
The works with which the Arabs first became acquainted
were those of Arya-Bhata and Brahmagupta, and it is doubtful
if they ever made much use of the great treatise of Bhaskara.
It is probable that the attention of the Arabs was called
to the works of the first two of these writers by the fact that
the Arabs adopted the Indian system of arithmetic, and were
thus led to look at the mathematical text-books of the Hindoos.
The Arabs had always had considerable commerce with India,
and with the establishment of their empire the amount of trade
naturally increased ; at that time, circ. 700, they found the
Hindoo merchants beginning to use the system of numeration
with which we are familiar and adopted it at once. This
immediate acceptance of it was made the easier as they had
no collection of science or literature written in another system,
and it is doubtful whether they then possessed any but the
most primitive system of notation for expressing numbers.
The earliest definite date assigned for the use in Arabia of the
decimal system of numeration is 773. In that year some
Indian astronomical tables were brought to Bagdad, and it is
almost certain that in these Indian numerals (including a zero)
were employed.
Tlie development of mathematics in Arabia*.
In the preceding sections of this chapter I have indicated
the two sources from which the Arabs derived their knowledge
of mathematics, and have sketched out roughly the amount of
knowledge obtained from each. We may sum the matter up
by saying that before the end of the eighth century the Arabs
were in possession of a good numerical notation and of
Brahmagupta s work on arithmetic and algebra ; while before
* A work by Baldi on the lives of several of the Arab mathematicians
! Tinted in Boncompagni s Bullctino di bibliogmjia, 1872, vol. v.,
!]). i-J7 S3 I.
I .. 11
162 THE MATHEMATICS OF THE ARABS.
the end of the ninth century they were acquainted with the
masterpieces of Greek mathematics in geometry, mechanics,
and astronomy. I have now to explain what use they made
of these materials.
Alkarismi. The first and in some respects the most illus
trious of the Arabian mathematicians was Mohammed ibn
Musa Abu Djefar Al-Khwdrizmi. There is no common agree
ment as to which of these names is the one by which he is to
be known : the last of them refers to the place where he was
born, or in connection with which he was best known, and I
am told that it is the one by which he would have been
usually known among his contemporaries. I shall therefore
refer to him by that name ; and shall also generally adopt the
corresponding titles to designate the other Arabian mathema
ticians. Until recently this was almost always written in the
corrupt form Alkarismi, and, though this way of spelling it is
incorrect, it has been sanctioned by so many writers that I
shall make use of it. We know nothing of Alkarismi s life
except that he was a native of Khorassan and librarian of the
caliph Al/Mamun; and that he accompanied a mission to
Afghan.istan, and possibly came back through India. On his
return, about 830, he wrote an algebra which is founded on
that of Brahmagupta, but in which some of the proofs rest on
the Greek method of representing numbers by lines : it was
published by Rosen, with an English translation, at London in
1831. Alkarismi also wrote a treatise on arithmetic : an
anonymous tract termed Algoritmi De Numero Indorum, which
is in the university library at Cambridge, is believed to be a
Latin translation of this treatise; tftis was published by B.
Boncompagni at Rome in 1857. Besides these two works
Alkarismi compiled some astronomical tables, with explanatory
remarks ; these included results taken from both Ptolemy and
Brahmagupta.
The algebra of Alkarismi holds a most important place in
the history of mathematics, for we may say that the subsequent
Arabian and the early mediaeval works on algebra were
ALKARISMI. 163
founded on it, and also that through it the Arabic or Indian
system of decimal numeration was introduced into the West.
The work is termed Al-gebr we I mukabala: al-gebr, from
which the word algebra is derived, may be translated by
the restoration and refers to the fact that any the same magni
tude may be added to or subtracted from both sides of an
equation ; al mukabala means the process of simplification
and is generally used in connection with the combination of
like terms into a single term. The unknown quantity is
termed either " the thing" or "the root" (i.e. of a plant)
and from the latter phrase our use of the word root as applied
to the solution of an equation is derived. The square of the
unknown is called " the power." All the known quantities
are numbers.
The work is divided into five parts. In the first Alkarismi
gives, without any proofs, rules for the solution of quadratic
equations, which he divides into six classes of the forms
ax 2 = bx, ax 2 = c, bx c, ax 2 + bx = c, ax 9 + c = bx, and ax 2 bx + c,
where a, b, c are positive numbers. He considers only real
and positive roots, but he recognizes the existence of two
roots, which as far as we know was never done by the Greeks.
It is somewhat curious that when both roots are positive he
generally takes only that root which is derived from the
negative value of the radical.
He next gives geometrical proofs of these rules in a
manner analogous to that of Euclid n. 4. For example, to
solve the equation x 2 + Wx= 39, or any equation of the form
x 3 +px = q, he gives two methods of which one is as follows.
Let A il represent the value of x, and construct on it the
square A BCD (see figure on next page). Produce DA to //
and DC to F so that AH --CF=.5 (or %p) ; and complete the
figure as drawn below. Then the areas AC, HB, and HF
represent the magnitudes x 2 , 5x, and 5x. Thus the left-hand
side of the equation is represented by the sum of the areas AC t
///>, and ttl \ that is, by the gnomon IfCG. To both sides of
the equation add the square KG, the area of which is 25 (or
112
164
THE MATHEMATICS OF THE ARABS.
\p*\ and we shall get a new square whose area is by hypo
thesis equal to 39 + 25, that is, to 64 (or q + ffi) and whose
side therefore is 8. The side of this square DH which is
equal to 8 will exceed AH which is equal to 5 by the value
of the unknown required, which therefore is 3.
In the third part of the book Alkarismi considers the
product of (xa) and (xb). In the fourth part he states
the rules for addition and subtraction of expressions which
involve the unknown, its square, or its square root; gives rules
for the calculation of square roots; and concludes with the
theorems that a*Jb = \ a 2 b and ija Jb \/ab. In the fifth
and last part he gives some problems, such, for example, as to
find two numbers whose sum is 10 and the difference of whose
squares is 40.
In all these early works there is no clear distinction between
arithmetic and algebra, and we find the account and explana
tion of arithmetical processes mixed up with algebra and treated
as part of it. It was from this book then that the Italians
first obtained not only the ideas of algebra but also of an arith-
me.ic founded on the decimal system. This arithmetic was
long known as algorism, or the art of Alkarismi, which served
to distinguish it from the arithmetic of Boethius ; and this
name remained in use till the eighteenth century.
Tabit ibn Korra. The work commenced by Alkarismi was
carried on by Tabit ibn Korra, born at Harran in 836 and died
TABIT IBN KOllllA. ALKAYAMI. 165
in 901, who was one of the most brilliant and accomplished
scholars produced by the Arabs. He issued translations of
the chief works of Euclid, Apollonius, Archimedes, and Ptolemy
(see above, p. 151). He also wrote several original works, all
of which are lost with the exception of a fragment on algebra,
consisting of one chapter on cubic equations, which are solved
by the aid of geometry in somewhat the same way as that given
later (see below, p. 228).
Algebra continued to develope very rapidly, but it re
mained entirely rhetorical. The problems with which the
Arabs were concerned were, either the solution of equations,
problems leading to equations, or properties of numbers. The
two most prominent algebraists of a later date were Omar
Alkayami and Alkarki, both of whom flourished at the
beginning of the eleventh century.
Alkayami. The first of these, Omar Alkayami, is notice
able for his geometrical treatment of cubic equations by which
he obtained a root as the abscissa of a point of intersection
of a conic and a circle. The equations he considers are of
the following forms, where a and c stand for positive integers,
(i) a; 3 + b 2 x = 6 2 c, whose root he says is the abscissa of a point
of intersection of x 2 = by and y 2 = x (c x) \ (ii) # 3 h ax 2 = c 3 ,
whose root he says is the abscissa of a point of intersection
of xy = c 2 and y 2 - c (x + a) ; (iii) x 3 ax 2 + b 2 x = b 2 c, whose
root he says is the abscissa of a point of intersection of
y* = (x a) (c - x) and x (by) = be. He gives one biquadratic,
namely, (100 - x 2 ) (10 - x) 2 = 8100, the root of which is deter
mined by the point of intersection of (10 x)y = 90 and
x* + y 2 = 100. It is sometimes said that he stated that it was
impossible to solve the equation x 3 + v/ 3 = s 3 in positive integers,
or in other words that the sum of two cubes can never be
a cube; though whether he gave an accurate proof, or whether,
as is more likely, the proposition (if enunciated at all) was the
result of a wide induction, it is now impossible to say ; but
the fact that such a theorem is attributed to him will serve to
illustrate the extraordinary progress the Arabs had made in
166 THE MATHEMATICS OF THE ARABS.
algebra. His treatise on algebra was published by Fr.
Woepcke, Paris, 1851.
Alkarki. The other mathematician of this time (circ.
1000) whom I mentioned was Alkarki. He gave expressions
for the sums of the first, second, and third powers of the first
n natural numbers ; solved various equations, including some
of the forms ax 2p =t bx p c = ; and discussed surds, shewing,
for example, that </8 + ^18 = ^50. His algebra was published
by Fr. Woepcke at Paris in 1853, and his arithmetic was
translated into German by Ad Hochheim at Halle in 1878.
Even where the methods of Arab algebra are quite general
the applications are confined in all cases to numerical problems,
and the algebra is so arithmetical that it is difficult to treat the
subjects apart. From their books on arithmetic and from the
observations scattered through various works on algebra we may
say that the methods used by the Arabs for the four fundamental
processes were analogous to, but more cumbrous than, those now
in use (see below, chapter XL); but the problems to which the
subject was applied were similar to those given in modern
books, and were solved by similar methods, such as rule of
three, &c. Some minor improvements in notation were intro
duced, such e.g. as the introduction of a line to separate the
numerator from the denominator of a fraction ; and hence a
line between two symbols came to -be used as a symbol of
division (see below, p. 244), Alhossein (980 1037) invented
the rule for testing the results of addition and multiplication
by " casting out the nines."
I am not concerned with the Arabian views of astronomy or
the value of their observations, but I may remark in passing
that the Arabs accepted the theory as laid down by Hippar-
chus and Ptolemy, and did not materially alter or advance it.
Albategni. Albuzjani. Like the Greeks, the Arabs never
used trigonometry except in connection with astronomy : but
they introduced the trigonometrical expressions which are now
current, and worked out the plane trigonometry of a single
angle. They were also acquainted with the elements of
ALBATEGNI. ALBUZJANI. ALHAZEN. ABD-AL-GEHL. 167
spherical trigonometry. The trigonometrical ratios seem to
have been the invention of Albategni, born at Batan in
Mesopotamia in 877 and died at Bagdad in 929, who was
among the earliest of the many distinguished Arabian astro
nomers. He wrote the Science of the Stars (published by
Regiomontanus at Nuremberg in 1537) and in it he determined
his angles by "the semi-chord of twice the angle," i.e. by
the sine of the angle (taking the radius vector as unity).
Hipparchus and Ptolemy, it will be remembered, had used the
chord. It is doubtful whether Albategni was acquainted
with the previous introduction of sines by Arya-Bhata and
Brahmagupta. Shortly after the death of Albategni, Albuzjani
who is also known as Abul- Wafa, born in 940 and died in 998,
introduced all the trigonometrical functions, and constructed
tables of tangents and cotangents. He was celebrated not
only as an astronomer but as one of the most distinguished
geometricians of his time.
Alhazen. Abd-al-gehl. The Arabs were at first content
to take the works of Euclid and Apollonius for their text-books
in geometry without attempting to comment on them, but
Alhazen (born at Bassora in 987 and died at Cairo in 1038)
issued in 1036 a collection of problems something like the
Data of Euclid, this was translated by Sedillot and published
at Paris in 1836. Besides commentaries on the definitions of
Euclid and on the Almagest Alhazen also wrote a work on
optics which shews that he was a geometrician of considerable
power : this was published at Bale in 1572, and served as the
foundation for Kepler s treatise. In it he gives, amongst
other things, a geometrical solution of the problem to find at
what point of a concave mirror a ray from a given point must
be incident so as to be reflected to another given point.
Another geometrician of a slightly later date was Abd-al-yehl
(circ. 1100) who wrote on conic sections, and was also the
author of three small geometrical tracts.
It was shortly after the last of the mathematicians mentioned
above that Bhaskara, the third great Hindoo mathematician,
168 THE MATHEMATICS OF THE ARABS.
flourished : there is every reason to believe that he was familiar
with the works of the Arab school as described above, and also
that his writings were at once known in Arabia.
The Arab schools continued to flourish until the fifteenth
century. But they produced no other mathematician of any
exceptional genius, nor was there any great advance on the
methods indicated above, and it is unnecessary for me to crowd
my pages with the names of a number of writers who did not
materially affect the progress of the science in Europe.
I have not alluded to a strange theory which has been
accepted by some writers, but which seems to me to be most
improbable. According to this theory there were two rival
schools of thought in Arabia, one of which derived its mathe
matics entirely from Greek sources and represented numbers by
lines, and the other from Hindoo sources and represented
numbers by abstract symbols each disdaining to make any use
of the authorities preferred by its rival.
From this rapid sketch it will be seen that the work of the
Arabs in arithmetic, algebra, and trigonometry was of a high
order of excellence. They appreciated geometry and the appli
cations of geometry to astronomy, but they did not extend the
bounds of the science. It may be also added that they made
no special progress in statics, or optics, or hydrostatics ; though
there is abundant evidence that they had a thorough knowledge
of practical hydraulics.
The general impression left on my mind is that the Arabs
were quick to appreciate the work of others notably of the
Greek masters and of the Hindoo mathematicians but, like
the ancient Chinese and Egyptians, they were unable to sys
tematically develope a subject to any considerable extent.
Their schools may be taken to have lasted in all for about
650 years, and if the work produced be compared with that
of Greek or modern European writers it is as a whole second-
rate both in quantfty and quality.
169
CHAPTER X.
THE INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
CIRC. 1150 1450.
IN the last chapter but one I discussed the development of
European mathematics to a date which corresponds roughly
with the end of the " dark ages " ; and in the last chapter
I traced the history of the mathematics of the Hindoos and
Arabs to the same date. The mathematics of the two or
three centuries that follow and are treated in this chapter are
characterized by the introduction of the Arabian mathematical
text-books and of Greek books derived from Arabian sources,
and the assimilation of the new ideas thus presented.
It was however from Spain and not from Arabia that
Arabian mathematics came into western Europe. The Moors
had established their rule in Spain in 747, and by the tenth or
eleventh century had attained a high degree of civilization.
Though their political relations with the caliphs at Bagdad
were somewhat unfriendly, they gave a ready welcome to the
works of the great Arabian mathematicians. In this way the
Arab translations of Euclid, Archimedes, Apollonius, Ptolemy,
and perhaps of other Greek writers, together with the works
of the Arabian algebraists, were read and commented on at
the three great Moorish universities or schools of Granada,
Cordova, and Seville. It seems probable that these works
indicate the full extent of Moorish learning, but, as all know-
170 INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
ledge was jealously guarded from Christians, it is impossible to
speak with certainty either on this point or on that of the time
when the Arab books were first introduced into Spain.
The eleventh century. The earliest Moorish writer of dis
tinction of whom I find mention is G-eber ibn Aphla, who was
born at Seville and died towards the latter part of the eleventh
century at Cordova. His works, which deal chiefly with astro
nomy and trigonometry, were translated into Latin by Gerard
and published at Nuremberg in 1533. He seems to have dis
covered the theorem that the sines of the angles of a spherical
triangle are proportional to the sines of the opposite sides.
Another Arab of about the same date was Arzachel*,
who was living at Toledo in 1080. He suggested that the
planets moved in ellipses, but his contemporaries with scientific
intolerance declined to argue about a statement which was
contrary to that made by Ptolemy in the Almagest.
The twelfth century. During the course of the twelfth
century copies of the books used in Spain were obtained in
western Christendom. The first step towards procuring a
knowledge of Arab and Moorish science was taken by an
English monk, Adelhard of Bath t> who, under the disguise of
a Mohammedan student, attended some lectures at Cordova
about 1120 and obtained a copy of Euclid s Elements. This
copy, translated into Latin, was the foundation of all the edi
tions known in Europe till 1533, when the Greek text was
recovered. How rapidly a knowledge of the work spread we
may judge when we recollect that before the end of the thir
teenth century Roger Bacon was familiar with it, while before
the close of the fourteenth century the first five books formed
part of the regular curriculum at some, if not all, universities.
The enunciations of Euclid seem to have been known before
* See his life by Baldi, circ. 1000, reprinted in Boneompagiii s
Bulletino di libliografia, 1872, vol. v, p. 508.
t On the influence of Adelhard and Ben Ezra, see the Abhandlungen
zur Geschichte der Mathematik in the Zeitschrift fiir Mathematik, vol.
xxv, 1880.
THE TWELFTH CENTURY. 171
Adelhard s time, and possibly as early as the year 1000,
though copies were rare. Adelhard also procured a manu
script of or commentary on Alkarismi s work, which he like
wise translated into Latin. He also issued a text-book on the
use of the abacus.
During the same century other translations of the Arab
text -books or commentaries on them were obtained. Amongst
those who were most influential in introducing Moorish learn
ing into Europe I may mention Abraham Ben Ezra*. Ben
Ezra was born at Toledo in 1097, and died at Rome in 1167.
He was one of the most distinguished Jewish rabbis who had
settled in Spain, where it must be recollected that they were
tolerated and even protected by the Moors on account of their
medical skill. Besides some astronomical tables and an astro
logy, Ben Ezra wrote an arithmetic, a short analysis of which
was published by O. Terquein in Liouville s Journal for 1841.
In this he explains the Arab system of numeration with nine
symbols and a zero, gives the fundamental processes of arith
metic, and explains the rule of three.
Another European who was induced by the reputation of
the Arab schools to go to Toledo was Gerard f who was born
at Cremona in 1114 and died in 1187. He translated the
Arab edition of the Almagest, the works of Alhazen, and the
works of Alfarabius whose name is otherwise unknown to us.
In this translation of Ptolemy s work which was made in
1136 the Arabic numerals are introduced. Gerard also wrote
a short treatise on algorism which exists in manuscript in
the Bodleian Library at Oxford. He was acquainted with
ne of the Aral) editions of Euclid s Elements, which he trans
lated into Latin.
Among, the contemporaries of Gerard was John Hispalensis
of Seville, who was originally a rabbi but was converted to
Christianity and baptized under the name given above. He
* See footnote on p. 170.
t Sec Boncompagni s />//,/ ritu < <1< lie open- di (iJierunio
Koine, 1851.
172 INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
made translations of several Arab and Moorish works, and
also wrote an algorism which contains the earliest examples of
the extraction of the square roots of numbers by the aid of
the decimal notation.
The thirteenth century. During the thirteenth century
there was a revival of learning throughout Europe, but the
new learning was I believe confined to a very limited class.
The early years of this century are memorable for the de
velopment of several universities, and for the appearance of
three remarkable mathematicians Leonardo of Pisa, Jordanus,
and Roger Bacon the Franciscan monk of Oxford.
Leonardo*. Leonardo Fibonacci (i.e. filius Bonaccii) gene
rally known as Leonardo of Pisa, was born at Pisa in 1175.
His father Boiiacci was a merchant, and was sent by his
fellow-townsmen to control the custom-house at Bugia in
Barbary ; there Leonardo was educated, and he thus became
acquainted with the Arabic system of numeration as also
with Alkarismi s work on algebra which was described in
the last chapter. It would seem that Leonardo was entrusted
with some duties in connection with the custom-house which
required him to travel. He returned to Italy about 1200,
and in 1202 published a work called Algebra et almuchabala
(the title being taken from Alkarismi s work) but generally
known as the Liber Abaci. He there explains the Arabic
system of numeration, and remarks on its great advantages
over the Roman system. He then gives an account of algebra,
and points out the convenience of using geometry to get rigid
demonstrations of algebraical formulae. He shews how to
solve simple equations, solves a few quadratic equations, and
states some methods for the solution of indeterminate equa
tions; these rules are illustrated by problems on numbers.
All the algebra is rhetorical. This work had a wide circu-
* See the Leben und Schriften Leonardos da Pisa by J. Giesing,
Dobeln, 1886 ; and Cantor, chaps. XLI., XLII. ; see also two articles by
Fr. Woepcke in the Atti delV Academia pontificia de nuovi Lincei for
1861, vol. xiv., pp. 342 348. Most of Leonardo s writings were edited
and published by B. Boncompagni between the years 1854 and 1862.
U lo.VAIMM) OF PISA. 173
lation, and for at least two centuries remained a standard
authority.
The Liber Abaci is especially interesting in the history of
arithmetic since it practically introduced the use of the Arabic
numerals into Christian Europe. The language of Leonardo
implies that they were previously unknown to his countrymen;
he says that having had to spend some years in Barbary he
there learnt the Arabic system which he found much more
convenient than that used in Europe; he therefore published
it "in order that the Latin* race might no longer be deficient
in that knowledge." Now Leonardo had read very widely,
and had travelled in Greece, Sicily, and Italy; and there is
therefore every presumption that the system was not then com
monly employed in Europe. Though Leonardo introduced its
use into commercial affairs, it is probable that a knowledge of
it as a method which was current in the East was previously
not uncommon among travellers and merchants, for the inter
course between Christians and Mohammedans was sufficiently
close for each to learn something of the language and common
practices of the other. We can also hardly suppose that the
Italian merchants were ignorant of the method of keeping ac
counts used by some of their best customers ; and we must recol
lect too that there were numerous Christians who had escaped
or been ransomed after serving the Mohammedans as slaves.
It was however Leonardo who brought the system into general
use, and by the middle of the thirteenth century a large pro
portion of the Italian merchants employed it by the side of the
old system.
The majority of mathematicians must have already known
of the system from the works of Ben Ezra, Gerard, and John
Hispalensis. But shortly after the appearance of Leonardo s
book Alphonso of Castile (in 1252) published some astronomical
* Dean Peacock says that the earliest known application of the word
Italians to describe the inhabitants of Italy occurs about the middle of
tlu thirteenth century : by the end of that century it was in common
174 INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
tables, founded on observations made in Arabia, which -were
computed by Arabs and which were expressed in Arabic nota
tion. Alphonso s tables had a wide circulation among men
of science and were largely instrumental in bringing these
numerals into universal use among mathematicians. By the
end of the thirteenth century it was generally assumed that all
scientific men would be acquainted with the system: thus
Roger Bacon writing in that century recommends the algorism
(that is, the arithmetic founded on the Arab notation) as a
necessary study for theologians who ought he says "to abound
in the power of numbering." We may then consider that by
the year 1300, or at the latest 1350, these numerals were
familiar both to mathematicians and to Italian merchants.
So great was Leonardo s reputation that the emperor
Frederick II. stopped at Pisa in 1225 in order to hold a sort
of mathematical tournament to test Leonardo s skill of which
he had heard such marvellous accounts. The competitors were
informed beforehand of the questions to be asked, some or
all of which were composed by John of Palermo who was one
of Frederick s suite. This is the first time that we meet
with an instance of those challenges to solve particular pro
blems which were so common in the sixteenth and seventeenth
centuries. The first question propounded was to find a number
of which the square when either increased or decreased by
5 would remain a square. Leonardo gave an answer, which
is correct, namely 41/12. The next question was to find by
the methods used in the tenth book of Euclid a line whose
length x should satisfy the equation ar* + 2x 2 + Wx = 20.
Leonardo shewed by geometry that the problem was im
possible, but he gave an approximate value of the root of this
equation, namely, 1 -22 7" 42 " 33"" 4 V 40 vi , which is equal to
1*3688081075..., and is correct to nine places of decimals*,
Another question was as follows. Three men A, B, C, possess
a sum of money u, their shares being in the ratio 3:2:1. A
takes away x, keeps half of it, and deposits the remainder with
* See Fr. Woepcke in Liouville s JmirnaJ for 1854, p. 401.
LEONARDO OF PISA. FREDERICK II. 175
D\ B takes away ?/, keeps two- thirds of it, and deposits the
remainder with D\ C takes away all that is left namely z,
keeps five-sixths of it, and deposits the remainder with D.
This deposit with D is found to belong to A, B, and C in
equal proportions. Find u t x, ?/, and z. Leonardo shewed
that the problem was indeterminate and gave as one solution
^ = 47, a; = 33, y^!3, z-\. The other competitors failed to
solve any of these questions.
The chief work of Leonardo is the Liber Abaci alluded to
above. This work contains a proof of the well-known result
(a 9 + b a )(c* 4- d*) = (ac + bdf + (be - ad) 2 = (ad + be) 2 + (bd - acf.
He also wrote a geometry termed Practica Geometriae which
was issued in 1220. This is a good compilation and some
trigonometry is introduced; among other propositions and
examples he finds the area of a triangle in terms of its sides.
Subsequently he published a Liber Quadratorum dealing with
problems similar to the first of the questions propounded at
the tournament*. He also issued a tract dealing with deter
minate algebraical problems: these are all solved by the rule
of false assumption in the manner explained above on p. 104.
Frederick II. The emperor Frederick II. who was born in
1194, succeeded to the throne in 1210, and died in 1250, was
not only interested in science, but did as much as any other
single man of the thirteenth century to disseminate a know
ledge of the works of the Arab mathematicians in western
Europe. The universities of Naples and Padua remain as
monuments of his munificence ; he having founded the former
in 1224, and the latter in 1238. I have already mentioned
that the presence of the Jews had Uvn tolerated in Spain on
account of their medical skill and scientific knowledge, and as
a matter of fact the titles of physician and algebraist!
* Fr. Woepcke in Liouville s Journal for 1855 (p. 51) has given an
analysis of Leonardo s method of treating problems on square numbers.
t For instance the reader may recollect that in Don (J it i. rote (part u.,
th. 1">), when Samson Carasco is thrown by the knight from his horse
and has his ribs broken, an tilfi,>l>ri*tii i- snninmiu d to bind up his wounds.
176 INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
for a long time nearly synonymous ; thus the Jewish physicians
were admirably fitted both to get copies of the Arab works and
to translate them. Frederick II. made use of this fact to engage
a staff of learned Jews to translate the Arab works which he
obtained, though there is no doubt that he gave his patronage
to them the more readily because it was singularly offensive to
the pope with whom he was then engaged in a quarrel. Afc
any rate by the end of the thirteenth century copies of Euclid,
Archimedes, Apollonius, Ptolemy, and some of the Arab works
on algebra were obtainable from this source, and by the end of
the next century were not uncommon. From this time then
we may say that the development of science in Europe was
independent of the aid of the Arabian schools.
Jordanus*. Among Leonardo s contemporaries was a
German mathematician, whose works were until the last few
years almost unknown. This was Jordanus Nemorarius,
sometimes called Jordanus de Saxonia or Teutonicus. Of the
details of his life we know but little, save that he was elected
general of the Dominican order in 1222.
Prof. Curtze, who has made a special study of the subject,
considers that the following works are due to Jordanus :
Geometria vel de Triangulis and De Similibus Arcubis, published
by M. Curtze in 1887 in vol. vi. of the Mitteilungen des Coper-
nicus-Vereins zu Thorn , De Isoperimetris ; Arithmetica De-
monstrata, published by Faber Stapulensis at Paris in 1496,
second edition, 1514; Alyoritkmus Demonstratus, published
by J. Schoner at Nuremberg in 1534; De Numeris Datis,
published by P. Treutlein in 1879 and edited in 1891 with
comments by M. Curtze in vol. xxxvi. of the Zeitschrift fur
Mathematikund Physik-, De Ponderibus, published by P. Apian
at Nuremberg in 1533, and re-issued at Venice in 1565 ; and
lastly two or three tracts on Ptolemaic astronomy. If we
assume, as Prof. Curtze does, that these works have not been
added to or improved by subsequent annotators, we must
* See Cantor, chaps. XLIII, XLIV, where the references to the autho
rities on Jordanus are collected.
JORDANUS. 177
esteem Jordanus as one of the most eminent mathematicians
of the middle ages.
His knowledge of geometry is illustrated by his De
TrianyaliS) De Siinilibus Arcubis, and De Isoperiiwtris. The
most important of these is the De Trianyulis which is divided
into four books. The first book, besides a few definitions,
contains 13 propositions on triangles which are based on
Euclid s Elements. The second book contains 19 propositions,
mainly on the ratios of straight lines and their application to
compare the areas of triangles ; for example, one problem is to
find a point inside a triangle so that the lines joining it to the
angular points may divide the triangle into three equal parts.
The third book contains 12 propositions, mainly concerning
arcs and chords of circles. The fourth book contains 28 propo
sitions, partly 011 regular polygons and partly on miscellaneous
questions such as the duplication and trisection problems.
The Algorithmic Demonstratus contains practical rules for
the four fundamental processes, and Arabic numerals are
generally (but not always) used. It is divided into ten books
dealing with properties of numbers, primes, perfect numbers,
polygonal numbers, &c., ratios, powers, and the progressions.
It would seem from it that Jordanus knew the general expres
sion for the square of any algebraic multinomial.
The De Numeris Da .is consists of four books containing
solutions of 115 problems. Some of these lead to simple or
quadratic equations involving more than one unknown quan
tity. He shews a knowledge of proportion ; but many of the
demonstrations of his general propositions are only numerical
illustrations of them.
In several of the propositions of the Algorithmus and De
A a. merits Datis letters are employed to denote both known
and unknown quantities, and they are used in the demonstra
tions of the rules of arithmetic as well as of algebra. As an
example of this I quote the following proposition (from the
De A anfris Datis, book i. prop. 3) the object of which is to
determine two quantities whose sum and product are known.
B. 12
178 INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
Dato numero per duo diuiso si, quod ex ductu unius in alterum pro-
ducitur, datum fuerit, et utrumque eorum datum esse necesse est.
Sit numerus datus abc diuisus in ab et c, atque ex ab in c fiat d datus,
itemque ex abc in se fiat e. Sumatur itaque quadruplum d, qui fit /, quo
dempto de e remaneat g, et ipse erit quadratum differentiae ab ad c.
Extrahatur ergo radix ex g, et sit h, eritque h differentia ab ad c, cumque
sic h datum, erit et c et ab datum.
Huius operatic facile constabit hoc modo. Verbi gratia sit x diuisus
in numeros duos, atque ex ductu unius eorum in alium fiat xxi ; cuius
quadruplum, et ipsum est LXXXIIII, tollatur de quadrato x, hoc est c, et
remanent xvi, cuius radix extrahatur, quae erit quatuor, et ipse est
differentia. Ipsa tollatur de x et reliquum, quod est vi, dimidietur,
eritque medietas in, et ipse est minor portio et maior vii.
It will be noticed that Jordanus, like Diophantus and the
Hindoos, denotes addition by juxtaposition. Expressed in
modern notation his argument is as follows. Let the numbers
be a + b (which I will denote by y) and c. Then y + c is
given; hence (y + c) 2 is known; denote it by e. Again yc is
given ; denote it by d ; hence 4yc, which is equal to 4c, is
known ; denote it by f. Then (y - c) 2 is equal to e -f, which
is known ; denote it by g. Therefore y c *Jg, which is
known ; denote it by h. Hence y + c and y - c are known,
and therefore y and c can be at once found. It is curious
that he should have taken a sum, like a + b for one of his
unknowns. In his numerical illustration he takes the sum to
be 10 and the product 21.
The above works are the earliest instances known in
European mathematics of syncopated algebra in which letters
are used for algebraical symbols. It is probable that the
Alyorithmus was not generally known until it was printed in
1534, and it is doubtful how far the works of Jordanus exercised
any considerable influence on the development of algebra. In
fact it constantly happens in the history of mathematics that
improvements in notation or discoveries are made long before
they are generally adopted or their advantages realized. Thus
the same thing may be discovered over and over again, and it
is not until the general standard of knowledge requires some
such improvement, or it is enforced by some one whose zeal or
JORDANUS. HOLY WOOD. 170
attainments compel attention, that it is adopted and becomes
part of the science. Jordan us in using letters or symbols to
represent any quantities which occur in analysis was far in
advance of his contemporaries. A similar notation was ten
tatively introduced by other and later mathematicians, but
it was not until it had been thus independently discovered
several times that it came into general use.
It is not necessary to describe in detail the mechanics,
optics, or astronomy of Jordanus. The treatment of mechanics
throughout the middle ages was generally unintelligent.
No mathematicians of the same ability as Leonardo and
Jordanus appear in the history of the subject for over two
hundred years. Their individual achievements must not be
taken to imply the standard of knowledge then current, but
their works were accessible to students in the following two
centuries though there were not many who seem to have
derived much benefit therefrom or who attempted to extend
the bounds of arithmetic and algebra as there expounded.
During the thirteenth century the most famous centres of
learning in western Europe -were Paris and Oxford, and I
must now refer to the more eminent members of those
schools.
Holywood. I will begin by mentioning John de Ilolywood.
whose name is perhaps better known in the latinized form of
Sacrobosco. Holywood was born in Yorkshire and educated
at Oxford, but after taking his master s degree he moved to
1 MIMS and taught there till his death in 1244 or 1246. His
liviures on algorism and algebra are the earliest of which I
can find mention. His work on arithmetic was for many
years a standard authority: it was printed at Paris in 1496,
and was re-issued in Halli well s Rura Mathematica, London,
1841. He also wrote a treatise on the sphere which was
made public in 1256: this had a wide circulation, and in
dicates how rapidly a knowledge of mathematics was spreading.
lea these, two pamphlets by him entitled respectively De
Compute / / xi txfico and De Astrolabio are still extant.
12-*
180 INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
Roger Bacon*. Another contemporary of Leonardo and
Jordaiius was lloger Bacon, who for physical science did work
somewhat analogous to what they did for arithmetic and
algebra. Roger Bacon was born near Ilchester in 1214 and
died at Oxford on June 11, 1294. He was the son of royal
ists, most of whose property had been confiscated at the end of
the civil wars : at an early age he was entered as a student at
Oxford, and is said to have taken orders in 1233. In 1234
he removed to Paris, then the intellectual capital of western
Europe, where he lived for some years devoting himself espe
cially to languages and physics ; and there he spent on books
and experiments all that remained of his family property and
his savings. He returned to Oxford soon after 1240, and
there for the following ten or twelve years he laboured in
cessantly, being chiefly occupied in teaching science. His
lecture room was crowded but everything that he earned was
spent in. buying manuscripts and instruments. He tells us
that altogether at Paris and Oxford he spent over 2000
in this way a sum which represents at least .20,000 now-a-
days.
Bacon strove hard to replace logic in the university curri
culum by mathematical and linguistic studies, but the influences
of the age were too strong for him. His glowing eulogy on
" divine mathematics " which should form the foundation of a
liberal education and which "alone can purge the intellect
and fit the student for the acquirement of all knowledge " fell
on deaf ears. We can judge how small was the amount of
geometry which was implied in the quadrivium when he tells
us that in geometry few students at Oxford read beyond Euc.
i. 5 ; though we might perhaps have inferred as much from
the character of the work of Boethius.
* See Roger Bacon, sa vie, ses ouvrages... by E. Charles, Paris, 1861 ;
and the memoir by J. S. Brewer, prefixed to the Opera Inedita, Rolls
Series, London, 1859 : a somewhat depreciatory criticism of the former of
these works is given in lloger Bacon eine Monographic by L. Schneider,
Augsburg, 1873.
K<X;KK P.ACON. 181
At last worn out, neglected, and ruined Bacon was per
suaded by his friend Grosseteste, the great bishop of Lincoln,
to renounce the world and take the Franciscan vows. The
society to which he now found himself confined was singularly
uncongenial to him, and he beguiled the time by writing on
scientific questions and perhaps lecturing. The superior of the
order heard of this, and in 1257 forbad him to lecture or
publish anything under penalty of the most severe punish
ments, and at the same time directed him to take up his
residence at Paris where he could be more closely watched.
Clement IV. when in England had heard of his abilities, and
in 1266 when he became pope he invited Bacon to write. The
Franciscan order reluctantly permitted him to do so, but they
refused him any assistance. With great difficulty Bacon ob
tained sufficient money to get paper and the loan of books, and
within the short space of fifteen months he produced in 1267
his Opus majus with two supplements which summarized all
that was then known in science, and laid down the principles
on which not only science, but philosophy and literature, should
be studied. He stated as the fundamental principle that the
study of natural science must rest solely on experiment ; and
in the fourth part he explained in detail how all sciences rest
ultimately on mathematics, and progress only when their fun
damental principles are expressed in a mathematical form.
Mathematics, he says, should be regarded as the alphabet of
all philosophy.
The results that he arrived at in this and his other works
are nearly in accordance with modern ideas, but were too far
in advance of that age to be capable of appreciation or perhaps
even of comprehension, and it was left for later generations to
rediscover his works, and give him that credit which he never
experienced in his lifetime. In astronomy he laid down the
principles for a reform of the calendar, explained the pheno
mena of shooting stars, and stated that the Ptolemaic system
was unscientific in so far as it rested on the assumption that
circular motion was the natural motion of a planet, while the
182 INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
complexity of the explanations required made it improbable
that the theory was true. In optics he enunciated the laws of
reflexion and in a general way of refraction of light, and
used them to give a rough explanation of the rainbow and of
magnifying glasses. Most of his experiments in chemistry
were directed to the transmutation of metals and led to no
result. He gave the composition of gunpowder, but there is
no doubt that it was nob his own invention, though it is
the earliest European mention of it. On the other hand some
of his results in these subjects appear to be guesses which
are more or less ingenious, while certain statements he makes
are certainly erroneous.
In the years immediately following the publication of his
Opus majus he wrote numerous works which developed in
detail the principles there laid down. Most of these have now
been published but I do not know of the existence of any com
plete edition. They deal only with applied mathematics and
physics.
Clement took no notice of the great work for which he had
asked, except to obtain leave for Bacon to return to England.
On the death of Clement, the general of the Franciscan order
was elected pope and took the title of Nicholas IV. Bacon s
investigations had never been approved of by his superiors,
and he was now ordered to return to Paris where we are told
he was immediately accused of magic : he was condemned in
1280 to imprisonment for life, and was released only about a
year before his death.
Campanus. The only other mathematician of this century
whom I need mention is Giovanni Campano, or in the latinized
form Campanus, a canon of Paris. A copy of Adelhard s
translation of Euclid s Elements fell into the hands of Campa
nus, who issued it as his own * ; he added a commentary thereon
in which he discussed the properties of a regular re-entrant
pentagon : this edition was printed by Ratdolt at Venice in
* On this work see J. L. Heiberg in the Zeitschrift fiir Mathematik,
vol. xxxv, 1890.
THE FOURTEENTH CENTURY. 183
1482. Besides some minor works Campanus wrote the Theory
of the Planets, which was a free translation of the Almagest.
The fourteenth century. The history of the fourteenth
century, like that of the one preceding it, is mostly concerned
with the introduction and assimilation of the Arabian mathe
matical text-books and the Greek books derived from Arabian
sources.
Bradwardine*. A mathematician of this time, who was
perhaps sufficiently influential to justify a mention here, is
Thomas Bradivardine, archbishop of Canterbury. Bradwardine
was born at Chichester about 1290. He was educated at
Merton College, Oxford, and subsequently lectured in that
university. From 1335 to the time of his death he was chiefly
occupied with the politics of the church and state : he took a
prominent part in the invasion of France, the capture of
Calais, and the victory of Cressy. He died at Lambeth in
1349. His mathematical works, which were probably written
when he was at Oxford, are (i) the Tractatus de Proportioni-
bus, printed at Paris in 1495 ; (ii) the Arithmetica Speculative^,
printed at Paris in 1502; (iii) the Geometria Speculative*^
printed at Paris in 1511; and (iv) the De Quadratures Circuli,
printed at Paris in 1495. They probably give a fair idea of
tin- nature of the mathematics then read at an English uni
versity.
Oresnmst. Nicholas Oresmus was another writer of the
fourteenth century who is said in most histories of mathematics
to have influenced the development of the subject. He was born
at ( Vien in 1323, became the confidential adviser of Charles V.
by whom he was made tutor to Charles VI., and subsequently
\vas appointed bishop of Lisieux, at which city he died on
Inly 11, 1382. He wrote the Algorismus Proportionum in
which the idea of fractional indices is introduced, and in the
* See my History of Mathematics at Cambridge, 1889, pp. C 7;
Cantor, vol. n. , p. 102 ct xr</.
t See ])<< matln inatixt-hi-n Srhrift,-n fa Nicok Orcsme by M. Curtzo,
Thorn. 1H70.
184 INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
eyes of his contemporaries was prominent as a mathematician
not less than as an economist and theologian ; but I do not
propose to discuss his writings. The treatise on which his
reputation chiefly rests deals with questions of coinage and
commercial exchange, from the mathematical point of view it
is noticeable only for the use of vulgar fractions and the intro
duction of symbols for them.
By the middle of this century Euclidean geometry (as ex
pounded by Campanus) and algorism were fairly familiar to
all professed mathematicians, and the Ptolemaic astronomy was
also generally known. About this time the almanacks began
to add to the explanation of the Arabic symbols the rules of
addition, subtraction, multiplication, and division, u de al-
gorismo." The more important calendars arid other treatises
also inserted a statement of the rules of proportion, illustrated
by various practical questions.
In the latter half of this century there was a general revolt
of the universities against the intellectual tyranny of the school
men. This was largely due to Petrarch, who to his own gene
ration was celebrated as a humanist rather than as a poet,
and who exerted all his power to destroy scholasticism, and
encourage scholarship. The result of these influences on
the study of mathematics may be seen in the changes then
introduced in the study of the quadrivium* The stimulus
came from the university of Paris, where a statute to that effect
was passed in 1366, and a year or two later similar regulations
were made at Oxford and Cambridge ; unfortunately no text
books are mentioned. We can however form a reasonable
estimate of the range of mathematical reading required, by
looking at the statutes of the universities of Prague founded
in 1348, of Vienna founded in 1365, and of Leipzig founded
in 1389.
By the statutes of Prague, dated 1384, candidates for the
bachelor s degree were required to have read Holywood s
* On the authorities for these statements, see my History of the Study
of Mathematics at Cambridge, Cambridge, 1889, p. 8 et seq.
MATHEMATICS IN THE UNIVERSITIES. 185
treatise on the sphere, and candidates for the master s degree
to be acquainted with the first six books of Euclid, optics,
hydrostatics, the theory of the lever, and astronomy. Lectures
were actually delivered on arithmetic, the art of reckoning with
the fingers, and the algorism of integers ; on almanacks, which
probably meant elementary astrology ; and on the Almagest,
that is, on Ptolemaic astronomy. There is however some reason
for thinking that mathematics received far more attention here
than was then usual at other universities.
At Vienna in 1389 the candidate for a master s degree was
required to have read five books of Euclid, common perspec
tive, proportional parts, the measurement of superficies, and
the Theory of the Planets. The book last named is the treatise
by Campanus which was founded on that by Ptolemy. This
was a fairly respectable mathematical standard, but I would
remind the reader that there was no such thing as " plucking"
in a mediaeval university. The student had to keep an act or
give a lecture on certain subjects, but whether he did it well or
badly he got his degree, and it is probable that it was only the
few students whose interests were mathematical who really
mastered the subjects mentioned above.
The fifteenth century. A few facts gleaned from the
history of the fifteenth century tend to shew that the regula
tions about the study of the quadrivium were not seriously
enforced. The lecture lists for the years 1437 and 1438 of the
university of Leipzig (the statutes of which are almost identical
with those of Prague as quoted above) are extant, and shew
that the only lectures given there on. mathematics in those
years were confined to astrology. The records of Bologna,
Padua, and Pisa seem to imply that there also astrology was
the only scientific subject taught in the fifteenth century, and
even as late as 1598 the professor of mathematics at Pisa was
required to lecture on the Quadripartitum, an astrological work
purporting (probably falsely) to have been written by Ptolemy.
The only mathematical subjects mentioned in the registers of
the university of Oxford as read there between the years 1449
186 INTRODUCTION OF ARABIAN WORKS INTO EUROPE.
and 1463 were Ptolemy s astronomy (or some commentary on
it) and the first two books of Euclid. Whether most students
got as far as this is doubtful. It would seem, from an edition
of Euclid published at Paris in 1536, that after 1452 candi
dates for the master s degree at that university had to take
an oath that they had attended lectures on the first six books
of Euclid s Elements.
Beldomandi. The only writer of this time that I need
mention here is Prodocimo Beldomandi of Padua, born about
1380, who wrote an algoristic arithmetic, published in 1410,
which contains the summation of a geometrical series; and
some geometrical works : for further details see Boncompagni s
Bulletino di bibliogrqfia, vols. xn., xvm.
By the middle of the fifteenth century printing was in
vented, and the facilities it gave for disseminating knowledge
were so great as to revolutionize the progress of science. We
have now arrived at a time when the results of Arab and
Greek science were known in Europe ; and this perhaps then
is as good a date as can be fixed for the close of this period
and the commencement of that of the renaissance. The mathe
matical history of the renaissance begins with the career of
Regiomontanus ; but before proceeding with the general history
it will be convenient to collect together the chief facts con
nected with the development of arithmetic during the middle
ages and the renaissance. To this the next chapter is devoted.
187
CHAPTER XL
THE DEVELOPMENT OF ARITHMETIC*.
CIRC. 13001637.
WE have seen in the last chapter that by the end of the
thirteenth century the Arabic arithmetic had been fairly intro
duced into Europe and was practised by the side of the older
arithmetic which was founded on the work of Boetbius. It will
be convenient to depart from the chronological arrangement
and briefly to sum up the subsequent history of arithmetic, but
I hope, by references in the next chapter to the inventions and
improvements in arithmetic here described, that I shall be able
to keep the order of events and discoveries quite clear.
The older arithmetic consisted of two parts : practical arith
metic or the art of calculation which was taught by means of
the abacus and possibly the multiplication table, and theoretical
arithmetic by which was meant the ratios and properties of
numbers taught according to Boethius a knowledge of the
latter being confined to professed mathematicians. The theo
retical part of this system continued to be taught till the
middle of the fifteenth century, ;md the practical part of it
* Sec the article on Arithmetic by G. Peacock in the Encyclopaedia
M "juilitana, vol. i., London, 1845; Arithmcticnl Html;* by A. De
m, London, 1847; and an article by P. Treutlein of Karlsruhe,
in the supplement (pp. 1 100) of tlir Ahhinnlliinin-ti :ur f/Yxr// /<///, //<;
ik, 1H77.
188 THE DEVELOPMENT OF ARITHMETIC. 1300 1637.
was used by the smaller tradesmen in England*, Germany,
and France till the beginning of the seventeenth century.
The new Arabian arithmetic was called algorism or the art
of Alkarismi to distinguish it from the old or Boethian arith
metic. The text-books on algorism commenced with the Arabic
system of notation, and began by giving rules for addition, sub
traction, multiplication, and division ; the principles of propor
tion were then applied to various practical problems, and the
books usually concluded with general rules for many of the
common problems of commerce. Algorism was in fact a mer
cantile arithmetic though at first it also included all that was
then known as algebra. Thus algebra has its origin in arith
metic; and to most people the term universal arithmetic by
which it was sometimes designated conveys a more accurate
impression of its objects and methods*than the more elaborate
definitions of modern mathematicians certainly better than the
definition of Sir William Hamilton as the science of pure time,
or that of De Morgan as the calculus of succession. No
o
doubt logically there is a marked distinction between arithmetic
and algebra, for the former is the theory of discrete magnitude
while the latter is that of continuous magnitude ; but a
scientific distinction such as this is of quite recent origin, and
the idea of continuity was not introduced into mathematics
before the time of Kepler. Of course the fundamental rules
of this algorism were not at first strictly proved that is the
work of advanced thought but until the middle of the seven
teenth century there was some discussion of the principles
involved ; since then very few arithmeticians have attempted
* See e.g. Chaucer, The Miller s Tale, v. 2225 ; Shakespeare, The
Winter s Tale, Act -iv. Sc. 2 ; Othello, Act i. Sc. 1. I am not sufficiently
familiar with early French or German literature to know whether they
contain any references to the use of the abacus. I believe that the
Exchequer division of the High Court of Justice derives its name from
the table before which the judges and officers of the court originally sat:
this was covered with black cloth divided into squares or chequers by
white lines, and apparently was used as an abacus.
ORIGIN OF Till-: AKAIJI J M MKKALS.
to justify or prove the processes used, or to do more than
enunciate rules and illustrate their use by numerical examples.
I have alluded frequently to the Arabic system of numeri
cal notation. I may therefore conveniently begin by a few
notes on the history of the symbols now current.
Their origin is obscure and has been much disputed*. On
the whole it seems probable that the symbols for the numbers
4, 5, 6, 7, and 9 (and possibly 8 too) are derived from the
initial letters of the corresponding words in the Indo-Bactrian
alphabet in use in the north of India perhaps 150 years before
Christ ; that the symbols for the numbers 2 and 3 are derived
respectively from two and three parallel penstrokes written
cursively; and similarly that the symbol for the number 1
represents a single penstroke. Numerals of this type were in
use in India before the end of the second century of our era
The origin of the symbol for zero is unknown ; it is not
impossible that it was originally a dot inserted to indicate a
blank space, or it may represent a closed hand, but these are
mere conjectures ; there is reason to believe that it was in
troduced in India towards the close of the fifth century of
our era, but the earliest writing now extant in which it occurs
is assigned to the eighth century.
The numerals used in India in and after the eighth century
are termed Devanagari numerals and their forms are shewn in
the first line of the table given on the next page. These forms
wi iv slightly modified by the eastern Arabs, and the resulting
symbols were again slightly modified by the western Arabs or
Moors. It is perhaps probable that at first the Spanish
Arabs discarded the use of the symbol for zero and only
re-inserted it when they found how inconvenient the omission
proved. The symbols finally used by the Arabs are termed
Gobur numerals, and an idea of the forms most commonly used
* See A. P. Pihan, Siyni-* ih numeration, Paris, 1860; Fr. Woepcke,
I.n propitiiat nm f/rx f/<////vs Imlii ms, Paris, 1863; A. C. Burnell, Stmth
Indian l\iUn o<iniphy, Mangalore, 1874; mul Is. Taylor, The Alphabet,
London, 1883; &\&o passim M. Cantor.
190 THE DEVELOPMENT OF ARITHMETIC. 1300 1637.
may be gathered from those printed in the second line of the
table given below. From Spain or Barbary the Gobar numerals
passed into western Europe. The further evolution of the
forms of the symbols to those with which we are familiar is
indicated below by facsimiles* of the numerals used at diffe
rent times. All the sets of numerals here represented are
written from left to right and in the order 1, 2, 3, 4, 5, 6,
7, 8, 9, 10.
Devaiiagari (Indian) nu
merals, circ. 950.
Gobar Arabic numerals,) \ *? <^ C (j V ^ Q Ck \
(7) i I ,C,7,7;T.,0,/,3,. V
circ. 1100(
From a missal, circ. 1385,
m a missal, circ. 1385, ) i">2ri//-" >vO/\
of German origin. $ /, *"> j>> <** ^ > ,A ,#, J ,
European (probably Italian)
numerals, circ. 1400.
From the Mirrour of the
World, printed by Cax-
ton in 1480.
From a Scotch calendar
for 1482, probably of
French origin.
From 1500 onwards the symbols employed are practically the
same as those now in use. t
The evolution of the symbols by the Arabs proceeded almost
independently of European influence. There are minute dif-
* The first, second, and fourth examples are taken from Is. Taylor s
Alphabet, London, 1883, vol. n., p. 266; the others are taken from
Leslie s Philosophy of Arithmetic, pp. 114, 115.
t See for example Tonstall s De Arte Supputandi, London, 1522 ;
or Eecord s Grounde of Artes, London, 1540, and Whetstone of Witte,
London, 1557.
INTRODUCTION OF THE AHAUlC NUMERALS. 191
ferences in the forms used by various writers and in some
cases alternative forms, without however entering into these
\ r r J*6n VA q i
details we may say that the numerals commonly employed finally
took the form shewn above, but the symbol there given for 4 is
at the present time generally written cursively.
Leaving now the history of the symbols I proceed to
discuss their introduction into general use and the develop
ment of algoristic arithmetic. I have already explained how
men of science, and particularly astronomers, had become
acquainted with the Arabic system by the middle of the
thirteenth century. The trade of Europe during the thirteenth
and fourteenth centuries was mostly in Italian hands, and the
obvious advantages of the algoristic system led to its general
adoption in Italy for mercantile purposes though not without
considerable opposition : thus, an edict was issued at Florence
in 1299 forbidding bankers to use Arabic numerals, and the
authorities of the university of Padua in 1348 directed that a
list should be kept of books for sale with the prices marked
" non per cifras sed per literas claras." The rapid spread of
the use of Arabic numerals and arithmetic through the rest of
Europe seems to have been quite as largely due to the makers
of almanacks and calendars as to merchants and men of science.
These calendars had a wide circulation in mediaeval times.
They were of two distinct types. Some of them were composed
with special reference to ecclesiastical purposes, and contained
the dates of the different festivals and fasts of the church
for a period of some seven or eight years in advance as well
as notes on church ritual. Nearly every monastery and
church of any pretensions possessed one of these, and numerous
specimens are still extant. Those of the second type were
written specially for the use of astrologers and physicians,
;uul the better specimens contained notes on various scien
tific subjects (especially medicine and astronomy); these were
192 THE DEVELOPMENT OF ARITHMETIC. 1300 1637.
not then uncommon, but, since it was only rarely that they
found their way into any corporate library, specimens are
now rather scarce. It was the fashion to use the Arabic
symbols in ecclesiastical works ; while their occurrence in all
astronomical tables and their oriental origin (which savoured
of magic) secured their use in calendars intended for scientific
purposes. Thus the symbols were generally employed in both
kinds of almanacks, and there are few, if any, specimens of
calendars issued after the year 1300 in which an explanation
of their use is not included. Towards the middle of the four
teenth century the rules of arithmetic de algorismo were also
added, and by the year 1400 we may consider that the Arabic
symbols were generally known throughout Europe, and were
used in most scientific and astronomical works. Most merchants,
outside Italy, continued however to keep their accounts in
Roman numerals till about 1550, and monasteries and colleges
till about 1650; though in both cases it is probable that
in and after the fifteenth century the processes of arithmetic
were performed in the algoristic manner. No instance of a
date or number being written in Arabic numerals is known
to occur in any English parish register or the court rolls of
any English manor before the sixteenth century ; but in the
rent roll of the St Andrews Chapter, Scotland, the Arabic
numerals are used in writing an entry for the year 1490. The
Arabic numerals were introduced into Constantinople by
Planudes at about the same time as into Italy (see above,
p. 119).
The history of mercantile arithmetic in Europe begins then
with its use by Italian merchants, and it is especially to the
Florentine traders and writers that we owe its early develop
ment and improvement. It was they who invented the system
of book-keeping by double entry. In this system every
transaction is entered on the credit side in one ledger, and
on the debtor side in another ; thus, if cloth be sold to A,
A s account is debited with the price, and the stock book con
taining the transactions in cloth is credited with the amount
IMPROVEMENTS INTRODUCED. 193
sold. It was they too who arranged the problems to which
arithmetic could be applied in different classes, such as rule of
three, interest, profit and loss, &c. They also reduced the
fundamental operations of arithmetic " to seven, in reverence "
says Pacioli "of the seven gifts of the Holy Spirit: namely,
numeration, addition, subtraction, multiplication, division,
raising to powers, and extraction of roots." Brahmagupta
had enumerated twenty processes besides eight subsidiary
ones, and had stated that "a distinct and several knowledge
of these" was "essential to all who wished to be calculators";
and whatever may be thought of Pacioli s reason for the
alteration the consequent simplification of the elementary pro
cesses was satisfactory.
The operations of algoristic arithmetic were at first very
cumbersome. The chief improvements subsequently intro
duced into the early Italian algorism were (i) the simplification
of the four fundamental processes : (ii) the introduction of
signs for plus, minus, and equality; and (though not so im
portant) for multiplication and division : (iii) the invention
of logarithms : and (iv) the use of decimals. I will consider
these in succession.
(i) In addition and subtraction the Arabs usually worked
from left to right. The modern plan of working from right
to left is shorter : it is said to have been introduced by an
Englishman named Garth, of whose life I can find no account.
The old plan continued in partial use till about 1600; even
now it would be more convenient in approximations where it
is necessary to keep only a certain number of places of decimals.
The Indians and Arabs had several systems of multipli
cation. These were all somewhat laborious, and were made
the more so as multiplication tables if not unknown were
at any rate used but rarely. The operation was regarded
as one of considerable difficulty, and the test of the accuracy
of the result by "casting out the nines" was invented by
the Arabs as a check on the correctness of the work. Various
other systems of multiplication were subsequently employed
B. 13
194 THE DEVELOrMENT OF ARITHMETIC. 1300 1637.
in Italy, of which several examples are given by Pacioli
and Tartaglia; and the use of the multiplication table at
least as far as 5 x 5 became common. From this limited
table the resulting product of the multiplication of all
numbers up to 10 x 10 can be deduced by what was termed
the regula ignavi. This is a statement of the identity
(5 + a) (5 + b) = (5 - a) (5 - b) + 10 (a + b). The rule was usually
enunciated in the following form. Let the number five be
represented by the open hand the number six by the hand
with one finger closed ; the number seven by the hand with two
fingers closed ; the number eight by the hand with three fingers
closed y and the number nine by the hand with four fingers
closed. To multiply one number by another let the multiplier
be represented by one hand, and the number multiplied by the
other, according to the above convention. Then the required
answer is the product of the number of fingers (counting the
thumb as a finger) open in the one hand by the number of
fingers open in the other together with ten times the total
number of fingers closed. The system of multiplication now
in use seems to have been first introduced at Florence.
The difficulty which all but professed mathematicians ex
perienced in the multiplication of large numbers led to the
invention of several mechanical ways of effecting the process.
Of these the most celebrated is that of Napier s rods invented
in 1617. In principle it is the same as a method which had
been long in use both in India and Persia, and which has
been described in the diaries of several travellers and notably
in the Travels of Svr John Chardin in Persia, London, 1686.
To use the method a number of rectangular slips of bone,
wood, metal, or cardboard are prepared, and each of them
divided by cross lines into nine little squares; a slip being
generally about three inches long and a third of an inch
across. In the top square one of the digits is engraved,
and the results of multiplying it by 2, 3, 4, 5, 6, 7, 8, and
9 are respectively entered in the eight lower squares : where
the result is a number of two digits, the ten-digit is written
PROCESSES OF MULTIPLICATION. NAPIER S RODS. 195
above and to the left of the unit-digit and separated from it
by a diagonal line. The slips are usually arranged in a box.
Figure i below represents nine such slips side by side : figure ii
1
2
3
4
5
G
7
8
9
/I
4
.- 6
8
1 o
-
tf
H
1 8
,-o
3
6
9
1 2
1 5
2 i
2 4
2 7
,-o
, 4
< 8
tf
1 6
2 u
K
2 8
3 2
3 6
5
. -&
5
2 6
2 G
3 o
3 5
4 o
4 5
6
6
, *
1 8
2 4
3 o
3 o
*2
4 8
5 4
7
1 4
2
2 8
3 5
4 2
4 o
w
6 3
-0
8
e
2 4
3 2
4 o
4 8
5 c
6 4
Ti
b
9
V 8
fr
3 6
4 5
5 4
3
7 2
8 1
,-b
Figure i.
Figure ii.
s^Ti
2
Figure iii.
shews the seventh slip, which is supposed to be taken out
of the box and put by itself. Suppose we wish to multiply
2985 by 317. The process as effected by the use of these slips
is as follows. The slips headed 2, 9, 8, and 5 are taken
out of the box and put side by side as shewn in figure iii
above. The result of multiplying 2985 by 7 may be written
thus
2985
7
35
56
63
14
20895
Now if the reader will look at the seventh line in figure iii,
he will see that the upper and lower rows of figures are respec
tively 1653 and 4365 ; moreover these are arranged by the
132
196 THE DEVELOPMENT OF ARITHMETIC. 1300 1637.
diagonals so that roughly the 4 is under the 6, the 3 under
the 5, and the 6 under the 3 ; thus
1653
4365.
The addition of these two numbers gives the required result.
Hence the result of multiplying by 7, 1, and 3 can be
successively determined in this way, and the required answer
(namely the product of 2985 and 713) is then obtained by
addition.
The whole process was written as follows.
2985
20895 / 7
2985 / 1
8955 /3
946245
The modification introduced by Napier in his Rabdologia,
published in 1617, consisted merely in replacing each slip by a
prism with square ends, which he called " a rod," each lateral
face being divided and marked in the same way as one of the
slips above described. These rods not only economized space,
but were easier to handle, and were arranged in such a way as
to facilitate the operations required.
If multiplication was considered difficult, division was at
first regarded as a feat which could be performed only by
skilled mathematicians. The method commonly employed by
the Arabs and Persians for the division of one number by
another will be sufficiently illustrated by a concrete instance.
Suppose we require to divide 17978 by 472. A sheet of
paper is divided into as many vertical columns as there
are figures in the number to be divided. The number to
be divided is written at the top and the divisor at the bottom ;
the first digit of each number being placed at the left hand
side of the paper. Then, taking the left hand column, 4 will
PROCESSES OF DIVISION.
197
go into 1 no times, hence the first figure in the dividend is 0,
which is written under the last figure of the divisor. This is
represented in figure i. Next (see figure ii) re-write the 472
1
4
7
7
9
2
7
8
1
7
9
7
8
1
2
5
9
7
8
2
1
3
8
7
8
6
3
8
1
8
4
7
2
X
*
X
3
1
1
7
2
9
7
8
5
2
9
1
7
8
3
8
7
6
8
3
3
8
2
1
8
6
5
1
6
8
5
1
8
6
4
2
4
4
7
4
7
2
7
2
2
3
8
Figure i.
Figure ii.
Figure iii.
immediately above its former position but shifted one place to
the right, and cancel the old figures. Then 4 will go into 17
four times ; but, as on trial it is found that 4 is too big for the
first digit of the dividend, 3 is selected ; 3 is therefore written
below the last digit of the divisor and next to the digit of the
dividend last found. The process of multiplying the divisor
by 3 and subtracting from the number to be divided is
indicated in figure ii, and shews that the remainder is 3818.
A similar process is then repeated, i.e. 472 is divided into
3818, shewing that the quotient is 38 and the remainder
42. This is represented in figure iii, which shews the whole
operation.
The method described above never found much favour
in Italy. The present system was in use there as early as the
198 THE DEVELOPMENT OF ARITHMETIC. 1300 1637.
beginning of the fourteenth century, but the method generally
employed was that known as the galley or scratch system.
The following example from Tartaglia, in which it is required
07
4 9
059Q
1 3 3 (15
844
8
to divide 1330 by 84, will serve to illustrate this method : the
arithmetic given by Tartaglia is shewn above, where numbers
in thin type are supposed to be scratched out in the course of
the work.
The process is as follows. First write the 84 beneath the
1330, as indicated below, then 84 will go into 133 once, hence
the first figure in the quotient is 1. Now 1 x8 = 8, which
subtracted from 13 leaves 5. Write this above the 13, and
cancel the 13 and the 8, and we have as the result of the
first step
5
1 330(1
84
Next, 1x4 = 4, which subtracted from 53 leaves 49. Insert
the 49, and cancel the 53 and the 4, and we have as the next
step
4
59
1330(1
8 4
which shews a remainder 490.
We have now to divide 490 by 84. Hence the next figure
in the quotient will be 5, and re- writing the divisor we have
4
59
1 3 3 ( 15
844
8
PROCESSES OF DIVISION. 199
Then 5 x 8 = 40, and this subtracted from 49 leaves 9. Insert
the 9, and cancel the 49 and the 8, and we have the following
result
49
5 9
1 3 3 ( 15
844
8
Next 5x4 = 20, and this subtracted from 90 leaves 70. Insert
the 70, and cancel the 90 and the 4, and the final result,
shewing a remainder 70, is
7
4 9
59Q
1 3 3 ( 15
844
8
The three extra zeros inserted in Tartaglia s work are un
necessary, but they do not affect the result, as it is evident that
a figure in the dividend may be shifted one or more places up
in the same vertical column if it be convenient to do so.
The mediaeval writers were acquainted with the method
now in use, but considered the scratch method more simple.
In some cases the latter is very clumsy as may be illustrated
by the following example take from Pacioli. The object is
to divide 23400 by 100. The result is obtained thus
040
03400
2 3 4 ( 234
10000
1 00
1
The galley method was used in India, and the Italians may
have derived it thence. In Italy it became obsolete some
where about 1600; but it continued in partial use for at least
200 THE DEVELOPMENT OF ARITHMETIC. 1300 1637.
another century in other countries. I should add that Napier s
rods can be, and sometimes were, used to obtain the result of
dividing one number by another.
(ii) The signs + and - to indicate addition and sub
traction occur in Widman s arithmetic published in 1489 (see
below, p. 210), but were first brought into general notice, at
any rate as symbols of operation, by Stifel in 1554 (see below,
p. 220). I believe I am correct in saying that Vieta in 1591
was the first well-known writer who used these signs consist
ently throughout his work, and it was not until the beginning
of the seventeenth century that they became recognized and
well-known symbols. The sign = to denote equality was in
troduced by Record in 1557 (see below, p. 218).
(iii) The invention of logarithms*, without which many
of the numerical calculations which have constantly to be
made would be practically impossible, was due to Napier of
Merchistoun (see below, p. 239). The first public announce
ment of the discovery was made in his Mirifici Logarithmorum
Canonis Descriptio, published in 1614, and of which an English
translation was issued in the following year; but he had
privately communicated a summary of his results to Tycho
Brahe as early as 1594. In this work Napier explains the
nature of logarithms by a comparison between corresponding
terms of an arithmetical and geometrical progression. He
illustrates their use, and gives tables of the logarithms of the
sines and tangents of all angles in the first quadrant, for differ
ences of every minute, calculated to seven places of decimals.
His definition of the logarithm of a quantity n was what we
should now express by 10 7 log e (I0 7 /n). This work is the more
interesting to us as it is the first valuable contribution to the
progress of mathematics which was made by any British writer.
The method by which the logarithms were calculated was ex
plained in the Conslructioj a posthumous work issued in 1619 :
it seems to have been very laborious and depended either on
* See the article on Logarithms in the Encyclopedia Britannica,
ninth edition.
INTRODUCTION OF LOGARITHMS. 201
direct involution and evolution or on the formation of geome
trical means. The method by finding the approximate value
of a convergent series was introdiH^d^by^Newtonj Cotes, and _
Euler^^Napier had determined to change the base to one ___
which was a power of 10, but died before he could effect it.
TEe rapid recognition throughout Europe of the advantages
of using logarithms in practical calculations was mainly due to
Briggs (see below, p. 240), who was one of the earliest to recognize
the value of Napier s invention. Briggs at once realized that
thji_J3ase to which Napier s logarithms were calculated" was ^
very inconvenient,! he accordingly visited Napier in 1616,
and urged the change to a decimal base, which was recognized
by Napier as an improvement. On his return Briggs im-
mediately^setTp^ work to calculate tables to a decimal base, and
in 1617 he brought out a table of logarrthnis ofThenumbers
from 1 to 1000 calculated to fourteen places of"" decimals.
He^uBsequently (in 1624) published tables of tlie^ogari thins
of additional numbers^a^d^oTvariQuijfigonometricar functions.
His logarithms of the natural numbers_are_equal tojbhose to
the base 10 when multiplied by 10 8 ^a,nd^of the sines of angles
to those toTEe baseTO when multiplied by 10 12 . A table of the
logarithms, to seven places of decimals, of the sines and tangents
of angles in the first quadrant had been brought out in 1620
by Edmund Gunter, one of the Gresham lecturers, who was
the inventor of the words cosine and cotangent. The calculation
of the logarithms of 70,000 numbers which had been omitted
by Briggs from his tables of 1624 was performed by Adrian
Vlacq and published in 1628 : with this addition the table gave
the logarithms of all numbers from 1 to 101,000. The Aritk-
metica Logarithmica of Briggs and Vlacq are substantially
the same as the existing tables : parts have at different times
been recalculated, but no tables of an equal range and fulness
entirely founded on fresh computations have been published
since. These tables were supplemented by Briggs s Trigono-
metrica Britannica, which contains tables not only of the
logarithms of the trigonometrical functions, but also of their
202 THE DEVELOPMENT OF ARITHMETIC. 1300 1637.
natural values: it was published posthumously in 1633. By
1630 tables of logarithms were in general use.
(iv) The introduction of the decimal notation for fractions
is also (in my opinion) due to Briggs. Stevinus had in 1585
used a somewhat similar notation, for he wrote a number
such as 25-379 either in the form 25, 3 7" 9" , or in the form
25379@; and Napier in 1617 in his essay on
rods had adopted the former notation. But these writers
had employed the notation only as a concise way of stating
results, and made no use of it as an operative form. The
same notation occurs however in the tables published by
Briggs in 1617, and would seem to have been adopted by
him in all his works; and, though it is difficult to speak
with absolute certainty, I have myself but little doubt that
he there employed the symbol as an operative form. In
Napier s posthumous Constructio published in 1619 it is
denned and used systematically as an operative form, and as
this work was written after consultation with Briggs, circ.
1615 6, and probably was revised by the latter before it was
issued, 1 think it confirms the view that the invention is due
to Briggs and was communicated by him to Napier. At any
rate it was not employed as an operative form by Napier in
1617, and, if Napier were then acquainted with it, it must be
supposed that he regarded its use as unsuitable in ordinary
arithmetic*. Before the sixteenth century fractions were
commonly written in the sexagesimal notation (ex. gr. see above
pp. 98, 102, 174).
In Napier s work of 1619 the point is written in the form
now adopted, but Briggs underlined the decimal figures, and
would have printed a number such as 25*379 in the form
25379. Subsequent writers added another line and would
have written it as 251379; nor was it till the beginning of the
eighteenth century that the notation now current was generally
employed.
* The claims of Napier to the invention are advocated by Dr Glaisher
in the Transactions of the British Association, 1873, pp. 1317.
203
CHAPTER XII.
THE MATHEMATICS OF THE RENAISSANCE*.
14501637.
THE last chapter is a digression from the chronological
arrangement to which as far as possible I have throughout
adhered, but I trust by references in this chapter to keep the
order of events and discoveries clear. I return now to the
general history of mathematics in western Europe. Mathe
maticians had barely assimilated the knowledge obtained from
the Arabs, including their translations of Greek writers, when
the refugees who escaped from Constantinople after the fall of
the eastern empire brought the original works and the tradi
tions of Greek science into Italy. Thus by the middle of the
fifteenth century the chief results of Greek and Arabian
mathematics were accessible to European students.
The invention of printing about that time rendered the dis
semination of discoveries comparatively easy. It is almost a
truism to remark that until printing was introduced a writer
appealed to a very limited class of readers, but we are perhaps
apt to forget that when a mediaeval writer " published" a
* For an account of the Italian mathematicians of this period for
win >m no special references are given, see Guil. Libri, Histoire des sciences
mathematiques en Itaiie, 4 vols., Paris, 1838 1841 ; and for the German
and other mathematicians of the renaissance for whom no references are
given, see parts xn, xni, and xiv of Cantor s Vorlt tmnfien fiber GeschicJttc
</( T Mathcmatik issued since the first edition of this work was published.
204 THE MATHEMATICS OF THE RENAISSANCE.
work the results were known to only a few of his contem
poraries. This had not been the case in classical times for
then and until the fourth century of our era Alexandria was
the recognized centre for the reception and dissemination of
new works and discoveries. In mediaeval Europe on the
other hand there was no common centre through which men of
science could communicate with one another, and to this cause
the slow and fitful development of mediaeval mathematics may
be partly ascribed.
The introduction of printing marks the beginning of the
modern world in science as in politics; for it was contempo
raneous with the assimilation by the indigenous European
school (which was born from scholasticism, and whose history
was traced in chapter vm.) of the results of the Indian and ?
Arabian schools (whose history and influence were traced in
chapters ix. and x.) and of the Greek schools (whose history
was traced in chapters u. to v.).
The last two centuries of this period of our history, which
may be described as the renaissance, were distinguished by
great mental activity in all branches of learning. The creation
of a fresh group of universities (including those in Scotland)
of a somewhat less complex type than the mediaeval univer
sities above described testify to the general desire for know
ledge. The discovery of America in 1492 and the discussions
that preceded the Reformation flooded Europe with new ideas
which by the invention of printing were widely disseminated ;
but the advance in mathematics was at least as well marked
as that in literature and that in politics.
During the first part of this time the attention of mathe
maticians was to a large extent concentrated on syncopated
algebra and trigonometry : the treatment of these subjects is
discussed in the first , section of this chapter, but the relative
importance of the mathematicians of this period is not very
easy to determine. The middle years of the renaissance were
distinguished by the development of symbolic algebra: this is
treated in the second section of this chapter. The close of
KEGIOMONTANUS. 205
the sixteenth century saw the creation of the science of dyna
mics: this forms the subject of the first section of chapter
xin. About the same time and in the early years of the
seventeenth century considerable attention was paid to pure
geometry : this forms the subject of the second section of
chapter xin.
The development of syncopated algebra and trigonometry.
Regiomontanus*. Amongst the many distinguished writers
of this time Johann Regiomontanus was the earliest and one of
the most able. He was born at Konigsberg on June 6, 1436,
and died at Rome on July 6, 1476. His real name was
Johannes Muller, but, following the custom of that time, he
issued his publications under a Latin pseudonym which in his
case was taken from his birthplace. To his friends, his
neighbours, and his tradespeople he may have been Johannes
Miiller, but the literary and scientific world knew him as
Regiomontanus, just as they knew Zepernik as Copernicus,
and Schwarzerd as Melanchthon. It seems to me as pedantic
as it is confusing to refer to an author by his actual name
when he is universally recognized under another: I shall there
fore in all cases as fa,r as possible use that title only, whether
latinized or not, by which a writer is generally known.
Regiomontanus studied mathematics at the university of
Vienna, then one of the chief centres of mathematical studies
in Europe, under Purbach who was professor there. His
first work, done in conjunction with Purbach, consisted of an
analysis of the Almagest. In this the trigonometrical functions
sine and cosine were used and a table of natural sines was
* His life was written by P. Gassendi, The Hague, second edition
1655. His letters, which afford much valuable information on the
mathematics of his time, were collected and edited by C. G. von Murr,
Nuremberg, 1786. An account of his works will be found in Eegiomon-
tanux, ciii fieixthjer Vorlihifer den Copernicus, by A. Ziegler, Dresden,
1874 : see also Cantor, chap. LV.
206 THE MATHEMATICS OF THE RENAISSANCE.
introduced. Pur bach died before the book was finished : it
was finally published at Venice, but not till 1496. As soon as
this was completed Regiomontanus wrote a work on astrology,
which contains some astronomical tables and a table of natural
tangents: this was published in 1490.
Leaving Vienna in 1462, Regiomontanus travelled for
some time in Italy and Germany; and at last in 1471 settled
for a few years at Nuremberg where he established an obser
vatory, opened a printing-press^ and probably lectured. Three
tracts on astronomy by him were written here. A mechanical
eagle, which flapped its wings and saluted the Emperor
Maximilian I. on his entry into the city, bears witness to
his mechanical ingenuity and was reckoned among the marvels
of the age. Thence Regiomontanus moved to Rome on an
invitation from Sixtus IV. who wished him to reform the
calendar. He was assassinated, shortly after his arrival, at
the age of 40.
Regiomontanus was among the first to take advantage of
the recovery of the original texts of the Greek mathematical
works in order to make himself acquainted with the methods
of reasoning and results there used ; the earliest notice in
modern Europe of the algebra of Diophantus is a remark of
his that he had seen a copy of it at the Vatican. He was
also well read in the works of the Arab mathematicians.
The fruit of this study was shewn in his De Triangulis
written in 1464. This is the earliest modern systematic
exposition of trigonometry, plane and spherical, though the
only trigonometrical functions introduced are those of the sine
and cosine. It is divided into five books. The first four are
given up to plane trigonometry, and in particular to determin
ing triangles from three given conditions. The fifth book is
devoted to spherical trigonometry. The work was printed in
five volumes at Nuremberg in 1533, nearly a century after the
death of Regiomontanus.
As an example of the mathematics of this time I quote one
of his propositions at length. It is required to determine a
REGIOMONTANUS.
207
triangle when the difference of two sides, the perpendicular on
the base, and the difference between the segments into which
the base is thus divided are given (book ii., prop. 23). The
following is the solution given by Regiomontanus.
Sit talis triangulus ABG, cujus duo latera AB et AG differentia
habeant nota HG, ductaque perpendicular! AD duorum casuum BD et
DG, differentia sit EG: hae duae differentiae sint datae, et ipsa perpen-
dicularis AD data. Dico quod omnia latera trianguli nota concludentur.
Per artem rei et census hoc problema absolvemus. Detur ergo differentia
laterum ut 3, differentia casuum 12, et perpendicularis 10. Pono pro
basi unam rem, et pro aggregate laterum 4 res, nae proportio basis ad
B D E G
congeriem laterum est ut H G ad GE, scilicet unius ad 4. Erit ergo BD
4 rei minus 6, sed AB erit 2 res demptis f . Duco AB in se, producuntur
4 census et 2^ demptis 6 rebus. Item BD in se facit census et 36
minus 6 rebus : huic addo quadratum de 10 qui est 100. Colliguntur
census et 136 minus 6 rebus aequales videlicet 4 censibus et 2 demptis
6 rebus. Eestaurando itaque defectus et auferendo utrobique aequalia,
quemadmodum ars ipsa praecipit, habemus census aliquot aequales
numero, unde cognitio rei patebit, et inde tria latera trianguli more suo
innotescet.
To explain the language of the proof I should add that
Regiomontanus always calls the unknown quantity res, and
its square census or zensus , but though he uses these technical
terms he writes the words in full. He commences by saying
that he will solve the problem by means of a quadratic equa
tion (per artem rei et census); and that he will suppose the
difference of the sides of the triangle to be 3, the difference
of tlif segments of the base to be 12, and the altitude of the
208 THE MATHEMATICS OF THE RENAISSANCE.
triangle to be 10. He then takes for his unknown quantity
(unam rem or x) the base of the triangle, and therefore the
sum of the sides will be x. Therefore ED will be equal to
!# 6 (| rei minus 6), and AB will be equal to 2x -f (2 res
demptis f); hence AB 2 (AB in se) will be 4=x 2 + 2|-6# (4 census
et 2J demptis 6 rebus), and BD 2 will be \x 2 + 36 - Qx. To BD 2
he adds AD 2 (quadratum de 10) which is 100, and states that
the sum of the two is equal to AB 2 . This he says will give
the value of x 2 (census), whence a knowledge of x (cognitio rei)
can be obtained, and the triangle determined.
To express this in the language of modern algebra we have
but by the given numerical conditions
AG-AB=3=\ (DG - DB),
AG+AB= (DG + DB) = x.
Therefore AB=2x-^ and BD = x-.
Hence (2x - 1) 2 - (x - 6) 2 + 1 00.
From which x can be found, and all the elements of the triangle
determined.
It is worth noticing that Regiomontanus merely aimed at
giving a general method, and the numbers are not chosen with
any special reference to the particular problem. Thus in his
diagram he does not attempt to make GE anything like four
times as long as GH, and, since x is ultimately found to be
equal to ^ V321, the point D really falls outside the base. The
order of the letters ABG, used to denote the triangle, is of
course derived from the Greek alphabet.
Some of the solutions which he gives are unnecessarily
complicated, but it must be remembered that algebra and
trigonometry were still only in the rhetorical stage of develop
ment, and when every, step of the argument is expressed in
words at full length it is by no means easy to realise all that
is contained in a formula.
REGIOMONTANUS. PURBACH. CUSA. 209
It will be observed from the above example that Regiomon-
tarius did not hesitate to apply algebra to the solution of geo
metrical problems. Another illustration of this is to be found
in his discussion of a question which appears in Brahmagupta s
Siddhanta. The problem was to construct a quadrilateral,
having its sides of given lengths, which should be inscribable
in a circle. The solution given by Regiomontanus was effected
by means of algebra and trigonometry: this was published by
0. G. von Murr at Nuremberg in 1786.
The Alyorithmus Demonstratus of Jordanus (see above, p.
176), which was first printed in 1534, was until recently uni
versally attributed to Regiomontanus. This work, which is
concerned with algebra and arithmetic, was known to Regio
montanus and it is possible that the text which has come down
to us contains additional matter contributed by him.
Regiomontanus was the most prominent mathematician of
his generation and I have dealt with his works in some detail
as typical of the most advanced mathematics of the time. Of
his contemporaries I shall do little more than mention the
names of a few of those who are best known; none were quite
of the first rank and I should sacrifice the proportion of the
parts of the subject were I to devote much space to them.
Purbach*. I may begin by mentioning Georg Furbach, first
the tutor and then the friend of Regiomontanus, born near
Linz on May 30, 1423 and died at Vienna on April 8, 1461,
who wrote a work on planetary motions which was published
in 1460; an arithmetic, published in 1511; a table of eclipses,
published in 1514; and a table of natural sines, published in
1541.
Cusa f. Next I may mention Nicolas von Cusa, who was
born in 1401 and died in 1464. Although the sou of a poor
fisherman and without influence, he rose rapidly in the church,
* His life was written by P. Gassendi, The Hague, second edition,
1655.
t His life was written by F. A. Scharpff, Tiibingen, 1871 ; and his
collected works, edited by H. Petri, were published at Bale in 1565.
B. 14
210 THE MATHEMATICS OF THE RENAISSANCE.
and in spite of being "a reformer before the reformation"
became a cardinal. His mathematical writings deal with the
reform of the calendar and the quadrature of the circle. He
argued in favour of the diurnal rotation of the earth.
Chuquet. I may also here notice a small treatise on
arithmetic, known as Le Triparty*, by Nicolas Ghuquet, a
bachelor of medicine in the university of Paris, which was
written in 1484. This work indicates that the extent of mathe
matics then taught was somewhat greater than was generally
believed a few years ago. It contains the earliest known use
of the radical sign with indices to mark the root taken, 2 for a
square-root, 3 for a cube-root, and so on; and also a definite
statement of the rule of signs. The words plus and minus are
denoted by the contractions p, m. The work is in French.
Introduction f of signs + and - . In England and Germany
algorists were less fettered by precedent and tradition than in
Italy, and introduced some improvements in notation which
were hardly likely to occur to an Italian. Of these the most
prominent were the introduction of the current symbols for ad
dition, subtraction, and equality.
The earliest instances of the use of the signs + and of
which we have any knowledge occur in the fifteenth century.
Johannes Widman of Eger, born about 1460, matriculated at
Leipzig in 1480, and probably by profession a physician, wrote
a Mercantile arithmetic, published at Leipzig in 1489: in this
book these signs are used, not however as symbols of opera
tion, but apparently merely as marks signifying excess or
deficiency ; the corresponding use of the word surplus or over
plus (see Levit. xxv. 27, and 1 Maccab. x. 41) is still retained
in commerce. It is noticeable that the signs generally occur
* See an article by A. Marre in Boncornpagni s Bulletino di biblio-
grafia for 1880, vol. xm., pp. 555659.
t See articles by P. Treutlein (Die deutsche Coss) in the Abhandlungen
zur Geschichte der Mathematik for 1879 ; by De Morgan in the Cambridge
Philosophical Transactions, 1871, vol. XL, pp. 203212 ; and by Bon-
compagni in the Bulletino di bibliografia for 1876, vol. ix., pp. 188210.
INTRODUCTION OF SIGNS + AND -. 211
only in practical mercantile questions : hence it has been con
jectured that they were originally warehouse marks. Some
kinds of goods were sold in a sort of wooden chest called a
layel, which when full was apparently expected to weigh
roughly either three or four centners ; if one of these cases
were a little lighter, say 5 Ibs., than four centners Widinan
describes it as 4c - 5 Ibs. : if it were 5 Ibs. heavier than the
normal weight it is described as 4c | 5 Ibs. : and there are
some slight reasons for thinking that these marks were chalked
on to the chests as they came into the warehouses. The
symbols are used as if they would be familiar to his readers.
It will be observed that the vertical line in the symbol for
excess printed above is somewhat shorter than the horizontal
line. This is also the case with Stifel and most of the early
writers who used the symbol : some presses continued to print
it in this, its earliest form, till the end of the seventeenth
century. Xy lander on the other hand in 1575 has the vertical
bar much longer than the horizontal line, and the symbol is
something like -)-. We infer that the more usual case was for
a chest to weigh a little less than its reputed weight, and, as
the sign - placed between two numbers was a common symbol
to signify some connection between them, that seems to have
been taken as the standard case, while the vertical bar was
originally a small mark superadded on the sign - to distinguish
the two symbols.
I am far from saying that this account of the origin of our
symbols for plus and minus is established beyond doubt, but it
i^ the most plausible that has been yet advanced. Another
suggested derivation is that + is a contraction of *$ the initial
letter in Old German of plus, while is the limiting form of m
(for minus) when written rapidly. De Morgan * proposed yet
another derivation. The Hindoos sometimes used a dot to
indicate subtraction, and this dot might he thought have been
elongated into a bar, and thus give the sign for minus ; while
* See p. 19 of his Arithmetical Hooks, London, 1847.
142
212 THE MATHEMATICS OF THE RENAISSANCE.
the origin of the sign for plus was derived from it by a super-
added bar as explained above : but I take it that at a later
time he abandoned this theory for what has been called the
warehouse explanation. Another conjecture, ingenious but
unsupported by any evidence, is that the symbol for plus is
derived from the Latin abbreviation & for et ; while that
for minus is obtained from the bar which is often written over
the contracted form of a word to signify that certain letters
have been left out.
I should perhaps here add that till the close of the six
teenth century the sign + connecting two quantities like a and
b was also used in the sense that if a were taken as the answer
to some question one of the given conditions would be too little
by b. This was a relation which constantly occurred in solu
tions of questions by the rule of false assumption (see ex. gr.
above, p. 104).
Lastly I would repeat again that these signs in Widman are
only abbreviations and not symbols of operation ; he attached
little or no importance to them, and no doubt would have
been amazed if he had been told that their introduction was
preparing the way for a revolution of the processes used in
algebra.
The Algorithmus of Jordanus was not published till 1534;
Widman s work was hardly known outside Germany ; and it
is to Pacioli that we owe the introduction into general use
of syncopated algebra ; that is, the use of abbreviations for
certain of the more common algebraical quantities and opera
tions, but where in using them the rules of syntax are observed.
Pacioli*. Lucas Pacioli, sometimes known as Lucas di
Bur go, and sometimes, but more rarely, as Lucas Paciolus, was
born at Burgo in Tuscany about the middle of the fifteenth
century. We know little of his life except that he was a
Franciscan friar; that he lectured on mathematics at Rome,
* See H. Staigmiiller in the Zeitschrift filr Mathematik, 1889, vol.
xxxiv.; also Libri, vol. in., pp. 133145; and Cantor, chap. LVII.
PACIOLI. 213
Pisa, Venice, and Milan; and that at the last named city he
was the first occupant of a chair of mathematics founded by
Sforza : he died at Florence about the year 1510.
His chief work was printed at Venice in 1494 and is
termed Summa de arithmetica, geometria, proporzioni e pro-
porzionalita. It consists of two parts, the first dealing with
arithmetic and algebra, the second with geometry. This was
the earliest printed book on arithmetic and algebra. It is
mainly based on the writings of Leonardo of Pisa, and its
importance in the history of mathematics is largely due to its
wide circulation.
In the arithmetic Pacioli gives rules for the four simple
processes, and a method for extracting square roots. He deals
pretty fully with all questions connected with mercantile
arithmetic, in which he works out numerous examples, and in
particular discusses at great length bills of exchange and the
theory of book-keeping by double entry. This part was the
first systematic exposition of algoristic arithmetic and has been
already alluded to in chapter xi. It and the similar work by
Tartaglia are the two standard authorities on the subject.
Most of the problems are solved by the method of false assump
tion (see above, p. 104), but there are several numerical mis
takes.
The following example will serve as an illustration of the
kind of arithmetical problems discussed.
I buy for 1440 ducats at Venice 2400 sugar loaves, whose nett weight
is 7200 lire ; I pay as a fee to the agent 2 per cent. ; to the weighers and
porters on the whole, 2 ducats ; I afterwards spend in boxes, cords,
canvas, and in fees to the ordinary packers in the whole, 8 ducats ; for
the tax or octroi duty on the first amount, 1 ducat per cent. ; afterwards
for duty and tax at the office of exports, 3 ducats per cent. ; for writing
directions on the boxes and booking their passage, 1 ducat ; for the bark
to Rimini, 13 ducats ; in compliments to the captains and in drink for
the crews of armed barks on several occasions, 2 ducats ; in expenses for
provisions for myself and servant for one month, 6 ducats ; for. expenses
for several short journeys over land here and there, for barbers, for
washing of linen, and of boots for myself and servant, 1 ducat ; upon my
arrival at Rimini I pay to the captain of the port for port dues in the
214 THE MATHEMATICS OF THE RENAISSANCE.
money of that city, 3 lire ; for porters, disembarkation on land, and
carriage to the magazine, 5 lire ; as a tax upon entrance, 4 soldi a load
which are in number 32 (such being the custom) ; for a booth at the fair,
4 soldi per load ; I further find that the measures used at the fair are
different to those used at Venice, and that 140 lire of weight are there
equivalent to 100 at Venice, and that 4 lire of their silver coinage are
equal to a ducat of gold. I ask therefore, at how much I must sell a
hundred lire Eimini in order that I may gain 10 per cent, upon my
whole adventure, and what is the sum which I must receive in Venetian
money?
In the algebra lie finds expressions for the sum of the
squares and the sum of the cubes of the first n natural numbers.
The larger part of this part of the book is taken up with simple
and quadratic equations, and problems on numbers which lead to
such equations. He mentions the Arabic classification of cubic
equations, but adds that their solution appears to be as im
possible as the quadrature of the circle. The following is the
rule he gives (edition of 1494, p. 145) for solving a quadratic
equation of the form x 2 + x = a : it is rhetorical and not synco
pated, and will serve to illustrate the inconvenience of that
method.
"Si res et census numero coaequantur, a rebus
dimidio sumpto censum prod u cere debes,
addereque numero, cujus a radice totiens
tolle semis rerum, census latusque redibit."
He confines his attention to the positive roots of equations.
Though much of the matter described above is taken from
Leonardo s Liber Abaci, yet the notation in which it is expressed
is superior to that of Leonardo. Pacioli follows the Arabs in
calling the unknown quantity the thing, in Italian cosa hence
algebra was sometimes known as the cossic art or in Latin
res, and sometimes denotes it by co or R or Rj. He calls
the square of it census or zensus and sometimes denotes it
by ce or Z ; similarly the cube of it, or cuba, is sometimes
represented by cu or C j the fourth power, or censo di censo,
is written either at length or as ce di ce or as ce ce. It
PACIOLI. 215
may be noticed that all his equations are numerical so that
he did not rise to the conception of representing known quan
tities by letters as Jordanus had done and as is the case in
modern algebra : but M. Libri gives two instances in which in
a proportion he represents a number by a letter. He indicates
addition and equality by the initial letters of the words plus
and aequalis, but he generally evades the introduction of a
symbol for minus by writing his quantities on that side of the
equation which makes them positive, though in a few places
he denotes it by m for minus or by de for demptus. This is a
commencement of syncopated algebra.
There is nothing striking in the results he arrives at in
the second or geometrical part of the work ; nor in two other
tracts on geometry which he wrote and which were printed
at Venice in 1508 and 1509. It may be noticed however
that like Regiornontanus he applied algebra to aid him in
investigating the geometrical properties of figures.
The following problem will illustrate the kind of geome
trical questions he attacked. The radius of the inscribed circle
of a triangle is 4 inches, and the segments into which one side
is divided by the point of contact are 6 inches and 8 inches
respectively. Determine the other sides. To solve this it is
sufficient to remark that rs = A = Js. (s a) (s b) (s c) which
gives 4s = Js x (s - 14) x 6 x 8, hence s - 21 ; therefore the
required sides are 21-6 and 21 -8, that is, 15 and 13. But
Pacioli makes no use of these formulae (with which he was
acquainted) but gives an elaborate geometrical construction
and then uses algebra to find the lengths of various segments
of the lines he wants. The work is too long for me to
reproduce here, but the following analysis of it will afford
sufficient materials for its reproduction. Let ABC be the
triangle, J9, E, F the points of contact of the sides, and
the centre of the given circle. Let H be the point of inter
section of OB and DF, and K that of OC and DE. Let L
and M be the feet of the perpendiculars drawn from E and
F on BC. Draw EP parallel to AB and cutting BC in P.
216 THE MATHEMATICS OF THE RENAISSANCE.
Then Pacioli determines in succession the magnitudes of the
following lines : (i) OB, (ii) 0(7, (iii) FD, (iv) FH, (v) ED,
(vi) EK. He then forms a quadratic equation from the
solution of which he obtains the values of MB and MD.
Similarly he finds the values of LG and LD. He now finds
in succession the values of EL, FM, EP and LP ; and then
by similar triangles obtains the value of AB which is 13.
This proof was, even sixty years later, quoted by Cardan as
"incomparably simple and excellent, and the very crown of
mathematics." I cite it as an illustration of the involved and
inelegant methods then current. The problems enunciated are
very similar to those in the De Triangulis of Regiomontanus.
Leonardo da Vinci. The fame of Leonardo da Vinci as an
artist has overshadowed his claim to consideration as a mathe
matician, but he may be said to have prepared the way for
a more accurate conception of mechanics and physics, while
his reputation and influence drew some attention to the sub
ject ; he was an intimate friend of Pacioli. Leonardo was
the illegitimate son of a lawyer of Vinci in Tuscany, was born
in 1452, and died in France in 1519 while on a visit to
Francis I. Several manuscripts by him were seized by the
French revolutionary armies at the end of the last century,
and "Venturi, at the request of the Institute, reported on those
concerned with physical or mathematical subjects*.
Leaving out of account Leonardo s numerous and important
artistic works, his mathematical writings are concerned chiefly
with mechanics, hydraulics, and optics his conclusions being
usually based on experiments. His treatment of hydraulics
and optics involves but little mathematics. The mechanics
contain numerous and serious errors ; the best portions are
those dealing with the equilibrium of a lever under any forces,
the laws of friction, the stability of a body as affected by the
position of its centre of gravity, the strength of beams, and
* Essai sur les ouvrages physico-mathdmatiques de Leonard de Vinci, by
J.-B. Venturi, Paris, 1797. See also the memoir by Fr. Woepcke, Rome,
1856.
LEONARDO DA VINCI. DURER. COPERNICUS. 217
the orbit of a particle under a central force ; he also treated a
few easy problems by virtual moments. A knowledge of the
triangle of forces is occasionally attributed to him, but I think
it is most probable that his views on the subject were some
what indefinite. Generally one may say that all his mathe
matical work is unfinished and consists largely of suggestions
which he had not the patience to verify or discuss in detail.
Diirer. Albrecht Diirer* was another artist of the same
time who was also known as a mathematician. He was born
at Nuremberg on May 21, 1471, and died there on April 6,
1528. His chief mathematical work was issued in 1525 and
contains a discussion of perspective, some geometry, and cer
tain graphical solutions : Latin translations of it were issued
in 1532, 1555, and 1605.
Copernicus. An account of Nicolaus Copernicus, born at
Thorn on Feb. 19, 1473 and died at Frauenberg on May 7,
1543, and his conjecture that the earth and planets all re
volved round the sun belong to astronomy rather than to
mathematics. I may however add that Copernicus wrote a
short text-book on trigonometry, published at Wittenberg in
1542, which is clear though it contains nothing new. It is
evident from this and his astronomy that he was well read in
the literature of mathematics, and was himself a mathematician
of considerable power. I describe his statement as to the
motion of the earth as a conjecture because he advocated it
only on the ground that it gave a simple explanation of natural
phenomena. Galileo in 1632 was the first to try to supply
anything like a proof of this hypothesis.
By the beginning of the sixteenth century the printing
press began to be active and many of the works of the earlier
mathematicians became now for the first time accessible to all
students. This stimulated inquiry, and before the middle of
the century numerous works were issued which, though they
did not include any great discoveries, introduced a variety
* See Diirer ah Mathematiker by H. Staigmiiller, Stuttgart, 1891.
218 THE MATHEMATICS OF THE RENAISSANCE.
of small improvements all tending to make algebra more
analytical.
Record. The sign now used to denote equality was in
troduced by Robert Record*. Record was born at Tenby in
Pembrokeshire about 1510 and died at London in 1558. He
entered at Oxford, and obtained a fellowship at All Souls
College in 1531 ; thence he migrated to Cambridge, where he
took a degree in medicine in 1545. He then returned to
Oxford and lectured there, but finally settled in London and
became physician to Edward VI. and to Mary. His prosperity
must have been short-lived, for at the time of his death he
was confined in the King s Bench prison for debt.
In 1540 he published an arithmetic, termed the Grounde of
Artes, in which he employed the signs + for plus and - for
minus ; " + whyche betokeneth too muche, as this line ,
plaine without a crosse line, betokeneth too little"; and
there are faint traces of his having used these signs as symbols
of operation and not as mere abbreviations. In this book the
equality of two ratios is indicated by two equal and parallel
lines whose opposite ends are joined diagonally, ex. gr. by ~z_ .
A few years later, in 1557, he wrote an algebra under the title
of the Whetstone of Witte. This is interesting as it contains
the earliest introduction of the sign = for equality, and he
says he selected that particular symbol because than two
parallel straight lines u noe 2 thynges can be moare equalle."
M. Charles Henry has however pointed out that this sign is a
not uncommon abbreviation for the word est in mediaeval
manuscripts ; and this would seem to indicate a more probable
origin. In this work Record shewed how the square root of an
algebraical expression could be extracted.
He also wrote an astronomy. These works give a clear
view of the knowledge of the time.
Rudolff. Riese. About the same time in Germany,
Rudolff and Riese took up the subjects of algebra and
* See pp. 1519 of my History of the Study of Mathematics at
Cambridge, Cambridge, 1889.
RUDOLFF. RIESE. STIFEL. 219
arithmetic. Their investigations form the basis of Stifel s well
known work. Christoff Rudolff* published his algebra in
1525 ; it is entitled Die Coss, and is founded on the writings
of Pacioli and perhaps of Jordanus. Rudolff introduced the
sign of ,J for the square root, the symbol being a corruption of
the initial letter of the word radix, similarly ,J *J J denoted
the cube root, and J J the fourth root. Adam Riese^ was born
near Bamberg, Bavaria, in 1489 of humble parentage, and after
working for some years as a miner set up a school; he died
at Annaberg on March 30, 1559. He wrote a treatise on
practical geometry, but his most important book was his well
known arithmetic (which may be described as algebraical)
issued in 1536 and founded on Pacioli s work. Riese used the
symbols + and .
Stifel + The methods used by Rudolff and Riese and their
results were brought into general notice through Stifel s work
which had a wide circulation in Germany. Michael Stifel,
sometimes known by the Latin name of Stiffelius, was born at
Esslingen in 1486 and died at Jena on April 19, 1567. He
was originally an Augustine monk, but he accepted the
doctrines of Luther of whom he was a personal friend. He
tells us in his algebra that his conversion was finally deter
mined by noticing that the pope Leo X. was the beast men
tioned in the Revelation. To shew this it was only necessary
to add up the numbers represented by the letters in Leo
decimus (the in had to be rejected since it clearly stood for
mysterUwn) and the result amounts to exactly ten less than 666,
thus distinctly implying that it was Leo the tenth. Luther
accepted his conversion, but frankly told him he had better
clear his mind of any nonsense about the number of the beast.
Unluckily for himself Stifel did not act on this advice. Be-
* See Wappler, Gi-xchii-Jiti tier dcutxcJifn Myebni it xv Jahrlutmlerte,
Zwickau, 1887.
t See two works by B. Berlot, Ueber Adum / />.<* , Annaberg, 1855;
mil />/ . COM i;m .1 1,1/n Ilicsc, Annaberg, lsro.
: The authorities on Stifd an- given by Cantor, chap. LXII.
220 THE MATHEMATICS OF THE RENAISSANCE.
lieving that he had discovered the true way of interpreting the
biblical prophecies, he announced that the world would come to
an end on Oct. 3rd, 1533. The peasants of Holzdorf, of which
place he was pastor, knowing of his scientific reputation ac
cepted his assurance on this point. Some gave themselves up to
religious exercises, others wasted their goods in dissipation, but
all abandoned their work. When the day foretold had passed,
many of the peasants found themselves ruined : furious at
having been deceived, they seized the unfortunate prophet, and
he was lucky in finding a refuge in the prison at Wittenberg,
from which he was after some time released by the personal
intercession of Luther.
Stifel wrote a small treatise on algebra, but his chief mathe
matical work is his Arithmetica Integra published at Nuremberg
in 1544, with a preface by Melanchthon.
The first two books of the Arithmetica Integra deal with
surds and incommensurables, and are Euclidean in form. The
third book is on algebra, and is noticeable for having called
general attention to the German practice of using the signs
+ and to denote addition and subtraction. There are faint
traces of these signs being occasionally employed by Stifel
as symbols of operation and not only as abbreviations; this
application of them was apparently new. He not only employed
the usual abbreviations for the Italian words which represent
the unknown quantity and its powers, but in at least one case-
when there were several unknown quantities he represented
them respectively by the letters A, B, C, &c. ; thus re-intro
ducing the general algebraic notation which had fallen into
disuse since the time of Jordanus. It used to be said that
Stifel was the real inventor of logarithms, but it is now certain
that this opinion was due to a misapprehension of a passage
in which he compares geometrical and arithmetical progressions.
Tartaglia. Piccolo Montana, generally known as Nicholas
Tartaglia, that is, Nicholas the stammerer, was born at Brescia
in 1500 and died at Venice on December 14, 1557. After the
capture of the town by the French in 1512 most of the inhabit-
TAKTAGLIA. 221
ants took refuge in the cathedral, and were there massacred
by the soldiers. His father, who was a postal messenger at
Brescia, was amongst the killed. The boy himself had his skull
split through in three places, while both his jaws and his palate
were cut open ; he was left for dead, but his mother got into
the cathedral, and finding him still alive managed to carry him
off. Deprived of all resources she recollected that dogs when
wounded always licked the injured place, and to that remedy
he attributed his ultimate recovery, but the injury to his palate
produced an impediment in his speech from which he received
his nickname. His mother managed to get sufficient money to
pay for his attendance at school for fifteen days, and he took
advantage of it to steal a copy-book from which he sub
sequently taught himself how to read and write ; but so poor
were they that he tells us he could not afford to buy paper, and
was obliged to make use of the tombstones as slates on which
to work his exercises.
He commenced his public life by lecturing at Yerona, but
he was appointed at some time before 1535 to a chair of mathe
matics at Venice where he was living when he became famous
through his acceptance of a challenge from a certain Antonio
del Fieri (or Florida). Fiori had learnt from his master, one
Scipione Ferreo (who died at Bologna in 1526), an empirical
solution of a cubic equation of the form x 3 + qx = r. This solu
tion was previously unknown in Europe, and it is probable that
Ferreo had found the result in an Arab work. Tartaglia, in
answer to a request from Colla in 1530, stated that he could
effect the solution of a numerical equation of the form x 3 +px*=r.
Fiori believing that Tartaglia was an impostor challenged him
to a contest. According to this challenge each of them was to
deposit a certain stake with a notary, and whoever could solve
the most problems out of a collection of thirty propounded by
the other was to get the stakes, thirty days being allowed for
the solution of the questions proposed. Tartaglia was aware
that his adversary was acquainted with the solution of a cubic
equation of some particular form, and suspecting that the
22 J THE MATHEMATICS OF THE RENAISSANCE.
questions proposed to him would all depend on the solution of
such cubic equations set himself the problem to find a general
solution, and certainly discovered how to obtain a solution of
some if not all cubic equations. His solution is believed to
have depended on a geometrical construction (see below, p. iJ-S),
but led to the formula which is often, but unjustly, described
as Cardan s,
When the contest took place all the questions proposed
to Tartaglia were as he had suspected reducible to the solution
of a cubic equation, and he succeeded within two hours in
bringing them to particular cases of the equation x a + qx /, of
which he knew the solution. His opponent failed to solve
any of the problems proposed to him, which as a matter of
fact were all reducible to numerical equations of the form
.r ;{ -f pjc 2 r. Tartaglia was therefore the conqueror; he sub
sequently composed some verses commemorative of his victory.
The chief works of Tartaglia are as follows, (i) His A Vow
9Ct6ttft, published in 15;>7 : in this he investigated the fall of
bodies under gravity ; and he determined the range of a pro
jectile, stating that it was a maximum when the angle of
projection was 45, but this seems to have been a lucky
guess, (ii) An arithmetic published in two parts in ir>.">i>.
(iii) A treatise on numbers, published in four parts in Ku >0,
and sometimes treated as a continuation of the arithmetic :
in this he shewed how the coefficients of a; in expansion of
(1-f.r)" could be calculated from those in the expansion of
(1 + x)*~ l for the cases when n is equal to 2, 3, 4, 5, or 6. It
is possible that he also wrote a treatise on algebra and the
solution of cubic equations, but if so no copy is now extant.
The other works were collected into a single edition and
re-published at Venice in 1GOG.
^ The treatise on arithmetic and numbers is one of the chief
authorities for our knowledge of the early Italian algorism. It
is verbose, but gives a clear account of the different arith
metical methods then in use, and has numerous historical
notes which, as far as we can judge, are reliable, and are the
TARTAGUA,
authorities for many of the statements in the last chapter.
It contains an immense number of questions on every kind
of problem which would be likely to occur in mercantile
arithmetic, and there are several attempts to frame algebraical
formulae suitable for particular problems.
These problems give incidentally a good deal of information
as to the ordinary life and commercial customs of the time.
Thus we find that the interest demanded on first class security
in Venice ranged from 5 to 12 per cent, a year; while the
interest on commercial transactions ranged from 20 per cent.
a year upwards. Tartaglia illustrates the evil effects of the
law forbidding usury by the manner in which it was evaded
in farming. Farmers who were in debt were forced by their
creditors to sell all their crops immediately after the harvest;
the market being thus glutted, the price obtained was very low,
and the money lenders purchased the corn in open market at
an extremely cheap rate. The farmers then had to borrow
their seed-corn on condition that they replaced an equal
quantity, or paid the then price of it, in the month of May,
when corn was dearest. Again, Tartaglia, who had been asked
by the magistrates at Verona to frame for them a sliding scale
by which the price of bread would be fixed by that of com,
enters into a discussion on the principles which it was then
supposed should regulate it. In another place he gives (lie
rules at that time current for preparing medicines.
Pacioli had given in his arithmetic some problems of an
amusing character, and Tartaglia imitated him by inserting a
large collection of mathematical pu/zles. He half apologi/es
for introducing them by saving that it was not uncommon at
dessert to propose; arithmetical questions to the company by
way of amusement, and he therefore adds some suitable
problems. I To gives several questions on how to guess a
number thought of by one of (.he company, or the relationships
caused by the marriage Of relatives, or diiliculties arising from
inconsistent bequests. Other pn/xles are such as t he following.
"There are three men, young, handsome, ami gullant, who have
224 THE MATHEMATICS OF THE RENAISSANCE.
three beautiful ladies for wives: all are jealous, as well the
husbands of the wives as the wives of the husbands. They
find on the bank of a river, over which they have to pass,
a small boat which can hold no more than two persons.
How can they pass so as to give rise to no jealousy?"
"A ship, carrying as passengers fifteen Turks and fifteen
Christians, encounters a storm; and the pilot declares that in
order to save the ship and crew one-half of the passengers
must be thrown into the sea. To choose the victims, the
passengers are placed in a circle, and it is agreed that every
ninth man shall be cast overboard, reckoning from a certain
point. In what manner must they be arranged, so that the
lot may fall exclusively upon the Turks ?" "Three men robbed
a gentleman of a vase containing 24 ounces of balsam. Whilst
running away they met in a wood with a glass-seller of whom
in a great hurry they purchased three vessels. On reaching a
place of safety they wish to divide the booty, but they find
that their vessels contain 5, 11, and 13 ounces respectively.
How can they divide the balsam into equal portions]"
These problems some of which are of oriental origin
form the basis of the collections of mathematical recreations
by Bachet de Meziriac, Ozanam, and Montucla.*
Cardanf. The life of Tartaglia was embittered by a quarrel
with his contemporary Cardan who, having under a pledge of
* Solutions of these and other similar problems are given in my
Mathematical Recreations and Problems, chaps, i., n. On Bachet, see
below, p. 306. Jacques Ozanam, born at Bouligneux in 1640 and died in
1717, left numerous works of which the only one worth mentioning is his
Recreations matJiematiques et physiques, 2 vols., Paris, 1696. Jean Etienne
Montucla, born at Lyons in 1725 and died in Paris in 1799, edited and
revised Ozanarn s mathematical recreations. His history of attempts to
square the circle, 1754, and history of mathematics to the end of the
seventeenth century in 2 volumes, 1758, are interesting and valuable works:
the second edition of the latter in 4 volumes, 1799, (the fourth volume
is by Lalande) forms the basis of most subsequent works on the subject.
t There is an admirable account of his life in the Nouvelle biographic
generate, by V. Sardou. Cardan left an autobiography of which an
analysis by H. Morley was published in two volumes in London in 1854.
CARDAN. 225
secrecy obtained Tartaglia s solution of a cubic equation,
published it. Girolamo Cardan was born at Pavia on Sept. 24,
1501, and died at Rome on Sept. 21, 1576. His career is an
account of the most extraordinary and inconsistent acts. A
gambler, if not a murderer, he was also the ardent student
of science, solving problems which had long baffled all investi
gation; at one time of his life he was devoted to intrigues
which were a scandal even in the sixteenth century, at another
he did nothing but rave on astrology, and yet at another he
declared that philosophy was the only subject worthy of man s
attention. His was the genius that was closely allied to
madness.
He was the illegitimate son of a lawyer of Milan, and was
educated at the universities of Pavia and Padua. After taking
his degree he commenced life as a doctor, and practised his
profession at Sacco and Milan from 1524 to 1550; it uas
during this period that he studied mathematics and publi>hr<]
his chief works. After spending a year or so in France,
Scotland, and England, be returned to Milan as professor of
science, and shortly afterwards was elected to a chair at Pavia.
Here he divided his time between debauchery, astrology, and
mechanics. His two sons were as wicked and passionate as him
self : the elder was in 1560 executed for poisoning his wife, and
about the same time Cardan in a fit of rage cut off the ears of
the younger who had committed some offence; for this scan
dalous outrage he suffered no punishment as the pope Gregory
XIII. took him under his protection. In 1562 Cardan moved
to Bologna, but the scandals connected with his name were so
great that the university took steps to prevent his lecturing,
and only gave way under pressure from Rome. In 1570 he
was imprisoned for heresy on account of his having published
the horoscope of Christ, and when released he found himself so
All Cardan s printed works were collected by Sponius, and published in
10 volumes, Lyons, H .ti.". ; tlic works on arithmetic and geometry are
contained in the fourth volume. It is said that there are in the Vatican
numerous manuscript note-books of his which have not been yet edited.
B. 15
226 THE MATHEMATICS OF THE RENAISSANCE.
generally detested that he determined to resign his chair. At
any rate he left Bologna in 1571, and shortly afterwards
moved to Rome. Cardan was the most distinguished astrologer
of his time, and when he settled at Rome he received a pension
in order to secure his services as astrologer to the papal court.
This proved fatal to him for, having foretold that he should
die on a particular day, he felt obliged to commit suicide in
order to keep up his reputation so at least the story runs.
The chief mathematical work of Cardan is the Ars Magna
published at Nuremberg in 1545. Cardan was much interested
in the contest between Tartaglia and Fiori, and as he had
already begun writing this book he asked Tartaglia to com
municate his method of solving a cubic equation. Tartaglia
refused, whereupon Cardan abused him in the most violent terms,
but shortly afterwards wrote saying that a certain Italian
nobleman had heard of Tartaglia s fame and was most anxious
to meet him, and begged him to come to Milan at once.
Tartaglia came, and though he found no nobleman awaiting
him at the end of his journey, he yielded to Cardan s impor
tunity and gave him the rule he wanted, Cardan on his side
taking a solemn oath that he would never reveal it, and would
not even commit it to writing in such a way that after his
death any one could understand it. The rule is given in
some doggerel verses which are still extant. Cardan asserts
that he was given merely the result, and that he obtained the
proof himself, but this is doubtful. He seems to have at once
taught the method, and one of his pupils Ferrari reduced the
equation of the fourth degree to a cubic and so solved it.
When the Ars Magna was published in 1545 the breach of
faith was made manifest. Tartaglia was not unnaturally very
angry, and after an acrimonious controversy he sent a challenge
to Cardan to take part in a mathematical duel. The pre
liminaries were settled, and the place of meeting was to be a
certain church in Milan, but when the day arrived Cardan
failed to appear, and sent Ferrari in his stead. Both sides
claimed the victory, though I gather that Tarfcaglia was the
CARDAN. 227
more successful ; at any rate his opponents broke up the
meeting, and he was fortunate in escaping with his life. Not
only did Cardan succeed in his fraud, but modern writers
generally attribute the solution to him, so that Tartaglia has
not even that posthumous reputation which is at least his
due.
The Ars Magna is a great advance on any algebra pre
viously published. Hitherto algebraists had confined their
attention to those roots of equations which were positive.
Cardan discussed negative and even imaginary roots, and
proved that the latter would always occur in pairs, though he
declined to commit himself to any explanation as to the
meaning of these "sophistic" quantities which he said were
ingenious though useless. Most of his analysis of cubic equa
tions seems to have been original ; he shewed that if the three
roots were real, Tartaglia s solution gave them in a form
which involved imaginary quantities. Except for the somewhat
similar researches of Bombelli a few years later (see below,
p. 231), the theory of imaginary quantities received little
further attention from mathematicians until Euler took the
matter up after the lapse of nearly two centuries. Gauss first
put the subject on a systematic and scientific basis, introduced
the notation of complex variables, and used the symbol i to
denote the square root of 1 : the modern theory is chiefly
based on his researches.
Cardan found the relations connecting the roots with the
coefficients of an equation. He was also aware of the principle
that underlies Descartes s " rule of signs/ but as he followed
the then general custom of writing his equations as the
equality of two expressions in each of which all the terms were
positive he was unable to express the rule concisely. He
gave a method of approximating to the root of a numerical
equation, founded on the tact that, if a function have opposite
signs when two numbers are substituted iu it, the equation
obtained by equating the function to zero will have a root
between these t\VO lUllllUerS.
152
228 THE MATHEMATICS OF THE RENAISSANCE.
Cardan s solution of a quadratic equation is geometrical
and substantially the same as that given by Alkarismi (see
above, p. 163). His solution of a cubic equation is also geo
metrical, and may be illustrated by the following case which
he gives in chapter xi. To solve the equation x 3 + 6x= 20 (or
any equation of the form x 3 + qx = r), take two cubes such that
the rectangle under their respective edges is 2 (or q) and the
difference of their volumes is 20 (or r). Then x will be equal
to the difference between the edges of the cubes. To verify
this he first gives a geometrical lemma to shew that, if from a
line AC a portion CB be cut off, then the cube on AB will be
less than the difference between the cubes on AC and BC by
three times the right parallelopiped whose edges are respec
tively equal to AC, BC, and AB this statement is equivalent
to the algebraical identity (a b) 3 = a 3 b 3 3ab (a - b) and
the fact that x satisfies the equation is then obvious. To obtain
the lengths of the edges of the two cubes he has only to solve
a quadratic equation for which the geometrical solution pre
viously given sufficed.
Like all previous mathematicians he gives separate proofs
of his rule for the different forms of equations which can fall
under it. Thus he proves the rule independently for equa
tions of the form x 3 + px = q, x 3 = px + q, x 3 + px + q = 0, and
x 3 + q=px. It will be noticed that with geometrical proofs
this was almost a necessity, but he did not suspect that the
resulting formulae were general. The equations he considers
are numerical, but in some of his analysis he uses literal
coefficients.
Shortly after Cardan came a number of mathematicians who
did good work in developing the subject, but who are hardly
of sufficient importance to require detailed mention here. Of
these the most celebrated are perhaps Ferrari and E/heticus.
Ferrari. Ludovico Ferraro usually known as Ferrari,
whose name I have already mentioned in connection with the
solution of a biquadratic equation, was born at Bologna on
Feb. 2, 1522 and died on Oct. 5, 1565. His parents were poor
FERRARI. RHETICUS. 229
and he was taken into Cardan s service as an errand boy, but
was allowed to attend his master s lectures, and subsequently
became his most celebrated pupil. He is described as " a neat
rosy little fellow, with a bland voice, a cheerful face, and an
agreeable short nose, fond of pleasure, of great natural powers "
but " with the temper of a fiend." His manners and numerous
accomplishments procured him a place in the service of the
cardinal Ferrando Gonzaga, where he managed to make a for
tune. His dissipations told on his health, and he retired in
1565 to Bologna where he began to lecture on mathematics.
He was poisoned the same year either by his sister, who seems
to have been the only person for whom he had any affection,
or by her paramour. Such work as he produced is incorporated
in Cardan s Ars Magna or Bombelli s Algebra, but nothing can
be definitely assigned to him except the solution of a biquad
ratic equation. Colla had proposed the solution of the equation
x 4 + Qx 2 + 36 = 60# as a challenge to mathematicians : this par
ticular equation had probably been found in some Arabic work.
Nothing is known about the history of this problem except
that Ferrari succeeded where Tartaglia and Cardan had failed.
Rheticus. Georg Joachim Rheticus, born at Feklkirch on
Feb. 15, 1514 and died at Kaschau on Dec. 4, 1576, was
professor at Wittenberg, and subsequently studied under
Copernicus whose works were produced under the direction of
Rheticus. Rheticus constructed various trigonometrical tables
some of which were published by his pupil Otho in 1596.
These were subsequently completed and extended by Yieta
and Pitiscus, and are the basis of those still in use. Rheticus
also found the values of sin 20 and sin 30 in terms of sin
and cos 0.
I add here the names of some other celebrated mathema
ticians of about the same time, though their works are now
of little value to any save antiquarians. Franciscus Mauro-
lycus, born at Messina of Greek parents in 1494 and died in
1575, translated numerous Latin and Greek mathematical
works, and discussed the conies regarded as sections of a cone :
230 THE MATHEMATICS OF THE RENAISSANCE.
his works were published at Venice in 1575. Jean Borrel,
born in 1492 and died at Grenoble in 1572, wrote an algebra,
founded on that of Stifel ; and a history of the quadrature of
the circle : his works were published at Lyons in 1559.
Wilhelm Xylander, born at Augsburg on Dec. 26, 1532 and
died on Feb. 10, 1576 at Heidelberg, where since 1558 he
had been professor, brought out an edition of the works of
Psellus in 1556; an edition of Euclid s Elements in 1562; an
edition of the Arithmetic of Diophantus in 1575; and some
minor works which were collected and published in 1577.
Federigo Commandino, born at Urbino in 1509 and died there
on Sept. 3, 1575, published a translation of the works of
Archimedes in 1558 ; selections from Apollonius, and Pappus
in 1566 ; Euclid s Elements in 1572 ; and selections from Ari-
starchus, Ptolemy, Hero, and Pappus in 1574 : all being
accompanied by commentaries. Jacques Peletier, born at le
Mans on July 25, 1517 and died at Paris in July 1582,
wrote several text-books on algebra and geometry : most of
the results of Stifel and Cardan are included in the former.
Adrian Romanus, born at Louvain on Sept. 29, 1561 and died
on May 4, 1625, professor of mathematics and medicine at the
university of Louvain, was the first to prove the usual formula
for sin (A + B). And lastly, Bartholomaus Pitiscus, born on
Aug. 24, 1561 and died at Heidelberg, where he was pro
fessor of mathematics, on July 2, 1613, published his Trigo
nometry in 1599 : this contains the expressions for sin (A B)
and cos (A B] in terms of the trigonometrical ratios of A and B.
About this time also several text-books were produced
which if they did not extend the boundaries of the subject
systematized it. In particular I may mention those of Ramus
and Bombelli.
Ramus*. Peter Ramus was born at Cuth in Picardy in
1515, and was killed at Paris at the massacre of St Bartho-
* See the monographs by Ch. Waddington, Paris, 1855 ; and by
C. Desmaze, Paris, 1864.
BOMBELLI. 231
lomew 011 Aug. 24, 1572. He was educated at the university
of Paris, and on taking his degree he astonished and charmed
the university with the brilliant declamation he delivered on
the thesis that everything Aristotle had taught was false. He
lectured for it will be remembered that in early days there
were no professors first at le Mans, and afterwards at Paris ;
at the latter he founded the first chair of mathematics.
Besides some works on philosophy he wrote treatises on
arithmetic, algebra, geometry (founded on Euclid), astronomy
(founded on the works of Copernicus), and physics which were
long regarded on the continent as the standard text-books on
these subjects. They are collected in an edition of his works
published at Bale in 1569.
Bombelli. Closely following the publication of Cardan s
great work, Rafaello Bombelli published in 1572 an algebra
which is a systematic exposition of what was then known
on the subject. In the preface he alludes to Diophantus who,
in spite of the notice of Regiomontanus, was still unknown in
Europe, and traces the history of the subject. He discusses
radicals, real and imaginary. He also treats the theory of
equations, and shews that in the irreducible case of a cubic
equation the roots are all real ; and he remarks that the
problem to trisect a given angle is the same as that of the
solution of a cubic equation. Finally he gave a large collection
of problems.
Bombelli is chiefly distinguished in connection with the
improvement in the notation of algebra which he introduced.
The symbols then ordinarily used for the unknown quantity
and its powers were letters which stood for abbreviations of
the words. Those most frequently adopted were R or Rj for
radix or res (x\ Z or C for zensus or census (or 2 ), C or K for
cid)us (x 3 ) y &c. Thus x* + 5x 4 would have been written
1 Z p. 5 R m. 4
where p stands for plus and in for minus. Xy lander, in his
edition of the Arithmetic of Diophantus in 1575, used other
232 THE MATHEMATICS OF THE RENAISSANCE.
letters and the symbols + and and would have written the
above expression thus
l() + 5^_4 :
a similar notation was sometimes used by Yieta and even by
Fermat. The advance made by Bombelli was that he intro
duced a symbol ^ for the unknown quantity, ^ for its square,
\& for its cube, and so on, and therefore wrote x 2 + 5x - 4 as
1 ^ p. 5 ^ m. 4.
Stevinus in 1586 employed , 0, , ... in a similar way ;
and suggested, though he did not use, a corresponding notation
for fractional indices (see below, p. 248). He would have
written the above expression as
1 + 5 - 4 .
But whether the symbols were more or less convenient they
were still only abbreviations for words, and were subject to
all the rules of syntax. They merely afforded a sort of short
hand by which the various steps and results could be expressed
concisely. The next advance was the creation of symbolic
algebra, and the chief credit of that is due to Vieta.
The development of symbolic algebra.
We have now reached a point beyond which any con
siderable development of algebra, so long as it was strictly
syncopated, could hardly proceed. It is evident that Stifel
and Bombelli and other writers of the sixteenth century had
introduced or were on the point of introducing some of the
ideas of symbolic algebra. But so far as the credit of in
venting symbolic algebra can be put down to any one man
we may perhaps assign it to Vieta, while we may say that
Harriot and Descartes did more than any other writers to
bring it into general use. It must be remembered however
that it took time before all these innovations became generally
known, and they were not familiar to mathematicians until the
lapse of some years after they had been published.
VIETA. 233
Vieta*. Franciscus Vieta (Francois Viete) was born in
1540 at Fontenay near la Rochelle and died in Paris in 1603.
He was brought up as a lawyer and practised for some time
at the Parisian bar; he then became a member of the pro
vincial parliament in Brittany; and finally in 1580 through
the influence of the duke de Rohan he was made master of
requests, an office attached to the parliament at Paris; the
rest of his life was spent in the public service. He was a
firm believer in the right divine of kings, and probably a zealous
catholic. After 1580 he gave up most of his leisure to mathe
matics, though his great work In Artem Analyticam Isagoye
in which he explained how algebra could be applied to the
solution of geometrical problems was not published till 1591.
His mathematical reputation was already considerable, when
one day the ambassador from the Low Countries remarked to
Henry IV. that France did not possess any geometricians capable
of solving a problem which had been propounded in 1593 by
his countryman Adrian Romanus (see above, p. 230) to all
the mathematicians of the world and which required the solu
tion of an equation of the 45th degree. The king thereupon
summoned Vieta, and informed him of the challenge. Vieta
saw that the equation was satisfied by the chord of a circle (of
unit radius) which subtends an angle 2?r/45 at the centre,
and in a few minutes he gave back to the king two solutions of
the problem written in pencil. In explanation of this feat I
should add that Vieta had previously discovered how to form
the equation connecting sin nO with sin and cos 0. Vieta
in his turn asked Romanus to give a geometrical construction
to describe a circle which should touch three given circles.
This was the problem which Apollonius had treated in his De
TactionibuSy a lost book which Vieta at a later time conjecturally
restored. Romanus solved the problem with the aid of the
conic sections, but failed to do it by Euclidean geometry. Vieta
gave a Euclidean solution which so impressed Romanus that
* An account of Vieta s works is given in vol. n. of C. Button s
Tracts, London, 181215.
234 THE MATHEMATICS OF THE RENAISSANCE.
he travelled to Fontenay, where the French court was then
settled, to make Yieta s acquaintance an acquaintanceship
which rapidly ripened into warm friendship.
Henry was much struck with the ability shewn by Yieta in
this matter. The Spaniards had at that time a cipher contain
ing nearly 600 characters which was periodically changed, and
which they believed it to be impossible to decipher. A despatch
having been intercepted, the king gave it to Yieta, and asked
him to try to read it and find the key to the system. Vieta
succeeded, and for two years the French used it, greatly to their
profit, in the war which was then raging. So convinced was
Philip II. that the cipher could not be discovered that when he
found his plans known he complained to the pope that the
French were using sorcery against him, " contrary to the prac
tice of the Christian faith."
Yieta wrote numerous works on algebra and geometry. The
most important are the In Artem Analyticam Isagoge, Tours,
1591; the Supplementum Geometriae and a collection of geome
trical problems, Tours, 1593; and the De Numerosa Potestatum
Resolutions, Paris, 1600 : all of these were printed for private
circulation only, but they were collected by F. van Schooten
and published in one volume at Leyden in 1646. Yieta also
wrote the De JEquationum Recognitions et Emendations which
was published after his death in 1615 by Alexander Anderson.
The In Artem is the earliest work on symbolic algebra. It
also introduced the use of letters for both known and unknown
quantities, a notation for the powers of quantities, and enforced
the advantage of working with homogeneous equations. To
this an appendix called Logistice Speciosa was added on ad
dition and multiplication of algebraical quantities, and on
the powers of a binomial up to the sixth. Yieta implies that
he knew how to form the coefficients of these six expansions
by means of the arithmetical triangle as Tartaglia had pre
viously done, but Pascal was the first to give the general rule
(see below, p. 285) for forming it for any order, which is equi
valent to saying that he could write down the coefficients of x
VI ETA.
in the expansion of (1 + x) n if those in the expansion of (1 + x)*~ l
\\rrr known; Newton was the first to give the general ex
pression for the coefficient of a,* in the expansion of (1 4- x) n .
Another appendix known as Zetetica on the solution of
equations was subsequently added to the In Artem.
The In Artem is memorable for two improvements in alge
braic notation which were introduced here, though it is probable
that Vieta took the idea of both from other authors.
One of these improvements was that he denoted the known
quantities by the consonants B, (7, D <fec. and the unknown
quantities by the vowels A, E, I &c. Thus in any problem
he was able to use a number of unknown quantities : in this
particular point he seems to have been forestalled by Jordanus
and by Stifel (see above, pp. 177, 220). The present custom of
using the letters at the beginning of the alphabet a, 6, c &c. to
represent known quantities and those towards the end, x, y, z
<fec. to represent the unknown quantities was introduced by
Descartes in 1637.
The other improvement was this. Till this time it had
been the custom to introduce new" symbols to represent the
square, cube, <kc. of quantities which had already occurred in
the equations ; thus, if R or N stood for x, Z or C or Q stood
for x 2 , and C or K for x 3 , &e. So long as this was the case the
chief advantage of algebra was that it afforded a concise state
ment of results every statement of which was reasoned out.
But when Vieta used A to denote the unknown quantity x, he
sometimes employed A quadratics, A cubus, ... to represent a; 2 ,
# 3 , ..., which at once shewed the connection between the dif
ferent powers : and later the successive powers of A w r ere
commonly denoted by the abbreviations Aq, Ac, Aqq, &c. Thus
Vieta would have written the equation
as B 3 in A quad. - D piano in A + A cubo aequatur Z solido.
It will be observed that the dimensions of the constants (B, D,
and Z) are chosen so as to make the equation homogeneous :
this is characteristic of all his work. It will be also noticed
that he does not use a sign for equality : and in fact the parti-
236 THE MATHEMATICS OF THE RENAISSANCE.
cular sign which we use to denote equality was employed by
him to represent a the difference between." Vieta s notation is
not so convenient as that previously used by Bombelli and
Stevinns, but it was more generally adopted ; occcasional in
stances of an approach to index notation, such as A q , are said
to occur in Vieta s works.
These two steps were almost essential to any further pro
gress in algebra. In both of them Yieta had been forestalled,
but it was his good luck in emphasizing their importance to
be the means of making them generally known at a time when
opinion was ripe for such an advance.
The De Mquationum Recognitions et Emendations is mostly
on the theory of equations. Vieta here shewed that the first
member of an algebraical equation <>(x) = Q could be resolved
into linear factors, and explained how the coefficients of x could
be expressed as functions of the roots. He also indicated how
from a given equation another could be obtained whose roots
were equal to those of the original increased by a given quan
tity or multiplied by a given quantity : and he used this
method to get rid of the cofficient of a? in a quadratic equation
and of the coefficient of x 2 in a cubic equation, and was thus
enabled to give the general algebraic solution of both.
His solution of a cubic equation is as follows. First reduce
the equation to the form X s + 3a 2 x = 2b 3 . Next let x = a 2 /y - y,
and we get y 6 + 2b 3 y 3 = a 6 which is a quadratic in y*. Hence y
can be found, and therefore x can be determined.
His solution of a biquadratic is similar to that known as
Ferrari s. He first got rid of the term involving x 3 , thus
reducing the equation to the form x 4 + a 2 x 2 + b 3 x = c 4 . He then
took the terms involving x 2 and x to the right-hand side of
the equation and added x 2 y 2 + \y* to each side, so that the
equation became (x 2 + |-y 2 ) 2 = x 2 (if a 2 } b 3 x + \y 4 + c 4 . He
then chose y so that the right-hand side of this equality is
a perfect square. Substituting this value of ?/, he was able
to take the square root of both sides, and thus obtain two
quadratic equations for x, each of which can be solved.
The De Numerosa Potestatum Resolutions deals with nume-
VIETA. 237
rical equations. In this a method for approximating to the
values of positive roots is given, but it is prolix and of little
use, though the principle (which is similar to that of Newton s
rule) is correct. Negative roots are uniformly rejected. This
work is hardly worthy of Vieta s reputation.
Vieta s trigonometrical researches are included in various
tracts which are collected in Schooten s edition. Besides some
trigonometrical tables he gave the general expression for the
sine (or chord) of an angle in terms of the sine and cosine of
its submultiples : Delambre considers this as the completion
of the Arab system of trigonometry. We may take it then
that from this time the results of elementary trigonometry
were familiar to mathematicians. Vieta also elaborated the
theory of right-angled spherical triangles.
Among Vieta s miscellaneous tracts will be found a proof
that each of the famous geometrical problems of the trisection
of an angle and the duplication of the cube depends on the
solution of a cubic equation. There are also some papers
connected with an angry controversy with Clavius, in 1594,
on the subject of the reformed calendar, in which Vieta was
not well advised.
Vieta s works on geometry are good but they contain
nothing which requires mention here. He applied algebra
and trigonometry to help him in investigating the properties
of figures. He also, as I have already said, laid great stress
on the desirability of always working with homogeneous
equations, so that if a square or a cube were given it should
be denoted by expressions like a 2 or b 3 and not by terms like
m or n which do not indicate the dimensions of the quantities
they represent. He had a lively dispute with Scaliger, on the
latter publishing a solution of the quadrature of the circle,
and succeeded in shewing the mistake into which his rival
had fallen. He gave a solution of his own which as far as it
goes is correct, and stated that the area of a square is to that
of the circumscribing circle as
238 THE MATHEMATICS OF THE RENAISSANCE.
This is one of the earliest attempts to find the value of TT by
means of an infinite series. He was well acquainted with the
extant writings of the Greek geometricians, and introduced the
curious custom, which during the seventeenth and eighteenth
centuries became fashionable, of restoring lost classical works.
He himself produced a conjectural restoration of the De
Tactionibus of Apollonius.
Girard. Yieta s results in trigonometry and the theory
of equations were extended by Albert Girard, a Dutch mathe
matician, who was born in Lorraine in 1590 and died in
1633.
In 1626 Girard published at the Hague a short treatise
on trigonometry, to which were appended tables of the values of
the trigonometrical functions. This work contains the earliest
use of the abbreviations sin, tan, sec for sine, tangent, and
secant. The supplemental triangles in spherical trigonometry
are also discussed and seem to have been discovered by Girard,
independently of Vie%a ; he also gave the expression for the
area of a spherical triangle in terms of the spherical excess
this was discovered independently by Cavalieri. In 1 627 Girard
brought out an edition of Maralois s Geometry with considerable
additions.
Girard s chief discoveries are contained in his Invention
nouvelle en I algebre published at Amsterdam in 1629 : this
contains the earliest use of brackets ; a geometrical interpre
tation of the negative sign ; the statement that the number of
roots of an algebraical equation is equal to its degree ; the
distinct recognition of imaginary roots ; and probably implies
also a knowledge that the first member of an algebraical equa
tion < (x) could be resolved into linear factors. Girard s
researches were unknown to most of his contemporaries, and
exercised no appreciable influence on the development of
mathematics.
The invention of logarithms by Napier of Merchistoun in
1614, and their introduction into England by Briggs and
others, have been already mentioned in chapter XT.
NAPIER. 239
Napier*. John Napier was born at Merchistoun in 1550
and died on April 4, 1617. He spent most of his time on the
family estate near Edinburgh, and took an active part in the
political and religious controversies of the day ; the business
of his life was to shew that the pope was antichrist, but his
favourite amusement was the study of mathematics and science.
As soon as the use of exponents became common in algebra
the introduction of logarithms would naturally follow, but
Napier reasoned out the result without the use of any symbolic
notation to assist him, and the invention of logarithms was so
far from being a sudden inspiration that it was the result of the
efforts of many years with a view to abbreviate the processes
of multiplication and division. It is likely that Napier s
attention may have been partly directed to the desirability
of facilitating computations by the stupendous arithmetical
efforts of some of his contemporaries, who seem to have taken
a keen pleasure in surpassing one another in the extent to
which they carried multiplications and divisions. The trigono
metrical tables by Rheticus, which were published in 1596 and
1613, were calculated in a most laborious way : Vieta himself
delighted in arithmetical calculations which must have taken
hours or days of hard work and of which the results often
served no useful purpose : L. van Ceulen (1539 1610) prac
tically devoted his life to finding a numerical approximation
to the value of TT, finally in 1610 obtaining it correct to 35
places of decimals : while, to cite one more instance, P. A.
Cataldi (1548 1626), who is chiefly known for his invention
in 1613 of the form of continued fractions (though he failed to
establish any of their properties), must have spent years in
numerical calculations.
In regard to Napier s other work I may again mention
(see above, p. 196) that in his Rabdologia, published in 1617,
he introduced an improved form of rod by the use of which
* See the Memoir* of Napier by Mark Napier, Edinburgh, 1834. An
edition of all his works was issiu-d at Edinburgh in 1839. A bibliography
of his writings is appended to a translation of the Conxtmrtio by W. K.
Macdonald, Edinburgh, ls--.i
240 THE MATHEMATICS OF THE RENAISSANCE.
the product of two numbers can be found in a mechanical way;
they can be also used for finding the quotient of one number
by another: he also invented two other rods called "virgulae" by
which square and cube roots can be extracted. I should add that
in spherical trigonometry he discovered certain formulae known
as Napier s analogies, and also enunciated a "rule of circular
parts " for the solution of right-angled spherical triangles.
Briggs. The earliest table of common logarithms was con
structed by Briggs and published in 1617 (see above, p. 201).
Henry Briggs* was born near Halifax in 1556. He was edu
cated at St John s College, Cambridge, took his degree in
1581, and obtained a fellowship in 1588. He was elected to
the Gresham professorship of geometry in 1594, and in 1619
became Savilian professor at Oxford, a chair which he held
until his death on Jan. 26, 1631. It may be interesting to
add that the chair of geometry founded by Sir Thomas
Gresham in 1596 was the earliest professorship of mathematics
established in Great Britain. Some twenty years earlier Sir
Henry Savile had given at Oxford open lectures on Greek
geometry and geometricians, and in 1619 he endowed the
chairs of geometry and astronomy in that university which are
still associated with his name. Both in London and at Oxford
Briggs was the first occupant of the chair of geometry. He
began his lectures at Oxford with the ninth proposition of the
first book of Euclid : that being the furthest point to which
Savile had been able to carry his audiences. ^At^Qanibridga
the Lucasian chair was established in 1663, the earliest occu
pants being Barrow andiN ewton.
~" The^aTmost iminediateacfoption throughout Europe of loga
rithms for astronomical and other calculations was mainly the
work of Briggs. Amongst others he convinced Kepler of the
advantages of Napier s discovery, and the spread of the use of
logarithms was rendered more rapid by the zeal and reputation
of Kepler who by his tables of 1625 and 1629 brought them
into vogue in Germany, while Cavalieri in 1624 and Edmund
* See pp. 27 30 of my History of the Study of Mathematics at Cam
bridge, Cambridge, 1889.
HARRIOT. OUGHTRED.
Wingatc iii 1626 did a similar service for Italian and F
mathematicians respectively.
Harriot. Thomas Harriot, who was born at Oxford in
1560, and died in London on July 2, 1621, di-. _ 1 deal
to extend and codify the theory of equations. The early part
of his life was spent in America with Sir Walter Raleigh :
while there he made the first survey of Virginia and North
Carolina, the maps of these being subsequently presented to
Queen Elizabeth. On his return to England he settled in
London, and gave up most of his time to mathematical studies.
The majority of the propositions I have assigned to Vieta
are to be found in Harriot s writings, but it is uncertain
whether they were discovered by him independently of Vieta
or not. In any case it is probable that Vieta had not fully
realized all that was contained in the propositions he had
enunciated. The full consequences of these, with numerous
extensions and a systematic exposition of the theory of equa
tions, were given by Harriot in his Artis Analyticae Praxis,
which was first printed in 1631. The Praxis does not differ
essentially from a good modern text-book; it is far more
analytical than any algebra that preceded it, and marks a
great advance both in symbolism and notation. It was widely
read and proved one of the most powerful instruments in
bringing analytical methods into general use. Harriot was I
believe the earliest writer who realized the advantage to be
obtained by taking all the terras of an equation to one side of
it. He was the first to use the signs > and < to represent
greater than and less than. When he denoted the unknown
quantity by a he represented a 2 by aa, a* by aaa, and so on.
This is a distinct improvement on Vieta s notation. The same
symbolism was used by Wallis as late as 1685, but concurrently
with the modern index notation which was introduced by
Descartes. Extracts from some of the other writings of
Harriot were published by Rigaud in 1833.
Oughtred. Among those who contributed to the general
adoption in England of these various improvements and ad-
B. 16
242 THE MATHEMATICS OF THE RENAISSANCE.
ditions to algorism and algebra was William Oughtred* , who
was born at Eton on March 5, 1574, and died at his vicarage
of Albury in Surrey on June 30, 1660 : it is usually said that
the cause of his death was the excitement and delight which
he experienced " at hearing the House of Commons had voted
the King s return," but a recent critic adds that it should be
remembered " by way of excuse that he [Oughtred] was then
eighty-six years old." Oughtred was educated at Eton and
King s College, Cambridge, of the latter of which colleges
he was a fellow and for some time mathematical lecturer.
His Clams Mathematica published in 1631 is a good sys
tematic text-book on arithmetic, and it contains practically all
that was then known on the subject. In this work he intro
duced the symbol x for multiplication, and the symbol : : in pro
portion; previously to his time a proportion such as a \b c id
was usually written as a - b c d, but he denoted it by
a . b : : c . d. Wallis says that some found fault with the
book on account of the style, but that they only displayed
their own incompetence, for Oughtred s " words be always full
but not redundant. 7 Pell makes a somew T hat similar remark.
Oughtred also wrote a treatise on trigonometry published in
1657, in which abbreviations for sine, cosine, &c. were employed.
This was really an important advance, but the works of Girard
and Oughtred, in which they were used, were neglected and
soon forgotten, and it was not until Euler reintroduced con
tractions for the trigonometrical functions that they were
generally adopted.
We may say roughly that henceforth elementary arith
metic, algebra, and trigonometry were treated in a manner
which is not substantially different from that now in use ; and
that the subsequent improvements introduced were additions to
the subjects as then known, and not a re-arrangement of them
on new foundations.
* See pp. 3031 of my History of the Study of Mathematics at
Cambridge, Cambridge, 1889. A complete edition of Oughtred s works
was published at Oxford in 1677.
ORIGIN OF COMMON SYMBOLS IN ALGEBRA. 243
The origin of the more common symbols in algebra.
It may be convenient if I collect here in one place the
scattered remarks I have made on the introduction of the
various symbols for the more common operations in algebra*.
The later Greeks (see p. 106), the Hindoos (see p. 159), and
Jordanus (see p. 178) indicated addition by mere juxtaposition.
It will be observed that this is still the custom in arithmetic,
where e.g. 2 J stands for 2 + J. The Italian algebraists, when
they gave up expressing every operation in words at full
length and introduced syncopated algebra, usually denoted
plus by its initial letter P or p, a line being sometimes drawn
through the letter to shew that it was a symbol of operation
and not a quantity : but the practice was not uniform ; Pacioli
for example sometimes denoted it by p, and sometimes by e,
and Tartaglia commonly denoted it by <. The German and
English algebraists on the other hand introduced the sign +
almost as soon as they used algorism, but they spoke of it as*
signum additorum and employed it only to indicate excess,
they also used it in the sense referred to above on p. 212.
Widman used it as an abbreviation for excess in 1489 (see
p. 210): by 1630 it was part of the recognized notation of
P 1 gebra, and was also used as a symbol of operation.
Subtraction was indicated by Diophaiitus by an inverted
and truncated ^ (see p. 106). The Hindoos denoted it by a
dot (see p. 159). The Italian algebraists when they introduced
syncopated algebra generally denoted minus by M or in, a line
being sometimes drawn through the letter: but the practice
\\ as not uniform ; Pacioli for example denoting it sometimes
by ni, and sometimes by de for demptus (see p. 215). The
German and English algebraists introduced the present symbol
which they described as signum subtractorum. It is most
likely that the vertical bar MI the symbol for plus was super-
See two articles by C. Henry in the June and July numbers of the
Revue Archeologiqite, 1879, vol. xxxvii., pp. 324333, vol. xxxvin. pp.
110.
162
244 THE MATHEMATICS OF THE RENAISSANCE.
imposed on the symbol for minus to distinguish the two. In.
origin both symbols were probably mercantile marks (see
p. 211). It may be noticed that Pacioli and Tartaglia found
the sign already used to denote a division, a ratio, or a
proportion indifferently (see p. 166 and p. 242). The present
sign was in general use by about the year 1630, and was then
employed as a symbol of operation.
Oughtred in 1631 used the sign x to indicate multiplica
tion: Harriot in 1631 denoted the operation by a dot:
Descartes in 1637 indicated it by juxtaposition. I am not
aware of any symbols for it which were in. previous use.
Leibnitz in 1686 employed the sign ^ to denote multiplica
tion, and ^ to denote division.
Division was ordinarily denoted by the Arab way of
writing the quantities in the form of a fraction by means of
a line drawn between them in any of the forms a b, a/b, or
j- . Oughtred in 1631 employed a dot to denote either division
or a ratio. I do not know when the colon (or symbol :) was
first introduced to denote a ratio, but it occurs in a work
by Clairaut published in 1760. I believe that the current
symbol for division -r is only a combination of the and the : ,
it was used by Johann Heinrich Rahn at Zurich in 1659, and
by John Pell in London in 1668.
The current symbol for equality was introduced by Record
in 1557 (see p. 218); Xylander in 1575 denoted it by two
parallel vertical lines; but in general till the year 1600 the
word was written at length ; and from then until the time of
Newton, say about 1680, it was more frequently represented by
oc or by DO than by any other symbol. Either of these latter
signs was used as a contraction for the first two letters of the
word aequalis. I may add that Yieta, Schooten, and others
employed the sign ~ to denote the difference between ; thus
a = b means with them what we denote by a - b.
The symbol :: to denote proportion, or the equality of two
ratios, was introduced by Oughtred in 1631, and was brought
ORIGIN OF COMMON SYMBOLS IN ALGEBRA. 245
into common use by Wallis in 1686. There is no object in
having a symbol to indicate the equality of two ratios which is
different from that used to indicate the equality of other things,
and it is better to replace it by the sign = .
The sign > for is greater than and the sign < for is less
than were introduced by Harriot in 1631, but Oughtred
simultaneously invented the symbols H and U for the same
purpose ; and these latter were frequently used till the begin
ning of the eighteenth century, e.g. by Barrow.
The symbols =j= for is not equal to, ^> is not greater than,
and < for is not less than are of recent introduction.
The vinculum was introduced by Vieta in 1591 ; and
brackets were first used by Girard in 1629.
The different methods of representing the power to which
a magnitude was raised have been already briefly alluded to.
The earliest known attempt to frame a symbolic notation was
made by Born belli in 1572 when he represented the unknown
quantity by ^, its square by vl;, its cube by ^, &c. (see p. 232).
In 1586 Stevinus used (T), @, (?) &c. in a similar way; and
suggested though he did not use a corresponding notation for
fractional indices (see p. 232 and p. 248). In 1591 Vieta im
proved on this by denoting the different powers of A by A,
AquacL, A cub., &c., so that he could indicate the powers of
different magnitudes (see p. 235); Harriot in 1631 further
improved on Vieta s notation by writing aa for a 2 , aaa for a 3 .
&c. (see p. 241), and this remained in use for fifty years
concurrently with the index notation. In 1634 P. Herigonus,
in his Cursus mathematicus published in five volumes at Paris
in 16341637, wrote a, a2, a3, ... for a, a 2 , a 3 .... The symbol
J to denote the square root was introduced by Rudolff in
1;VJC>; a similar notation had been used by Bhaskjira (see
p. 160).
The idea of using exponents to mark the power to which
a quantity was raised was due to Descartes, and was intro
duced bj him in 1637: but lie used only positive integral
indices a\ a\ a\.... Wallis in 1659 explained the mean-
246 THE MATHEMATICS OF THE RENAISSANCE.
ing of negative and fractional indices in expressions such
as x~\ ar, &c. (see p. 290) : the latter conception having been
foreshadowed by Oresmus (see p. 183) and perhaps by Stevinus.
Finally the idea of an index unrestricted in magnitude, such as
the n in the expression a n , is I believe due to Newton and was
introduced by him in connection with the binomial theorem in
the letters for Leibnitz written in 1676.
The symbol <x> for infinity was first employed by Wallis in
1655 in his Arithmetica Infinitorum ; but does not occur
again until 1713 when it is used in James Bernoulli s Ars
Conjectandi. This sign was sometimes employed by the
Romans to denote the number 1000, and it has been conjec
tured that this led to its being applied to represent any very
large number.
There are but few special symbols in trigonometry, I may
however add here the following note which contains all that I
have been able to learn on the subject. The current sexagesimal
division of angles is derived from the Babylonians through the
Greeks. The Babylonian unit angle was the angle of an equi
lateral triangle; following their usual practice (see p. 5) this
was divided into sixty equal parts or degrees, a degree was sub
divided into sixty equal parts or minutes, and so on. The word
sine was used by Regiomoiitanus and was derived from the Arabs :
the terms secant and tangent were introduced by Thomas Finck
(born in Denmark in 1561 and died in 1646) in his Geometriae
Rotundi, Bale, 1583 : the word cosecant was (I believe) first used
by Rheticus in his Opus Palatinum, 1596 : the terms cosine and
cotangent were first employed by E. Gunter in his Canon
Triangulormn, London, 1620. The abbreviations sin, tan, sec
were used in 1626 by Albert Girard, and those of cos and
cot by Oughtred in 1657 ; but these contractions did not come
into general use till Euler re-introduced them in 1748. The
idea of trigonometrical functions originated with John Bernoulli,
and this view of the subject was elaborated in 1748 by Euler
in his Introductio in Analysin Infinitorum.
247
CHAPTER XIII.
THE CLOSE OF THE RENAISSANCE.
CIRC. 158G 1637.
THE closing years of the renaissance were marked by a
revival of interest in nearly all branches of mathematics and
science. As far. as pure mathematics is concerned we have
already seen that during the last half of the sixteenth century
there had been a great advance in algebra, theory of equations,
and trigonometry ; and we shall shortly see (in the second sec
tion of this chapter) that in the early part of the seventeenth
century some new processes in geometry were invented. If how
ever we turn to applied mathematics it is impossible not to be
struck by the fact that even as late as the middle or end of
the sixteenth century no marked progress in the theory had been
made from the time of Archimedes. Statics (of solids) and
hydrostatics remained in much the state in which he had left
them, while dynamics as a science did not exist. It was
Stevinus who gave the first impulse to the renewed study
of statics, and Galileo who laid the foundation of dynamics ;
and to their works the first section of this chapter is devoted.
The development of mechanics and experimental methods.
Stevinus*. Simon Stemnus was born at Bruges in 1548,
and died at the Hague early in the seventeenth century. We
* An analysis of his works is given in the Histoire des sciences
maikfmatiguet ttphyriqwt chcz /ex LY ///*. $ by L. A. J. Quetelet, Brussels,
248 THE CLOSE OF THE RENAISSANCE.
know very little of his life save that he was originally a
merchant s clerk at Antwerp, and at a later period of his life
was the friend of Prince Maurice of Orange by whom he was
made quarter- master-general of the Dutch army.
To his contemporaries he was best known for his works on
fortifications and military engineering, and the principles he
laid down are said to be in accordance with those which are
now usually accepted. To the general populace he was also well
known on account of his invention of a carriage which was pro
pelled by sails \ this ran on the sea-shore, carried twenty-eight
people, and easily outstripped horses galloping by the side : his
model of it was destroyed in 1802 by the French when they
invaded Holland. It was chiefly owing to the influence of
Stevinus that the Dutch and French began a proper system
of book-keeping in the national accounts.
I have already alluded (see above, p. 232) to the intro
duction in his Arithmetic, published in 1585, of exponents to
mark the power to which quantities were raised : he is said to
have suggested the use of fractional (but not negative) expo
nents. For instance he wrote 3x 2 -5x+l as 30-5(7) + ! (7).
His notation for decimal fractions was of a similar character
(see above, p. 202). In the same book he likewise suggested
a decimal system of weights and measures.
He also published a geometry which is ingenious though it
does not contain many results which were not previously known.
It is however on his Statics and Hydrostatics published (in
Flemish) at Leyden in 1586 that his fame will rest. In this
work he enunciated the triangle of forces a theorem which
some think was first propounded by Leonardo da Vinci (see
above, p. 217). Stevinus regarded this as the fundamental
proposition of the subject; previous to the publication of his
1866, pp. 144 168 : see also Notice historique sur la vie et les ouvrages
de Stevinus by J. V. Gothals, Brussels, 1841 ; and Les travaux de Stevinus
by M. Steichen, Brussels, 1846. The works of Stevinus were collected
by Snell, translated into Latin and published at Leyden in 1605 under
the title llypomnemata.
STEVINUS. GALILEO. 249
work the science of statics had rested on the theory of the
lever, but since then it has been usual to commence by
proving the possibility of representing forces by straight lines,
and so of reducing many theorems to geometrical propositions,
and in particular to obtaining in that way a proof of the
parallelogram (which is equivalent to the triangle) of forces.
Stevinus is not clear in his arrangement of the various proposi
tions and discussion of their sequence, and the new treatment
of the subject was not definitely established before the ap
pearance in 1687 of Varignon s work on mechanics. Stevinus
also found the force which must be exerted along the line of
greatest slope to support a given weight on an inclined plane
a problem the solution of which had been long in dispute. He
further distinguished between stable and unstable equilibrium.
In hydrostatics he discussed the question of the pressure which
a fluid can exercise, and explained the so-called hydrostatic
paradox. Stevinus was somewhat dogmatic in his statements,
and allowed no one to differ from his conclusions, "and
those," says he, in one place, " who cannot see this, may the
Author of nature have pity upon their unfortunate eyes, for
the fault is not in the thing, but in the sight which we are
not able to give them."
Galileo*. Just as the modern treatment of statics originates
with Stevinus, so the foundation of the science of dynamics is
due to Galileo. Galileo Galilei was born at Pisa on Feb. 18,
1564, and died near Florence on Jan. 8, 1642. His father, a
poor descendant of an old and noble Florentine house, was
himself a fair mathematician and a good musician. Galileo was
educated at the monastery of Yallombrosa where his literary
* See the biography of Galileo, by T. H. Martin, Paris, 1868. There
is also a life by Sir David Brewster, London, 1841 ; and a long notice by
Libri in the fourth volume of his Histoire dcs sciences mathematitjues en
Itiiiir. An edition of Galileo s works was issued in 16 volumes by
E. AlbSri, Florence, 18421856. A good many of his letters on various
mathematical subjects have been since discovered, and a new and com
plete edition is now being prepared by Antonio Favaro of Padua for the
Italian Government.
250 THE CLOSE OF THE KENAISSANC&
ability and mechanical ingenuity attracted considerable atten
tion. He was persuaded to become a novitiate of the order in
1580, but his father, who intended him to be a doctor, at once
removed him, and sent him in 1581 to the university of Pisa
to study medicine. It was there that he noticed that the great
bronze lamp, which still hangs from the roof of the cathedral,
performed its oscillations in equal times, quite independently
of whether the oscillations were large or small a fact which
he verified by counting his pulse. He had been hitherto
purposely kept in ignorance of mathematics, but one day, by
chance hearing a lecture on geometry, he was so fascinated by
the science that he thenceforward devoted all his spare time to
its study, and finally he got leave to discontinue his medical
studies. He left the university in 1586, and almost im
mediately commenced his original researches.
He published in 1587 an account of the hydrostatic balance,
and in 1588 an essay on the centre of gravity in solids. The
fame of these works secured for him the appointment to the
mathematical chair at Pisa the stipend, as was the case with
most professorships, being very small. During the next three
years he carried on from the leaning tower that series of ex
periments on falling bodies which established the first principles
of dynamics. Unfortunately the manner in which he pro
mulgated his discoveries and the ridicule he threw on those
who opposed him gave not unnatural offence, and in 1591
he was obliged to resign his position.
At this time he seems to have been much hampered by
want of money. Influence was however exerted on his behalf
with the Venetian senate, and he was appointed professor at
Padua, a chair which he held for eighteen years (1592 1610).
His lectures there seem to have been chiefly on mechanics and
hydrostatics, and the substance of them is contained in his
treatise on mechanics which was published in 1612. In these
lectures he repeated his Pisan experiments, and demonstrated
that falling bodies did not (as was then believed) descend with
velocities proportional amongst other things to their weights.
GALILEO. 251
He further shewed that, if it were assumed that they descended
with a uniformly accelerated motion, it was possible to deduce
the relations connecting velocity, space, and time which did
actually exist. At a later date, by observing the times of
descent of bodies sliding down inclined planes, he shewed that
this hypothesis was true. He also proved that the path of a
projectile was a parabola, and in doing so implicitly used the
principles laid down in the first two laws of motion as
enunciated by Newton. He gave an accurate definition of
momentum which some writers have thought may be taken to
imply a recognition of the truth of the third law of motion.
The laws of motion are however nowhere enunciated in a
precise and definite form, and Galileo must be regarded rather
as prepaiing the way for Newton than as being himself the
creator of the science of dynamics.
In statics he laid down the principle that in machines what
was gained in power was lost in speed, and in the same ratio.
In the statics of solids he found the force which can support a
given weight on an inclined plane ; in hydrostatics he pro
pounded the more elementary theorems on pressure and on
floating bodies; while among hydrostatical instruments he
invented the thermometer, though in a somewhat imperfect
form.
It is however as an astronomer that most people regard
Galileo, and though strictly speaking his astronomical researches
lie outside the subject-matter of this book it may be interest
ing to give the leading facts. It was in the spring of 1609
that Galileo heard that a tube containing lenses had been made
by Lippershey in IJolland which served to magnify objects seen
through it. This gave him the clue, and he constructed a
telescope of that kind which still bears his name, and of which
an ordinary opera-glass is an example. Within a few months
he had produced instruments which were capable of magnifying
thirty-two diameters, and within a year he had made and pub
lished observations on the solar spots, the lunar mountains,
Jupiter s satellites, the phases of Venus, and Saturn s ring.
252 THE CLOSE OF THE RENAISSANCE.
Honours and emoluments were showered on him, and he was
enabled in 1610 to give up his professorship and retire to
Florence. In 1611 he paid a temporary visit to Rome, and
exhibited in the gardens of the Vatican the new worlds revealed
by the telescope.
It would seem that Galileo had always believed in the
Copernican system, but was afraid of promulgating it on
account of the ridicule it excited. The existence of Jupiter s
satellites seemed however to make its truth almost certain, and
he now boldly preached it. The orthodox party resented his
action, and on Feb. 24, 1616, the Inquisition declared that to
suppose the sun the centre of the solar system was absurd,
heretical, and contrary to Holy Scripture. The edict of March
5, 1616, which carried this into effect has never been repealed
though it has been long tacitly ignored. It is well known that
towards the middle of the seventeenth century the Jesuits
evaded it by treating the theory as an hypothesis from which,
though false, certain results would follow.
In January 1632 Galileo published his dialogues on the
system of the world in which in clear and forcible language
he expounded the Copernican theory. In these, apparently
through jealousy of Kepler s fame, he does not so much as
mention Kepler s laws (the first two of which had been pub
lished in 1609 and the third in 1619) and he rejects Kepler s
hypothesis that the tides are caused by the attraction of the
moon. He rests the proof of the Copernican hypothesis on
the absurd statement that it would cause tides because different
parts of the earth would rotate with different velocities. He
was more successful in shewing that mechanical principles
would account for the fact that a stone thrown straight up
would fall again to the place from which it was thrown a
fact which had previously been one of the chief difficulties in
the way of any theory which supposed the earth to be in motion.
The publication of this book was certainly contrary to the
edict of 1616. Galileo was at once summoned to Rome, forced
to recant, do penance, and was only released on good behaviour.
GALILEO. FRANCIS BACON. 253
The documents recently printed shew that he was threatened
with the torture, but that there was no intention of carrying
the threat into effect.
When released he again took up his work on mechanics,
and by 1636 had finished a book which was published under
the title Discorsi intorno a due nuove scienze at Leyden in 1638.
In 1637 he lost his sight, but with the aid of pupils he con
tinued his experiments on mechanics and hydrostatics, and in
particular on the possibility of using a pendulum to regulate a
clock, and on the theory of impact.
An anecdote of this time has been preserved, which may
or may not be true, but is sufficiently interesting to bear
repetition. According to one version of the story, Galileo
was one day interviewed by some members of a Florentine
guild who wanted their pumpsKalterei^ as to raise water to a
height which was greater than thirty feet; and thereupon he
remarked that it might be desirable to first find out why the
water rose at all. A bystander interfered and said there was
110 difficulty about that because nature abhorred a vacuum.
Yes, said Galileo, but apparently it is only a vacuum which is
less than thirty feet. His favourite pupil Torricelli was
present, and thus had his attention directed to the question
which he subsequently elucidated.
Galileo s work may I think be fairly summed up by saying
that his researches on mechanics are deserving of high praise,
and that they are memorable for clearly enunciating the fact
that science must be founded on laws obtained by experiment;
his astronomical observations and his deductions therefrom
were also excellent, and were expounded with a literary
and scientific skill which leaves nothing to be desired, but
though he produced some of the evidence which placed the
Copernican theory on a satisfactory basis he did not himself
make any special advance in the theory of astronomy.
Francis Bacon*. The necessity of an experimental founda-
* See his life by J. Spedding, London, 187274. The best edition of
his works is that by Ellis, Spedding, and Heath in 7 volumes, London,
second edition, 1870.
254 THE CLOSE OF THE RENAISSANCE.
tion for science was advocated with even greater effect by
Galileo s contemporary frauds Bacon (Lord Verulam), who
was born at London on Jan. 22, 1561, and died on April 9,
1626. He was educated at Trinity College, Cambridge. His
career in politics and at the bar culminated in his becoming
lord chancellor with the title of Lord Verulam : the story of
his subsequent degradation for accepting bribes is well known.
His chief work is the Novum Organum, published in 1620,
in which he lays down the principles which should guide those
who are making experiments on which they propose to found
a theory of any branch of physics or applied mathematics. He
gave rules by which the results of induction could be tested,
hasty generalizations avoided, and experiments used to check
one anothei*. The influence of this treatise in the eighteenth
century was great, but it is probable that during the preceding
century it was little read, and the remark repeated by several
French writers that Bacon and Descartes are the creators of
modern philosophy rests on a misapprehension of Bacon s
influence on his contemporaries: any detailed account of this
book belongs however to the history of scientific ideas rather
than to that of mathematics.
Before leaving the subject of applied mathematics I may
add a few words on the writings of Guldinus, Wright, and
Snell.
Guldinus. Ilabakkuk Guldinus, born at St Gall on June
12, 1577, and died at Gratz on Nov. 3, 1643, was of Jewish
descent but was brought up as a protestant: he was converted
to Roman Catholicism and became a Jesuit when he took the
Christian name of Paul, and it was to him that the Jesuit
colleges at Rome and Gratz owed their mathematical reputa
tion. The two theorems known by the name of Pappus (to
which I alluded on p. 101) were published by Guldinus in the
fourth book of his De Centra Gravitatis, Vienna, 1635 1642.
Not only were the rules in question taken without acknow
ledgment from Pappus, but (according to Montucla) the proof
of them given by Guldinus was faulty, though he was success-
WRIGHT. 255
ful in applying them to the determination of the volumes and
surfaces of certain solids. The theorems were however pre
viously unknown, and their enunciation excited considerable
interest.
Wright*. I may here also refer to Edward Wright, who
is worthy of mention for having put the art of navigation
on a scientific basis. Wright was born in Norfolk about 1560,
and died in 1615. He was educated at Caius College, Cam
bridge, of which society he was subsequently a fellow. He
seems to have been a good sailor and he had a special talent
for the construction of instruments. About 1600 he was
elected lecturer on mathematics by the East India Company ;
he then settled in London, and shortly afterwards was ap
pointed mathematical tutor to prince Henry of Wales, the son
of James I. His mechanical ability may be illustrated by an
orrery of his construction by which it was possible to predict
eclipses for over seventeen thousand years in advance : it was
shewn in the Tower as a curiosity as late as 1675.
In the maps in use before the time of Gerard Mercator a
degree, whether of latitude or longitude, had been represented
in all cases by the same length, and the course to be pursued
by a vessel was marked on the map by a straight line joining
the ports of arrival and departure. Mercator had seen that
this led to considerable errors, and had realized that to make
this method of tracing the course of a ship at all accurate the
space assigned on the map to a degree of latitude ought
gradually to increase as the latitude increased. Using this
principle, he had empirically constructed some charts, which
were published about 1560 or 1570. Wright set himself the
problem to determine the theory on which such maps should
be drawn, and succeeded in discovering the law of the scale of
the maps, though his rule is strictly correct for small arcs only.
The result was published in the second edition of Blundeville s
Exercises.
* See pp. 25 27 of my History of the Study of Mathematics at Cam
bridge, Cambridge, 1889.
256 THE CLOSE OF THE RENAISSANCE.
In 1599 Wright published his Certain errors in navigation
detected and corrected, in which he explained the theory and
inserted a table of meridional parts. The reasoning shews con
siderable geometrical power. In the course of the work he
gives the declinations of thirty- two stars, explains the pheno
mena of the dip, parallax, and refraction, and adds a table
of magnetic declinations, but he assumes the earth to be
stationary. In the following year he published some maps
constructed on his principle. In these the northernmost point
of Australia is shewn: the latitude of London is taken to be
51 32 .
Snell. A contemporary of Guldinus and Wright was
Willebrod Snell, whose name is still well known through his
discovery in 1619 of the law of refraction in optics. Snell
was born at Leydeii in 1591, occupied a chair of mathematics
at the university there, and died there on Oct. 30, 1626.
He was one of those infant prodigies who occasionally appear,
and at the age of twelve he was acquainted with the standard
mathematical works. I will here only add that in geodesy
he laid down the true principles for measuring the arc of
a meridian, and in spherical trigonometry he discovered the
properties of the pola/- or supplemental triangle.
Revival of interest in pure geometry.
The close of the sixteenth century was marked not only by
the attempt to found a theory of dynamics based on laws
derived from experiment, but also by a revived interest in
geometry. This was largely due to the influence of Kepler.
Kepler*. Johann Kepler, one of the founders of modern
astronomy, was born of humble parents near Stuttgart on
* See Johann Keppler s Leben und Wirken.loy J. L. E. von Breitschwert,
Stuttgart, 1831 ; and E. Wolf s Geschichte der Astronomic, Munich, 1871.
A complete edition of Kepler s works was published by C. Frisch at
Frankfort in 8 volumes 1858 71 ; and an analysis of the mathematical
part of his chief work, the Harmonice mundi, is given by Chasles in his
Aperqu historique.
KEPLER. 257
Dec. 27, 1571, and died at Ratisbon on Nov. 15, 1630. He
was educated under Maestlin at Tubingen; in 1593 he was
appointed professor at Gratz, where he made the acquaintance
of a wealthy and beautiful widow whom he married, but
found too late that he had purchased his freedom from
pecuniary troubles at the expense of domestic happiness. In
1599 he accepted an appointment as assistant to Tycho Brahe,
and in 1601 succeeded his master as astronomer to the emperor
Rudolph II. But his career was dogged by bad luck; first his
stipend was not paid; next his wife went mad and then died;
and though he married again in 1611 this proved an even more
unfortunate venture than before, for though, to secure a better
choice, he took the precaution to make a preliminary selection
of eleven girls whose merits and demerits he carefully analysed
in a paper which is still extant, he finally selected a wrong
one; while to complete his discomfort he was expelled from
his chair, and narrowly escaped condemnation for heterodoxy.
During this time he depended for his income on telling
fortunes and casting horoscopes, for as he says "nature which
has conferred upon every animal the means of existence lias
designed astrology as an adjunct and ally to astronomy." He
seems however to have had no scruple in charging heavily for
his services, and to the surprise of his contemporaries was
found at his death to have a considerable hoard of money.
He died while on a journey to try and recover for the benefit
of his children some of the arrears of his stipend.
In describing Galileo s work I alluded briefly to the three
laws in astronomy that Kepler had discovered, and in connec
tion with which his name will be always associated ; and I
have already mentioned the prominent part he took in bring
ing logarithms into general use on the continent. These are
familiar facts, but it is not known so generally that Kepler was
also a geometrician and algebraist of considerable power ; and
that he, Desargues, and perhaps Galileo may be considered as
forming a connecting link between the mathematicians of the
renaissance and those of modern times.
B. 17
258 THE CLOSE OF THE RENAISSANCE.
Kepler s work in geometry consists rather in certain general
principles which he laid down and illustrated by a few cases
than in any systematic exposition of the subject. Tn a short
chapter on conies inserted in his Paralipomena, published in
1604, he lays down what has been called the principle of
continuity ; and gives as an example the statement that a
parabola is at once the limiting case of an ellipse and of a
hyperbola; he illustrates the same doctrine by reference to
the foci of conies (the word focus was introduced by him); and
he also explains that parallel lines should be regarded as meet
ing at infinity.
In his Stereometria^ which was published in 1615, he deter
mines the volumes of certain vessels and the areas of certain
surfaces, by means of infinitesimals instead of by the long and
tedious method of exhaustions. These investigations as well
as those of 1604 arose from a dispute with a wine merchant as
to the proper way of gauging the contents of a cask. This
use of infinitesimals was objected to by Guldinus and other
writers as inaccurate, but though the methods of Kepler are
not altogether free from objection he was substantially correct,
and by applying the law of continuity to infinitesimals he
prepared the way* for Cavalieri s method of indivisibles, and
the infinitesimal calculus of Newton and Leibnitz.
Kepler s work on astronomy lies outside the scope of this
book. I will mention only that it was founded on the ob
servations of Tycho Brahe f whose assistant he was. His three
laws of planetary motion were the result of many and laborious
efforts to reduce the phenomena of the solar system to certain
simple rules. The first two were published in 1609, and stated
that the planets describe ellipses round the sun, the sun
being in a focus ; and that the line joining the sun to any
planet sweeps over equal areas in equal times. The third was
published in 1619, and stated that the squares of the periodic
* See Cantor, chap. LXXVIII.
f For an account of Tycho Brahe, born at Knudstrup in 1546 and
died at Prague in 1601, see his life by J. L. E. Dreyer, Edinburgh, 1890.
KEPLER. DESARGUES. 259
times of the planets are proportional to the cubes of the major
axes of their orbits. I ought to add that he attempted to
explain why these motions took place by a hypothesis which
is not very different from Descartes s theory of vortices.
Kepler also devoted considerable time to the elucidation of the
theories of vision and refraction in optics.
While the conceptions of the geometry of the Greeks were
being extended by Kepler, a Frenchman, whose name until
recently was almost unknown, was inventing a new method
of investigating the subject a method which is now known
as projective geometry. This was the discovery of Desargues
whom I put (with some hesitation) at the close of this period,
and not among the mathematicians of modern times.
Desargues*. Gerard Desargues, born at Lyons in 1593,
and died in 1662, was by profession an engineer and architect,
but he gave some courses of gratuitous lectures in Paris from
1626 to about 1630 which made a great impression upon his
contemporaries. Both Descartes and Pascal had a high opinion
of his work and abilities, and both made considerable use of the
theorems he had enunciated.
In 1636 Desargues issued a work on perspective ; but most
of his researches were embodied in his Brouillon proiect on
conies, published in 1639, a copy of which was discovered
by Chasles in 1845. I take the following summary of it from
Ch. Taylor s work on conies. Desargues commences with a
statement of the doctrine of continuity as laid down by
Kepler : thus the points at the opposite ends of a straight
line are regarded as coincident, parallel lines are treated as
meeting at a point at infinity, and parallel planes on a line at
infinity, while a straight line may be considered as a circle
whose centre is at infinity. The theory of involution of six
points, with its special cases, is laid down, and the projective
property of pencils in involution is established. The theory of
polar lines is expounded, and its analogue in space suggested.
* See Oeuvres de Desargues by M. Poudra, 2 vols., Paris, 1864; and
a note in the Bibliotheca Mathematica, 1885, p. 90.
172
260 THE CLOSE OF THE RENAISSANCE.
b
A tangent is defined as the limiting case of a secant, and an
asymptote as a tangent at infinity. Desargues shews that the
lines which join four points in a plane determine three pairs
of lines in involution on any transversal, and from any conic
through the four points another pair of lines can be obtained
which are in involution with any two of the former. He
proves that the points of intersection of the diagonals and
the two pairs of opposite sides of any quadrilateral inscribed
in a conic are a conjugate triad with respect to the conic, and
when one of the three points is at infinity its polar is a
diameter ; but he fails to explain the case in which the quad
rilateral is a parallelogram, although he had formed the con
ception of a straight line which was wholly at infinity. The
book therefore may be fairly said to contain the fundamental
theorems on involution, homology, poles and polars, and per
spective.
The influence exerted by the lectures of Desargues on
Descartes, Pascal, and the French geometricians of the seven
teenth century was considerable ; but the subject of projective
geometry soon fell into oblivion, chiefly because the analytical
geometry of Descartes was so much more powerful as a method
of proof or discovery.
The researches of Kepler and Desargues will serve to
remind us that as the geometry of the Greeks was not capable
of much further extension, mathematicians were now beginning
to seek for new methods of investigation, and were extending
the conceptions of geometry. The invention of analytical
geometry and of the infinitesimal calculus temporarily diverted
attention from pure geometry, but at the beginning of the
present century there was a revival of interest in it, and since
then it has been a favourite subject of study with many
mathematicians.
THE CLOSE 0V THE RENAISSANCE. 261
Mathematical knowledge at the close of the renaissance.
Thus by the beginning of the seventeenth century we may
say that the fundamental principles of arithmetic, algebra,
theory of equations, and trigonometry had been laid down, and
the outlines of the subjects as we know them had been traced.
It must be however remembered that there were no good
elementary text-books on these subjects ; and a knowledge of
them was therefore confined to those who could extract it from
the ponderous treatises in which it lay buried. Though much of
the modern algebraical and trigonometrical notation had been
introduced, it was not familiar to mathematicians, nor was it
even universally accepted ; and it was not until the end of the
seventeenth century that the language of these subjects was
definitely fixed. Considering the absence of good text- books I
am inclined rather to admire the rapidity with which it came
into universal use, than to cavil at the hesitation to trust to it
alone which many writers shewed.
If we turn to applied mathematics we find on the other
hand that the science of statics had made but little advance in
the eighteen centuries that had elapsed since the time of
Archimedes, while the foundations of dynamics were laid by
Galileo only at the close of the sixteenth century. In fact, as
we shall see later, it was not until the time of Newton that the
science of mechanics was placed on a satisfactory basis. The
fundamental conceptions of mechanics are difficult, but the
ignorance of the principles of the subject shewn by the mathe
maticians of this time is greater than would have been antici
pated from their knowledge of pure mathematics.
With this exception we may say that the principles of
analytical geometry and of the infinitesimal calculus were
needed before there was likely to be much further progress.
The former was employed by Descartes in 1637, the latter was
invented by Newton (and possibly independently by Leibnitz)
some thirty or forty years later: and their introduction may be
taken as marking the commencement of the period of modern
mathematics.
262
THIRD PERIOD.
Jtflatfcnnatfcs.
This period begins with the invention of analytical geometry
and the infinitesimal calculus. The mathematics is far more
complex than that produced in either of the preceding periods :
but it may be generally described as characterized by the de
velopment of analysis, and its application to the phenomena of
nature.
263
I continue the chronological arrangement of the subject.
Chapter xv. contains the history of the forty years from 1635
to 1675, and an account of the mathematical discoveries of
Descartes, Cavalieri, Pascal, Wallis, Fermat, and Huygens.
Chapter xvi. is given up to a discussion of Newton s researches.
Chapter xvn. contains an account of the works of Leibnitz and
his followers during the first half of the eighteenth century
(including D Alembert), and also of the contemporary English
school to the death of Maclaurin. The works of Euler, La-
grange, Laplace, and their contemporaries form the subject-
matter of chapter xvm. Lastly in chapter xix. I have added
some notes on a few of the mathematicians of recent times ;
but I exclude all detailed reference to living writers, and
partly because of this, partly for other reasons there given, the
account of contemporary mathematics does not profess to be
exhaustive or complete. I may remind the reader that the
lives of the mathematicians considered at the end of one
chapter generally overlap the lives of some of those who are
mentioned in the next chapter ; and that the close of a chapter
is not a sign of any abrupt change in the history of the
subject, though it generally indicates a point when new
methods of analysis or new subjects were coming into promi
nence.
264
CHAPTER XIV.
FEATURES OF MODERN MATHEMATICS.
THE division between this period and that treated in the
last six chapters is by no means so well defined as that which
separates the history of Greek mathematics from the mathe
matics of the middle agea. The methods of analysis used in
the seventeenth century and the kind of problems attacked
changed but gradually; and the mathematicians at the begin
ning of this period were in immediate relations with those at
the end of that last considered. For this reason some writers
have divided the history of mathematics into two parts only,
treating the schoolmen as the lineal successors of the Greek
mathematicians, and dating the creation of modern mathe
matics from the introduction of the Arab text-books into
Europe. The division I have given is I think more con
venient, for the introduction of analytical geometry and of
the calculus completely revolutionized the development of
the subject, and it therefore seems preferable to take their in
vention as marking the commencement of modern mathematics.
The time that has elapsed since these methods were in
vented has been a period of incessant intellectual activity in
all departments of knowledge, and the progress made in mathe
matics has been immense. The greatly extended range of
knowledge and the rapid intercommunication of ideas due to
printing increase the difficulties of a historian ; while the mass
of materials which has to be mastered, the absence of per-
FEATURES OF MODERN MATHEMATICS. 265
spective, and even the echoes of old controversies combine to
make it very difficult to give a clear and just account of the
development of the subject. As however the leading facts
are generally known, and the works published during this
time are accessible to any student, I may deal more concisely
with the lives and writings of modern mathematicians than
with those of their predecessors, and confine myself more
strictly than before to those who have materially affected the
progress of the subject.
Roughly speaking we may say that five distinct stages in
the history of this period can be discerned.
First of all there is the invention of analytical geometry by
Descartes in 1637; and almost at the same time the intro
duction of the method of indivisibles, by the use of which
areas, volumes, and the positions of centres of mass can be
determined by summation in a manner analogous to that
effected now-a-days by the aid of the integral calculus. The
method of indivisibles was soon superseded by the integral
calculus. Analytical geometry however maintains its position
as part of the necessary training of every mathematician, and
is incomparably more potent than the geometry of the ancients
for all purposes of research. The latter is still no doubt
an admirable intellectual training, and it frequently affords
an elegant demonstration of some proposition the truth of
which is already known, but it requires a special procedure
for every particular problem attacked. The former on the
other hand lays down a few simple rules by which any
property can be at once proved or disproved.
In the second place, we have the invention of the fluxional
or differential calculus about 1666 (and possibly an indepen
dent invention of it in 1674). Wherever a quantity changes
according to some continuous law (and most things in nature
do so change) the differential calculus enables us to measure its
rate of increase or decrease; and, from its rate of increase or
decrease, the integral calculus enables us to find the original
quantity. Formerly every separate function of x such as
266 FEATURES OF MODERN MATHEMATICS.
(I+x) n , log(l+#), sin a;, tan~ 1 ,^, &c., could be expanded in
ascending powers of x only by means of such special procedure
as was suitable for that particular problem ; but, by the aid of
the calculus, the expansion of any function of x in ascending
powers of x is in general reducible to one rule which covers
all cases alike. So again the theory of maxima and minima,
the determination of the lengths of curves, and the areas en
closed by them, the determination of surfaces, of volumes, and
of centres of mass, and many other problems are each reducible
to a single rule. The theories of differential equations, of the
calculus of variations, of finite differences, &c. are the develop
ments of the ideas of the calculus.
These two subjects analytical geometry and the calculus
became the chief instruments of further progress in mathe
matics. In both of them a sort of machine was constructed :
to solve a problem, it was only necessary to put in the parti
cular function dealt with, or the equation of the particular
curve or surface considered, and on performing certain simple
operations the result came out. The validity of the process
was proved once for all, and it was no longer requisite to
invent some special method for every separate function, curve,
or surface.
In the third place, Huygens laid the foundation of a satis
factory treatment of dynamics, and Newton reduced it to an
exact science. The latter mathematician proceeded to apply
the new analytical methods not only to numerous problems in
the mechanics of solids and fluids on the earth but to the solar
system: the whole of mechanics terrestrial and celestial was
thus brought within the domain of mathematics. There is no
doubt that Newton used the calculus to obtain many of his re
sults, but he seems to have thought that, if his demonstrations
were established by the aid of a new science which was at that
time generally unknown, his critics (who would not understand
the fluxional calculus) would fail to realize the truth and im
portance of his discoveries. He therefore determined to give
geometrical proofs of all his results. He accordingly cast the
FEATURES OF MODERN MATHEMATICS.
Principia into a geometrical form, and thus presented it to the
world in a language which all men could then understand.
The theory of mechanics was extended and was systematized
into its modern form by Laplace and Lagrange towards the end
of the eighteenth century.
In the fourth place, we may say that during this period
the chief branches of physics have been brought within the
scope of mathematics. This extension of the domain of mathe
matics was commenced by Huygens and Newton when they
propounded their theories of light; but it was not until the
beginning of this century that sufficiently accurate observations
were made in most physical subjects to enable mathematical
reasoning to be applied to them. From the results of the
observations and experiments which have been since published,
numerous and far-reaching conclusions have been obtained by
the use of mathematics, but we now want some more simple
hypotheses from which we can deduce those laws which at
present form our starting-point. If, to take one example, we
could say in what electricity consisted, we might get some
simple laws or hypotheses from which by the aid of mathe
matics all the observed phenomena could be deduced, in the
same way as Newton deduced all the results of physical astro
nomy from the law of gravitation. All lines of research seem
moreover to indicate that there is an intimate connection be
tween the different branches of physics, e.g. between light, heat,
electricity, and magnetism. The ultimate explanation of this
and of the leading facts in physics seems to demand a study
of molecular physics; a knowledge of molecular physics in its
turn seems to require some theory as to the constitution of
matter; it would further appear that the key to the constitu
tion of matter is to be found in chemistry or chemical physics.
So the matter stands at present. Helmholtz in Germany, and
Maxwell and Lord Kelvin (Sir William Thomson) in Great
Britain, have done a great deal in applying mathematics to
physics; but the connection between the different branches of
physics, and the fundamental laws of those branches (if there
268 FEATURES OF MODERN MATHEMATICS.
be any simple ones), are riddles which are yet unsolved. This
history does not pretend to treat of problems which are now
the subject of investigation, and though mathematical physics
forms a large part of "modern mathematics" I shall not dis
cuss it in any detail.
Fifthly, this period has seen an immense extension of pure
mathematics. Much of this is the creation of comparatively
recent times, and I regard the details of it as outside the limits
of this book though in chapter xix. I have allowed myself to
mention some of the subjects discussed. The most striking
features of this extension are the developments of higher
geometry, of higher arithmetic or the theory of numbers,
of higher algebra (including the theory of forms), and of
the theory of equations, also the discussion of functions of
double and multiple periodicity, and notably the creation of
a theory of functions.
269
CHAPTER XV.
HISTORY OF MATHEMATICS FROM DESCARTES TO HUYGENS.
CIRC. 16351675.
I PROPOSE in this chapter to consider the history of mathe
matics during the forty years in the middle of the seventeenth
century. I regard Descartes, Cavalieri, Pascal, Wallis, Fermat,
and Huygens as the leading mathematicians of this time.
I shall treat them in that order, and I shall conclude with
a brief list of the more eminent remaining mathematicians
of the same date.
I have already stated that the mathematicians of this
period and the remark applies more particularly to Descartes,
Pascal, and Fermat were largely influenced by the teaching
of Kepler and Desargues, and I would repeat again that I
regard these latter and Galileo as forming a connecting link
between the writers of the renaissance and those of modern
times. I should also add that the mathematicians considered
in this chapter were contemporaries, and, although I have tried
to place them roughly in such an order that their chief works
shall come in a chronological arrangement, it is essential to
remember that they were in relation one with the other, and
in general were acquainted with one another s researches as
soon as these were published.
270 MATHEMATICS FROM DESCARTES TO HUYGENS.
Descartes*. Subject to the above remarks we may con
sider Descartes as the first of the modern school of mathe
matics. Rene Descartes was born near Tours on March 31,
1596, and died at Stockholm on Feb. 11, 1650 : he was thus a
contemporary of Galileo and Desargues. His father, who as
the name implies was of a good family, was accustomed to
spend half the year at Henries when the local parliament
in which he held a commission as councillor was in session,
and the rest of the time on his family estate of les Cartes
at la Haye. Rene, the second of a family of two sons and one
daughter, was sent at the age of eight years to the Jesuit
School at la Fleche, and of the admirable discipline and
education there given he speaks most highly. On account
of his delicate health he was permitted to lie in bed till late in
the mornings ; this was a custom which he always followed, and
when he visited Pascal in 1647 he told him that the only way
to do good work in mathematics and to preserve his health was
never to allow anyone to make him get up in the morning
before he felt inclined to do so : an opinion which I chronicle
for the benefit of any schoolboy into whose hands this work
may fall.
On leaving school in 1612 Descartes went to Paris to be
introduced to the world of fashion. Here through the medium
of the Jesuits he made the acquaintance of Mydorge and
renewed his schoolboy friendship with Father Mersenne, and
together with them he devoted the two years of 1615 and
1616 to the study of mathematics. At that time a man of
position usually entered either the army or the church; Descartes
chose the former profession, and in 1617 joined the army of
* See La vie de Descartes by A. Baillet, 2 vols., Paris, 1691, which
is summarized in vol. i. of K. Fischer s Geschichte der neuern Philosophic,
Munich, 1878. A tolerably complete account of his mathematical and
physical investigations is given in Ersch and Gruber s Encyclopadie,
and is the authority for most of the statements here contained. The
most complete edition of his works is that by Victor Cousin in 11 vols.
Paris, 1824 6. Some minor papers subsequently discovered were printed
by F. de Careil, Paris, 1859.
DESCARTES. 271
Prince Maurice of Orange then at Breda. Walking through the
streets he saw a placard in Dutch which excited his curiosity,
and stopping the first passer asked him to translate it into either
French or Latin. The stranger, who happened to be Isaac
Beeckman, the head of the Dutch College at Dort, offered to do
so if Descartes would answer it : the placard being in fact a
challenge to all the world to solve a geometrical problem there
given. Descartes worked it out within a few hours, and a warm
friendship between him and Beeckman was the result. This
unexpected test of his mathematical attainments made the
uncongenial life of the army distasteful to him, but under
family influence and tradition he remained a soldier, and was
persuaded at the commencement of the thirty years war to
volunteer under Count de Bucquoy in the army of Bavaria.
He continued all this time to occupy his leisure with mathe
matical studies, and was accustomed to date the first ideas of
his new philosophy and of his analytical geometry from three
dreams which he experienced on the night of Nov. 10, 1619, at
Neuberg when campaigning on the Danube. He regarded
this as the critical day of his life, and one which determined
his whole future.
He resigned his commission in the spring of 1621, and
spent the next five years in travel, during most of which time
he continued to study pure mathematics. In 1626 we find
him settled at Paris "a little well-built figure, modestly clad
in green taffety, and only wearing sword and feather in token of
his quality as a gentleman." During the first two years there
he interested himself in general society and spent his leisure in
the construction of optical instruments ; but these pursuits were
merely the relaxations of one who failed to find in philosophy
that theory of the universe which he was convinced finally
awaited him. In 1628 Cardinal de Berulle, the founder of the
Oratorians, met Descartes, and was so much impressed by his
conversation that he urged on him the duty of devoting his
life to the examination of truth. Descartes agreed, and the
better to secure himself from interruption moved to Holland
272 MATHEMATICS FROM DESCARTES TO HUYGENS.
then at the height of its power. There for twenty years he
lived, giving up all his time to philosophy and mathematics.
Science, he says, may be compared to a tree, metaphysics is the
root, physics is the trunk, and the three chief branches are me
chanics, medicine, and moi*als, these forming the three applica
tions of our knowledge, namely, to the external world, to the
human body, and to the conduct of life : and with these sub
jects alone his writings are concerned. He spent the first four
years, 1629 to 1633, of his stay in Holland in writing Le
Monde which embodies an attempt to give a physical theory
of the universe; but finding that its publication was likely to
bring on him the hostility of the church, and having no desire
to pose as a martyr, he abandoned it : the incomplete manu
script was published in 1664. He then devoted himself to
composing a treatise on universal science ; this was published
at Ley den in 1637 under the title Discours de la methode
pour bien conduire sa raison et chercher la verite dans les
sciences, and was accompanied with three appendices (which
possibly were not issued till 1638) entitled La Dioptrique,
Les Meteores, and La Geometric: it is from the last of these
that the invention of analytical geometry dates. In 1641 he
published a work called Meditationes in which he explained
at some length his views of philosophy as sketched out in
the Discours. In 1644 he issued the Principia Philosophiae,
the greater part of which was devoted to physical science,
especially the laws of motion and the theory of vortices. In
1647 he received a pension from the French court in honour
of his discoveries. He went to Sweden on the invitation of
the Queen in 1649, and died a few months later of inflam
mation of the lungs.
In appearance, Descartes was a small man with large head,
projecting brow, prominent nose, and black hair coming down
to his eyebrows. His voice was feeble. Considering the range
of his studies he was by no means widely read, and he de
spised both learning and art unless something tangible could
be extracted therefrom. In disposition he was cold and selfish.
DESCARTES. 273
He never married and left no descendants, though he had one
illegitimate daughter who died young.
As to his philosophical theories, it will be sufficient to say
that he discussed the same problems which have been debated
for the last two thousand years. It is hardly necessary to say
that the problems themselves are of great interest, but from
the nature of the case no solution ever offered is capable either
of proof or of disproof, and whenever a philosopher like
Descartes believes that he has at last finally settled a question
it has been easy for his successors to point out the fallacy in
his assumptions. All that can be effected is to make one
explanation somewhat more probable than another. I have
read somewhere that philosophy has always been chiefly en
gaged with the inter-relations of God, Nature, and Man. The
earliest philosophers were Greeks who occupied themselves
mainly with the relations between God and Nature, and dealt
with Man separately. The Christian Church was so absorbed
in the relation of God to Man as to entirely neglect Nature.
Finally modern philosophers concern themselves chiefly with
the relations between Man and Nature. Whether this is a
correct historical generalization of the views which have been
successively prevalent I do not care to discuss here, but the
statement as to the scope of modern philosophy marks the
limitations of Descartes s writings, and these may be taken
as the commencement of the modern school.
Descartes s chief contributions to mathematics were his
analytical geometry and his theory of vortices, and it is on
his researches in connection with the former of these subjects
that his reputation rests.
Analytical geometry does not consist merely (as is some
times loosely said) in the application of algebra to geometry :
that had been done by Archimedes and many others, and had
become the usual method of procedure in the works of the
mathematicians of the sixteenth century. The great advance
made by Descartes was that he saw that a point in a plane
could be completely determined if its distances, say x and y,
B. 18
274 MATHEMATICS FROM DESCARTES TO HUYGENS.
from two fixed lines drawn at right angles in the plane were
given, with the convention familiar to us as to the interpre
tation of positive and negative values ; and that though an
equation f (x, 2/) = was indeterminate and could be satisfied by
an infinite number of values of x and y, yet these values of x and
y determined the co-ordinates of a number of points which form
a curve of which the equation f (x, y) = expresses some geo
metrical property, that is, a property true of the curve at every
point on it. Descartes asserted that a point in space could be
similarly determined by three coordinates, but he confined his
attention to plane curves.
It was at once seen by Descartes and his successors that
in order to investigate the properties of a curve it was sufficient
to select any characteristic geometrical property as a definition,
and to express it by means of an equation between the (current)
coordinates of any point on the curve, that is, to translate the
definition into the language of analytical geometry. The equa
tion so obtained contains implicitly every property of the
curve, and any particular property can be deduced from it
by ordinary algebra without troubling about the geometry of
the figure. The points in which two curves intersect can
be determined by finding the roots common to their two equa
tions. I need not go further into details, for nearly every
one to whom the above is intelligible will have read analytical
geometry, and be able to appreciate the value of its invention.
Descartes s Geometrie is divided into three books : the
first two of these treat of analytical geometry, and the third in
cludes an analysis of the algebra then current. It is some
what difficult to follow the reasoning, but the obscurity was
intentional and due to the jealousy of Descartes. " Je n ai
rien omis," says he, " qu a dessein...j avois prevu que cer-
taines gens qui se vantent de S9avoir tout n auroient pas
manque de dire que je n avois rien ecrit qu ils n eussent sgu
auparavant, si je me fusse rendu assez intelligible pour eux."
The first book commences with an explanation of the prin
ciples of analytical geometry, and contains a discussion of a
DESCARTES. 275
certain problem which had been propounded by Pappus in the
seventh book of his Swaywy?; and of which some particular
cases had been considered by Euclid and Apollonius. The
general theorem had baffled previous geometricians, and it was
in the attempt to solve it that Descartes was led to the inven
tion of analytical geometry. The full enunciation of the
problem is rather involved, but the most important case is to
find the locus of a point such that the product of the perpen
diculars on m given straight lines shall be in a constant ratio to
the product of the perpendiculars on n other given straight lines.
The ancients had solved this geometrically for the case ?n,= l,
n 1 , and the case m = 1 , n = 2. Pappus had further stated
that, if m = n 2, the locus was a conic, but he gave no proof ;
Descartes also failed to prove this by pure geometry, but he
shewed that the curve was represented by an equation of the
second degree, that is, was a conic ; subsequently Newton gave
an elegant solution of the problem by pure geometry.
In the second book Descartes divides curves into two
classes ; namely, geometrical and mechanical curves. He de
fines geometrical curves as those which can be generated by
the intersection of two lines each moving parallel to one co
ordinate axis with "commensurable" velocities, by which he
meant that dyjdx was an algebraical function, as for example
is the case in the ellipse and the cissoid. He calls a curve
mechanical when the ratio of the velocities of these lines is
"incommensurable," by which he meant that dyjdx was a
transcendental function, as for example is the case in the
cycloid and the quadratrix. Descartes confined his discussion
to algebraical curves, and did not treat of the theory of me
chanical curves. The classification into algebraical and transcen
dental curves now usual is due to Newton (see below, p. 346).
Descartes also paid particular attention to the theory of
the tangents to curves as perhaps might be inferred from
his system of classification just alluded to. The then current
definition of a tangent at a point was a straight line through
the point such that between it and the curve no other straight
182
276 MATHEMATICS FROM DESCARTES TO HUYGENS.
line could be drawn, i.e. the straight line of closest contact.
Descartes proposed to substitute for this that the tangent was
the limiting position of the secant ; Fermat, and at a later
date Maclaurin and Lagrange, adopted this definition. Barrow,
followed by Newton and Leibnitz, considered a curve as the
limit of an inscribed polygon when the sides become indefinitely
small, and stated that a side of the polygon when produced
became in the limit a tangent to the curve. Roberval on the
other hand defined a tangent at a point as the direction of
motion at that instant of a point which was describing the curve.
The results are the same whichever definition is selected, but
the controversy as to which definition was the correct one was
none the less lively. Descartes illustrated his theory by giving
the general rule for drawing tangents and normals to a roulette.
The method used by Descartes to find the tangent or
normal at any point of a given curve was substantially as
follows. He determined the centre and radius of a circle
which should cut the curve in two consecutive points there.
The tangent to the circle at that point will be the required
tangent to the curve. In modern text-books it is usual to
express the condition that two of the points in which a straight
line (such as y = mx + c) cuts the curve shall coincide with the
given point : this enables us to determine m and c, and thus
the equation of the tangent there is determined. Descartes
however did not venture to do this, but selecting a circle as
the simplest curve and one to which he knew how to draw a
tangent, he so fixed his circle as to make it touch the given
curve at the point in question and thus reduced the problem
to drawing a tangent to a circle. I should note in passing that
he only applied this method to curves which are symmetrical
about an axis, and he took the centre of the circle on the axis.
Much of the reasoning in these two books is not easy to
follow ; but a Latin translation of them, with explanatory
notes, was prepared by F. de Beaune, and an edition of this
with a commentary by F. van Schooten was issued in 1659,
and had a wide circulation.
DESCARTES. 277
The third book of the Geometrie contains an analysis of
the algebra then current, and it has affected the language
of the subject by fixing the custom of employing the letters at
the beginning of the alphabet to denote known quantities, and
those at the end of the alphabet to denote unknown quantities*.
Descartes further introduced the system of indices now in use,
but I would here remind the reader that the suggestion had
been made by previous writers, though it had not been generally
adopted; but very likely it was original on the part of Descartes.
I think also that Descartes was the first to realize that his
letters might represent any quantities, positive or negative, and
that it was sufficient to prove a proposition for one general case
(compare the old procedure as illustrated above on p. 163). In
this book he made use of the rule for determining a limit to
the number_of_positive and of negative roots of an algebraical
equation, which is still known by his name ; and introduced the
method of indeterminate coefficients for the solution of equations.
He believed that he had given a method by which algebraical
equations of any order could be solved, but in this he was mis
taken. He made use of the method of indeterminate coefficients.
Of the two other appendices to the Discours one was
devoted to optics. The chief interest of this consists in the
statement given of the law of refraction. This appears to have
been taken from SnelPs work (see above, p. 256), but not only
is there no acknowledgment of the source from which it was
obtained, but it is enunciated in such a way as to lead a
careless reader to suppose that it is due to the researches of
Descartes. Descartes would seem to have repeated Snell s
experiments when in Paris in 1626 or 1627, and it is possible
that he subsequently forgot how much he owed to the earlier
investigations of Snell. A large part of the optics is devoted
to determining the best shape for the lenses of a telescope, but
the mechanical difficulties in grinding a surface of glass to a
* On the origin of the custom of using x to represent an unknown
example, see a note by G. Enestrom in the Bibliotheca Mathematica,
1885, p. 43.
278 MATHEMATICS FROM DESCARTES TO HUYGENS.
required form are so great as to render these investigations of
little practical use. Descartes seems to have been doubtful
whether to regard the rays of light as proceeding from the eye
and so to speak touching the object, as the Greeks had done, or
as proceeding from the object, and so affecting the eye ; but,
since he considered the velocity of light to be infinite, he did
not deem the point particularly important.
The other appendix, on meteors, contains an explanation
of numerous atmospheric phenomena, including the rainbow ;
Descartes was unacquainted with the unequal refrangibility
of rays of light of different colours, and the explanation of
the latter is necessarily incomplete.
Descartes s physical theory of the universe, embodying most
of the results contained in his earlier and unpublished Le
Monde, was given in his Principia, 1644, and rests on a ineta-
Rhysical basjs. He commences with_jt discussion an_motion ;
and thenjays dowrTten law s ofliature, of which the first two
are almost identical with the tirst two laws of motion as
given by Newton fsee below, p. 337) ; the remaining eight
laws are inaccurate^ li!e next proceeds to discuss the nature
of matter which he regards as uniform in kind though there
are three forms of it. He assumes that the matter of the
universe must be in motion, and that the motion must result
in a number of vortices. He states that the sun is the centre
of an immense whirlpool of this matter, in which the planets
float and are swept round like straws in a whirlpool of water.
Each planet is supposed to be the centre of a secondary whirl
pool by which its satellites are carried : these secondary whirl
pools are supposed to produce variations of density in the
surrounding medium which constitute the primary whirlpool,
and so cause the planets to move in ellipses and not in circles.
All these assumptions are arbitrary and unsupported by any
investigation. It is not difficult to prove that on his hypotheses
the sun would be in the centre of these ellipses and not at a
focus (as Kepler had shewn was the case), and that the weight
of a body at every place on the surface of the earth except the
CAVALIERI. 279
equator would act in a direction which was not vertical ; but
it will be sufficient here to say that Newton in the second book
of his Principia, 1687, considered the theory in detail, and
shewed that its consequences are not only inconsistent with
each of Kepler s laws and with the fundamental laws of
mechanics, but are also at variance with the ten laws of nature
assumed by Descartes. Still, in spite of its crudeness and its
inherent defects, the theory of vortices marks a fresh era in
astronomy, for it was an attempt to explain the phenomena of
the whole universe by the same mechanical laws which ex
periment shews to be true on the earth.
Cavalieri*. Almost contemporaneously with the publica
tion in 1637 of Descartes s geometry, the principles of the
integral calculus, so far as they are concerned with summation,
were being worked out in Italy. This was effected by what
was called the principle of indivisibles, and was the invention
of Cavalieri. It was applied to numerous problems connected
with the quadrature of curves and surfaces, the determination
of volumes, and the positions of centres of mass to the com
plete exclusion of the tedious method of exhaustions used by
the Greeks. In principle the methods are the same, but the
notation of indivisibles is more concise and convenient. It
was in its turn superseded at the beginning of the eighteenth
century by the integral calculus, but its use will be familiar to
all mathematicians who have read any commentary on the first
section of the first book of Newton s Principia in the appli
cation of lemmas 2 and 3 to the determination of areas,
volumes, &c.
Bonaventura Cavalieri was born at Milan in 1598, and died
at Bologna on Nov. 27, 1647. He became a Jesuit at an early
age ; on the recommendation of the Order he was in 1629 made
professor, of mathematics at Bologna ; and he continued to
* Cavalieri s life has been written by P. Frisi, Milan, 1778; by F.
Predari, Milan, 1843 ; by Gabrio Piola, Milan, 1844 ; and by A. Favaro,
Bologna, 1888. An analysis of his works is given in Marie s Histoire,
vol. iv., pp. 6990.
280 MATHEMATICS FROM DESCARTES TO HUYGENS.
occupy the chair there until his death. I have already
mentioned Cavalieri s name for the part that he took in in
troducing the use of logarithms into Italy. He was one of the
most influential mathematicians of his time, but his subsequent
reputation rests mainly on his invention of the principle of
indivisibles.
The principle of indivisibles had been used by Kepler (see
above, p. 258) in 1604 and 1615 in a somewhat crude form.
It was first stated by Cavalieri in 1629, but he did not publish
his results till 1635. In his early enunciation of the principle
in 1635 Cavalieri asserted that a line was made up of an
infinite number of points (each without magnitude), a surface
of an infinite number of lines (each without breadth), and a
volume of an infinite number of surfaces (each without thick
ness). To meet the objections of Guldinus and others the
statement was recast, and in its final form as used by the
mathematicians of the seventeenth century it was published in
Cavalieri s Exercitationes Geometricae Sex in 1647, the third of
which is devoted to a defence of the theory. These exercises
contain the first rigid demonstration of the properties of
Pappus (see above, pp. 101, 254). Cavalieri s works on the
subject were reissued with his later corrections in 1653.
The method of indivisibles is simply that any magnitude
may be divided into an infinite number of small quantities
which can be made to bear any required ratios (e.g. equality)
one to the other. The analysis given by Cavalieri is hardly
worth quoting except as being one of the first steps taken
towards the formation of an infinitesimal calculus. One
example will suffice. Suppose it be required to find the area
of a right-angled triangle. Let the base contain n points and
the other side na points, then the ordinates at the successive
points of the base will contain a, 2a, . . . , na points. Therefore
the number of points in the figure is a + 2a + . . . + na ; the
sum of which is ^n 2 a + \na. Since n is very large, we may
neglect the \na as inconsiderable compared with the %n*a, and
the area is J (na) n, that is, \ altitude x base. There is no diffi-
CAVALIERI. 281
culty in criticizing such a proof, but, although the form in which
it is presented is indefensible, the substance of it is correct.
It would be misleading to give the above as the only
specimen of the method of indivisibles, and I therefore quote
another example, taken from a later writer, which will fairly
illustrate the use of the method when modified and corrected
by the method of limits. Let it be required to find the area
bounded by the parabola A PC the tangent at A, and any
diameter DC. Complete the parallelogram A BCD. Divide
AD into n equal parts, let AM contain r of them, and let
B
MN be the (r + l)th part. Draw MP and NQ parallel to AB,
and draw PR parallel to AD. Then, when n becomes in
definitely large, the curvilinear area A PCD will be the limit of
the sum of all parallelograms like PN. Now
area PN : area BD = MP . MN : DC .AD.
But by the properties of the parabola
MP : DC = AM 2 : AD 2 = r 2 : ri\
and MN : AD = I : n.
Hence MP . MN : DC . AD = r 2 : n 3 .
Therefore area PN : area BD = r 2 : n 3 .
Therefore ultimately
area APCD : area BD= I 2 + 2 2 + ... + (n - I) 2 : n 3
= *n(n
which, in the limit, =1:3.
282 MATHEMATICS FROM DESCARTES TO HUYGENS.
It is perhaps worth noticing that Cavalieri and his suc
cessors always used the method to find the ratios of two areas,
volumes, or magnitudes of the same kind and dimensions, that
is, they never thought of an area as containing so many units
of area. The idea of comparing a magnitude with a unit of the
same kind seems to have been due to Wallis.
It is evident that in its direct form the method is appli
cable to only a few curves. Cavalieri proved that, if m be
a positive integer, then the limit, when n is infinite, of
... .
^TI is -- -- , which is equivalent to saying that
n m + 1
he found the integral to x of x m from x = to x = 1 ; he also
discussed the quadrature of the hyperbola.
Pascal*. Among the contemporaries of Descartes none
displayed greater natural genius than Pascal, but his reputa
tion rests more on what he might have done than on what
he actually effected, as during a considerable part of his life
he deemed it his duty to devote his whole time to religious
exercises.
Blaise Pascal was born at Clermont on June 19, 1623, and
died at Paris on Aug. 19, 1662. His father, a local judge at Cler
mont and himself of some scientific reputation, moved to Paris
in 1631, partly to prosecute his own scientific studies, partly
to carry on the education of his only son who had already
displayed exceptional ability. Pascal was kept at home in
order to ensure his not being overworked, and with the same
object it was directed that his education should be at first con
fined to the study of languages and should not include any
mathematics. This naturally excited the boy s curiosity, and
one day being then twelve years old he asked in what geometry
consisted. His tutor replied that it was the science of con-
* See Pascal by J. Bertrand, Paris, 1891. Pascal s life, written by
his sister Mme P6rier, was edited by A. P. Faugere, Paris, 1845, and
has formed the basis for several works. An edition of his writings was
published in 5 vols. at the Hague in 1779, second edition, Paris, 1819;
some additional pamphlets and letters were published by Lahure in
3 vols. at Paris in 1858.
PASCAL. 283
str acting exact figures and of determining the proportions
between their different parts. Pascal, stimulated no doubt by
the injunction against reading it, gave up his play-time to this
new study, and in a few weeks had discovered for himself
many properties of figures, and in particular the proposition
that the sum of the angles of a triangle is equal to two right
angles. I have read somewhere, but I cannot lay my hand on
the authority, that his proof merely consisted in turning the
angular points of a triangular piece of paper over so as to
meet in the centre of the inscribed circle : a similar demon
stration can be got by turning the angular points over so as
to meet at the foot of the perpendicular drawn from the biggest
angle to the opposite side. His father struck by this display
of ability gave him a copy of Euclid s Elements, a book which
Pascal read with avidity and soon mastered.
At the age of fourteen he was admitted to the weekly
meetings of Roberval, Mersenne, Mydorge, and other French
geometricians; from which the French Academy ultimately
sprung, being created by ordinance of Louis XIV. on Dec. 22,
16G6. At sixteen Pascal wrote an essay on conic sections;
and in 1641, at the age of eighteen, he constructed the first
arithmetical machine, an instrument which eight years later
he further improved and patented. His correspondence with
Fermat about this time shews that he was then turning his
attention to analytical geometry and physics. He repeated
Torricelli s experiments, -by which the pressure of the atmo
sphere could be estimated as a weight, and he confirmed his
theory of the cause of barometrical variations by obtaining at
the same instant readings at different altitudes on the hill of
Puy-de-D6me.
In 1G50, when in the midst of these researches, Pascal
suddenly abandoned his favourite pursuits to study religion, or
as he says in his Pensees u to contemplate the greatness and the
misery of man " ; and about the same time he persuaded the
younger of his two sisters to enter the Port Royal society.
In 1653 he had to administer his father s estate. He now
284 MATHEMATICS FROM DESCARTES TO HUYGENS.
took up his old life again, and made several experiments on the
pressure exerted by gases and liquids : it was also about this
period that he invented the arithmetical triangle, and together
with Fermat created the calculus of probabilities. He was
meditating marriage when an accident again turned the current
of his thoughts to a religious life. He was driving a four-in-
hand on Nov. 23, 1654, when the horses ran away; the two
leaders dashed over the parapet of the bridge at Neuilly, and
Pascal was only saved by the traces breaking. Always some
what of a mystic, he considered this a special summons to
abandon the world. He wrote an account of the accident on
a small piece of parchment, which for the rest of his life he
wore next to his heart to perpetually remind him of his cove
nant ; and shortly moved to Port Royal where he continued
to live until his death in 1662. Always delicate, he had
injured his health by his incessant study ; from the age of
seventeen or eighteen he suffered from insomnia and acute
dyspepsia, and at the time of his death was completely worn
out.
His famous Provincial Letters directed against the Jesuits,
and his Pensees, were written towards the close of his life, and
are the first example of that finished form which is characte
ristic of the best French literature. The only mathematical
work that he produced after retiring to Port Royal was the
essay on the cycloid in 1658. He was suffering from sleepless
ness and tooth -ache when the idea occurred to him, and to his
surprise his teeth immediately ceased to ache. Regarding this
as a divine intimation to proceed with the problem, he worked
incessantly for eight days at it, and completed a tolerably full
account of the geometry of the cycloid.
I now proceed to consider his mathematical works in
rather greater detail.
His early essay on the geometry of conies, written in 1639
but not published till 1779, seems to have been founded on
the teaching of Desargues. Two of the results are important
as well as interesting. The first of these is the theorem known
PASCAL.
285
now as "Pascal s theorem," namely, that if a hexagon be
inscribed in a conic, the points of intersection of the opposite
sides will lie in a straight line. The second, which is really due
to Desargues, is that if a quadrilateral be inscribed in a conic,
and a straight line be drawn cutting the sides taken in order
in the points A, B, C, and />, and the conic in P and Q, then
PA . PC : PB . PD = QA . QC : QB . QD.
Pascal s Arithmetical triangle was written in 1653, but
not printed till 1665. The triangle is constructed as in the
11111
12345
1 3 6 /10 15
10 20 35
5 15 35 70
annexed figure, each horizontal line being formed from the one
above it by making every number in it equal to the sum of those
above and to the left of it in the row immediately above ; e.g. in
the 4th line 20 = 1 + 3 + 6 + 10. Then Pascal s arithmetical
triangle (to any required order) is got by drawing a diagonal
downwards from right to left as in the figure. These num
bers are what are now called jiyurate numbers. Those in
the first line are called numbers of the first order; those
in the second line, natural numbers or numbers of the second
order; those in the third lino numbers of the third order,
and so on. It is easily -li< \vn that the >//th number in the ?tth
row is (m + n - 2) ! / (m - 1) ! (n -1)1
The numbers in any diagonal give the coefficients of the
expansion of a binomial : for example, the figures in the
286 MATHEMATICS FROM DESCARTES TO HUYGENS.
fifth diagonal namely, 1, 4, 6, 4, 1, are the coefficients in the
expansion (a + b) 4 . Pascal used the triangle partly for this
purpose and partly to find the numbers of combinations of .
ra things taken n at a time, which he stated (correctly) to
be (n + 1) (n + 2) (n + 3) . . . m / (ra - n) !
Perhaps as a mathematician Pascal is best known in
connection with his correspondence with Fermat in 1654 in
which he laid down the principles of the theory of probabilities.
This correspondence arose from a problem proposed by a
gamester, the Chevalier de Mere, to Pascal who communicated
it to Fermat. The problem was this. Two players of equal
skill want to leave the table before finishing their game. Their
scores and the number of points which constitute the game
being given, it is desired to find in what proportion should they
divide the stakes. Pascal and Fermat agreed on the answer,
but gave different proofs. The following is a translation of
Pascal s solution. That of Fermat is given later.
The following is my method for determining the share of each player,
when, for example, two players play a game of three points and each
player has staked 32 pistoles.
Suppose that the first player has gained two points, and the second
player one point ; they have now to play for a point on this condition,
that, if the first player gain, he takes all the money which is at stake,
namely, 64 pistoles ; while, if the second player gain, each player has two
points, so that they are on terms of equality, and, if they leave off play
ing, each ought to take 32 pistoles. Thus, if the first player gain, then
64 pistoles helong to him, and, if he lose, then 32 pistoles belong to him.
If therefore the players do not wish to play this game, but to separate
without playing it, the first player would say to the second "I am certain
of 32 pistoles even if I lose this game, and as for the other 32 pistoles
perhaps I shall have them and perhaps you will have them ; the chances
are equal. Let us then divide these 32 pistoles equally, and give me also
the 32 pistoles of which I am certain." Thus the first player will have
48 pistoles and the second 16 pistoles.
Next, suppose that the first player has gained two points and the
second player none, and that they are about to play for a point; the
condition then is that, if the first player gain this point, he secures the
game and takes the 64 pistoles, and, if the second player gain this point,
then the players will be in the situation already examined, in which the
first player is entitled to 48 pistoles and the second to 16 pistoles.
PASCAL. 287
Thus, if they do not wish to play, the first player would say to the second
"If I gain the point, I gain 64 pistoles; if I lose it, I am entitled to
48 pistoles. Give me then the 48 pistoles of which I am certain, and
divide the other 16 equally, since our chances of gaining the point are
equal." Thus the first player will have 56 pistoles and the second player
8 pistoles.
Finally, suppose that the first player has gained one point and the
second player none. If they proceed to play for a point, the condition is
that, if the first player gain it, the players will be in the situation first
examined, in which the first player is entitled to 56 pistoles ; if the first
player lose the point, each player has then a point, and each is entitled
to 32 pistoles. Thus, if they do not wish to play, the first player would
say to the second * Give me the 32 pistoles of which I am certain and
divide the remainder of the 56 pistoles equally, that is, divide 24 pistoles
equally." Thus the first player will have the sum of 32 and 12 pistoles,
that is, 44 pistoles, and consequently the second will have 20 pistoles.
Pascal proceeds next to consider the similar problem when
the game is won by whoever first obtains m -f n points, and one
player has m while the other has n points. The answer is ob
tained by using the arithmetical triangle. The general solution
(in which the skill of the players is unequal) is given in many
modern text-books on algebra and agrees with Pascal s result,
though of course the notation of the latter is different and
less convenient.
Pascal made a most illegitimate use of the new theory in
the seventh chapter of his Pensees. He practically puts his
argument that, as the value of eternal happiness must be infi
nite, then, even if the probability of a religious life ensuring
eternal happiness be very small, still the expectation (which is
measured by the product of the two) must be of sufficient
magnitude to make it worth while to be religious. The argu
ment, if worth anything, would apply equally to any religion
which promised eternal happiness to those who accepted its
doctrines. If any conclusion may be drawn from the statement
it is the undesirability of applying mathematics to questions of
morality of which some of the data are necessarily outside the
range of an exact science. It is only fair to add that no one
had more contempt than Pascal for those who changed their
288 MATHEMATICS FROM DESCARTES TO HUYGENS.
opinions according to the prospect of material benefit, and this
isolated passage is at variance with the spirit of his writings.
The last mathematical work of Pascal was that on the
cycloid in 1658. The cycloid is the curve traced out by a
point on the circumference of a circular hoop which rolls along
a straight line. Galileo, in 1630, nad been the first to call
attention to this curve, and had suggested that the arches of
bridges should be built in the form of it: it is a graceful
curve, but the only bridge with cycloidal arches of which
I have heard is the one built by Essex in the grounds of
Trinity College, Cambridge. Four years later, in 1634,
Roberval found the area of the cycloid ; Descartes thought
little of this solution and defied him to find its tangents, the
same challenge being also sent to Fermat who at once solved
the problem. Several questions connected with the curve, and
with the surface and volume generated by its revolution about
its axis, base, or the tangent at its vertex were then proposed
by various mathematicians. These and some analogous ques
tions, as well as the positions of the centres of the mass of the
solids formed, were solved by Pascal in 1658, and the results
were issued as a challenge to the world. Wallis succeeded in
solving all the questions except those connected with the centre
of mass. Pascal s own solutions were effected by the method
of indivisibles, and are similar to those which a modern
mathematician would give by the aid of the integral calculus.
He obtained by summation what are equivalent to the follow
ing integrals
/sin < dfa /sin 2 < c/<, /</> siii </> d<j>,
one limit being either or JTT. He also investigated the
geometry of the Archimedean spiral, These researches ac
cording to D Alembert form a connecting link between the geo
metry of Archimedes and the infinitesimal calculus of Newton.
Wallis*. John Wallis was born at Ashford on Nov. 22,
* See my History of the Study of Mathematics at Cambridge, pp. 41
46. An edition of Wallis s mathematical works was published in three
volumes at Oxford, 169398.
WALLIS. 289
1616, and died at Oxford on Oct. 28, 1703. When fifteen
years old he happened to see a book of arithmetic in the hands
of his brother; struck with curiosity at the odd signs and
symbols in it he borrowed the book, and in a fortnight had
mastered the subject. It was intended that he should be a
doctor, and he was sent to Emmanuel College, Cambridge.
While there he kept an " act " on the doctrine of the circulation
of the blood this is said to have been the first occasion in
Europe on which this theory was publicly maintained in a
disputation. His interests however centred on mathematics.
He was elected to a fellowship at Queens College, Cam
bridge, and subsequently took orders, but on the whole
adhered to the Puritan party to whom he rendered great
assistance in deciphering the royalist despatches. He however
joined the moderate Presbyterians in signing the remonstrance
against the execution of Charles I., by which he incurred the
lasting hostility of the Independents. In spite of their oppo
sition, he was appointed in 1649 to the Savilian chair of
geometry at Oxford, where he lived until his death on Oct. 28,
1703. Besides his mathematical works he wrote on theology,
logic, and philosophy ; and was the first to devise a system for
teaching deaf-mutes. I confine myself to a few notes on his
more important mathematical writings. They are notable partly
for the introduction of the use of infinite series as an ordinary
part of analysis, and partly for the fact that they revealed and
explained to all students the principles of those new methods
which distinguish modern from classical mathematics.
The most important of Wallis s works was his Amthmetica
Infinitorum, which was published in 1656. In this treatise
the methods of analysis of Descartes and Cavalieri were
systematized and greatly extended, but their logical exposition
is open to criticism. It at once became the standard book
on the subject, and is constantly referred to by subsequent
writers. It is prefaced by a short tract on conic sections
which was subsequently expanded into a separate treatise.
He commences by proving the law of indices ; shews that
B. 19
290 MATHEMATICS FROM DESCARTES TO HUYGENS.
05, x~\ x~ 2 ... represent 1, l/x, l/x 2 ... ; that x* represents the
square root of x, that x* represents the cube root of x 2 , and
generally that x~ n represents the reciprocal of x n and that
x represents the qth root of x p .
Leaving the numerous algebraical applications of this dis
covery he next proceeds to find, by the method of indivisibles,
the area enclosed between the curve y = x m , the axis of x, and
any ordinate x=-h m } and he proves that the ratio of this area
to that of the parallelogram on the same base and of the
same altitude is equal to the ratio 1 : m + 1 . He apparently
assumed that the same result would be true also for the
curve y = ax m , where a is any constant, and m any number
positive or negative ; but he only discusses the case of the
parabola in which m = 2, and that of the hyperbola in which
m 1 : in the latter case his interpretation of the result is
incorrect. He then shews that similar results might be
written down for any curve of the form y = 2<ax m ; and hence
that, if the ordinate y of a curve can be expanded in powers
of the abscissa x, its quadrature can be determined : thus he
said that, if the equation of a curve were y x + x 1 + x 2 + . . . ,
its area would be x + ^x 2 + ^x 3 + ... . He then applies this
to the quadrature of the curves y (x- x 2 ) , y (x a? 2 ) 1 ,
y (x x 2 ) 2 , y = (x x 2 ) 3 , &c. taken between the limits x and
x 1 ; and shews that the areas are respectively 1, ^, ^, T ^,
i
&c. He next considers curves of the form y = x m and estab
lishes the theorem that the area bounded by the curve, the axis
of x, and the ordinate x = 1, is to the area of the rectangle on
the same base and of the same altitude as m : m + 1. This is
C ~
equivalent to finding the value of / x m dx. He illustrates
this by the parabola in which m 2. He states, but does not
prove, the corresponding result for a curve of the form y x p
This work contains also one of the earliest investigations of
the formation and properties of continued fractions, a dis-
WALLIS. 291
cussion that was suggested by Brouncker s use of these fractions
(see below, p. 314).
Wallis shewed considerable ingenuity in reducing the equa
tions of curves to the forms given above, but, as he was
unacquainted with the binomial theorem, he could not effect
the quadrature of the circle, whose equation is y = (x-x*)^>
since he was unable to expand this in powers of x. He laid
down however the principle of interpolation. Thus, as the ordi-
nate of the circle y = (x-x 2 )* is the geometrical mean between
the ordinates of the curves y (x x 2 ) and y (x a; 2 ) 1 , it might
be supposed that, as an approximation, the area of the semi
circle / (x - x 2 ) dx, which is |-TT, might be taken as the geometri-
J o
cal mean between the values of I (x-x 2 )dx and I (x x*) l dx,
Jo Jo
that is, 1 and J ; this is equivalent to taking 4 J^ or 3 *26 . . .
as the value of TT. But, Wallis argued, we have in fact a
series 1, , -$, T |^, ... , and therefore the term interpolated
between 1 and ^ ought to be so chosen as to obey the law
of this series. This, by an elaborate method, which I need
not describe in detail, leads to a value for the interpolated
term which is equivalent to taking
2.2.4.4.6.6.8.8..
=2
1.3.3.5.5.7.7.9
The subsequent mathematicians of the seventeenth century
constantly used interpolation to obtain results which we should
attempt to obtain by direct analysis.
A few years later, in 1659, Wallis published a tract con
taining the solution of the problems on the cycloid which had
been proposed by Pascal (see above, p. 288). In this he
incidentally explained how the principles laid down in his
Arithmetics Infinitorum could be used for the rectification of
algebraic curves ; and gave a solution of the problem to rectify
the semi-cubical parabola x 3 = ay*, which had been discovered
iu 1657 by his pupil William Neil. This was the first case
192
292 MATHEMATICS FROM DESCARTES TO HUYGENS.
in which the length of a curved line was determined by
mathematics, and since all attempts to rectify the ellipse
and hyperbola had been (necessarily) ineffectual, it had
been previously supposed that no curves could be rectified,
. as indeed Descartes had definitely asserted to be the case.
The cycloid was the second curve rectified ; this was done by
Wren in 1658. Early in 1658 a similar discovery, independ
ent of that of Neil, was made by van Heuraet*, and this was
published by van Schooten in his edition of Descartes s Geometria
in 1659. Van Heuraet s method is as follows. He supposes
the curve to be referred to rectangular axes ; if this be so,
and if (x, ?/) be the coordinates of any point on it, and n the
length of the normal, and if another point whose coordinates
are (x, rj) be taken such that f]\li n\y^ where h is a con
stant ; then, if ds be the element of the length of the required
curve, we have by similar triangles ds : dx = n\y. Therefore
hds = rjdx. Hence, if the area of the locus of the point (x, 77)
can be found, the first curve can be rectified. In this way
van Heuraet effected the rectification of the curve y 3 = ax 2 ;
and added that the rectification of the parabola y 2 = ax is
impossible since it requires the quadrature of the hyperbola.
The solutions given by Neil and Wallis are somewhat similar
to that given by van Heuraet, but no general rule is enunciated,
and the analysis is clumsy. A third method was suggested
by Fermat in 1660, but it is both inelegant and laborious.
In 1665 Wallis published the first systematic treatise on
analytical conic sections. I have already mentioned that the
Geometric of Descartes is both difficult and obscure, and to
many of his contemporaries, to whom the method was new, it
must have been incomprehensible. Wallis made the method
intelligible to all mathematicians. This is the earliest book in
which these curves are considered and defined as curves of the
second degree and not as sections of a cone on a circular base.
* On van Heuraet, see the Bibliotheca Mathematica, 1887, vol. i.,
pp. 7680.
WALLLS. FERMAT. 293
The theory of the collision of bodies was propounded by
the Royal Society in 1668 for the consideration of mathe
maticians. Wallis, Wren, and Huygens sent correct and
similar solutions, all depending on what is now called the
conservation of momentum ; but, while Wren and Huygens
confined their theory to perfectly elastic bodies, Wallis con
sidered also imperfectly elastic bodies. This was followed in
1669 by a work on statics (centres of gravity), and in 1670 by
one on dynamics : these provide a convenient synopsis of what
was then known on the subject.
In 1685 Wallis published an Algebra, preceded by a his
torical account of the development of the subject, which
contains a great deal of valuable information. The second
edition, issued in 1693 and forming the second volume of his
Opera, is considerably enlarged. This algebra is noteworthy
as containing the first systematic use of formulae. A given
magnitude is here represented by the numerical ratio which
it bears to the unit of the same kind of magnitude: thus, when
Wallis wants to compare two lengths he regards each as con
taining so many units of length. This perhaps will be made
clearer if I say that the relation between the space described
in any time by a particle moving with a uniform velocity would
be denoted by Wallis by the formula s = vt, where s is the
number representing the ratio of the space described to the
unit of length ; while previous writers would have denoted the
same relation by stating what is equivalent to the proposition
*i :S 2 = V J\ : V 2*2 : ( see e -9- Newton s Principia, bk. I. sect. I.
lemma 10 or 11). It is curious to note that Wallis rejected
as absurd the now usual idea of a negative number as being
less than nothing, but accepted the view that it is something
greater than infinity. The latter opinion may be right and
consistent with the former, but it is hardly a more simple one.
Fennat. While Descartes was laying the foundations of
analytical geometry, the same subject was occupying the
attention of another and hardly less distinguished Frenchman.
This was Fermat. Pierre de Fermat, who was born near
294 MATHEMATICS FROM DESCARTES TO HUYGENS.
Montauban in 1601, and died at Castres on Jan. 12, 1665,
was the son of a leather-merchant; he was educated at
home; in 1631 he obtained the post of councillor for the local
parliament at Toulouse, and he discharged the duties or the
office with scrupulous accuracy and fidelity. There, devoting
most of his leisure to mathematics, he spent the remainder of
his life a life which, but for a somewhat acrimonious dispute
with Descartes on the validity of analysis used by the latter,
was unruffled by any event which calls for special notice.
The dispute was due chiefly to the obscurity of Descartes,
but the tact and courtesy of Fermat brought it to a friendly
conclusion. Fermat was a good scholar and amused himself by
conjecturally restoring the work of Apollonius on plane loci.
Except a few isolated papers Fermat published nothing
in his lifetime, and gave no systematic exposition of his
methods. Some of the most striking of his results were found
after his death on loose sheets of paper or written in the
margins of works which he had read and annotated, and are
unaccompanied by any proof. It is thus somewhat difficult to
estimate the dates and originality of his work. After his death
his papers and correspondence were printed by his nephew
at Toulouse in two volumes, 1670 and 1679 : a summary of it
with notes was published by Brassine at Toulouse in 1853,
and a reprint of it was issued at Berlin in 1861 : anew edition
is now being issued by the French government, which will
include some letters on his discoveries and methods in the theory
of numbers recently found at Leyden by M. Charles Henry.
Fermat was constitutionally modest and retiring, and does not
seem to have intended his papers to be published. It is
probable that he revised his notes as occasion required, and
that his published works represent the final form of his
researches, and therefore cannot be dated much earlier than
1660. I shall consider separately (i) his investigations in the
theory of numbers ; (ii) his use in geometry of analysis and
of infinitesimals ; arid (iii) his method of treating questions of
probability.
FERMAT. 295
(i) The theory of numbers appears to have been the
favourite study of Fermat. He prepared an edition of Dio-
phantus, and the notes and comments thereon contain numerous
theorems of considerable elegance : this forms the first of the two
volumes of his works. Most of the proofs of Fermat are lost,
and it is possible that some of them were nob rigorous an
induction by analogy and the intuition of genius sufficing to
lead him to correct results. The following examples will
illustrate these investigations.
(a) If p be a prime and a be prime to p, then a p ~ l - 1 is
divisible by p, that is, a p ~ l -1 = (mod. p). A proof of this,
first given by Euler, is well known. A more general theorem
is that a<M w ) 1 = (mod. ri), where a is prime to n and < (n)
is the number of integers less than n and prime to it.
(6) A prime (greater than 2) can be expressed as the
difference of two square integers in one and only one way.
Fermat s proof is as follows. Let n be the prime, and suppose
it equal to x 2 - y 2 , that is, to (x + y] (x-y). Now, by hypo
thesis, the only integral factors of n are n and unity, hence
x + y n and x y \. Solving these equations we get
x = ^ (n + 1 ) and y \ (n 1 ).
(c) He gave a proof of the statement made by Diophantus
(quoted above on p. Ill) that the sum of the squares of two
integers cannot be of the form n - 1 ; and he added a corollary
which I take to mean that it is impossible that the product
of a square and a prime of the form 4/i - 1 [even if mul
tiplied by a number prime to the latter], can be either a
square or the sum of two squares. For example, 44 is a
multiple of 11 (which is of the form 4x3-1) by 4, hence
it cannot be expressed as the sum of two squares. He also
stated that a number of the form 2 + 6 2 , where a is prime
to by cannot be divided by a prime of the form 4n - 1.
(d) Every prime of the form 4n + 1 is expressible, and that
in one way only, as the sum of two squares. This problem was
first solved by Euler who shewed that a number of the form
2 m (4/i + 1) can be always expressed as the sum of two squares.
296 MATHEMATICS FROM DESCARTES TO HUYGENS.
(e) If a, b, c be integers, such that a 2 + 6 2 = c 2 , then ab
cannot be a square. Lagrange gave a solution of this.
(/) The determination of a number x such that x 2 n + 1 may
be a square, where n is a given integer which is not a square.
(g) There is only one integral solution of the equation
x 2 + 2 = y 3 ; and there are only two integral solutions of the
equation x 2 + 4 = y 3 . The required solutions are evidently for
the first equation x = 5, and for the second equation x = 2 and
a? = 11. This question was issued as a challenge to the English
mathematicians Wallis and Digby.
(h) No integral values of x, y, z can be found to satisfy
the equation x n + y n = z n , if n be an integer greater than 2.
This proposition* has acquired extraordinary celebrity from
the fact that no general demonstration of it has been given,
but there is no reason to doubt that it is true.
Probably Fermat discovered its truth first for the case
n= 3, and then for the case n = 4. His proof for the former of
these cases is lost, but that for the latter is extant, and a
similar proof for the case of n 3 was given by Euler. These
proofs depend upon shewing that, if three integral values of
x, y, z can be found which satisfy the equation, then it will be
possible to find three other and smaller integers which also
satisfy it : in this way finally we shew that the equation must
be satisfied by three values which obviously do not satisfy it.
Thus no integral solution is possible. It would seem that this
method is inapplicable to any cases except those of n = 3 and
11=4.
Fermat s discovery of the general theorem was made later.
An easy demonstration can be given on the assumption that a
number can be resolved into prime (complex) factors in one
and only one way. The assumption has been made by some
writers, but it is not universally true. It is possible that
Fermat made some such supposition though it is perhaps more
likely that he discovered a rigorous demonstration.
* On this curious proposition, see my Mathematical Recreations and
Problems, pp. 2730.
FERMAT. 297
In 1823 Legeridre obtained a proof for the case of n = 5 ,
in 1832 Lejeune Dirichlet gave one for n= 14, and in 1840
Lame and Lebesgue gave proofs for n = 7. The proposition
appears to be true universally, and in 1849 Kummer, by means
of ideal primes, proved it to be so for all numbers except those
(if any) which satisfy three conditions. It is not certain whether
any number can be found to satisfy these conditions, but there
is 110 number less than 100 which does so. The proof is com
plicated and difficult, and there can be no doubt is based on
considerations unknown to Fermat. I may add that, to prove
the truth of the proposition when n is greater than 4, it obvi
ously is sufficient to confine ourselves to cases where n is a
prime, and the first step in Rummer s demonstration is to
shew that in such cases one of the numbers #, y, z must be
divisible by n.
The following extracts, from a letter* now in the univer
sity library at Leyden, will give an idea of Fermat s methods ;
the letter is undated, but it would appear that, at the time
Fermat wrote it, he had proved the proposition (A) above
only for the case when n = 3.
Je ne m en servis au commencement que pour demontrer les propo
sitions negatives, comme par exemple, qu il n y a aucu nombre moindre
de I unit6 qu un multiple de 3 qui soit compost d un quarre et du triple
d un autre quarre. Qu il n y a aucun triangle rectangle de nombres dont
1 aire soit un nombre quarr6. La preuve se fait par a.Tra.yuyT)v rty ek
aduvarov en cette maniere. S il y auoit aucun triangle rectangle en
nombres entiers, qui eust son aire esgale & un quarrc*, il y auroit un
autre triangle moindre que celuy la qui auroit la mesme propriete. S il
y en auoit un second moindre que le premier qui eust la mesme pro
priete il y en auroit par un pareil raisonnement un troisieme moindre
que ce second qui auroit la mesme proprie te et enfin un quatrieme, un
cinquieme etc. a 1 infini en descendant. Or est il qu estant donntS un
nombre il n y en a point infinis en descendant moindres que celuy la,
j entens parler tousjours des nombres entiers. D ou on conclud qu il est
done impossible qu il y ait aucun triangle rectangle dont 1 aire soit
quarre. Vide foliu post sequens....
* The letter is printed at length in Boncompagni s Bullettino di
bibliografia for 1879, pp. 737740.
298 MATHEMATICS FROM DESCARTES TO HUYGENS.
Je fus longtemps sans pouuoir appliquer ma methode aux questions
affirmatiues, parce que le tour et le biais pour y venir est beaucoup plus
malaise que celuy dont je me sers aux negatives. De sorte que lors qu il
me falut demonstrer que tout nombre premier qui surpasse de I unite un
multiple de 4, est compose de deux quarrez je me treuuay en belle peine.
Mais enfin une meditation diverses fois reiteree me donna les lumieres
qui me manquoient. Et les questions affirmatiues passerent par ma
methode a 1 ayde de quelques nouueaux principes qu il y fallust joindre
par necessity. Ce progres de mon raisonnement en ces questions affir
matives estoit tel. Si un nombre premier pris a discretion qui surpasse
de I unite un multiple de 4 n est point compose de deux quarrez il y aura
un nombre premier de mesme nature moindre que le donn6 ; et ensuite
un troisieme encore moindre, etc. en descendant a Finfini jusques a ce
que uous arriviez au nombre 5, qui est le moindre de tous ceux de cette
nature, lequel il s en suivroit n estre pas compose de deux quarrez, ce
qu il est pourtant d ou on doit inferer par la deduction a 1 impossible que
tous ceux de cette nature sont par consequent composez de 2 quarrez.
II y a infinies questions de cette espece.
Mais il y en a quelques autres qui demandent de nouveaux principes
pour y appliquer la descente, et la recherche en est quelques fois si mal
aistje, qu on n y peut venir qu auec une peine extreme. Telle est la ques
tion suiuante que Bachet sur Diophante avoiie n avoir jamais peu demon
strer, sur le suject de laquelle M. r Descartes fait dans une de ses lettres
la mesme declaration, jusques la qu il confesse qu il la juge si difficile,
qu il ne voit point de voye pour la resoudre. Tout nombre est quarre*,
ou compost de deux, de trois, ou de quatre quarrez. Je 1 ay enfin rang6e
sous ma methode et je demonstre que si un nombre donne n estoit point
de cette nature il y en auroit un moindre qui ne le seroit pas non plus,
puis un troisieme moindre que le second &c. a 1 infini, d ou Ton infere
que tous les nombres sont de cette nature....
J ay ensuite considere certaines questions qui bien que negatives ne
restent pas de receuoir tres-grande difficulte la methode pour y pratiquer
la descente estant tout a fait diuerse des precedentes comme il sera aise
d esprouuer. Telles sont les suiuantes. II n y a aucun cube diuisible
en deux cubes. II n y a qu un seul quarr6 en entiers qui augmente du
binaire fasse un cube ledit quarr6 est 25. II n y a que deux quarrez en
entiers lesquels augmentes de 4 fassent cube, lesdits quarrez sont 4
et 121....
Apr6s auoir couru toutes ces questions la pluspart de diuerses (sic)
nature et de differente facon de demonstrer, j ay passe a 1 inuention
des regies generales pour resoudre les equations simples et doubles de
Diophante. On propose par exemple 2 quarr. + 7967 esgaux a un quarre*
(hoc est 2## + 7967 x quadr.) J ay une regie generale pour resoudre
cette equation si elle est possible, ou decouvrir son impossibility Et
FERMAT. 299
ainsi en tons les cas et en tous nombres tant des quarrez que des unitez.
On propose cette equation double 2x + 3 et 3x + 5 esgaux chacun a un
quarre. Bachet se glorifie en ses commentaires sur Diophante d auoir
trouve une regie en deux cas particuliers. Je la donne generale en toute
sorte de cas. Et determine par regie si elle est possible ou non....
Voila sommairement le conte de mes recherches sur le suject des
nombres. Je ne 1 ay escrit que parce que j apprehende que le loisir
d estendre et de mettre au long toutes ces demonstrations et ces metliodes
me manquera. En tout cas cette indication seruira aux scauants pour
trouver d eux mesmes ce que je n estens point, principalement si M. r de
Carcaui et Fr6nicle leur font part de quelques demonstrations par la
descente que je leur ay enuoyees sur le suject de quelques propositions
negatiues. Et peut estre la posterite me scaura gre de luy avoir fait
connoistre que les anciens n ont pas tout sceu, et cette relation pourra
passer dans I esprit de ceux qui viendront apres moy pour traditio lam-
padis ad filios, comme parle le grand Chancelier d Angleterre, suiuant le
sentiment et la deuise duquel j adjousteray, multi pertransibunt et auge-
bitur scientia.
(ii) I next proceed to mention Fermat s use in geometry
of analysis and of infinitesimals. It would seem from his
correspondence that he had thought out the principles of
analytical geometry for himself before reading Descartes s
Geometric and had realized that from the equation (or as he
calls it, the " specific property ") of a curve all its properties
could be deduced. His extant papers on geometry deal how
ever mainly with the application of infinitesimals, to the
determination of the tangents to curves, to the quadrature of
curves, and to questions of maxima and minima ; probably
these papers are a revision of his original manuscripts (which
he destroyed) and were written about 1663, but there is no
doubt that he was in possession of the general idea of his
method for finding maxima and minima as early as 1628 or
1629.
He obtained the subtangent to the ellipse, cycloid, cissoid,
conchoid, and quadratrix by making the ordinates of the curve
and a straight line the same for two points whose abscissae
were x and x - e ; but there is nothing to indicate that he was
aware that the process was general, and, though in the course
300 MATHEMATICS FROM DESCARTES TO HUYGENS.
of his work he used the principle, it is probable that he never
separated it, so to speak, from the symbols of the particular
problem he was considering. The first definite statement of
the method was due to Barrow and was published in 1669
(see below, p. 312),
Fermat also obtained the areas of parabolas and hyperbolas
of any order, and determined the centre of mass of a few
simple curves and of a paraboloid of revolution. As an ex
ample of his method of solving these questions I will quote
his solution of the problem to find the area between the
parabola y^py^^ the axis of #, and the line x a. He says
that, if the several ordinates at the points for which x is
equal to a, a (1 - e), a(l-e) 2 , ... be drawn, then the area
will be split into a number of little rectangles whose areas are
respectively
afl(pa )* ae(l - e){pa (l - ) }* , ... .
The sum of these is p* a? e/{l -(I - e)" 8 ); and by a subsidiary
proposition (for of course he was not acquainted with the
binomial theorem) he finds the limit of this when e vanishes
to be %p*a*. The theorems last mentioned were published
only after his death; and probably they were not written till
after he had read the works of Cavalieri and Wallis.
Kepler had remarked that the values of a function imme
diately adjacent to and on either side of a maximum (or
minimum) value must be equal. Fermat applied this principle
to a few examples. Thus, to find the maximum value of
x (a x), his method is essentially equivalent to taking a con
secutive value of x, namely x e where e is very small, and
putting x(a-x]-(x-e) (a x+e). Simplifying, and ultimately
putting e = 0, we get x = ^a. This value of x makes the given
expression a maximum.
(iii) Fermat must share with Pascal the honour of having
founded the theory of probabilities. I have already mentioned
(see above, p. 286) the problem proposed to Pascal, and which
he communicated to Fermat, and have there given Pascal s
FEKMAT. 301
solution. Fermat s solution depends on the theory of com
binations and will be sufficiently illustrated by the following
example the substance of which is taken from a letter dated
Aug. 24, 1654, which occurs in the correspondence with Pascal.
Fermat discusses the case of two players, and supposes that the
first wants two points to win and the second three points.
The game will be then certainly decided in the course of four
trials. Take the letters a and b and write down all the com
binations that can be formed of four letters. These combi
nations are the following, 16 in number :
a a
a
a
a
b a
a
b
a a
a
b
b
a a
a a
a
b
a
b a
b
b
a a
b
b
b
a b
a a
b
a
a
b b
a
b
a b
a
b
b
b a
a a
b
b
a
b b
b
b
a b
b
b
b
b b
Now let A denote the player who wants two points, and B the
player who wants three points. Then in these 16 combinations
every combination in which a occurs twice or oftener represents
a case favourable to A, and every combination in which b
occurs three times or oftener represents a case favourable to B.
Thus, on counting them, it will be found that there are 11 cases
favourable to A , and 5 cases favourable to B ; and, since these
cases are all equally likely, A s chance of winning the game is
to It s chances as 11 is to 5.
The only other problem on this subject which as far as
I know attracted the attention of Fermat was also proposed to
him by Pascal and was as follows. A person undertakes to
throw a six with a die in eight throws; supposing him to have
made three throws without success, what portion of the stake
should he be allowed to take on condition of giving up his
fourth throw? Fermat s reasoning is as follows. The chance
of success is , so that he should be allowed to take ^ of the
stake on condition of giving up his throw. But, if we wish to
estimate the value of the fourth throw before any throw is
made, then the first throw is worth of the stake ; the second
is worth of what remains, that is, ^ of the stake ; the third
302 MATHEMATICS FROM DESCARTES TO HUYGENS.
throw is worth i of what now remains, that is, ^yV ^ the
stake ; the fourth throw is worth i of what now remains, that
is T V 2 w f *ke stake.
Fermat does not seem to have carried the matter much
further, but his correspondence with Pascal shews that his
views on the fundamental principles of the subject were ac
curate : those of Pascal were not altogether correct.
Fermat s reputation is quite unique in the history of
science. The problems on numbers which he had proposed
long defied all efforts to solve them, and many of them yielded
only to the skill of Euler. One still remains unsolved. This
extraordinary achievement has overshadowed his other work,
but in fact it is all of the highest order of excellence, and we
can only regret that he thought fit to write so little.
Huygens*. Christian Huygens was born at the Hague
on April 14, 1629, and died in the same town on June 8, 1695.
He generally wrote his name as Hugens, but I follow the usual
custom in spelling it as above : it is also sometimes written
as Huyghens. His life was uneventful and is a mere record of
the dates of his various works.
In 1651 he published an essay in which he shewed the fallacy
in a system of quadratures proposed by Gregoire de Saint-
Vincent (see below, p. 309) who was well versed in the geo
metry of the Greeks but had not grasped the essential points
in the more modern methods. This essay was followed by tracts
on the quadrature of the conies and the approximate rectification
of the circle.
In 1654 his attention was directed to the improvement of
the telescope. In conjunction with his brother he devised
a new and better way of grinding and polishing lenses.
As a result of these improvements he was able during the
* The works of Huygens were collected and published in six volumes ;
four at Leyden in 1724 and two at Amsterdam in 1728 : a life by s Grave-
sande is prefixed to the first volume. His scientific correspondence was
published at the Hague in 1833. A new edition of all his works is now
being issued at the Hague.
HUYGENS. 303
following two years, 1655 and 1656, to resolve numerous
astronomical questions ; as for example the nature of Saturn s
appendage.
His astronomical observations required some exact means
of measuring time, and he was thus led in 1656 to invent
the pendulum clock, as described in his tract Horologium,
1658. The time-pieces previously in use had been balance-
clocks.
In the year, 1657, Huygens wrote a small work on the
calculus of probabilities founded on the correspondence of
Pascal and Fermat. He spent a couple of years in England
about this time. His reputation was now so great that
in 1665 Louis XIV. offered him a pension if he would
live in Paris, which accordingly then became his place of
residence.
In 1668 he sent to the Royal Society of London, in answer to
a problem they had proposed, a memoir in which (simultane
ously with Wallis and Wren) he proved by experiment that
the momentum in a certain direction before the collision of two
bodies is equal to the momentum in that direction after the
collision. This was one of the points in mechanics on which
Descartes had been mistaken.
The most important of Huygens s work was his Horolo-
gium Oscillatorium published at Paris in 1673. The first
chapter is devoted to pendulum clocks. The second chapter
contains a complete account of the descent of heavy bodies
under their own weights in a vacuum, either vertically down
or on smooth curves. Amongst other propositions he shews
that the cycloid is tautochronous. In the third chapter he
defines evolutes and involutes, proves some of their more
elementary properties, and illustrates his methods by finding
the evolutes of the cycloid and the parabola. These are the
earliest instances in which the envelope of a moving line was
determined. In the fourth chapter he solves the problem of
the compound pendulum, and shews that the centres of oscil
lation and suspension are interchangeable. In the fifth and
304 MATHEMATICS FROM DESCARTES TO HUYGENS.
last chapter he discusses again the theory of clocks, points out
that if the bob of the pendulum were made by means of cy-
cloidal checks to oscillate in a cycloid the oscillations would be
isochronous ; and finishes by shewing that the centrifugal force
on a body which moves in a circle of radius r with a uniform
velocity v varies directly as v 2 and inversely as r. This work
contains the first attempt to apply dynamics to bodies of finite
size and not merely to particles.
In 1675 Huygens proposed to regulate the motion of
watches by the use of the balance spring, in the theory of
which he had been perhaps anticipated in a somewhat am
biguous and incomplete statement made by Hooke in 1G58.
Watches or portable clocks had been invented early in the
sixteenth century and by the end of that century were not
very uncommon, but they were clumsy and unreliable, being
driven by a main spring and regulated by a conical pulley and
verge escapement; moreover until 1687 they had only one hand.
The first watch whose motion was regulated by a balance spring
was made at Paris under Huygens s directions, and presented
by him to Louis XIV. The increasing intolerance of the
Catholics led to his return to Holland in 1681, and after
the revocation of the edict of Nantes he refused to hold any
further communication with France. He now devoted himself
to the construction of lenses of enormous focal length : of these
three of focal lengths 123ft., 180ft., and 210ft. were sub
sequently given by him to the Royal Society of London in
whose possession they still remain. It was about this time
that he discovered the achromatic eye-piece (for a telescope)
which is known by his name. In 1689 he came from Holland
to England in order to make the acquaintance of Newton
whose Principia had been published in 1687, the extraordinary
merits of which Huygens had at once recognized.
On his return in 1690 Huygens published his treatise on
light in which the undulatory theory was expounded and ex
plained. Most of this had been written as early as 1678.
The general idea of the theory had been suggested by Robert
HUYGENS. 305
Hooke in 1664, but he had not investigated its consequences
in any detail. This publication falls outside the years con
sidered in this chapter, but here it may be briefly said that
according to the wave or undulatory theory space is filled with
an extremely rare ether, and light is caused by a series of
waves or vibrations in this ether which are set in motion by
the pulsations of the luminous body. From this hypothesis
Huygens deduced the laws of reflexion and refraction, explained
the phenomena of double refraction, and gave a construction
for the extraordinary ray in biaxal crystals ; while he found
by experiment the chief phenomena of polarization.
The immense reputation and unrivalled powers of Newton
led to disbelief in a theory which he rejected, and to the
general adoption of Newton s emission theory (see below,
p. 326) ; but it should be noted that Huygens s explanation
of some phenomena, such as the colours of thin plates, was
inconsistent with the results of experiments, nor was it until
Young and Wollaston at the beginning of this century revived
the undulatory theory and modified some of its details and
Fresnel elaborated their views that its acceptance could be fully
justified.
Besides these works Hnygeus took part in most of the
controversies and challenges which then played so large a part
in the mathematical world, and wrote several minor tracts.
In one of these he investigated the form and properties of the
catenary. In another he stated in general terms the rule for
finding maxima and minima of which Fermat had made use,
and shewed that the sub tangent of an algebraical curve
f ( x > y) = was equal to #/],//*., where f y is the derived function
of f fa y) regarded as a function of y. In some posthumous
works, issued at Leyden in 1703, he further shewed how from
the focal lengths of the component lenses the magnifying
power of a telescope could be determined ; and explained some
of the phenomena connected with halos and parhelia.
I should add that almost all his demonstrations, like those
of Newton, are rigidly geometrical, and lie would seem to have
B. 20
306 MATHEMATICS FROM DESCARTES TO HUYGENS.
made no use of the differential or fluxional calculus, though he
admitted the validity of the methods used therein. Thus, even
when first written, his works were expressed in an archaic
language, and perhaps received less attention than their intrinsic
merits deserved.
I have now traced the development of mathematics for a
period which we may take roughly as dating from 1635 to
1675 under the influence of Descartes, Cavalieri, Pascal, Wallis,
Fermat, and Huygens. The life of Newton partly overlaps
this period: his works and influence are considered in the next
chapter.
I may dismiss the remaining mathematicians of this time
whom I desire to mention with comparatively slight notice. The
following is an alphabetical list of the more remarkable among
them : the dates given are those of the birth and death of the
mathematician to whose name they are appended. Backet,
1581 1638: Barrow, 1630 1677 : Brouncker, 16201684:
Collins, 1625 1683: Courtier, 16041692: de Beaune,
16011652: de Laloubere, 16001664: Frenicle, 1605
1670: Jas. Gregory, 16381675: Hooke, 16351703: Hudde,
16331704: Kinckkuysen, 16301679: Nich. Mercator, 1620
-1687: Mersenne, 1588 1648: Mydorge, 15851647:
Pell, 16101685: Ricci, 16191692: Roberval, 16021675 :
Roemer, 1644 1710: Saint- Vincent, 1584; 1667 : Sluze, 1622
1685: Torricelli, 1608 1647: Tsckirnkausen, 1631 1708:
van Schooten, died in 1661: and Wren, 1632 1723. In the
following notes I have arranged the above-mentioned mathe
maticians so that as far as possible their chief contributions
shall come in chronological order.
Bachet. Claude Gaspard Backet de Meziriac was born at
Bourg in 1581, and died in 1638. He wrote the ProUemes
plaisants, 1612, second and enlarged edition 1624, which con
tains an interesting collection of arithmetical tricks and ques
tions many of which are quoted in chapter i. of my Mathe
matical Recreations and Problems ; also Les elements arith-
BACHET. MYDORGE. MERSENNE. 307
metiques, which exists in manuscript ; and a translation of the
Arithmetic of Diophantus. Bachet was the earliest writer who
discussed the solution of indeterminate equations by means
of continued fractions.
Mydorge. Claude Mydorge, born at Paris in 1585 and
died in 1647, belonged to a distinguished " family of the robe,"
and was himself a councillor at Chatelet, and then treasurer
to the local parliament at Amiens. He published some works
on optics of which one, issued in 1631, is extant, and in
1641 a treatise on conic sections. He also left a manuscript
containing solutions of over a thousand geometrical problems,
many of which are. said to be ingenious : the enunciations
were published by M. Charles Henry in 1882.
Mersenne. Marin Mersenne, born in 1588 and died at Paris
in 1648, was a Franciscan friar, who made it his business to be
acquainted and correspond with the French mathematicians of
that date and many of their foreign contemporaries. In 1634
he published a translation of Galileo s mechanics ; in 1644 he
issued his Cogitata Physico-Mathernatica, by which he is best
known, containing an account of some experiments in physics ; he
also wrote a synopsis of mathematics, which was printed in 1664.
The preface to the Cogitata contains a statement (probably
due to Fermat), that in order that 2 P 1 may be prime, the
only values of p, not greater than 257, which are possible are 1,
2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 : to which list Herr
Seelhoff has shewn we must add 61. With this addition, the
statement has been verified for all except twenty-three values
of p: namely, 67, 71, 89, 101, 103, 107, 109, 127, 137, 139,
149, 157, 163, 167, 173, 181, 193, 197, 199, 227, 229, 241,
and 257. Of these values, Mersenne asserted that p = 67,
p= 127, and /?=257 make 2 P - 1 a prime, and that the other
.values make 2 P 1 a composite number. It is most likely that
these results are particular cases of some general theorem
on the subject which remains to be discovered. The number
2 61 - 1 contains 19 digits, and is the highest number at present
known to be a prime : its value is 2,305843,009213,693951.
202
308 MATHEMATICS FROM DESCARTES TO HUYGENS.
The theory of perfect numbers depends directly on that of
Mersenne s numbers. It is probable that all perfect numbers
are included in the formula 2 p ~ l (2 P 1), where 2 P - 1 is a prime.
Euclid proved that any number of this form is perfect;
Euler shewed that the formula includes all even perfect
numbers ; and there is reason to believe though a rigid
demonstration is wanting that an odd number cannot be
perfect. If we assume that the last of these statements is
true, then every perfect number is of the above form. Thus,
if p = 2, 3, 5, 7, 13, 17, 19, 31, 61, then by Mersenne s rule
the corresponding values of 2^-1 are prime; they are 3, 7, 31,
127,8191, 131071,524287,2147483647,2305843009213693951 ;
and the corresponding perfect numbers are 6, 28, 496, 8128,
33550336, 8589869056, 137438691328,2305843008139952128,
and 2658455991569831744654692615953842176.
De Beaune. Florimond de Beaune, born at Blois in 1601
and died in 1652, wrote explanatory notes on the obscure and
difficult analytical geometry of Descartes. He also discussed
the superior and inferior limits to the roots of an equation ;
this was not published till 1659.
Roberval. Gilles Personier (de) Roberval, born at Roberval
in 1602 and died at Paris in 1675, described himself from the
place of his birth as de Roberval, a seignorial title to which he
had no right. He discussed the nature of the tangents to
curves (see above, p. 276), solved some of the easier questions
connected with the cycloid, generalized Archimedes s theorems
on the spiral, wrote on mechanics, and on the method of indi
visibles which he rendered more precise and logical. He was
a professor in the university of Paris, and in correspondence
with nearly all the leading mathematicians of his time. A
complete edition of his works was included in the old
Memoires of the Academy of Sciences published in 1693.
Van Schooten. Frans van Schooten, to whom we owe an
edition of Yieta s works, succeeded his father (who had taught
mathematics to Huygens, Hudde, and Sluze) as professor at
Leyden in 1646 : he brought out in 1659 a Latin translation
SAINT-VINCENT. TOKRICELLI. HUDDE. 309
of Descartes s Geometric ; and in 1657 a collection of mathe
matical exercises in which he recommended the use of co
ordinates in space of three dimensions : he died in 1661.
Saint- Vincent. Gregoire de Saint- Vincent, a Jesuit, born at
Bruges in 1584 and died at Ghent in 1667, discovered the
expansion of log(l + x) in ascending powers of x. Although
a circle-squarer he is worthy of mention for the numerous
theorems of interest which he discovered in his search after
the impossible, and Montucla ingeniously remarks that "no
one ever squared the circle with so much ability or (except for
his principal object) with so much success." He wrote two
books on the subject, one published in 1647 and the other in
1668, which cover some two or three thousand closely printed
pages : the fallacy in the quadrature was pointed out by
Huygens. In the former work he used indivisibles ; an earlier
work entitled Theoremata Mathematica published in 1624 con
tains a clear account of the method of exhaustions, which is
applied to several quadratures, notably that of the hyperbola.
For further details of Saint- Vincent s life and works, see
L. A. J. Quetelet s Histoire des sciences chez les Beiges, Brussels,
1866.
Torricelli. Evangelista Torricelli, born at Faenza on Oct.
15, 1608 and died at Florence in 1647, wrote on the quadra
ture of the cycloid and conies; the theory of the barometer;
the value of gravity found by observing the motion of two
weights connected by a string passing over a fixed pulley ;
the theory of projectiles ; and the motion of fluids. His
mathematical writings were published in 1644.
Hudde. Jokann Hudde, burgomaster of Amsterdam, was
born there in 1633 and died in the same town in 1704. He
wrote two tracts in 1659 : one was on the reduction of equa
tions which have equal roots ; in the other he stated what
is equivalent to the proposition that, \ff(x, y) =0 be the alge
braical equation of a curve, then the subtangent is -y I- ~ ;
cyl ox
but being ignorant of the notation of the calculus his enuncia
tion is involved.
310 MATHEMATICS FROM DESCARTES TO HUYGENS.
Fre*nicle. Bernard Fr&nicle de Bessy, born in Paris circ.
1605 and died in 1670, wrote numerous papers on combina
tions and on the theory of numbers, also on magic squares.
It may be interesting to add that he challenged Huygens to
solve the following system of equations in integers, x 2 + y 2 = z 2 ,
x* = u 2 + v 2 , x y = u v: a solution was given by M. Pepin
in 1880. Frenicle s miscellaneous works, edited by De la
Hire, were published in the Memoir es de VAcademie, vol. v,
1691.
De Laloubere* Antoine de Laloubere, a Jesuit, born in Laii-
guedoc in 1600 and died at Toulouse in 1664, is chiefly cele
brated for an incorrect solution of Pascal s problems on the
cycloid, which he gave in 1660, but he has a better claim
to distinction in having been the first mathematician to study
the properties of the helix.
Kinckhuysen. Gerard Kinckhuysen, born in Holland in
1630 and died in 1679, wrote in 1660 a text-book on analytical
conies, in 1661 an algebra, and in 1669 formed a collection of
geometrical problems solved by analytical geometry.
Courcier. Pierre Courcier^ a Jesuit, born at Troyes in 1604
and died at Auxerre in 1692, wrote on the curves of intersection
of a sphere with a cylinder or cone, also on spherical polygons :
the latter work was published in 1663.
Ricci. Michel- Ange Ricci, born in 1619, made a cardinal
in 1681, and died at Rome in 1692, wrote in 1666 a treatise
in which he solved by Greek geometry those problems on
maxima and minima and on tangents to curves which had been
considered by Descartes, Pascal, and Fermat.
N. Mercator. Nicholas Mercator (sometimes known as
Kauffmann) was born in Holstein about 1620, but resided
most of his life in England : he went to France in 1683,
where he designed and constructed the fountains at Versailles,
but when they were finished Louis XIV. refused to make him
the payment agreed on unless he would turn Catholic : he died
of vexation and poverty in Paris in 1687. He wrote a treatise
on logarithms entitled Logorithmotechnia published in 1668,
NICHOLAS MERCATOR. BARROW. 311
and discovered the series
log(l +x)^x-x* + lx*-\x*+ ...;
he proved this by writing the equation of a hyperbola in the
form
form
1 +x
to which Wallis s formula (see above, p. 290) could be applied.
The same series had been independently discovered by Saint-
Vincent. For further details see C. Button s Mathematical
Tracts.
Barrow. Isaac Barrow was born in London in 1630, and
died at Cambridge in 1677. He went to school first at
Charterhouse (where he was so troublesome that his father was
heard to pray that if it pleased God to take any of his children
he could best spare Isaac), and subsequently to Fel stead. He
completed his education at Trinity College, Cambridge; after
taking his degree in 1648, he was elected to a fellowship in
1649, he then resided for a few years in college, but in 1655 he
was driven out by the persecution of the Independents. He
spent the next four years in the East of Europe, and after
many adventures returned to England in 1659. He was
ordained the next year, and appointed to the professorship of
Greek at Cambridge. In 1662, he was made professor of
geometry at Gresham College, and in 1663, was selected as the
first occupier of the Lucasian chair at Cambridge. He resigned
the latter to his pupil Newton in 1669 whose superior abilities
he recognized and frankly acknowledged. For the remainder
of his life he devoted himself to the study of divinity. He was
appointed master of Trinity College in 1672, and held the post
until his death.
He is described as "low in stature, lean, and of a pale
complexion," slovenly in his dress, and an inveterate smoker.
He was noted for his strength and courage, and once when
travelling in the East he saved the ship by his own prowess
from capture by pirates. A ready and caustic wit made him a
favourite of Charles II., and induced the courtiers to respect
312 MATHEMATICS FHOM DESCARTES TO HUYGENS.
even if they did not appreciate him. He wrote with a sus
tained and somewhat stately eloquence, and with his blameless
life and scrupulous conscientiousness was an impressive person
age of the time.
His earliest work was a complete edition of the Elements of
Euclid which he issued in 1655, he published an English
translation in 1660, and in 1657 an edition of the Data. His
lectures, delivered in 1664, 1665, and 1666, were published in
1683 under the title Lectiones Mathematicae: these are mostly
on the metaphysical basis for mathematical truths. His
lectures for 1667 were published in the same year, and suggest
the analysis by which Archimedes was led to his chief results.
In 1669 he issued his Lectiones Opticae et Geometricae; it is
said in the preface that Newton revised and corrected these
lectures adding matter of his own, but it seems probable from
Newton s remarks in the fluxional controversy that the
additions were confined to the parts which dealt with optics :
this, which is his most important work in mathematics, was
republished with a few minor alterations in 1674. In 1675
he published an edition with numerous comments on the first
four books of the Conies of Apollonius, and of the extant works
of Archimedes and Theodosius.
In the optical lectures many problems connected with the
reflexion and refraction of light are treated with great ingenuity.
The geometrical focus of a point seen by reflexion or refraction
is defined; and it is explained that the image of an object is the
locus of the geometrical foci of every point on it. Barrow also
worked out a few of the easier properties of thin lenses; and con
siderably simplified the Cartesian explanation of the rainbow.
The geometrical lectures contain some new ways of deter
mining the areas and tangents of curves. The most celebrated
of these is the method given for the determination of tangents
to curves, and this is sufficiently important to require a detailed
notice because it illustrates the way in which Barrow, Hudde,
and Sluze were working on the lines suggested by Fermat
towards the methods of the differential calculus. Fermat had
UAlMtoNV.
observed that the tangent at a point P on a curve was detc
if one other point besides P on it were known ; hence, i.
length of the subtangent MT could be found (thus determim.
the point T\ then the line TP would be the required tangent.
Now Barrow remarked that if the abscissa and ordinate at a
point Q adjacent to P were drawn, he got a small triangle PQR
(which he called the differential triangle, because its sides PR
and PQ were the differences of the abscissas and ordinates of P
and (?), so that
TM : MP = QR : RP.
To find QR : RP he supposed that x, y were the coordinates of
P, and x e, y a those of Q (Barrow actually used p for x
and m for y but I alter these to agree with the modern practice).
Substituting the coordinates of Q in the equation of the curve,
and neglecting the squares and higher powers of e and a as
compared with their first powers, he obtained e : a. The ratio
a/6 was subsequently (in accordance with a suggestion made
by Sluze) termed the angular coefficient of the tangent at the
point.
Barrow applied this method to the curves (i) o(&+jf)=l*jfi
(ii) o; 3 -f ?/ 3 -? 3 ; (iii) x 3 + y 3 = rxy, called la galande ; (iv)
y=(r x) tan 7rx/2r, the quadratrix ; and (v) y = r tan 7rx/ 2r.
It will be sufficient here if I take as an illustration the simpler
case of the parabola y 2 = px. Using the notation given
above, we have for the point P, y* = px 9 and for the point
Q, (y - of = p (x e). Subtracting we get 2ay a 2 = pe. But,
MATHEMATICS FROM DESCARTES TO HUYGENS.
if a be an infinitesimal quantity, a 2 must be infinitely smaller
and therefore may be neglected : hence e : a = 2y : p. There
fore TM : y = e : a = Zy : p. That is, TM = 2y 2 /p = 2x. This
is exactly the procedure of the differential calculus, except that
we there have a rule by which we can get the ratio a/e or dyjdx
directly without the labour of going through a calculation similar
to the above for every separate case.
Brouncker. William, Viscount Brouncker, one of the
founders of the Royal Society of London, born in 1620 and
died on April 5, 1684, was among the most brilliant mathe
maticians of this time, and was in intimate relations with
Wallis, Fermat, and other leading mathematicians. I men
tioned on p. 155 his curious reproduction of Brahmagupta s
solution of a certain indeterminate equation. Brouncker
proved (Phil. Trans. 1668, No. 34) that the area enclosed
between the equilateral hyperbola xy = \, the axis of x, and
the ordinates x 1 and x = 2, is equal either to
111 111
: + - + __+..., or to 1-- +o - 7 +....
1.2 T 3.4 5.6^ 234
He also worked out other similar expressions for different
areas bounded by the hyperbola and straight lines (Phil. Trans.
1672). He wrote on the rectification of the parabola (Phil.
Trans. 1673) and of the cycloid (Phil. Trans. 1678). It is
noticeable that he used infinite series to express quantities
whose values he could not otherwise determine. In answer to
a request of Wallis to attempt the quadrature of the circle he
shewed that the ratio of the area of a circle to the area of the
circumscribed square, that is, the ratio TT : 4 is equal to the ratio
i r 3 2 5 2 r . 1
1+2+2 + 2+2 +... :
Continued fractions* had been introduced by Cataldi in his
* On the history of continued fractions see papers by S. Giinther and
A. Favaro in Boncompagni s Bulletino di bibliografia, Rome, 1874, vol.
vii., pp. 213, 451, 533.
BROUNCKER. JAMES GREGORY. WREN. 315
treatise on finding the square roots of numbers published at
Bologna in 1613, but he treated them as common fractions
(see above, p. 239); Brouncker was the first writer who in
vestigated or made any use of their properties. For further
details see C. Hutton s Mathematical Dictionary.
James Gregory. James Gregory, born at Drumoak near
Aberdeen in 1638 and died at Edinburgh in October, 1675, was
successively professor at St Andrews and Edinburgh. In 1660
he published his Optica Promota in which the reflecting
telescope known by his name is described. In 1667 he issued
his Vera Circuli et Hyperbolae Quadratura in which he shewed
how the areas of the circle and hyperbola could be obtained in
the form of infinite convergent series, and here (I believe for
the first time) we find a distinction drawn between convergent
and divergent series. This work contains a remarkable geo
metrical proposition to the effect that the ratio of the area of
any arbitrary sector to that of the inscribed or circumscribed
regular polygons is not expressible by a finite number of alge
braical terms. Hence he inferred that the quadrature was
impossible : this was accepted by Montucla, but it is not con
clusive, for it is conceivable that some particular sector might
be squared, and this particular sector might be the whole circle.
This book contains also the earliest enunciation of the expansions
in series of sin a;, cos#, sin" 1 a;, and cos" 1 x. It was reprinted
in 1668 with an appendix, Geometriae Pars, in which Gregory
explained how the volumes of solids of revolution could be
determined. In 1671, or perhaps earlier, he established the
theorem that
= tan - $ tan 3 + i- tan 5 - ... ,
the result being true only if 6 lie between -%TT and JTT. This
is the theorem on which the work of most of the subsequent
calculation of approximations to the numerical value of TT has
been based. For further details see C. Hutton s Mathematical
Dictionary.
Wren. Sir Christopher Wren was born at Knoyle in
316 MATHEMATICS FROM DESCARTES TO HUYGENS.
1632, and died in London in 1723. Wren s reputation as a
mathematician has been overshadowed by his fame as an
architect, but he was Savilian professor of astronomy at
Oxford from 1661 to 1673, and for some time president of the
Royal Society. Together with Wallis and Huygens he in
vestigated the laws of collision of bodies (Phil. Trans. 1669);
he also discovered the two systems of generating lines on
the hyperboloid of one sheet, though it is probable that he
confined his attention to a hyperboloid of revolution (Phil.
Trans. 1669). Besides these he communicated papers on the
resistance of fluids, and the motion of the pendulum. He was a
friend of Newton and (like Huygens, Hooke, Halley, and
others) had made attempts to shew that the force under which
the planets move varies inversely as the square of the distance
from the sun.
Wallis, Brouncker, Wren, and Boyle (the last-named being
a chemist and physicist rather than a mathematician) were the
leading philosophers who founded the Royal Society of London.
The society arose from the self-styled "indivisible college" in
London in 1645; most of its members moved to Oxford during
the civil war, where Hooke, who was then an assistant in
Boyle s laboratory, joined in their meetings; the society was
formally constituted in London in 1660; and was incorporated
on July 15, 1662.
Hooke. Robert Hooke , born at Freshwater on July 18,
1635 and died in London on March 3, 1703, was educated at
Westminster, and Christ Church, Oxford, and in 1665 became
professor of geometry at Gresham College, a post which he
occupied till his death. He is still known by the law which
he discovered that the tension exerted by a stretched string is
(within certain limits) proportional to the extension, or as it
is better stated that the stress is proportional to the strain.
He invented and discussed the conical pendulum, and was the
first to state explicitly that the motions of the heavenly bodies
were merely dynamical problems. He was as jealous as he was
vain and irritable, and accused both Newton and Huygens of
HOOKE. COLLINS. PELL. SLUZE. 317
unfairly appropriating his results. Like Huygens, Wren, and
Halley he made efforts to find the law of force under which
the planets move about the sun, and he believed the law to be
that of the inverse square of the distance. He, like Huygens,
discovered that the small oscillations of a coiled spiral spring
were practically isochronous, and was thus led to recommend
(possibly in 1658) the use of the balance -spring in watches; he
had a watch of this kind made in London in 1675, it was
finished just three months later than the one made under the
directions of Huygens in Paris.
Collins. John Collins, born near Oxford on March 5, 1625
and died in London on Nov. 10, 1683, was a man of great
natural ability but of slight education. Being devoted to
mathematics he spent his spare time in correspondence with
the leading mathematicians of the time for whom he was
always ready to do anything in his power, and he has been
described not inaptly as the English Merseime. To him
we are indebted for much information on the details of the
discoveries of the period. See the Commercium Epistolicum,
and Rigaud s Correspondence of Scientific Men of the Seventeenth
Century.
Pell. Another mathematician who devoted a considerable
part of his time to making known the discoveries of others, and
to correspondence with leading mathematicians was John Pell.
Pell was born in Sussex on March 1, 1610, and died in London
on Dec. 10, 1685. He was educated at Trinity College,
Cambridge; he occupied in succession the mathematical chairs
at Amsterdam and Breda; he then entered the English
diplomatic service; but finally settled in 1661 in London where
he spent the last twenty years of his life. His chief works
were ah edition, with considerable new matter, of the Algebra
by Branker and Rhonius, London, 1668; and a table of square
numbers, London, 1672. For further details see my History
of Mathematics at Cambridge.
Sluze. Rene Francois Walther de Sluze (Slusius), canon of
Liege, born on July 7, 1622 and died on March 19, 1685, found
318 MATHEMATICS FROM DESCARTES TO HUYGENS.
for the subtangent of a curve f (x, y) an expression which is
equivalent to y ^- / - ; he wrote numerous tracts, and in par-
By I dx
ticular discussed at some length spirals and points of inflexion.
Some of his papers were published by Le Paige in vol. xvn. of
Boncompagni s Bulletino di bibliografia, Rome, 1884.
Tschirnhausen. Ehrenfried Walther von Tschirnhausen
was born at Kislingswalde on April 10, 1631, and died at
Dresden on Oct. 11, 1708. In 1682 he worked out the theory
of caustics by reflexion, or as they were usually called cata-
caustics, and shewed that they were rectifiable. This was the
second case in which the envelope of a moving line was deter
mined (see above, p. 303). He constructed burning mirrors of
great power. The transformation by which he removed certain
intermediate terms from a given algebraical equation is well
known: it was published in the Acta Eruditorum for 1683.
Roemer. Olof Roemer, born at Aarhuus on Sept. 25, 1644
and died at Copenhagen on Sept. 19, 1710, was the first to
measure the velocity of light : this was done in 1675 by means
of the eclipses of Jupiter s satellites. He brought the transit
and mural circle into common use, the altazimuth having been
previously generally employed, and it was on his recommenda
tion that astronomical observations of stars were subsequently
made in general on the meridian. He was also the first to
introduce micrometers and reading microscopes into an observa
tory. He also deduced from the properties of epicycloids the
form of the teeth in toothed -wheels best fitted to secure a
uniform motion.
319
CHAPTER XVI.
THE LIFE AND WORKS OF NEWTON*.
THE mathematicians considered in the last chapter com
menced the creation of those processes which distinguish
modern mathematics. The extraordinary abilities of Newton
enabled him within a few years to perfect the more elementary
of those processes, and to distinctly advance every branch of
mathematical science then studied, as well as to create some
new subjects. Newton was the contemporary and friend of
Wallis, Huygens, and others of those mentioned in the last
chapter, but, though most of his mathematical work was done
between the years 1665 and 1686, the bulk of it was not
printed at any rate in book-form till some years later.
I propose to discuss the works of Newton somewhat more
fully than those of other mathematicians, partly because of the
intrinsic importance of his discoveries, and partly because this
book is mainly intended for English readers and the develop
ment of mathematics in Great Britain was for a century
entirely in the hands of the Newtonian school.
Isaac Newton was born in Lincolnshire near Grantham on
Dec. 25, 1642, and died at Kensington, London, on March 20,
* Newton s life and works are discussed in The Memoirs of Newton, by
D. Brewster, 2 volumes, Edinburgh, second edition, 1860. An edition of
most of Newton s works was published by S. Horsley in 5 volumes,
London, 1779 85; and a bibliography of them was issued by G. J.
Gray, Cambridge, 1888. The larger portion of the Portsmouth Collec
tion of Newton s papers has been recently presented to the university of
Cambridge, a catalogue of this was published at Cambridge in 1888.
320 THE LIFE AND WORKS OF NEWTON.
1727. He was educated at Trinity College, Cambridge, and
lived there from 1661 till 1696 during which time he produced
the bulk of his work in mathematics ; in 1696 he was appointed
to a valuable Government office, and moved to London where
he resided till his death.
His father, who had died shortly before Newton was born,
was a yeoman farmer, and it was intended that Newton should
carry on the paternal farm. He was sent to school at Grantham,
where his learning and mechanical proficiency excited some
attention; and as one instance of his ingenuity I may mention
that he constructed a clock worked by water which kept very
fair time. In 1656 he returned home to learn the business of
a farmer under the guidance of an old family servant. Newton
however spent most of his time solving problems, making
experiments, or devising mechanical models ; his mother
noticing this sensibly resolved to find some more congenial
occupation for him, and his uncle, having been himself
educated at Trinity College, Cambridge, recommended that
he should be sent there.
In 1661 Newton accordingly entered as a subsizar at Trinity
College, where for the first time he found himself among
surroundings which were likely to develope his powers. He
seems however to have had but little interest for general society
or for any pursuits save science and mathematics, and he
complained to his friends that he found the other under
graduates disorderly. Luckily he kept a diary, and we can
thus form a fair idea of the course of education of the most
advanced students at an English university at that time. He
had not read any mathematics before coming into residence,
but was acquainted with Sanderson s Logic, which was then
frequently read as preliminary to mathematics. At the be
ginning of his first October term he happened to stroll down
to Stourbridge Fair, and there picked up a book on astrology,
but could not understand it on account of the geometry and
trigonometry, He therefore bought a Euclid, and was sur
prised to find how obvious the propositions seemed. He
THE LIFE AND WORKS OF NEWTON. 321
thereupon read Oughtred s Clavis and Descartes s Geometrie,
the latter of which he managed to master by himself though
with some difficulty. The interest he felt in the subject led
him to take up mathematics rather than chemistry as a
serious study. His subsequent mathematical reading as an
undergraduate was founded on Kepler s Optics, the works of
Yieta, van Schooten s Miscellanies, Descartes s Geometric, and
Wallis s Arithmetica Infinitorum : he also attended Barrow s
lectures. At a later time on reading Euclid more carefully
he formed a high opinion of it as an instrument of education,
and he used to express his regret that he had not applied
himself to geometry before proceeding to algebraic analysis.
There is a manuscript of his, dated May 28, 1665, written
in the same year as that in which he took his B.A. degrep^-
which is the earliest documentary proof of his invention of
fluxions. It was about the same time that he discovered the
binomial theorem (see below, pp. 328 ; 348). J^
On account of the plague the college was asai down in the
summer of 1665, and for a large part of the next year and a half
Newton lived at home. This period was crowded with brilliant
discoveries. He thought out the fundamental principles of his
theory of gravitation, namely, that every particle of matter
attracts every other particle, and he suspected that the attrac
tion varied as the product of their masses and inversely as the
square of the distance between them. He also worked out the
fluxional calculus tolerably completely : thus in a manuscript
dated Nov. 13, 1665, he used fluxions to find the tangent
and the radius of curvature at any point on a curve, and in
October, 1666, he applied them to several problems in the
theory of equations. Newton communicated these results to
his friends and pupils from and after 1669, but they were not
published in print till many years later. It was also while
staying at home at this time that he devised some instruments
for grinding lenses to particular forms other than spherical,
and perhaps hr decomposed solar light into different colours.
Leaving out details and taking round numbers only, his
B. 21
322
THE LIFE AND WORKS OF NEWTON.
reasoning at this time on the theory of gravitation seems
to have been as follows. He suspected that the force which
retained the moon in its orbit about the earth was the
same as terrestrial gravity, and to verify this hypothesis he
proceeded thus. He knew that, if a stone were allowed to
fall near the surface of the earth, the attraction of the
earth (that is, the weight of the stone) caused it to move
through 16 feet in one second. The moon s orbit relative to
o
the earth is nearly a circle ; and as a rough approximation
taking it to be so, he knew the distance of the moon, and
therefore the length of its path ; he also knew the time the
moon took to go once round it, namely, a month. Hence he
could easily find its velocity at any point such as M. He
could therefore find the distance MT through which it would
move in the next second if it were not pulled by the earth s
attraction. At the end of that second it was however at M ,
and therefore the earth must have pulled it through the dis
tance TM in one second (assuming the direction of the earth s
pull to be constant). Now he and several physicists of the
time had conjectured from Kepler s third law that the
attraction of the earth on a body would be found to decrease
as the body was removed further away from the earth in a
proportion inversely as the square of the distance from the
\ S VIEWS ON GRAVITY, 16G6. 323
centre of the earth*; if this were the actual law and gravity
were the sole force which retained the moon in its orbit, then
TM should be to 16 feet in a proportion which was inversely
as the square of the distance of the moon from the centre of
the earth to the square of the radius of the earth. In 1679,
when he repeated the investigation, TM was found to have the
value which was required by the hypothesis, and the verification
was complete; but in 1666 his estimate of the distance of the
moon was inaccurate, and when he made the calculation he
found that TM was about one-eighth less than it ought to
have been on his hypothesis.
This discrepancy does not seem to have shaken his faith in
the belief that gravity extended to the moon and varied in
versely as the square of the distance ; but, from Whiston s *
notes of a conversation with Newton, it would seem that
Newton inferred that some other force probably Descartes s
vortices acted on the moon as well as gravity. This state
ment is confirmed by Pemberton s account of the investigation.
It seems moreover that Newton already believed firmly in the
principle of universal gravitation, that is, that every particle
of matter attracts every other particle, and suspected that the
attraction varied as the product of their masses and inversely
as the square of the distance between them : but it is certain
that he did not then know what the attraction of a spherical
mass on any external point would be, and did not think it
likely that a particle would be attracted by the earth as if
the latter were concentrated into a single particle at its centre.
On his return to Cambridge in 1667 Newton was elected
to a fellowship at his college, and permanently took up his
residence there. In the early part of 1669, or perhaps in 1668,
he revised Barrow s lectures for him (see above, p. 312). The
* The argument was as follows. If v be the velocity of a planet,
r the radius of its orbit taken as a circle, and T its periodic time,
v = 2irr/T. But, if/ be the acceleration to the centre of the circle, we
have/=t7 2 /r. Therefore, substituting the above value of v, f=Tr 2 rfT 2 .
Now by Kepler s third law /"- v;iri< i < a< r ; ; hence /varies inversely as r-.
212
324 THE LIFE AND WORKS OF NEWTON.
end of Lecture xiv. is known to have been written by Newton,
but how much of the rest is due to his suggestions cannot now
be determined. As soon as this was finished he was asked
by Barrow and Collins to edit and add notes to a translation
of Kinckhuysen s Algebra; which he consented to do, but on
condition that his name should not appear iu the matter. In
1670 he also began a systematic exposition of his analysis by
infinite series, the object of which was to express the ordinate
of a curve in an infinite algebraical series every term of which
could be integrated by Wallis s rule (see above, p. 290), his
results on this subject had been communicated to Barrow,
Collins, and others in 1669. This was never finished: the
fragment was published in 1711, but the substance of it had
been printed as an appendix to the Optics in 1704. These
works were only the fruit of Newton s leisure ; most of his time
during these two years being given up to optical researches.
In October, 1669, Barrow resigned the Lucasian chair in
I O
favour of Newton. During his tenure of the professorship,
it was Newton s practice to lecture publicly once a week, for
from half-an-hour to an hour at a time, in one term of each
year, probably dictating his lectures as rapidly as they could
be taken down ; and in the week following the lecture to
devote four hours to appointments which he gave to students
who wished to come to his rooms to discuss the results of the
previous lecture. He never repeated a course, which usually
consisted of nine or ten lectures, and generally the lectures of
one course began from tlie point at which the preceding course
had ended. The manuscripts of his lectures for seventeen out
of the first eighteen years of his tenure are extant.
When first appointed Newton chose optics for the subject
of his lectures and researches, and before the end of 1669 he
had worked out the details of his discovery of the decompo
sition of a ray of white light into rays of different colours by
means of a prism. The complete explanation of the theory of
the rainbow followed from this discovery. These discoveries
formed the subject-matter of the lectures which he delivered
NEWTON S GEOMETRICAL OPTICS, lt)72. :>^">
as Lucasian professor in the years 16G9, 1670, and 1671. The
chief new results were embodied in a paper communicated
to the Royal Society in February, 1672, and subsequently
published in the Philosophical Transactions. The manuscript
of his original lectures was printed in 1729 under the title
Lectiones Opticae. This work is divided into two books,
the first of which contains four sections and the second five.
The first section of the first book deals with the decomposition
of solar light by a prism in consequence of the unequal re-
frangibility of the rays that compose it, and a description
of his experiments is added. The second section contains an
account of the method which Newton invented for the deter
mining the coefficients of refraction of different bodies. This
is done by making a ray pass through a prism of the material
so that the deviation is a minimum ; and he proves that, if the
angle of the prism be i and the deviation of the ray be 8, the
refractive index will be sin i (i + 8) cosec \ i. The third section
is on refractions at plane surfaces ; he here shews that if a ray
pass through a prism with minimum deviation, the angle of
incidence is equal to the angle of emergence most of this
section is devoted to geometrical solutions of different problems.
The fourth section contains a discussion of refractions at curved
surfaces. The second book treats of his theory of colours and
of the rainbow.
By a curious chapter of accidents Newton failed to correct
the chromatic aberration of two colours by means of a couple
of prisms. He therefore abandoned the hope of making a
refracting telescope which should be achromatic, and instead
designed a reflecting telescope, probably on the model of a
small one which he had made in 1668. The form he used is that
still known by his name ; the idea of it was naturally suggested
by Gregory s telescope. In 1672 he invented a reflecting
microscope, and some years later he invented the sextant
which was re-discovered by Hadley in 1731.
His professorial lectures from 1673 to 1683 were on algebra
and the theory of equations, and are described below; but much
326 THE LIFE AND WORKS OF NEWTON.
of his time during these years was occupied with other investi
gations, and I may remark that throughout his life Newton
must have devoted at least as much attention to chemistry and
theology as to mathematics, though his conclusions are not of
sufficient interest to require mention here. His theory of colours
and his deductions from his optical experiments were attacked
with considerable vehemence by Pardies in France, Linus and
Lucas at Liege, Hooke in England, and Huygens in Paris ; but
his opponents were finally refuted. The correspondence which
this entailed on Newton occupied nearly all his leisure in the
years 1672 to 1675, and proved extremely distasteful to him.
Writing on Dec. 9, 1675, he says, "I was so persecuted with
discussions arising out of iny theory of light, that I blamed my
own imprudence for parting with so substantial a blessing as
my quiet to run after a shadow." Again on Nov. 18, 1676, he
observes, " I see I have made myself a slave to philosophy ; but,
if I get rid of Mr Liims s business, I will resolutely bid adieu
to it eternally, excepting what I do for my private satisfaction,
or leave to come out after me ; for I see a man must either
resolve to put out nothing new, or to become a slave to defend
it." The unreasonable dislike to have his conclusions doubted
or to be involved in any correspondence about them was a
prominent trait in Newton s character.
He next set himself to examine the problem as to how
light was really produced, and by the end of 1675 he had
worked out the corpuscular or emission theory a theory to
which he was perhaps led by his researches on the theories of
attraction. Only three ways have been suggested in which
light can be produced mechanically. Either the eye may be
supposed to send out something which, so to speak, feels the
object (as the Greeks believed) ; or the object perceived may
send out something which hits or affects the eye (as assumed
in the emission theory) ; or there may be some medium between
the eye and the object, and the object may cause some change
in the form or nature of this intervening medium and thus
affect the eye (as Hooke and Huygens supposed in the wave
NEWTON S PHYSICAL OPTICS, 1675. 327
or undulatory theory). It will be enough here to say that on
either of the two latter theories all the obvious phenomena of
geometrical optics such as reflexion, refraction, &c., can be
accounted for. Within the present century crucial experiments
have been devised which give different results according as one
or the other theory is adopted ; all these experiments agree
with the results of the undulatory theory and differ from the
results of the Newtonian theory : the latter is therefore un
tenable, but whether the former represents the whole truth and
nothing but the truth is still an open question. Until however
the theory of interference suggested by Young, was worked out
by Fresnel, the hypothesis of Huygens failed to account for all
the facts and was open to more objections than that of Newton.
It should be noted that Newton nowhere expresses an opinion
that the corpuscular theory is true, but always treats it as an
hypothesis from which, if true, certain results would follow : it
would moreover seem that he believed the wave theory to be
intrinsically more probable, and it was only the difficulty of
explaining diffraction on that theory that led him to reject
it. His remarks on other physical subjects shew a similar
caution.
Newton s corpuscular theory was expounded in memoirs
communicated to the Royal Society in December, 1675, which
are substantially reproduced in his Optics, published in 1704.
In the latter work he dealt in detail with his theory of fits of
easy reflexion and transmission, and the colours of thin plates
to which he added an explanation of the colours of thick plates
(bk. ii. part 4) and observations on the inflexion of light
(bk. in.).
Two letters written by Newton in the year 1676 are
sufficiently interesting to justify an allusion to them. Leibnitz,
who had been in London in 1673, had communicated some
results to the Royal Society which he had supposed to be new,
but which it was pointed out to him had been previously proved
by Mouton. This led to a correspondence with Oldenburg,
the secretary of the Society. In 1674 Leibnitz wrote saying
o2cS THE LIFE AND WORKS OF NEWTON.
that he possessed " general analytical methods depending on
infinite series." Oldenburg in reply told him that Newton
and Gregory had used such series in their work. In answer
to a request for information Newton wrote on June 13, 1676,
giving a brief account of his method, but adding the expansions
of a binomial (i.e. the binomial theorem) and of sin" 1 x\ from
the latter of which he deduced that of sin x, this seems to
be the earliest known instance of a reversion of series. He
also inserted an expression for the rectification of an elliptic
arc in an infinite series.
Leibnitz wrote oil Aug. 27 asking for fuller details ; and
Newton in a long but interesting reply, dated Oct. 24, 1676,
and sent through Oldenburg, gives an account of the way in
which he had been led to some of his results.
In this letter, Newton begins by saying that altogether he
had used three methods for expansion in series. His first was
arrived at from the study of the method of interpolation by
which Wallis had found expressions for the area of a circle
and a hyperbola. Thus, by considering the series of expressions
(1- a?*)*, (1 - x 2 )%, (1 - x*y,..., he deduced by interpolations the
law which connects the successive coefficients in the expansions
1 3
of (1 x 2 )", (1 a; 2 ) ,... ; and then by analogy obtained the ex
pression for the general term in the expansion of a binomial,
i.e. the binomial theorem. He says that he proceeded to test
this by forming the square of the expansion of (1 05 2 ) a which
reduced to 1 x 2 ; and he proceeded in a similar way with
other expansions. He next tested the theorem in the case
of ( 1 x 2 ) * by extracting the square root of 1 a; 2 , more
arithitietico. He also used the series to determine the areas of
the circle and the hyperbola in infinite series, and found that the
results were the same as those he had arrived at by other means.
Having established this result, he then discarded the
method of interpolation in series, and employed his binomial
theorem to express (when possible) the ordinate of a curve in
an infinite series in ascending powers of the abscissa, and thus
LETTER TO LEIBNITZ, 1676. 329
by Wallis s method he obtained expressions in an infinite
series for the areas and arcs of curves in the manner described
in the appendix to his Optics and his De Analysi per Equationes
Numero Terminorum Infinitorum (see below, p. 348). He
states that he had employed this second method before the
plague in 1665 66, and goes on to say that he was then obliged
to leave Cambridge, and subsequently (i.e. presumably on his
return to Cambridge) he ceased to pursue these ideas as he
found that Nicholas Mercator had employed some of them in
his Loga/rith/rnotecJvnM^ published in 1668; and he supposed
that the remainder had been or would be found out before he
himself was likely to publish his discoveries.
Newton next explains that he had also a third method, of
which (he says) he had about 1669 sent an account to Barrow
and Collins, illustrated by applications to areas, rectification,
cubature, tkc. This was the method of fluxions ; but Newton
gives no description of it here, though he adds some illustrations
of its use. The first illustration is on the quadrature of the
curve represented by the equation
y - ax m (b + cx") p ,
which he says can be effected as a sum of (m+ l)/n terms if
(m+ l)/n be a positive integer, and which he thinks cannot
otherwise be effected except by an infinite series*. He also
gives a list of other forms, which are immediately integrable,
of which the chief are
x mn ~ l ajO+i)-i
"
> -
a + bx* + cx 2n a + bx
x mn ~ l (a + btff*(c + <br)-\ x mn -"- 1 (a + bx n )* (c + dx")~ * ;
where m is a positive integer and n is any number whatever.
Lastly he points out that the area of any curve can be easily
determined approximately by the method of interpolation
described below (see p. 349) in discussing his Metkodus Differ-
entialis.
* This is not so, the integration i- p>ible if p + (m + l)ln be an
integer.
330 THE LIFE AND WORKS OF NEWTON.
At the end of his letter Newton alludes to the solution of
the "inverse problem of tangents," a subject on which Leibnitz
had asked for information. He gives formulae for reversing
any series, but says that besides these formulae he has two
methods for solving such questions which for the present he
will not describe except by an anagram which being read is
as follows, " Una methodus consistit in extractione fluentis
quail titatis ex aequatione simul involvente fluxioiiem ejus :
altera tantum in assumption e seriei pro quantitate qualibet
incognita ex qua caetera commode derivari possunt, et in
collatione terminorum homologorum aequationis resultantis, ad
eruendos terminos assumptae seriei."
He implies in this letter that he is worried by the questions
he is asked and the controversies raised about every new
matter which he produces, which shew his rashness in publishing
" quod umbram captando eatenus perdideram quietem meam,
rern prorsus substantialem."
Leibnitz did not reply to this letter till June 21, 1677. In
his answer he explains his method of drawing tangents to
curves, which he says proceeds "not by fluxions of lines but
by the differences of numbers"; and he introduces his notation
^f dx and dy for the infinitesimal differences between the co
ordinates of two consecutive points on a curve. He also gives
a solution of the problem to find a curve whose subtangent
is constant, which shews that he could integrate.
In 1679 Hooke, at the request of the Royal Society, wrote
to Newton expressing a hope that he would make further com
munications to the Society and informing him of various facts
then recently discovered. Newton replied saying that he had
abandoned the study of philosophy, but he added that the
earth s diurnal motion might be proved by the experiment of
observing the deviation from the perpendicular of a stone
dropped from a height to the ground an experiment which
was subsequently made by the Society and succeeded. Hooke
in his letter mentioned Picard s geodetical researches ; in
these Picard used a value of the radius of the earth which is
DISCOVERIES IN 1679.
substantially correct. This led Newton to repeat, with Picard s
data, his calculations of 1666 on the lunar orbit, and he
found the verification of his view was complete. He then
proceeded to the general theory of motion under a centripetal
force, and demonstrated (i) the equable description of areas,
(ii) that if an ellipse were described about a focus under a
centripetal force the law was that of the inverse square of the
distance, (iii) and conversely, that the orbit of a particle pro
jected under the influence of such a force was a conic (or, it
may be, he thought only an ellipse). Obeying his rule to
publish nothing which could land him in a scientific contro
versy these results were locked up in his note-books, and it
was only a specific question addressed to him five years later
that led to their publication.
The Universal Arithmetic, which is on algebra, theory of
equations, and miscellaneous problems, contains the substance
of Newton s lectures during the years 1673 to 1683. His
manuscript of it is still extant ; Whiston * extracted a some
what reluctant permission from Newton to print it, and it was
published in 1707. Amongst several new theorems on various
points in algebra and the theory of equations Newton here
enunciated the following important results. He explained that
the equation whose roots are the solution of a given problem
will have as many roots as there are different possible cases ;
and he considered how it happened that the equation to which
a problem led might contain roots which did not satisfy the
original question. He extended Descartes s rule of signs to
give limits to the number of imaginary roots. He used the
* William Whi^tnn^ born in Leicestershire 011 Dec. 9, 1607, educated
at Clare College, Cambridge, of which society he was a fellow, and died
in London on Aug. 22, 17-VJ, wrote several works on astronomy. He
acted as Newton s deputy in the Lucasian chair from 1690, and in 1703
succeeded hirn as professor, but he was expelled in 1711, mainly for
theological reasons. He was succeeded by Nicholas Saunderson, the
blind mathematician, who was born in Yorkshire in 1682 and died at
Christ s College, Cambridge, on April 19, 1739.
332 THE LIFE AND WORKS OF NEWTON.
principle of continuity to explain how two real and unequal
roots might become imaginary in passing through equality,
and illustrated this by geometrical considerations ; thence
he shewed that imaginary roots must occur in pairs. Newton
also here gave rules to find a superior limit to the positive
roots of a numerical equation, and to determine the approxi
mate values of the numerical roots. He further enunciated
the theorem known by his name for finding the sum of the
nth powers of the roots of an equation, and laid the foundation
of the theory of symmetrical functions of the roots of an
equation.
The most interesting theorem contained in the work is
his attempt to find a rule (analogous to that of Descartes for
real roots) by which the number of imaginary roots of an
equation can be determined. He knew that the result which
he obtained was not universally true, but he gave no proof and
did not explain what were the exceptions to the rule. His
theorem is as follows. Suppose the equation to be of the nth
degree arranged in descending powers of x (the coefficient of
x n being positive), and suppose the n + I fractions
n 2 n-I 3 n p+l p+ I 2 n
1 n^l P n~2 2 " ~~r^-~p p 9 " In-l
to be formed and written below the corresponding terms of
the equation, then, if the square of any term when multiplied
by the corresponding fraction be greater than the product
of the terms on each side of it, put a plus sign above it : other
wise put a minus sign above it, and put a plus sign above
the first and last terms. Now consider any two consecutive
terms in the original equation, and the two symbols written
above them. Then we may have any one of the four following
cases : (a) the terms of the same sign and the symbols of the
same sign ; (/?) the terms of the same sign and the symbols of
opposite signs ; (y) the terms of opposite signs and the symbols
of the same sign; (S) the terms of opposite signs and the symbols
of opposite signs. Then it has been shewn that the number of
DISCOVERIES IN 1684. 333
negative roots will not exceed the number of cases (a), and the
number of positive roots will not exceed the number of cases (y);
and therefore the number of imaginary roots is not less than
the number of cases (/3) and (8). In other words the number
of changes of signs in the row of symbols written above the
equation is an inferior limit to the number of imaginary roots.
Newton however asserted that "you may almost know how
many roots are impossible" by counting the changes of sign
in the series of symbols formed as above. That is to say
he thought that in general the actual number of positive,
negative arid imaginary roots could be got by the rule and
not merely superior or inferior limits to these numbers. But
though he knew that the rule was not universal he could
o
not find what were the exceptions to it : this theorem was
subsequently discussed by Campbell, Maclaurin, Euler, and
other writers ; at last in 1865 Sylvester succeeded in proving
the general result*.
In August 1684, Halley came to Cambridge in order to
consult Newton about the law of gravitation. Hooke, Huygens,
Halley, and Wren had all conjectured that the force of the
attraction of the sun or earth on an external particle varied
inversely as the square of the distance. These writers seem to
have independently shewn that, if Kepler s conclusions were
rigorously true, as to which they were uot quite certain, the
law of attraction must be that of the inverse square, but they
could not deduce from the law the orbits of the planets.
Halley explained that their investigations were stopped by
their inability to solve this problem, and asked Newton if he
could find out what the orbit of a planet would be if the law
of attraction were that of the inverse square. Newton imme
diately replied that it was an ellipse, and promised to send or
write out afresh the demonstration of it which he had found
in 1679. This was sent in November, 1684.
Instigated by Halley, Newton now returned to the problem
* See the Proceedings of the London Mathematical Society, 1865,
vol. i., no. 2.
334 THE LIFE AND WORKS OF NEWTON.
of gravitation; and before the autumn of 1684, he had worked
out the substance of propositions 1 19, 21, 30, 32 35 in the
first book of the Principia. These, together with notes on the
laws of motion and various lemmas, were read for his lectures
in the Michaelmas Term, 1684.
In November Halley received Newton s promised com
munication, which probably consisted of the substance of props.
1, 11, and either 17 or Cor. 1 of 13; and thereupon he again
went to Cambridge where he saw "a curious treatise, De Motu,
drawn up since August." Most likely this contained Newton s
manuscript notes of the lectures above alluded to : these notes
are now in the University Library and are headed u De Motu
Corporum" Halley begged that the results might be pub
lished, and finally secured a promise that they shou!4 be sent
to the Royal Society: they were accordingly communicated to
the Society not later than February, 1685, in the paper De
Motu^ which contains the substance of the following propo
sitions in the Principia, book i., props. 1, 4, 6, 7, 10, 11, 15,
17, 32 ; book n., props. 2, 3, 4.
It seems also to have been due to the influence and tact of
Halley at this visit in November, 1684, that Newton under
took to attack the whole problem of gravitation, and practically
pledged himself to publish his results. As yet Newton had
not determined the attraction of a spherical body on an ex
ternal point, nor had he calculated the details of the planetary
motions even if the members of the solar system could be re
garded as points. The first problem was solved in 1685,
probably either in January or February. "No sooner," to
quote from Dr Glaisher s address on the bicentenary of the
publication of the Principia, "had Newton proved this superb
theorem and we know from his own words that he had no
expectation of so beautiful a result till it emerged from his
mathematical investigation than all the mechanism of the
universe at once lay spread before him. When he discovered
the theorems that form the first three sections of book i.,
when he gave them in his lectures of 1684, he was unaware
THE PRINCIPIA. 335
that the sun and earth exerted their attractions as if they
were but points. How different must these propositions have
seemed to Newton s eyes when he realized that these results,
which he had believed to be only approximately true when
applied to the solar system, were really exact ! Hitherto they
had been true only in so far as he could regard the sun as
a point compared to the distance of the planets, or the earth
as a point compared to the distance of the moon a distance
amounting to only about sixty times the earth s radius but
now they were mathematically true, excepting only for the
slight deviation from a perfectly spherical form of the sun,
earth and planets. We can imagine the effect of this sudden
transition from approximation to exactitude in stimulating
Newton s mind to still greater efforts. It was now in his
power to apply mathematical analysis with absolute precision
to the actual problems of astronomy."
Of the three fundamental principles applied in the Principia
we may say that the idea that every particle attracts every
other particle in the universe was formed at least as early as
1666 ; the law of equable description of areas, its consequences,
and the fact that if the law of attraction were that of the
inverse square the orbit of a particle about a centre of force
would be a conic were proved in 1679 ; and lastly the discovery
that a sphere, whose density at any point depends only on the
distance from the centre, attracts an external point as if the
whole mass were collected at its centre was made in 1685.
It was this last discovery that enabled him to apply the first
two principles to the phenomena of bodies of finite size.
The draft of the first book of the Principia was finished
before the summer of 1685, but the corrections and additions
took some time, and the book was not presented to the Royal
Society until April 28, 1686. This book is given up to the
consideration of the motion of particles or bodies in free space
either in known orbits, or under the action of known forces,
or under their mutual attraction. In it Newton generalizes
the law of attraction into a statement that every particle of
336 THE LIFE AND WORKS OF NEWTON.
matter in the universe attracts every other particle with a
force which varies directly as the product of their masses and
inversely as the square of the distance between them ; and he
thence deduces the law of attraction for spherical shells of
constant density. The book is prefaced by an introduction on
the science of dynamics.
The second book of the Principia was completed by the
summer of 1686. This book treats of motion in a resisting
medium, and of hydrostatics and hydrodynamics, with special
applications to waves, tides, and acoustics. He concludes it
by shewing that the Cartesian theory of vortices was incon
sistent both with the known facts and with the laws of motion.
The next nine or ten months were devoted to the third
book. Probably for this he had originally no materials ready.
In it the theorems obtained in the first book are applied to the
chief phenomena of the solar system, the masses and distances
of the planets and (whenever sufficient data existed) of their
satellites are determined. In particular the motion of the
moon, the various inequalities therein, and the theory of the
tides are worked out in detail. He also investigates the
theory of comets, shews that th^y belong to the solar system,
explains how from three observations the orbit can be de
termined, and illustrates his results by considering certain
special comets. The third book as we have it is but little more
than a sketch of what Newton had finally proposed to himself
to accomplish ; his original scheme is among the " Portsmouth
papers, 7 and his notes shew that he continued to work at it
for some years after the publication of the first edition of the
Principia : the most interesting of his memoranda are those
in which by means of fluxions he has carried his results beyond
the point at which he was able to translate them into
geometry*.
* I take this opportunity of saying that I hope shortly to publish a
memoir on the history and compilation of the Principia. The following
brief summary of the contents of the work will give the reader a general
idea of its arrangement. The Principia is preceded by a preface in which
THE PRINCIPIA. 337
The demonstrations throughout the work are geometrical,
but to readers of ordinary ability are rendered unnecessarily
difficult by the absence of illustrations and explanations, and
by the fact that no clue is given to the method by which
Newton says that his object is to apply mathematics to the phenomena of
nature. Among these phenomena motion is one of the most important.
Now motion is the effect of force, and, though he does not know what is
the nature or origin of force, still many of its effects can be measured ;
and it is these that form the subject-matter of the work. The work begins
therefore naturally with an introduction on dynamics or the science of
motion. This commences with eight definitions of various terms such as
mass, momentum, &c. Newton then lays down three laws of motion
which are incapable of exact proof, but are confirmed partly by direct
experiments, partly by the agreement with observation of the deductions
from them. From these he deduces six fundamental principles of
mechanics, and addsan^ apporirHy nn the_motiorL ..of. .falling
projectiles, oscill^ti^s^nj^iiLct^-aiid 4he fimtu&i attractions oJLtwo bodies.
The most important deduction is that of the parallelogram of velocities,
accelerations, and forces.
The first book of the Principia is on the motion of bodies in free
space, and is divided into fourteen sections.
The first section consists of eleven preliminary lemmas treated by the
method of prime and ultimate ratios, and not by that of indivisibles.
The second section commences by shewing that, if a body (such as a
planet) revolve in an orbit subject to a force tending to a fixed point
(such as the sun), the areas swept^o.uJLJt yJ^dii -drawja.4foca -fee-body- t
the pom^rejrj^pjia-plaiie and are p/ojgortional to the times_of jles
them ; and conversely, if the areas be proportional to the times, the force
acting on the body musf be directed: ttr the point. Newton then shews
how, if the orbit be known and the centre of force be given, the law of
force can be determined ; and he finds the law for various curves.
In the third section he applies these propositions to a body which
describes a conic section about a focus, and proves that the force must
vary inversely as the square of the distance, and that Kepler s third law
would necessarily be true of such a system. Conversely he proves that,
if a body were projected in any way and subject to a central force
which varied according to this law, then it must move in a conic section
having tue centre of force in a focus. He concludes (prop. 17, cors. 3
and 4) with a suggestion as to how the effects of disturbing forces
should be calculated : this was first done by the brilliant investigations
of Laplace and Lagrange ; and Laplace says (Mccaniqne celeste, book xv.,
chap, i.) that Lagrange s paper in the Berlin memoirs for 1786 on which
B. 22
338 THE LIFE AND WORKS OF NEWTON.
Newton arrived at his results. The reason why it was pre
sented in a geometrical form appears to have been that the
the modern treatment of the subject is founded was suggested by these
remarks of Newton.
The fourth and fifth sections are devoted to the geometry of conic
sections, especially to the construction of conies which satisfy five con
ditions. In section four one of the conditions is that the focus is given ;
this includes the problem of finding the path of a comet from three
observations which Newton says he found the most difficult problem of
any which he had to solve : curiously enough he gave a second solution
of this problem in book in. prop. 41, which he recommended as more
simple but which is inapplicable in practice.
The sixth section is devoted to determining what at any given time is
the velocity and what is the position of a body which is describing a
given conic about a centre of attraction in a focus : together with various
converse problems. To effect this Newton had to find the area of a
sector of a conic. This is easily done for the parabola. He then
endeavours to shew that exact quadrature of any closed oval curve
having no infinite branches (such as the ellipse) is impossible : the proof
is not correct as it stands, since the result is not true for ovals of the
form yi = (2n) 2m x t2m ( 2n -V (a? n -x 2n ), where m and n are positive inte
gers ; Newton seems himself to have felt some doubt about inserting it,
though he believed the result to be true. An exact quadrature being
impossible, he proceeds to give three ways, two arithmetical and one
geometrical, of approximating to the sectorial area of an ellipse as closely
as is desired.
The seventh section is given up to the discussion of motion in a
straight line under a force which varies inversely as the square of the
distance, and its comparison with motion in a conic under the same
force. He concludes by giving a general solution for all the problems
considered in this section for any law of force. He here determines
geometrically what is equivalent to finding the integral of x (ax - x 2 )-?.
The eighth section contains general solutions for any orbit described
under any central force of some of the problems previously considered. In
proposition 40 he states that the kinetic energy acquired by a body in
moving from one point to another point is equal to the total work done
by the force between those two points.
In the ninth section he discusses the case where the orbit is in
motion in its own plane round the centre of force, and treats in detail of
the motion of the apse-line, and the forces by which a given motion
would be produced. Newton applied this reasoning (prop. 45, cor. 2) to
the case of the moon, but the resulting motion of the apses only came out
THE PRINCIPIA.
infinitesimal calculus was then unknown, and, had Newton
used it to demonstrate results which were in themselves
about one-half of the actual amount. The approximation was in fact not
carried to a sufficiently high order. Newton was aware of the discrepancy,
and as he explained the similar difficulty in the case of the nodes it had
been long suspected (ex. (jr. Godfray s Lunar Theory, 2nd edition, art. 68)
that the scholium in the first edition to book in. prop. 35 meant that he
had found the explanation. Nowhere in the Principia does he however
give any hint as to how this was effected, and the true explanation of a
difference which had long formed an obstacle to the universal acceptance
of the Newtonian system was first given by Clairaut in 1752. The
Portsmouth papers contain Newton s original work, and shew that he
had obtained the true value by carrying the approximation to a sufficiently
high order. It also seems clear from these papers that Newton gave the
corollary to book i. prop. 45 as a mere illustration of the motion of the
apses in orbits which are nearly circular and did not mean it to apply to
the moon, but by an inadvertence in the .second and third editions a
reference to it as an authority for a result connected with the moon was
added which would naturally deceive any reader. Newton left most of the
revision of the second edition to Gotes and it is probable that the mistake
is due to a blunder of the editor. Other questions connected with lunar
and planetary irregularities are also discussed in this proposition, but the
extreme conciseness of Newton misled all the early commentators, and
even Laplace in his Systeme du nwnde published in 1796 speaks of Newton
as having only roughly sketched out this part of the subject, leaving it to
be completed when the calculus should be further perfected ; but in the
last volume of his Mecanique celeste published in 1825 he says that on
more careful reading he has no hesitation in regarding it as among the
most profound parts of the work.
The tenth section is devoted to the consideration of the motion of
bodies along given surfaces, but not in planes passing through the centre
of force ; with special reference to the vibration of pendulums, and the
determination of the accelerating effect of gravity. In connection with
the latter problem Newton investigates the chief geometrical properties
of cycloids, epicycloids, and hypocycloids.
In the eleventh section are considered the problems connected with
motion in orbits where the centre of force is disturbed, or where the
moving body is disturbed by other forces. Until the calculus of variations
was invented by Lagrange in 1755 it was impossible to do more than
sketch out the principles on which the problem should be solved, and
Laplace in his Mecanique celeste was the first to work out most of the
questions in any detail. Newton commences by considering the dis-
99 9
ttft *
340 THE LIFE AND WORKS OF NEWTON.
opposed to the prevalent philosophy of the time, the contro
versy as to the truth of his results would have been hampered
tnrbance produced by the mutual action of two bodies revolving round
one another. He then proceeds to consider the problem of three or more
bodies which mutually attract one another. He first solves the question
completely if the force of attraction varies directly as the distance. He
next takes the case of three bodies moving under their mutual attractions
as in nature. This problem has not been yet solved generally, but in
Newton s day it was beyond any analysis of which he had the command :
he contrived however to work out roughly the chief effects of the dis
turbing action of the sun on the motion of the moon (prop. 66) : this
proposition was singled out by Lagrange as the most striking single
illustration of the genius of Newton. To this proposition twenty-two
corollaries are appended in which it is applied to determine the motion
in longitude, in latitude, the annual equation, the motion of the apse
line, and of the nodes, the evection, the change of inclination of the
plane of the lunar orbit, the precession of the equinoxes, and the theory
of the tides. The greater part of the third book consists of the numerical
application of these principles to the case of the moon and the earth.
Lastly Newton shewed how from the motion of the nodes the interior
constitution of the body could be roughly determined.
Up to this point Newton had generally treated the bodies with which
he dealt as if they were particles. He now proceeds in section twelve to
consider the attractions of spherical masses which are either of uniform
density, or whose density at any point is a single-valued function of the
distance of the point from the centre of the sphere. These are worked
out for any law of attraction.
In section thirteen he gives some general theorems 011 the theory of
attractions and some propositions dealing with the attractions of solids
of revolution, but these problems are almost insoluble without the aid of
the infinitesimal calculus, and the Newtonian account of them is in
complete.
The fourteenth section contains a statement of some theories and
experiments in physical optics ; and a solution by geometry of some
problems in geometrical optics, particularly on the form of aplanatic
refracting surfaces of revolution.
The second book of the Principia is concerned with hydromechanics,
and especially with motion in a resisting medium. These questions are
not worked out so completely as those treated in the first book ; and,
though this book provided the basis on which much of the subsequent
work of Daniel Bernoulli, Clairaut, D Alembert, Euler, and Laplace was
erected, it is not of the same epoch-making character as the first book.
THE PRINCIPIA. 341
by a dispute concerning the validity of the methods used
in proving them. He therefore cast the whole reasoning
This book is divided into nine sections. The motion of bodies in a
medium where the resistance varies directly as the velocity is considered
in the first section. The motion where the resistance varies as the
square of the velocity is discussed in the second section. The motion
where the resistance can be expressed as the sum of two terms, one of
which varies as the velocity and the other as the square of the velocity,
is dealt with in the third section. The second section contains (prop. 25)
a construction for the shape of the solid of least resistance. No proof is
given and it had been long somewhat of a mystery to know how
Newton had contrived to solve the problem without the use of the
calculus of variations. Newton s demonstrations (there are two of them)
have been recently discovered in the Portsmouth collection.
The fourth section is devoted to spiral motion in a resisting medium.
The fifth to the theory of hydrostatics and elastic fluids. The sixth to
the motion of pendulums in a resisting medium. The seventh to hydro
dynamics, and especially to the motion of projectiles in air and other
fluids. The eighth to the theory of waves, including the principles from
which the chief effects of the wave hypotheses in light and sound are
calculated, and in particular the velocity of sound is determined.
In the ninth section Newton discusses the Cartesian theory of vortices
(see above, p. 278). He begins by shewing that, if there were no internal
friction, the motion would be impossible. He must therefore assume
some law of friction, and as a working hypothesis he supposes that "the
resistance arising from want of lubricity in the parts of a fluid is,
cater is paribus, proportional to the velocity with which the parts of the
fluid are separated from each other." This hypothesis, as he himself
remarks, is probably not altogether correct, but he thinks that it will
give a general idea of the motion. He next proves that on this hypo
thesis the motion would be unstable. He must therefore suppose that
some constraining force prevents this catastrophe, and he then shews
that in that case Kepler s third law could not be true. Lastly he shews
by independent reasoning that the hypothesis must lead to results which
are inconsistent with Kepler s other two laws, and that both the vortices
and the motion of the planets would be necessarily unstable. Several
continental mathematicians made attempts to modify the Cartesian
hypothesis so as to avoid these conclusions, but they could never
explain one phenomenon without introducing fresh difficulties. It may
be taken that by 1750 the Cartesian theory was finally abandoned.
The third book is headed On the system of the world and is concerned
chiefly with the application of the results of the first book to the solar
system. It is introduced by certain rules of philosophizing, and a list of
342 THE LIFE AND WORKS OF NEWTON.
into a geometrical shape which, if somewhat longer, can at
any rate be made intelligible to all mathematical students. So
closely did he follow the lines of Greek geometry that he con-
certain data obtained from astronomical observations. The rules are
(i) we may only assume as the possible causes of phenomena such causes
as are sufficient to explain them and are also verae causae, a vera
causa being one which is capable of detection and such that its con
nection with the phenomenon can be ultimately shewn by independent
evidence ; (ii) effects of a similar kind must have similar causes;
(iii) whatever properties of bodies are found by experience to be in
variable should be assumed to be so in places where direct experiments
cannot be made.
Newton commences by illustrating the universality of the law of
gravitation, and sketches out the principles which lead him to think that
the solar system is necessarily stable : he determines the mass of the
moon, the masses of the planets, their distances from the sun, and their
figures. In the first edition he estimated (prop. 37) that the ratio of the
mass of the moon to that of the earth was approximately that of 1 : 26,
in the second and third editions this was altered to a ratio which is
nearly that of 1 : 40 ; but except for the mass of the moon he approximates
to the results now known with astonishing closeness. He finds the
disturbing force exerted by the sun on the moon, and considers the five
chief irregularities in the orbit of the moon. He next discusses the solar
and lunar tides ; determines the precession of the equinoxes ; and finally
shews how the elements of a comet can be determined by three obser
vations, and applies his results to certain comets : before this time it had
been commonly believed that comets had nothing to do with the solar
system, though in 1681 Dorffel had shewn that the path of the great
comet of 1680 was a parabola having the sun at its focus.
Lastly the Principia is concluded by a general scholium containing
reflections on the constitution of the universe, and on "the eternal, the
infinite, and perfect Being" by whom it is governed.
The chief alterations in the second edition, published in 1713, were
the substitution of simpler proofs for some of the propositions in the
second section of the first book; a more full and accurate investigation
(founded on some fresh experiments made by Newton about the year
1690) of the resistance of fluids in the seventh section of the second
book; and the addition of a detailed examination of the causes of
the precession of the equinoxes and the theory of comets in the third book.
The chief alterations in the third edition, published in 1726, were in
the scholium on fluxions ; and the addition of a new scholium on the
motion of the moon s nodes (book in., prop. 53).
THE PRINCIPIA. 343
stantly used graphical methods, and represented forces, velocities,
and other magnitudes in the Euclidean way by straight lines
(ex. gr. book I., lemma 10), and not by a certain number of units.
The latter and modern method had been introduced by Wallis,
and must have been familiar to Newton. The effect of his
confining himself rigorously to classical geometry is that the
Principia is written in a language which is archaic, even if
not unfamiliar.
The adoption of geometrical methods in the Principia for
purposes of demonstration does not indicate a preference on
Newton s part for geometry over analysis as an instrument /
of research, for it is known now that Newton used the fluxioriaf^
calculus in the first instance in finding some of the theorems,
especially those towards the end of book I. and in book n. ;
and in fact one of the most important uses of that calculus-
is stated in book n., lemma 2. But it is only just to remark
that, at the time of its publication and for nearly a century
afterwards, the differential and fluxional calculus were not fully-
developed and did not possess the same superiority over the
method he adopted which they do now ; and it is a matter for
astonishment that when Newton did employ the calculus he
was able to use it to so good an effect. The ability shewn in
the translation in a few months of theorems so numerous and
of so great complexity into the language of the geometry of
Archimedes and Apollonius is I suppose unparalleled in the
history of mathematics.
The printing of the work was slow and it was not finally
published till the summer of 1687. The whole cost was borne
by Halley who also corrected the proofs and even put his own
researches on one side to press the printing forward. The
conciseness, absence of illustrations, and synthetical character
of the book restricted the numbers of those who were able to
appreciate its value ; and, though nearly all competent critics
admitted the validity of the conclusions, some little time
elapsed before it affected the current beliefs of educated men.
I should be inclined to say (but on this point opinions differ
344 THE LIFE AND WORKS OF NEWTON.
widely) that within ten years of its publication it was gene
rally accepted in Britain as giving a correct account of the
laws of the universe; it was similarly accepted within about
twenty years on the continent, except in France where the
Cartesian hypothesis held its ground until Voltaire in 1738
took up the advocacy of the Newtonian theory.
The manuscript of the Principia was finished by 1686.
Newton devoted the remainder of that year to his paper on
physical optics, the greater part of which is given up to the
subject of diffraction (see above, p. 327).
In 1687 James II. having tried to force the university to
admit as a master of arts a Roman Catholic priest who refused
to take the oaths of supremacy and allegiance, Newton took
a prominent part in resisting the illegal interference of the
king, and was one of the deputation sent to London to protect
the rights of the university. The active part taken by
Newton in this affair led to his being in 1689 elected member
for the university. This parliament only lasted thirteen months,
and on its dissolution he gave up his seat. He was subse
quently returned in 1701, but he never took any prominent
part in politics.
On his coining back to Cambridge in 1690 he resumed his
mathematical studies and correspondence. If he lectured at
this time (which is doubtful), it was on the subject-matter of
the Principia. The two letters to Wallis, in which he explained
his method of fluxions and fluents, were written in 1692 and
published in 1693. Towards the close of 1692 and throughout
the two following years Newton had a long illness, suffering
from insomnia and general nervous irritability. Perhaps he
never quite regained his elasticity of mind, and, though after
his recovery he shewed the same power in solving any question
propounded to him, he ceased thenceforward to do original
work on his own initiative, and it was somewhat difficult to
stir him to activity in new subjects.
In 1694 Newton began to collect data connected with the
irregularities of the moon s motion with the view of revising
PUBLICATION OF THE OPTICS, 1704. 345
the part of the Principia which dealt with that subject. To
render the observations more accurate he forwarded to Flam-
steed* a table of corrections for refraction which he had
previously made. This was not published till 1721, when
Halley communicated it to the Royal Society. The original
calculations of Newton and the papers connected with it are
in the Portsmouth collection, and shew that Newton obtained
it by finding the path of a ray by means of quadratures in a
manner equivalent to the solution of a differential equation.
As an illustration of Newton s genius I may mention that even
as late as 1754 Euler failed to solve the same problem. In
1782 Laplace gave a rule for constructing such a table, and
his results agree substantially with those of Newton.
I do not suppose that Newton would in any case have
produced much more original work after his illness ; but his
appointment in 1696 as warden, and his promotion in 1699
to the mastership of the Mint at a salary of 1 500 a year,
brought his scientific investigations to an end, though it was
only after this that many of his previous investigations were
published in the form of books. In 1696 he moved to London,
in 1701 he resigned the Lucasian chair, and in 1703 he was
elected president of the Royal Society.
In 1704 Newton published his Optics which contains the
results of the papers already mentioned (see above, p. 327).
To the first edition of this book were appended two minor
works which have no special connection with optics ; one being
on cubic curves, the other on the quadrature of curves and on
fluxions. Both of them were old manuscripts with which
* John Flamsteed, born at Derby in 1646 and died at Greenwich in
1719, was one of the most distinguished astronomers of this age, and
the first astronomer-royal. Besides much valuable work in astronomy
he invented the system (published in 1680) of drawing maps by pro
jecting the surface of the sphere on an enveloping cone, which can then
be unwrapped. His life by B. F. Baily was published in London in
1835, but various statements in it should be read side by side with
those in Brewster s life of Newton. Flamsteed was succeeded as as
tronomer-royal by Edmund Halley (see below, p. 387).
346 THE LIFE AND WORKS OF NEWTON.
his friends and pupils were familiar, but they were here pub
lished urbi et orbi for the first time.
The first of these appendices is entitled jBnumeratio Linea-
rum Tertii Ordinis* , the object seems to be to illustrate the
use of analytical geometry, and as the application to conies was
well known Newton selected the theory of cubics.
He begins with some general theorems, and classifies
curves according as to whether their equations are alge
braical or transcendental : the former being cut by a straight
line in a number of points (real or imaginary) equal to the
degree of the curve, the latter being cut by a straight line in
an infinite number of points. Newton then shews that many
of the most important properties of conies have their analogues
in the theory of cubics, and he discusses the theory of asymp
totes and curvilinear diameters.
After these general theorems he commences his detailed
examination of cubics by pointing out that a cubic must have
at least one real point at infinity. If the asymptote or tangent
at this point be at a finite distance, it may be taken for the axis
of y. This asymptote will cut the curve in three points alto
gether, of which at least two are at infinity. If the third
point be at a finite distance, then (by one of his general theorems
on asymptotes) the equation can be written in the form
xy* + hy = ax ?l + bx 2 4- ex + d,
where the axes of x and y are the asymptotes of the hyperbola
which is the locus of the middle points of all chords drawn
parallel to the axis of y ; while, if the third point in which
this asymptote cuts the curve be also at infinity, the equation
can be written in the form
xy - ax 3 + bx* + ex + d.
Next he takes the case where the tangent at the real point
at infinity is not at a finite distance. A line parallel to the
* On this work and its bibliography, see my memoir in the Transactions
of the London Mathematical Society, 1891, vol. xxn., pp. 104143.
CLASSIFICATION OF CUBIC CURVES. 347
direction in which the curve goes to infinity may be taken
as the axis of y. Any such line will cut the curve in three
points altogether, of which one is by hypothesis at infinity, and
one is necessarily at a finite distance. He then shews that, if
the remaining point in which this line cuts the curve be at a
finite distance, the equation can be written in the form,
y* = ax 3 + bx* + ex + d ;
while, if it be at an infinite distance, the equation can be written
in the form
y = ax 3 + bx 2 + cx + d.
Any cubic is therefore reducible to one of four charac
teristic forms. Each of these forms is then discussed in detail,
and the possibility of the existence of double points, isolated
ovals, &c. is worked out. The final result is that in all there
are seventy-eight possible forms which a cubic may take. Of
these Newton enumerated only seventy-two ; four of the re
mainder were mentioned by Stirling in 1717, one by Nicole
in 1731, and one by Nicholas Bernoulli about the same time.
In the course of the work Newton states the remarkable
theorem that, just as the shadow of a circle (cast by a luminous
point on a plane) gives rise to all the conies, so the shadows of
the curves represented by the equation y 2 = ax 3 + bx 2 + cx + d
give rise to all the cubics. This remained an unsolved puzzle
until 1731, when Nicole and Clairaut gave demonstrations of
it : a better proof is that given by Murdoch in 1740, which
depends on the classification of these curves into five species
according as to whether their points of intersection with the
axis of x are real and unequal, real and two of them equal (two
cases), real and all equal, or finally two imaginary and one
real.
In this tract Newton also discusses double points in the
plane and at infinity, the description of curves satisfying given
conditions, and the graphical solution of problems by the use
of curves.
The second appendix to the Ojifir* is entitled De Qundrn-
348 THE LIFE AND WORKS OF NEWTON.
tura Curvarum. Most of it had been communicated to Barrow
in 1668 or 1669, and probably was familiar to Newton s pupils
and friends from that time onwards. It consists of two parts.
The bulk of the first part is a statement of Newton s
method of effecting the quadrature and rectification of curves
~~by means of infinite series (see above, p. 329) : it is noticeable
as containing the earliest use in print of literal indices, and also
the first printed statement of the binomial theorem, but these
are introduced only incidentally. The main object is to give
rules for developing a function of a? in a series in ascending
powers of x, so as to enable mathematicians to effect the
quadrature of any curve in which the ordinate y can be ex
pressed as an explicit algebraical function of the abscissa x.
Wallis had shewn how this quadrature could be found when y
was given as a sum of a number of multiples of powers of x,
and Newton s rules of expansion here established rendered
possible the similar quadrature of any curve whose ordinate
can be expressed as the sum of an infinite number of such
terms. In this way he effects the quadrature of the curves
but the results are of course expressed as infinite series. He
then proceeds to curves whose ordinate is given as an implicit
function of the abscissa ; and he gives a method by which y
can be expressed as an infinite series in ascending powers of 05,
but the application of the rule to any curve demands in general
such complicated numerical calculations as to render it of little
value. He concludes this part by shewing that the rectification
of a curve can be effected in a somewhat similar way. His
process is equivalent to finding the integral with regard to x
of (l+y 2 )* in the form of an infinite series. I should add
that Newton indicates the importance of determining whether
the series are convergent an observation far in advance of
his time but he knew of no general test for the purpose ;
and in fact it was not until Gauss and Cauchy took up the
NEWTON S THEORY OF FLUXIONS.
question that the necessity of such limitations were commonly
recognized.
The part of the appendix which I have just described is
practically the same as Newton s manuscript De Analysi per
Equationes Numero Terminorum InfinitaSj which was subse
quently printed in 1711. It is said that this was originally
intended to form an appendix to Kinckhuysen s Algebra (see
above, p. 324). The substance of it was communicated to
Barrow, and by him to Collins, in letters of July 31 and Aug.
12, 1669; and a summary of part of it was included in the
letter of Oct. 24, 1676, sent to Leibnitz.
It should be read in connection with Newton s Metliodus
Differentialis, published in 1736. Some additional theorems
are there given, and he discusses his method of interpolation,
which had been briefly described in the letter of Oct. 24, 1676.
The principle is this. If y = <$> (x) be a function of x and if
when x is successively put equal to a 1? ,..., the values of y
be known and be b l9 6 2 ,..., then a parabola whose equation is
y = p + qx + rx 2 + . . . can be drawn through the points (a 1? &,),
(a 2 , 6 2 ), ..., and the ordinate of this parabola may be taken as
an approximation to the ordinate of the curve. The degree
of the parabola will of course be one less than the number of
given points. Newton points out that in this way the areas
of any curves can be approximately determined.
The second part of this appendix to the Optics contained a
description of Newton s method of fluxions. This is best con
sidered in connection with Newton s manuscript on the same
subject which was published by John Colsoii in 1736, and of
which it is a summary.
The fiuxional calculus is one form of the infinitesimal
calculus expressed in a certain notation, just as the differential
calculus is another aspect of the same calculus expressed in a
different notation. Newton assumed that all geometrical mag
nitudes might be conceived as generated by continuous motion;
thus a line may be considered as generated by the motion of a
point, a surface by that of a line, a solid by that of a surface,
350 THE LIFE AND WORKS OF NEWTON.
j.
fc
a plane angle by the rotation of a line, and so on. The quantity
thus generated was denned by him as the fluent or flowing
quantity. The velocity of the moving magnitude was denned
the fluxion of the fluent. This seems to be the earliest
definite recognition of the idea of a continuous function, though
it had been foreshadowed in some of Napier s papers.
The following is a summary of Newton s treatment of
fluxions. There are two kinds of problems. The object of the
first is to find the fluxion of a given quantity, or more generally
"the relation of the fluents being given, to find the relation of
their fluxions.^ This is equivalent to differentiation. The object
of the second or inverse method of fluxions is from the fluxion
or some relations involving it to determine the fluent, or more
generally " an equation being proposed exhibiting the relation
of the fluxions of quantities, to find the relations of those quan
tities, or fluents, to one another*. " This is equivalent either to
integration which Newton termed the method of quadrature,
or to the solution of a differential equation which was called
by Newton the inverse method of tangents. The methods
for solving these problems are discussed at considerable length.
Newton then went on to apply these results to questions
connected with the maxima and minima of quantities, the
method of drawing tangents to curves, and the curvature of
curves (namely, the determination of the centre of curvature,
the radius of curvature, and the rate at which the radius of
curvature increases). He next considered the quadrature of
curves, and the rectification of curvesf. In finding the maxi
mum and minimum of functions of one variable we regard
the change of sign of the difference between two consecutive
values of the function as the true criterion : but his argument
is that when a quantity increasing has attained its maximum
it can have no further increment, or when decreasing it has
attained its minimum it can have no further decrement ; conse
quently the fluxion must be equal to nothing.
* Colson s edition of Newton s manuscript, pp. xxi. xxii.
f Ibid., pp. xxii. xxiii.
NEWTON S TIIKOKY OF FLUXIONS. 351
It has been remarked that neither Newton nor Leibnitz
produced a calculus, that is a classified collection of rules ; and *
that the problems they discussed were treated from first prin
ciples. That no doubt is the usual sequence in the history of
such discoveries, though the fact is frequently forgotten by
subsequent writers. In this case I think the statement, so far
as Newton s treatment of the differential or fluxional part of
the calculus is concerned, is incorrect, as the foregoing account
sufficiently shews.
If a flowing quantity or fluent were represented by x,
Newton denoted its fluxion by x, the fluxion <t>f x or second
fluxion of x by x, and so on. Similarly the fluentbf x was de
noted by [ccj, or sometimes by x or []. The infinitely small
part by which a fluent such as x increased in a small interval of
time measured by o was called the moment of the fluent ; and
its value was shewn * to be xo. Newton adds the important
remark that thus we may in any problem neglect the terms
multiplied by the second and higher powers of o, and we can
always find an equation between the coordinates x, y of a
point on a curve and their fluxions x, y. It is an application of
this principle which constitutes one of the chief values of the
calculus ; for if we desire to find the effect produced by
several causes on a system, then, if we can find the effect pro
duced by each cause when acting alone in a very small time,
the total effect produced in that time will be equal to the sum
of the separate effects. I should here note the fact that Vince
and other English writers in the eighteenth century used x to
denote the increment of x and not the velocity with which it
increased ; that is, x in their writings stands for what Newton
would have expressed by xo and what Leibnitz would have
written as dx.
I need not discuss in detail the manner in which Newton
treated the problems above mentioned. I will only add that,
in spite of the form of his definition, the introduction into
* Colson s edition of Newton s manuscript, p. 24.
352 THE LIFE AND WORKS OF NEWTON.
geometry of the idea of time was evaded by supposing that
some quantity (e.g. the abscissa of a point on a curve) increased
equably; and the required results then depend on the rate at
which other quantities (e.g. the ordinate or radius of curvature)
increase relatively to the one so chosen*. The fluent so chosen
is what we now call the independent variable ; its fluxion was
termed the "principal fluxion;" and of course, if it were denoted
by x, then x was constant, and consequently x = 0.
There is no question that Newton used the method of
fluxions in 1666, and it is practically certain that accounts of it
were communicated in manuscript to friends and pupils from
and after 1669. The manuscript, from which most of the
above summary has been taken, is believed to have been written
between 1671 and 1677, and to have been in circulation at
Cambridge from that time onwards. It was unfortunate that
it was not published at once. Strangers at a distance naturally
judged of the method by the letter to Wallis in 1692, or by the
Tractatus de Quadratures Curvarum, and were not aware that
it had been so completely developed at an earlier date. This
was the cause of numerous misunderstandings.
At the same time it must be added that all mathematical
analysis was leading up to the ideas and methods of the infi
nitesimal calculus. Foreshado wings of the principles and
even of the language of that calculus can be found in the
writings of Napier, Kepler, Cavalieri, Pascal, Fermat, Wallis,
and Barrow. It was Newton s good luck to come at a time
when everything was ripe for the discovery, and his ability
_ enabled him to construct almost at once a complete calculus.
The notation of the fluxional calculus is for most purposes
less convenient than that of the differential calculus. The
latter was invented by Leibnitz in 1675, and published in
1684 some nine years before the earliest printed account of
Newton s method of fluxions. But the question whether the
general idea of the calculus expressed in that notation was
obtained by Leibnitz from Newton or whether it was invented
* Colson s edition of Newton s manuscript, p. 20.
LIST OF NEWTON S WORKS. 353
independently gave rise to a long and bitter controversy. The
leading facts are given in the next chapter. The question is
one of considerable difficulty, but I will here only say that
from what I have read of the voluminous literature on the
question, I think on the whole it points to the fact that
Leibnitz obtained the idea of the differential calculus from a
manuscript of Newton s which he saw in 1675. I believe how
ever that the prevalent opinion is that the inventions were
independent.
The remaining events of Newton s life require little or no
comment. In 1705 he was knighted. From this time onwards
he devoted much of his leisure to theology, and wrote at great
length on prophecies and predictions, subjects which had always
been of interest to him. His Universal Arithmetic was pub
lished by Whiston in 1707, and his Analysis by Infinite Series
in 1711 ; but Newton had nothing to do with the preparation
of either of these for the press. His evidence before the House
of Commons in 1714 on the determination of longitude at sea
marks an important epoch in the history of navigation.
The dispute with Leibnitz as to whether he had derived
the ideas of the differential calculus from Newton or invented
it independently originated about 1708, and occupied much of
Newton s time, especially between the years 1709 and 1716.
In 1709 Newton was persuaded to allow Cotes to prepare
the long-talked-of second edition of the Principia : it was
issued in March 1713. A third edition was published in 1726
under the direction of Henry Pemberton. In 1725 Newton s
health began to fail. He died on March 20, 1727, and eight
days later was buried with great state in Westminster Abbey.
His chief works, taking them in their order of publication,
are the Principia, published in 1687 ; the Optics (with appen
dices on cubic curves, the quadrature and rectification of curves
by tJie use of infinite series, and the method of fluxions),
published in 1704; the Universal Arithmetic, published in
1707; the Analysis per Series, Flaxiones, &c., published in
1711; the Lectiones Opticae, published in 1729; the Method
B. 23
354 THE LIFE AND WORKS OF NEWTON.
9
of Fluxions, &c. (i.e. Newton s manuscript on fluxions), trans
lated by J. Colson and published in 1736; and the Methodus
Differentials, also published in 1736.
In appearance Newton was short, and towards the close of
his life rather stout, but well set, with a square lower jaw,
brown eyes, a very broad forehead, and rather sharp features.
His hair turned grey before he was thirty, and remained thick
and white as silver till his death.
As to his manners, he dressed slovenly, was rather languid,
and was often so absorbed in his own thoughts as to be
anything but a lively companion. Many anecdotes of his
extreme absence of mind when engaged in any investigation
have been preserved. Thus once when riding home from
Gran th am he dismounted to lead his horse up a steep hill,
when he turned at the top to remount he found that he had
the bridle in his hand, while his horse had slipped it and gone
away. Again on the few occasions when he sacrificed his time
to entertain his friends, if he left them to get more wine or for
any similar reason, he would as often as not be found after the
lapse of some time working out a problem, oblivious alike of
his expectant guests and of his errand. He took no exercise,
indulged in no amusements, and worked incessantly, often
spending eighteen or nineteen hours out of the twenty-four in
writing.
In character he was religious and conscientious, with an
exceptionally high standard of morality, having, as Bishop
Burnet said, "the whitest soul" he ever knew. Newton was
always perfectly straightforward and honest, but in his con
troversies with Leibnitz, Hooke, and others, though scrupulously
just, he was not generous; and it would seem that he frequently
took offence at a chance expression when none was intended.
He modestly attributed his discoveries largely to the admirable
work done by his predecessors ; and once explained that, if he
had seen farther than other men, it was only because he had
stood on the shoulders of giants. He summed up his own
estimate of his work in the sentence, " I do not know what I
CHARACTER OF NEWTON. 355
may appear to the world; but to myself I seem to have been
only like a boy, playing on the sea-shore, and diverting myself,
in now and then finding a smoother pebble, or a prettier shell
than ordinary, whilst the great ocean of truth lay all undis
covered before me." He was morbidly sensitive to being in
volved in any discussions. I believe that, with the exception of
his papers on optics, every one of his works was published only
under pressure from his friends and against his own wishes.
There are several instances of his communicating papers and
results on condition that his name should not be published :
thus when in 1669 he had at Collins s request solved some
problems on harmonic series and on annuities which had
previously baffled investigation, he only gave permission that
his results should be published " so it be," as he says, " without
my name to it : for I see not what there is desirable in public
esteem, were I able to acquire and maintain it : it would
perhaps increase my acquaintance, the thing which I chiefly
study to decline."
In intellect he has never been surpassed and probably never
been equalled. Of this his extant works are the only proper
test. Perhaps the most wonderful single illustration of his
powers was the composition in seven months of the first book
of the Principia.
As specific illustrations of his ability I may mention his so
lutions of the problem of Pappus, of John Bernoulli s challenge,
and of the question of orthogonal trajectories. The problem
of Pappus is to find the locus of a point such that the rectangle
under its distances from two given straight lines shall be in a
given ratio to the rectangle under its distances from two other
given straight lines. Many geometricians from the time of
Apollonius had tried to find a geometrical solution and had
failed, but what had proved insuperable to his predecessors
seems to have presented little difficulty to Newton who gave
an elegant demonstration that the locus was a conic. Geometry,
said Lagrange when recommending the study of analysis to
his pupils, is a strong bow, but it is one which only a Newton
232
356 THE LIFE AND WORKS OF NEWTON.
can fully utilize. As another example I may mention that in
1696 John Bernoulli challenged mathematicians (i) to deter
mine the brachistochrone, and (ii) to find a curve such that
if any line drawn from a fixed point cut it in P and Q
then OP n + OQ n would be constant. Leibnitz solved the first
of these questions after an interval of rather more than six
months, and then suggested they should be sent as a challenge
to Newton and others. Newton received the problems on
Jan. 29, 1697, and the next day gave the complete solutions
of both, at the same time generalizing the second question.
An almost exactly similar case occurred in 1716 when Newton
was asked to find the orthogonal trajectory of a family of
curves. In five hours Newton solved the problem in the form
in which it was propounded to him and laid down the prin
ciples for finding trajectories.
It is almost impossible to describe the effect of Newton s
writings without being suspected of exaggeration. But, if
the state of mathematical knowledge in 1669 or at the death
of Pascal or Fermat be compared with what was known
in 1687, it will be seen how immense was the advance. In
fact we may say that it took mathematicians half a century or
more before they were able to assimilate the work which
Newton had produced in those twenty years.
In pure geometry, Newton did not establish any new
methods, but no modern writer has shewn the same power
in using those of classical geometry. In algebra and the
theory of equations, he introduced the system of literal
indices, established the binomial theorem, and created no in
considerable part of the theory of equations : one rule which
he enunciated in this subject remained till a few years ago as
an unsolved riddle which had overtaxed the resources of
succeeding mathematicians. In analytical geometry, he intro
duced the modern classification of curves into algebraical and
transcendental ; and established many of the fundamental
properties of asymptotes, multiple points, and isolated loops,
illustrated by a discussion of cubic curves. The fluxional or
NEWTON S DISCOVERIES. 357
infinitesimal calculus was invented by Newton in or before
the year 1666, and circulated in manuscript amongst his
friends in and after the year 1669, though no account of the
method was printed till 1693. The fact that the results are
now-a-days expressed in a different notation has led to Newton s
investigations on this subject being somewhat overlooked.
Newton further was the first to place dynamics on a
satisfactory basis, and from dynamics he deduced the theory of
statics : this was in the introduction to the Principia pub
lished in 1687. The theory of attractions, the application of
the principles of mechanics to the solar system, the creation of
physical astronomy, and the establishment of the law of
universal gravitation are wholly due to him and were first
published in the same work. The particular questions con
nected with the motion of the earth and moon were worked
out as fully as was then possible. The theory of hydro
dynamics was created in the second book of the Principia,
and he added considerably to the theory of hydrostatics which
may be said to have been first discussed by Pascal. The
theory of the propagation of waves, and in particular the
application to determine the velocity of sound, is due to
Newton and was published in 1687. In geometrical optics,
he explained amongst other things the decomposition of light
and the theory of the rainbow ; he invented the reflecting
telescope known by his name, and the sextant. In physical
optics, he suggested and elaborated the emission theory of light.
The above list does not exhaust the subjects he investigated,
but it will serve to illustrate how marked was his influence on
the history of mathematics. On his writings and on their
effects, it will be enough to quote the remarks of two or three
of those who were subsequently concerned with the subject-
matter of the Principia. Lagrange described the Principia as
the greatest production of the human mind, and said he felt
dazed at such an illustration of what man s intellect might be
capable. In describing the effect of his own writings and
those of Laplace it was a favourite remark of his that Newton
358 THE LIFE AND WOKKS OF NEWTON.
was not only the greatest genius that had ever existed but he
was also the most fortunate, for as there is but one universe, it
can happen but to one man in the world s history to be the
interpreter of its laws. Laplace, who is in general very sparing
of his praise, makes of Newton the one exception, and the
words in which he enumerates the causes which "will always
assure to the Principia a pre-eminence above all the other pro
ductions of the human intellect" have been often quoted. Not
less remarkable is the homage rendered by Gauss : for other
great mathematicians or philosophers, he used the epithets
magnus, or clarus, or clarissimus ; for Newton alone he kept
the prefix summus. Finally Biot, who had made a special
study of Newton s works, sums up his remarks by saying,
" comme geometre et comme experimentateur Newton est sans
egal ; par la reunion de ces deux genres de genies a leur plus
haut degre, il est sans exemple."
CHAPTER XVII.
LEIBNITZ AND THE MATHEMATICIANS OF THE FIRST
HALF OF THE EIGHTEENTH CENTURY.
I HAVE briefly traced in the last chapter the nature and
extent of Newton s contributions to science. Modern analysis
is however derived directly from the works of Leibnitz and the
elder Bernoullis ; and it is immaterial to us whether the funda
mental ideas of it were obtained by them from Newton, or
discovered independently. The English mathematicians of the
years considered in this chapter continued to use the language
and notation of Newton : they are thus somewhat distinct from
their continental contemporaries, and I have therefore grouped
them together in a section by themselves.
Leibnitz and the Bernoullis.
Leibnitz*. Gottfried Willvelrti Leibnitz (or Leibniz) was
born at Leipzig on June 21 (O. S.), 1646, and died at Hanover
on Nov. 14, 1716. His father died before he was six, and the
teaching at the school to which he was then sent was ineffi
cient, but his industry triumphed over all difficulties ; by the
time he was twelve he had taught himself to read Latin easily,
and had begun Greek ; and before he was twenty he had
* See the life of Leibnitz by G. E. Guhrauer, 2 volumes and a supple
ment, Breslau, 1842 and 1846. Leibnitz s mathematical papers have
been collected and edited by C. J. Gerhardt in 7 volumes, Berlin
and Halle, 184963.
360 LEIBNITZ.
mastered the ordinary text-books on mathematics, philo
sophy, theology, and law. Refused the degree of doctor of
laws at Leipzig by those who were jealous of his youth and
learning, he moved to Nuremberg. An essay which he there
wrote on the study of law was dedicated to the elector of
Mainz, and led to his appointment by the elector on a commis
sion for the revision of some statutes, from which he was
subsequently promoted to the diplomatic service. In the
latter capacity he supported (unsuccessfully) the claims of the
German candidate for the crown of Poland. The violent
seizure of various small places in Alsace in 1670 excited
universal alarm in Germany as to the designs of Louis XIY. ;
and Leibnitz drew up a scheme by which it was proposed to
offer German co-operation, if France liked to take Egypt and
use the possession of that country as a basis for attack against
Holland in Asia, on the condition that Germany was to be
left undisturbed by France. This bears a curious resemblance
to the similar plan by which Napoleon I. proposed to attack
England. In 1672 Leibnitz went to Paris on the invitation
of the French government to explain the details of the scheme,
but nothing came of it.
At Paris he met Huygens who was then residing there,
and their conversation led him to study geometry, which he
described as opening a new world to him, though he had as a
matter of fact previously written some tracts on various minor
points in mathematics ; the most important of them being a
paper on combinations written in 1668, and a description of a
new calculating machine. In January, 1673, he was sent on a
political mission to London, where he stopped some months
and made the acquaintance of Oldenburg, Collins, and others :
it was at this time that he communicated the memoir to the
Royal Society in which he was found to have been forestalled
by Mouton, (see above, p. 327).
In 1673 the elector of Mainz died, and in the following year
Leibnitz entered the service of the Brunswick family; in 1676
he again visited London, and then moved to Hanover, where
LEIBNITZ. 361
till his death he occupied the well-paid post of librarian in
the ducal library. His pen was thenceforth employed in all the
political matters which affected the Hanoverian family, and his
services were recognized by honours and distinctions of various
kinds : his memoranda on the various political, historical, and
theological questions which concerned the dynasty during the
forty years from 1673 to 1713 form a valuable contribution to
the history of that time. His appointment in the Hanoverian
service gave him increased leisure for his favourite pursuits.
Leibnitz used to assert that as the first-fruit of his increased
leisure he invented the differential and integral calculus in 1674, *
but the earliest traces of the use of it in his extant note-books
do not occur till 1675, and it was not till 1677 that we
find it developed into a consistent system : it was not pub
lished till 1634. Nearly all his mathematical papers were
produced within the ten years from 1682 to 1692, and most of
them in a journal, called the Acta Eruditorum, which he and
Otto Mencke had founded in 1682, and which had a wide
circulation on the continent.
Leibnitz occupies at least as large a place in the history of
philosophy as he does in the history of mathematics. Most of
his philosophical writings were composed in the last twenty or
twenty-five years of his life ; and the point as to whether his
views were original or whether they were appropriated from
Spinoza, whom he visited in 1676, is still in question among
philosophers, though the evidence seems to point to the origin
ality of Leibnitz. As to Leibnitz s system of philosophy it will
be enough to say that he regarded the ultimate elements of the
universe as individual percipient beings whom he called monads.
According to him the monads are centres of force, and substance
is force, while space, matter, and motion are merely pheno
menal : finally the existence of God is inferred from the existing
harmony among the monads. His services to literature were
almost as considerable as those to philosophy; in particular
I may single out his overthrow of the then prevalent belief
that Hebrew was the primaeval language of the human race.
362 LEIBNITZ.
In 1700 the Academy of Berlin was created on his advice,
and he drew up the first body of statutes for it. On the
accession in 1714 of his master George I. to the throne of
England, Leibnitz was practically thrown aside as a useless
tool ; he was forbidden to come to England ; and the last
two years of his life were spent in neglect and dishonour.
He died at Hanover in 1716. He was overfond of money
and personal distinctions; was unscrupulous, as might be
expected of a professional diplomatist of that time; but pos
sessed singularly attractive manners, and all who once came
under the charm of his personal presence remained sincerely
attached to him. His mathematical reputation was largely aug
mented by the eminent position that he occupied in diplomacy,
philosophy, and literature; and the power thence derived was
considerably increased by his influence in the management
of the A eta Eruditorum which I believe was the only private
scientific journal of the time.
The last years of his life from 1709 to 1716 were em
bittered by the long controversy with John Keill, Newton,
and others as to whether he had discovered the differential
calculus independently of Newton s previous investigations or
whether he had derived the fundamental idea from Newton
and merely invented another notation for it. The controversy*
occupies a place in the scientific history of the early years of
the eighteenth century quite disproportionate to its true
importance, but it so materially affected the history of mathe
matics in western Europe, that I feel obliged to give the
* The case in favour of the independent invention by Leibnitz is
stated in Gerhardt s Leibnizens mathematische Schriften, and in Biot and
Lefort s edition of the Commercium Epistolicum, Paris, 1856. The
arguments on the other side are given in H. Slornan s Leibnitzens
Anspruch auf die Erfindung der Differ enzialreclmung, Leipzig, 1857,
of which an English translation, with additions by Dr Slomau, was
published at Cambridge in 1860. The history of the invention of the
Jculus is given in an article on it in the ninth edition of the Encyclo
paedia Britannica, and in P. Mansion s Esquisse de Vhistoire du calcul
infinitesimal, Gand, 1887.
DISPUTE AS TO ORIGIN OF THE CALCULUS. 363
leading facts, though I am reluctant to take up so much space
with questions of a personal character.
The ideas of the infinitesimal calculus can be expressed
either in the notation of fluxions or in that of differentials.
The former was used by Newton in 1666, and communicated
in manuscript to his friends and pupils from 1669 onwards,
but no distinct account of it was printed till 1693. The
earliest use of the latter in the note-books of Leibnitz is dated
1675, it was employed in the letter sent to Newton in 1677,
and an account of it was printed in the memoir of 1684
described below. There is no question that the differential
notation is due to Leibnitz, and the sole question is as to
whether the general idea of the calculus was taken from
Newton or discovered independently.
The case in favour of the independent invention by
Leibnitz rests on the ground that he published a description of
his method some years before Newton printed anything on
fluxions, that he always alluded to the discovery as being his
own invention, and that for many years this statement was
unchallenged ; while of course there must be a strong pre
sumption that he acted in good faith. To rebut this case it is
necessary to shew (i) that he saw some of Newton s papers on
the subject in or before 1675 or at least 1677, and (ii) that he
thence derived the fundamental ideas of the calculus. The
fact that his claim was unchallenged for some years is in my
opinion in the particular circumstances of the case immaterial.
That Leibnitz saw some of Newton s manuscripts was
always intrinsically probable; but when, in 1849, C. J.
Gerhardt* examined Leibnitz s papers he found among them
a manuscript copy, the existence of which had been previously
unsuspected, in Leibnitz s handwriting of extracts from
Newton s De Analysi per Equationes Numero Terminorum
Infinitas (which was printed in the De Quadratura Curvarum
in 1704, see above, p. 348), together with notes on their
* Gerhardt, Leibnizem mathematischc Schriften, vol. i., p. 7.
364 LEIBNITZ.
expression in the differential notation. The question of the
date at which these extracts were made is therefore all
important. It is known that a copy of Newton s manuscript
had been sent to Tschirnhausen in May, 1675, and as in that
year he and Leibnitz were engaged together on a piece of
work, it is not impossible that these extracts were made then*.
It is also possible that they may have been made in 1676, for
Leibnitz discussed the question of analysis by infinite series
with Collins and Oldenburg in that year, and it is a priori
probable that they would have then shewn him the manuscript
of Newton on that subject, a copy of which was possessed by
one or both of them. On the other hand it may be supposed
that Leibnitz made the extracts from the printed copy in or
after 1704. Leibnitz shortly before his death admitted in a
letter to Conti that in 1676 Collins had shewn him some
Newtonian papers, but implied that they were of little or no
value presumably he referred to Newton s letters of June 13
and Oct. 24, 1676, and to the letter of Dec. 10, 1672 on the
method of tangents, extracts from which accompanied f the
letter of June 13 but it is curious that, on the receipt of
these letters, Leibnitz should have made no further inquiries,
unless he was already aware from other sources of the method
followed by Newton.
Whether Leibnitz made no use of the manuscript from
which he had copied extracts, or whether he had previously
invented the calculus are questions on which at this distance
of time no direct evidence is available. It is however worth
noting that the unpublished Portsmouth papers shew that,
when, in 1711, Newton went carefully into the whole dispute,
he picked out this manuscript as the one which had probably
somehow fallen into the hands of Leibnitz J. At that time
there was no direct evidence that Leibnitz had seen this
manuscript before it was printed in 1704, and accordingly
* Sloman, English translation, p. 34.
t Gerhardt, vol. i., p. 91.
J Catalogue of Portsmouth papers, pp. xvi, xvii, 7, 8.
DISPUTE AS TO ORIGIN OF THE CALCULUS. 365
Newton s conjecture was not published ; but Gerhardt s dis
covery of the copy made by Leibnitz tends to confirm the
accuracy of Newton s judgment in the matter. It is said by
some that to a man of Leibnitz s ability the manuscript,
especially if supplemented by the letter of Dec. 10, 1672,
would supply sufficient hints to give him a clue to the methods
of the calculus, though as the fluxional notation is not em
ployed in it anyone who used it would have to invent a
notation; but this is denied by others.
There was at first no reason to suspect the good faith of
Leibnitz; and it was not until the appearance in 1704 of an
anonymous review of Newton s tract on quadrature, in which
it was implied that Newton had borrowed the idea of the
fluxional calculus from Leibnitz, that any responsible mathe
matician* questioned the statement that Leibnitz had invented
the calculus independently of Newton. It is universally
admitted that there was no justification or authority for the
statements made in this review, which was rightly attributed
to Leibnitz. But the subsequent discussion led to a critical
examination of the whole question, and doubt was expressed
as to whether Leibnitz had not derived the fundamental idea
from Newton. The case against Leibnitz as it appeared to
Newton s friends was summed up in the Commercium Episto-
licum issued in 1712. The evidence there collected may be
inconclusive, but at any rate detailed references are given for
all the facts mentioned.
No such summary (with facts, dates, and references) of
the case for Leibnitz was issued by his friends; but John
Bernoulli attempted to indirectly weaken the evidence by
attacking the personal character of Newton : this was in a
letter dated June 7, 1713. The charges were false, and,
when pressed for an explanation of them, Bernoulli most
solemnly denied having written the letter. In accepting the
* In 1699 Duillier had accused Leibnitz of plagiarism from Newton,
hut Duillier was not a person of much importance.
366 LEIBNITZ.
denial Newton added in a private letter to him the following
remarks which are interesting as giving Newton s account of
why he was at last induced to take any part in the con
troversy. "I have never," said he, "grasped at fame among
foreign nations, but I am very desirous to preserve my cha
racter for honesty, which the author of that epistle, as if by
the authority of a great judge, had endeavoured to wrest from
me. Now that I am old, I have little pleasure in mathematical
studies, and I have never tried to propagate my opinions over
the world, but have rather taken care not to involve myself
in disputes on account of them."
Leibnitz s defence or explanation of his silence is given in
the following letter, dated April 9, 1716, from him to Conti.
"Pour repondre de point en point a Pouvrage public centre
moi, il falloit un autre ouvrage aussi grand pour le moins que
celui-la : il falloit entrer dans un grand detail de quantite de
minuties passees il y a trente a quarante ans, dont je ne me
souvenois guere : il me falloit chercher mes vieilles lettres,
dont plusieurs se sont perdues, outre que le plus souvent je
n ai point garde les minutes des miennes : et les autres sont
ensevelies dans un grand tas de papiers, que je ne pouvois
debrouiller qu avec du temps et de la patience ; mais je n en
avois guere le loisir, etant charge presentement d occupations
d une toute autre nature."
The death of Leibnitz in 1716 only put a temporary stop
to the controversy which was bitterly debated for many years
later. The question is one of great difficulty ; the evidence is
conflicting and circumstantial ; and every one must form for
themselves the opinion which seems most probable. I think
the majority of modern writers would accept the view that
probably Leibnitz s invention of the calculus was independent
of that of Newton, and everyone will hope that they are right.
For myself I cannot however but think it probable that
Leibnitz read Newton s manuscript De Analysi before 1677,
and was materially assisted by it. His unacknowledged
possession of a copy of part of one of Newton s manuscripts
DISPUTE AS TO ORIGIN OF THE CALCULUS. 367
may be explicable, but the admitted fact that on more than
one occasion he deliberately altered or added to important
documents (ex. gr. the letter of June 7, 1713, in the Charta
Volans, and that of April 8, 1716, in the Acta Eruditorum)
before publishing them seems to me to make his own testimony
of little value. In mitigation of his conduct I can only say
that it must be recollected that what he is alleged to have
received was rather a series of hints than an account of the
calculus ; and it seems to me that the facts that he did not
publish his results of 1677 until 1684, and that the notation
and subsequent development of it were all of his own invention
may have led him thirty years later to minimize any assistance
which he obtained originally and finally consider that it was
immaterial.
If we must confine ourselves to one system of notation
then there can be no doubt that that which was invented by
Leibnitz is better fitted for most of the purposes to which the "
infinitesimal calculus is applied than that of fluxions, and
for some (such as the calculus of variations) it is indeed
almost essential. It should be remembered however that at
the beginning of the eighteenth century the methods of the
infinitesimal calculus had not been systematized, and either
notation was equally good. The development of that calculus
was the main work of the mathematicians of the first half of
the eighteenth century. The differential form was adopted by
continental mathematicians. The application of it by Euler.
Lagrange, and Laplace to the principles of mechanics laid
down in the Principia was the great achievement of the last
half of that century, and finally demonstrated the superiority
of the differential to the fluxional calculus. The translation of
the Principia into the language of modern analysis and the
filling in of the details of the Newtonian theory by the aid of
that analysis were effected by Laplace.
The controversy with Leibnitz was regarded in England as
an attempt by foreigners to defraud Newton of the credit of
his invention, and the question was complicated on both sides
368 LEIBNITZ.
by national jealousies. It was therefore natural though it was
unfortunate that in England the geometrical and fluxional
methods as used by Newton were alone studied and employed.
For more than a century the English school was thus out
of touch with continental mathematicians. The consequence
was that, in spite of the brilliant band of scholars formed by
Newton, the improvements in the methods of analysis gradually
effected on the continent were almost unknown in Britain.
It was not until 1820 that the value of analytical methods was
fully recognized in England, and that Newton s countrymen
again took any large share in the development of mathematics.
Leaving now this long controversy I come to the discussion
of the mathematical papers produced by Leibnitz, all the more
important of which were published in the Acta Eruditorum.
They are mainly concerned with applications of the infinitesimal
calculus and with various questions on mechanics.
The only papers of first-rate importance which he produced
are those on the differential calculus. The earliest of these
was one published in the Acta Eruditorum for October, 1684,
in which he enunciated a general method for finding maxima
and minima, and for drawing tangents to curves. One in
verse problem, namely, to find the curve whose subtangent
is constant, was also discussed. The notation is the same as
that with which we are familiar, and the differential co
efficients of x n and of products and quotients are determined.
In 1686 he wrote a paper on the principles of the new
calculus. In both of these papers the principle of continuity
is explicitly assumed, while his treatment of the subject is
based on the use of infinitesimals and not on that of the
limiting value of ratios. In answer to some objections which
were raised in 1694 by Bernard Nieuwentyt who asserted that
dyjdx stood for an unmeaning quantity like 0/0, Leibnitz
explained, in the same way as Barrow had previously done,
that the value of dyjdx in geometry could be expressed as the
ratio of two finite quantities. I think that Leibnitz s statement
of the objects and methods of the infinitesimal calculus as
LEIBNITZ. 369
contained in these papers, which are the three most important
memoirs on it that he produced, is somewhat obscure, and his
attempt to place the subject on a metaphysical basis did not
tend to clearness; but the notation he introduced is superior
to that of Newton, and the fact that all the results of modern
mathematics are expressed in the language invented by Leib
nitz has proved the best monument of his work.
In 1686 and 1692 he wrote papers on osculating curves.
These however contain some bad blunders ; as, for example,
the assertion that an osculating circle will necessarily cut
a curve in four consecutive points : this error was pointed
out by John Bernoulli, but in his article of 1692 Leibnitz
defended his original assertion, and insisted that a circle could
never cross a curve where it touched it.
In 1692 Leibnitz wrote a memoir in which he laid the
foundation of the theory of envelopes. This was further
developed in another paper in 1694, in which he introduced
for the first time the terms " coordinates" and "axes of co
ordinates."
Leibnitz also published a good many papers on mechanical
subjects ; but some of them contain mistakes which shew
that he did not understand the principles of the subject.
Thus, in 1685, he wrote a memoir to find the pressure exerted
by a sphere of weight W placed between two inclined planes
of complementary inclinations, placed so that the lines of
greatest slope are perpendicular to the line of the intersection
of the planes. He asserted that the pressure on each plane
must consist of two components, " unum quo decliviter de-
scendere tendit, alterum quo planum declive premit." He
further said that for metaphysical reasons the sum of the two
pressures must be equal to W. Hence, if R and R be the
required pressures, and a and -J-?r a the inclinations of the
planes, he finds that
R - ^ W(\- sin a + cos a) and R = W (1 - cos a + sin a).
The true values are R = W cos a and R - W sin a. Never-
B. 24
370 LEIBNITZ.
theless some of his papers on mechanics are valuable. Of these
the most important were two, in 1689 and 1694, in which he
solved the problem of finding an isochronous curve; one, in
1697, on the curve of quickest descent (this was the problem
sent as a challenge to Newton); and two, in 1691 and 1692, in
which he stated the intrinsic equation of the curve assumed by
a flexible rope suspended from two points, i.e. the catenary, but
gave no proof. This last problem had been originally proposed
by Galileo.
In 1689, that is, two years after the Principia had been
published, he wrote on the movements of the planets which
he stated were produced by a motion of the ether. Not only
were the equations of motion which he obtained wrong, but
his deductions from them were not even in accordance with
his own axioms. In another memoir in 1706, that is, nearly
twenty years after the Principia had been written, he admitted
that he had made some mistakes in his former paper but
adhered to his previous conclusions, and summed the matter
up by saying "it is certain that gravitation generates a
new force at each instant to the centre, but the centrifugal
force also generates another away from the centre. . . . The
centrifugal force may be considered in two aspects according
as the movement is treated as along the tangent to the curve
or as along the arc of the circle itself." It seems clear from
this paper that he did not really understand the manner in
which Newton had reduced dynamics to an exact science. It
is hardly necessary to consider his work on dynamics in further
detail. Much of it is vitiated by a constant confusion between
momentum and kinetic energy: when the force is " passive"
he uses the first, which he calls the vis mortua, as the
measure of a force ; when the force is " active " he uses the
latter, the double of which he calls the vis viva.
The series quoted by Leibnitz comprise those for e?, log (1 +x),
sin a?, verso;, and tan" 1 ^; all of these had been previously
published, and he rarely, if ever, added any demonstrations.
Leibnitz (like Newton) recognized the importance of James
LEIBNITZ. 371
Gregory s remarks on the necessity of examining whether
infinite series are convergent or divergent, and proposed a test
to distinguish series whose terms are alternately positive and
negative. In 1693 he explained the method of expansion by
indeterminate coefficients, though his applications were not
free from error.
To sum the matter up briefly, it seems to me that Leibnitz s
work exhibits great skill in analysis, but much of it is un
finished, and when he leaves his symbols and attempts to
interpret his results he frequently commits blunders. No
doubt the demands of politics, philosophy, and literature on his
time may have prevented him from elaborating any scientific
subject completely or writing any systematic exposition of his
views, though they are no excuse for the mistakes of principle
which occur so frequently in his papers. Some of his memoirs
contain suggestions of methods which have now become valu
able means of analysis, such as the use of determinants and of
indeterminate coefficients : but when a writer of manifold
interests like Leibnitz throws out innumerable suggestions,
some of them are likely to turn out valuable ; and to enumerate
these (which he never worked out) without reckoning the others,
which are wrong, gives a false impression of the value of his
work. But in spite of this, his title to fame rests on a sure
basis, for it was he who brought the differential calculus into
general use, and his name is inseparably connected with one of
the chief instruments of analysis, just as that of Descartes
another philosopher is with analytical geometry.
Leibnitz was only one amongst several continental writer*
whose papers in the Ada Eruditorum familiarized mathe
maticians with the use of the differential calculus. The most
important of these were James and John Bernoulli, both of
whom were warm friends and admirers of Leibnitz, and to
their devoted advocacy his reputation is largely due. Not
only did they take a prominent part in nearly every mathe
matical question then discussed, but nearly all the leading
mathematicians on the continent for the first half of the
372 JAMES BERNOULLI.
eighteenth century caine directly or indirectly under the
influence of one or both of them.
The Bernoullis (or as they are sometimes, and perhaps
more correctly, called the Bernouillis) were a family of Dutch
origin, who were driven from Holland by the Spanish persecu
tions, and finally settled at Bale in Switzerland. The first
member of the family who attained any marked distinction in
mathematics was James.
James Bernoulli*. Jacob or James Bernoulli was born at
Bale on Dec. 27, 1654; in 1687 he was appointed to a chair
of mathematics in the university there ; and occupied it until
his death on Aug. 16, 1705.
He was one of the earliest to realize how powerful as an
instrument of analysis was the infinitesimal calculus, and he
applied it to several problems, but he did not himself invent
any new processes. His great influence was uniformly and
successfully exerted in favour of the use of the differential cal
culus, and his lessons on it, which were written in the form
of two essays in 1691 and are published in volume n. of his
works, shew how completely he had even then grasped the
principles of the new analysis. These lectures, which contain
the earliest use of the term integral, were the first published
attempt to construct an integral calculus ; for Leibnitz had
- treated each problem by itself, and had not laid down any
general rules on the subject.
The most important discoveries of James Bernoulli were
his solution of the problem to find an isochronous curve; his
proof that the construction for the catenary which had been
given by Leibnitz was correct, and his extension of this to
strings of variable density and under a central force ; his de
termination of the form taken by an elastic rod fixed at one
end and acted on by a given force at the other, the elastica \
* See the eloge by B. de Fontenelle, Paris, 1766 ; also Montucla s
Histoire, vol. n. A collected edition of the works of James Bernoulli was
published in two volumes at Geneva in 1744, and an account of his life
is prefixed to the first volume.
JOHN BERNOULLI. 373
also of a flexible rectangular sheet with two sides fixed hori
zontally and filled with a heavy liquid, the lintearia ; and
lastly of a sail filled with wind, the velaria. In 1696 he offered
a reward for the general solution of isoperi metrical figures, i.e.
the determination of a figure of a given species which should
include a maximum area, its perimeter being given : his own
solution, published in 1701, is correct as far it goes. In 1698
he published an essay on the differential calculus and its applica
tions to geometry. He here investigated the chief properties
of the equiangular spiral, and especially noticed the manner in
which various curves deduced from it reproduced the original
curve : struck by this fact he begged that, in imitation of
Archimedes, an equiangular spiral should be engraved on his
tombstone with the inscription eadem numero mutata resurgo.
He also brought out in 1695 an edition of Descartes s
Geometric. In. his Ars Conjectandi, published in 1713, he
established the fundamental principles of the calculus of pro
babilities ; in the course of the work he defined the numbers
known by his name* and explained their use, he also gave
some theorems on finite differences. His higher lectures were
mostly on the theory of series; these were published by
Nicholas Bernoulli in 1713.
John Bernoulli t. Johann Bernoulli, the brother of James
Bernoulli, was born at Bale on Aug. 7, 1667, and died
there on Jan. 1, 1748. He occupied the chair of mathe
matics at Groningen from 1695 to 1705 ; and at Bale, where
he succeeded his brother, from 1705 to 1748. To all who
did not acknowledge his merits in a manner commensurate
with his own view of their importance he behaved most un-
* A bibliography of Bernoulli s Numbers has been given by G. S. Ely,
in the American Journal of Mathematics, 1882, vol. v., pp. 228 235.
t D Alembert wrote a eulogistic eloge on the work and influence of
John Bernoulli, but he explicitly refused to deal with his private life or
quarrels ; see also Montucla s Histoire, vol. n. A collected edition of the
works of John Bernoulli was published at Geneva in four volumes in 1742,
and his correspondence with Leibnitz was published in two volumes at
the same place in 1745.
374 JOHN BERNOULLI.
justly : as an illustration of his character it may be mentioned
that he attempted to substitute for an incorrect solution of his
own on isoperimetrical curves another stolen from his brother
James, while he expelled his son Daniel from his house for
obtaining a prize from the French Academy which he had
expected to receive himself. After the deaths of Leibnitz and
THospital he claimed the merit of some of their discoveries ;
these claims are now known to be false. He was however the
most successful teacher of his age, arid had the faculty of
inspiring his pupils with almost as passionate a zeal for mathe
matics as he felt himself. The general adoption on the conti
nent of the differential rather than the fluxional notation was
largely due to his influence.
Leaving out of account his innumerable controversies, the
chief discoveries of John Bernoulli were the exponential cal
culus, the treatment of trigonometry as a branch of analysis,
the conditions for a geodesic, the determination of orthogonal
trajectories, the solution of the brachistochrone, the statement
that a ray of light traversed such a path that ^ds was a
minimum, and the enunciation of the principle of virtual work.
I believe that he was the first to denote the accelerating effect
of gravity by an algebraical sign g, and he thus arrived at the
formula v 2 2gh : the same result would have been previously
expressed by the proportion v* : v 2 2 = h l : h 2 . The notation
<(>x to indicate a function of x was introduced by him in 1718,
and displaced the notation X or proposed by him in 1698 :
but the general adoption of symbols like f, F, <, \f/, . . . to
represent functions, seems to be mainly due to Euler and
Lagrange.
Several members of the same family, but of a younger
generation, enriched mathematics by their teaching and writings.
The most important of these were the three sons of John ;
namely, Nicholas, Daniel, and John the younger ; and the two
sons of John the younger, who bore the names of John and
James. To make the account complete I add here their respec
tive dates. Nicholas Bernoulli, the eldest of the three sons of
L HOSPITAL. 375
John, was born on Jan. 27, 1695, and was drowned at St
Petersburg where he was professor on July 26, 1726. Daniel
Bernoulli, the second son of John, was born on Feb. 9, 1700,
and died on March 17, 1782 ; he was professor first at St
Petersburg and afterwards at Bale, and shares with Euler the
unique distinction of having gained the prize proposed annually
by the French Academy no less than ten times : I refer to
him again a few pages later. John Bernoulli, the younger, a
brother of Nicholas and Daniel, was born on May 18, 1710,
and died in 1790; he also was a professor at Bale. He left
two sons, John and James : of these, the former, who was born
on Dec. 4, 1744, and died on July 10, 1807, was astronomer
royal and director of mathematical studies at Berlin ; while the
latter, who was born on Oct. 17, 1759, and died in July 1789,
was successively professor at Bale, Verona, and St Petersburg.
The development of analysis on the continent.
u
Leaving for a moment the English mathematicians of the
first half of the eighteenth century we come next to a number of
continental writers who barely escape mediocrity, and to whom
it will be necessary to devote but few words. Their writings
mark the steps by which analytical geometry and the diffe
rential and integral calculus were perfected and made familiar
to mathematicians. Nearly all of them were pupils of one
or other of the two elder Bernoullis, and they were so nearly
contemporaries that it is difficult to arrange them chrono
logically. The most eminent of them are Cramer, de Qua,
de Montmort, Fagnano, I 9 Hospital, Nicole, Parent, Riccati,
Saurin, and Varignon.
L Hospital. Guillaume Francois Antoine V Hospital, Mar
quis de St-Mesme, born at Paris in 1661, and died there on
Feb. 2, 1704, was among the earliest pupils of John Bernoulli,
who, in 1691, spent some months at FHospitaPs house in
Paris for the purpose of teaching him the new calculus. It
seems strange but it is substantially true that a knowledge of
376 L HOSPITAL. VARIGNON.
the infinitesimal calculus and the power of using it was then
confined to Newton, Leibnitz, and the two elder Bernoullis
and it will be noticed that they were the only mathematicians
who solved the more difficult problems then proposed as chal
lenges. There was at that time no text-book on the subject,
and the credit of putting together the first treatise which
explained the principles and use of the method is due to
1 Hospital : it was published in 1696 under the title Analyse des
infiniment petits. This contains a partial investigation of
the limiting value of the ratio of functions which for a certain
value of the variable take the indeterminate form : 0, a
problem solved by John Bernoulli in 1704. This work had
a wide circulation, it brought the differential notation into
universal use in France, and helped to make it generally known
in Europe. A supplement, containing a similar treatment of
the integral calculus, together with additions to the differential
calculus which had been made in the following half century,
was published at Paris, 1754 6, by L. A. de Bougainville.
L Hospital took part in most of the challenges issued
by Leibnitz, the Bernoullis, and other continental mathe
maticians of the time; in particular he gave a solution of
the brachistochrone, and investigated the form of the solid
of least resistance of which Newton in the Principia had
stated the result. He also wrote a treatise on analytical
conies which was published in 1707, and for nearly a century
deemed a standard work on the subject.
Varignon. Pierre Varignon, born at Caen in 1654, and
in Paris on Dec. 22, 1722, was an intimate friend of
Newton, Leibnitz, and the Bernoullis, and, after 1 Hospital, was
the earliest and most powerful advocate in France of the use of
the differential calculus. He realized the necessity of obtaining
a test for examining the convergency of series, but the
analytical difficulties were beyond his powers. He simplified
the proofs of many of the leading propositions in mechanics,
and in 1687 recast the treatment of the subject, basing it on
the composition of forces (see above, p. 249). His works were
BE MONTMORT. NICOLE. PARENT. SAURIN. DE GUA. 877
published at Paris in 1725. For further details see the eloge
by B. de Fontenelle, Paris, 1766.
De Montmort, Pierre Raymond de Montmort, born at Paris
on Oct. 27, 1678, and died there on Oct. 7, 1719, was
interested in the subject of finite differences. He determined
in 1713 the sum of n terms of a finite series of the form
n(n-\) . n(n-l)(n-2) .
tia+ -iT2" Aa+ ITS rhr Aa+ - ;
a theorem which seems to have been independently re-discovered
by Chr. Goldbach in 1718.
Nicole. Franqois Nicole, who was born at Paris on Dec. 23,
1683, and died there on Jan. 18, 1758, was the first to publish a
systematic treatise on finite differences. Taylor had regarded
the differential coefficient, i.e. the ratio of two infinitesimal
differences, as the limiting value of the ratio of two finite
differences, a method which is still used by many English
writers though it has been generally abandoned on the con
tinent, and thus had been led to give a sketch of the subject in
his Methodus published in 1715 (see below, p. 389). Nicole s
Traite du calcul des differences finies was published in 1717 :
it is a well-arranged book, and contains rules both for forming
differences and for effecting the summation of given series.
Besides this, in 1706, he wrote a work on roulettes, especially
spherical epicycloids: and in 1729 and 1731 he published
memoirs on Newton s essay on curves of the third degree.
Parent. Antoine Parent, born at Paris on Sept. 16, 1666,
and died there on Sept. 26, 1716, wrote in 1700 on analytical
geometry of three dimensions. His works were collected and
published in three volumes at Paris in 1713.
Saurin. Joseph Saurin, born at Courtaison in 1659, and
died at Paris on Dec. 29, 1737, was the first to shew how the
tangents at the multiple points of curves could be deter
mined by analysis.
De G-ua. Jean Paul de Gua de Malves, was born at Car
cassonne in 1713, and died at Paris on June 2, 1785. He
378 DE GUA. CRAMER. RICCATI. FAGNANO.
published in 1740 a work on analytical geometry in which he
applied it, without the aid of the differential calculus, to find
the tangents, asymptotes, and various singular points of an
algebraical curve ; and he further shewed how singular points
and isolated loops were affected by conical projection. He
gave the proof of Descartes s rule of signs which is to be
found in most modern works : it is not clear whether Descartes
ever proved it strictly, and Newton seems to have regarded it
as obvious.
Cramer. Gabriel Cramer, born at Geneva in 1704, and
died at Bagnols in 1752, was professor at Geneva. The work
by which he is best known is his treatise on algebraic
curves, published in 1750, which, as far as it goes, is fairly
complete ; it contains the earliest demonstration that a curve
of the -H/th degree is in general determined if ^n (n + 3) points
on it be given :\ this work is still sometimes read^) Besides
this, he edited the works of the two elder Bernoullis ; and
wrote on the physical cause of the spheroidal shape of the
planets and the motion of their apses (1730), and on Newton s
treatment of cubic curves (1746).
Riccati. Jacopo Francesco, Count Riccati, born at Venice
on May 28, 1676, and died at Treves on April 15, 1754, did a
great deal to disseminate a knowledge of the Newtonian
philosophy in Italy. Besides the equation known by his
name, certain cases of which he succeeded in integrating, he
discussed the question of the possibility of lowering the order
of a given differential equation. His works were published at
Treves in four volumes in 1758. He had two sons who wrote
on several minor points connected with the integral calculus
and differential equations, and applied the calculus to several
mechanical questions : these were Vincenzo, who was born in
1707 and died in 1775, and Giordano, who was born in 1709
and died in 1790.
Fagnano. Giulio Carlo, Count Fagnano, and Marquis de
Toschi, born at Sinigaglia on Dec. 6, 1682, and died on Sept. 26,
1766, may be said to have been the first writer who directed
FAGNANO. VIVIANI. DE LA HIRE. 879
attention to the theory of elliptic functions. Failing to rectify
the ellipse or hyperbola, Fagnano attempted to determine arcs
whose difference should be rectifiable. He also pointed out
the remarkable analogy existing between the integrals which
represent the arc of a circle and the arc of a lemniscate.
Finally he proved the formula
,r = 2tlog{(l-i)/(l+t)}
where i stands for v 1. His works were collected and
published in two volumes at Pesaro in 1750.
It was inevitable that some mathematicians should object
to methods of analysis founded on the infinitesimal calculus.
The most prominent of these were Viviani, De la Hire, and
Rolle. Chronologically they come here but they flourished
half a century after the date to which their writings properly
belong.
Viviani. Vincenzo Viviani, a pupil of Galileo and Tor-
ricelli, born at Florence on April 5, 1622, and died there on
Sept. 22, 1703, brought out in 1659 a restoration of the lost
book of Apollonius on conic sections; and in 1701 a restoration
of the work of Aristseus. He explained in 1677 how an
angle could be trisected by the aid of the equilateral hyperbola
or the conchoid. In 1692 he proposed the problem to con
struct four windows in a hemispherical vault so that the
remainder of the surface can be accurately determined : a
celebrated problem of which analytical solutions were given by
Wallis, Leibnitz, David Gregory, and James Bernoulli.
De la Hire. Philippe De la Hire (or Lahire), born in Paris
on March 18, 1640, and died there on April 21, 1719, wrote on
graphical methods, 1673; on the conic sections, 1685; a trea
tise on epicycloids, 1694; one on roulettes, 1702; and lastly
another on conchoids, 1708. His works on conic sections and
epicycloids were founded on the teaching of Desargues, whose
favourite pupil he was. He also translated the essay of
Moschopulus on magic squares, and collected many of the
theorems on them which were previously known : this was
published in 1705.
380 ROLLE. CLAIRA.UT.
Rolle. Michel Rolle, born at Ambert on April 21, 1652,
and died in Paris on Nov. 8, 1719, wrote an algebra in 1689
which contains the theorem on the position of the roots of an
equation which is known by his name. He published in 1696
a treatise on the solution of equations, whether determinate or
indeterminate, and he produced several other minor works.
He taught that the differential calculus was nothing but a
collection of ingenious fallacies.
So far no one of the school of Leibnitz and the two elder
Bernoullis had shewn any exceptional ability, but by the action
of a number of second-rate writers the methods and language
of analytical geometry and the differential calculus had become
well known by about 1740. The close of this school is
marked by the appearance of Clairaut, D Alembert, and Daniel
Bernoulli. Their lives overlap the period considered in the
next chapter, but, though it is difficult to draw a sharp dividing
line which shall separate by a definite date the mathematicians
there considered from those whose writings are discussed in
this chapter, I think that on the whole the works of these three
writers are best treated here.
Clairaut. Alexis Claude Clairaut was born at Paris on
May 13, 1713, and died there on May 17, 1765. He belongs
to the small group of children who, though of exceptional
precocity, survive and maintain their powers when grown up.
As early as the age of twelve he wrote a memoir on four
geometrical curves, but his first important work was a
treatise on tortuous curves published when he was eighteen
a work which procured for him immediate admission to the
French Academy. In 1731 he gave a demonstration of the
fact noted by Newton that all curves of the third order were
projections of one of five parabolas.
In 1741 Clairaut went on a scientific expedition to measure
the length of a meridian degree on the earth s surface, and
on his return in 1743 he published his Theorie de la figure
de la terre. This is founded on a paper by Maclaurin, where
CLAIRAUT. 381
it had been shewn that a mass of homogeneous fluid set in
rotation about a line through its centre of mass would, under
the mutual attraction of its particles, take the form of a
spheroid. This work of Clairaut treated of heterogeneous
spheroids and contains the proof of his formula for the accele
rating effect of gravity in a place of latitude /, namely,
where G is the value of equatorial gravity, m the ratio of the
centrifugal force to gravity at the equator, and c the ellipticity
of a meridian section of the earth. In 1849 Prof. Stokes*
shewed that the same result was true whatever was the in
terior constitution or density of the earth provided the surface
was a spheroid of equilibrium of small ellipticity.
Impressed by the power of geometry as shewn in the writ
ings of Newton and Maclaurin, Clairaut abandoned analysis,
and his next work, the Theorie de la lune, published in 1752,
is strictly Newtonian in character. This contains the expla
nation of the motion of the apse which had previously puzzled
astronomers (see above, p. 339), and which Clairaut had at first
deemed so inexplicable that he was on the point of publishing
a new hypothesis as to the law of attraction when it occurred
to him to carry the approximation to the third order, and he
thereupon found that the result was in accordance with the
observations. This was followed in 1754 by some lunar tables;
Clairaut subsequently wrote various papers on the orbit of
the moon, and on the motion of comets as affected by the
perturbation of the planets, particularly on the path of Hal ley s
comet.
His growing popularity in society hindered his scientific
work : " engage/ 7 says Bossut, " a des soupers, a des veilles,
entraine par un gout vif pour les femmes, voulant allier le
plaisir a ses travaux ordinaires, il perdit le repos, la sante,
enfin la vie a 1 age de cinquante-deux ans."
* See Cambridge Philosophical Transactions, vol. vin. pp. 672 695.
382 D ALEMBERT.
D Alembert*. Jean-le-Rond D Alembert, was born at Paris
on Nov. 16, 1717, and died there on Oct. 29, 1783. He was
the illegitimate child of the chevalier Destouches. Being
abandoned by his mother on the steps of the little church of
St Jean-le-Rond which then nestled under the great porch of
Notre Dame, he was taken to the parish commissary, who,
following the usual practice in such cases, gave him the
Christian name of Jean-le-Rond : I do not know by what title
he subsequently assumed the right to prefix de to his name.
He was boarded out by the parish with the wife of a glazier
in a small way of business who lived near the cathedral, and
here he seems to have found a real home though a very
humble one. His father appears to have looked after him,
and paid for his going to a school where he obtained a fair
mathematical education. An essay written in 1738 on the
integral calculus, and another in 1740 on " ducks and drakes"
or ricochets attracted some attention, and in the same year
he was elected a member of the French Academy; this
was probably due to the influence of his father. It is to
his credit that he absolutely refused to leave his adopted
mother with whom he continued to live until her death in
1757. It cannot be said that she sympathized with his
success for, at the height of his fame, she remonstrated
with him for wasting his talents on such work: "vous ne
serez jamais qu un philosophe," said she, "et qu est-ce qu un
philosophe ? c est un fou qui se tourmente pendant sa vie, pour
qu on parle de lui lorsqu il n y sera plus."
Nearly all his mathematical works were produced within
the years 1743 to 1754. The first of these was his Traite de
dynamique, published in 1743, in which he enunciates the
principle known by his name, namely, that the " internal
forces of inertia 7 (i.e. the forces which resist acceleration) must
* Condorcet and J. Bastien have left sketches jdi D Alembert s life:
his literary works have been published, but there is no complete edition
of his scientific writings. Some papers and letters recently discovered
were published by C. Henry at Paris in 1887.
D ALEMBERT. 383
be equal and opposite to the forces which produce the accelera
tion. This is a particular case of Newton s second reading of
his third law of motion, but the full consequence of it had not
been realized previously. The application of this principle
enables us to obtain the differential equations of motion of any
rigid system.
In 1744 D Alembert published his Traite de I equilibre
et du mouvement des fluides, in which he applies his principle
to fluids : this led to partial differential equations which he
was then unable to solve. In 1745 he developed that part
of the subject which dealt with the motion of air in his
Theorie generate des vents, and this again led him to partial
differential equations : a second edition of this in 1746 was
dedicated to Frederick the Great of Prussia, and procured an
invitation to Berlin and the offer of a pension; he declined the
former, but subsequently, after some pressing, pocketed his
pride and the latter. In 1747 he applied the differential cal
culus to the problem of a vibrating string, and again arrived at
a partial differential equation.
His analysis had three times brought him to an equation
of the form
and he now succeeded in shewing that it was satisfied by
u = </> (x 4- 1) + \l/ (x t),
where < and ^ are arbitrary functions. It may be interesting
to give his solution which was published in the transactions
of the Berlin Academy for 1747. He begins by saying that, if
- be denoted by p and r- by a. then
dx ot J ^
But, by the given equation, -j- = i- , and therefore pdt + qdx is
also an exact differential : denote it by dv.
384 D ALEMBERT.
Therefore dv=pdt + qdx.
Hence du + dv = (pdx + qdt) + (pdt + qdx) = (p + q) (dx + dt),
and du-dv = (pdx + qdt) - (pdt + qdx) = (p-q) (dx - dt).
Thus u + v must be a function of x + 1, and M v must be a
function of # t. We may therefore put
u + v - 2</> (x -t- ),
and u v %\l/ (x-t).
Hence M = (sc + 2) + ^ (a; - 2).
D Alembert added that the conditions of the physical
problem of a vibrating string demand that, when x - 0, u should
vanish for all values of t. Hence identically
*(t) + *(-*)-0.
Assuming that both functions can be expanded in integral
powers of t, this requires that they should contain only odd
powers. Hence
* (-0=-* ()=*(-<)
Therefore u = <f> (x + t) + < (# ).
Euler now took the matter up and shewed that the equation
of the form of the string was -= - a 2 ^ , and that the general
dt 2 dx 2
integral was u = <f> (x at)+\l/ (x + at), where <^> and \j/ are
arbitrary functions.
The chief remaining contributions of D Alembert to mathe
matics were on physical astronomy ; especially on the pre
cession of the equinoxes, and on variations in the obliquity of
the ecliptic. These were collected in his Systeme du monde
published in three volumes in 1754.
During the latter part of his life he was mainly occupied
with the great French encyclopaedia. For this he wrote the
introduction, and numerous philosophical and mathematical
articles : the best are those on geometry arid on probabilities.
DANIEL BERNOULLI. 385
His style is brilliant, but not polished, and faithfully reflects
his character which was bold, honest, and frank. He defended
a severe criticism which he had offered on some mediocre work
by the remark, "j aime mieux etre incivil qu ennuye"; and
with his dislike of sycophants and bores it is not surprising
that during his life he had more enemies than friends.
Daniel Bernoulli*. Daniel Bernoulli, whose name I men
tioned above, and who was by far the ablest of the younger
"Bernoullis, was a contemporary and intimate friend of Euler,
whose works are mentioned in the next chapter. Daniel
Bernoulli was born on Feb. 9, 1700, and died at Bale, where
he was professor of natural philosophy, on March 17, 1782.
He went to St Petersburgh in 1724 as professor of mathe
matics, but the roughness of the social life was distasteful to
him, and he was not sorry when a temporary illness in 1733
allowed him to plead his health as an excuse for leaving. He
then returned to Bale, and held successively chairs of medicine,
metaphysics, and natural philosophy there.
His earliest mathematical work was the Eocercitationes
published in 1724 : these contain a theory of the oscillations
of rigid bodies, and a solution of the differential equation pro
posed by Riccati. Two years later he pointed out for the first
time the frequent desirability of resolving a compound motion
into motions of translation and motions of rotation. His chief
work is his Hydrodynamique published in 1738 : it resembles
* The only account of Daniel Bernoulli s life with which I am
acquainted is the eloge by his friend Condorcet. Marie Jean Antoinc
Nicolas Caritat, Marquis de Condorcet, was born in Picardy on Sept. 17,
17 i3, and fell a victim to the republican terrorists on March 28, 1794.
He was secretary to the Academy and is the author of numerous eloges.
He is perhaps more celebrated for his studies in philosophy, literature,
and politics than iii mathematics, but his mathematical treatment of
probabilities, and his discussion of differential equations and finite dif
ferences, shew an ability which might have put him in the first rank if he
had concentrated his attention on mathematics. He sacrificed himself
in a vain effort to guide the revolutionary torrent into a constitutional
channel.
B. 25
386 MATHEMATICIANS OF THE ENGLISH SCHOOL.
Lagrange s Mecanique analytique in being arranged so that all
the results are consequences of a single principle, namely, in
this case, the conservation of energy. This was followed by a
memoir on the theory of the tides to which, conjointly with
memoirs by Euler and Maclaurin, a prize was awarded by the
French Academy : these three memoirs contain all that was
done on this subject between the publication of Newton s
Principia and the investigations of Laplace. Bernoulli also
wrote a large number of papers on various mechanical ques
tions, especially on problems connected with vibrating strings,
and the solutions given by Taylor and by D Alembert. He is
the earliest writer who attempted to formulate a kinetic theory
of gases, and he applied the idea to explain the law associated
with the names of Boyle and Mariotte.
The English mathematicians of the eighteenth century.
I have reserved a notice of the English mathematicians
who succeeded Newton in order that the members of the
English school may be all treated together. It was almost a
matter of course that the English should at first have adopted
the notation of Newton in the infinitesimal calculus in pre
ference to that of Leibnitz, and the English school would
consequently in any case have developed on somewhat different
lines to that on the continent where a knowledge of the in
finitesimal calculus was derived solely from Leibnitz and the
Bernoullis. But this separation into two distinct schools
became very marked owing to the action of Leibnitz and
John Bernoulli, which was naturally resented by Newton s
friends : and so for forty or fifty years, to the mutual disad
vantage of both sides, the quarrel raged. The leading members
of the English school were Cotes, Demoivre, Ditton, David
Gregory, Halley, Maclaurin, Simjjson, and Taylor. I may
however again remind my readers that as we approach modern
times the number of capable mathematicians in Britain,
France, Germany, and Italy becomes very considerable, but
DAVID GREGORY. HALLEY. 387
that in a popular sketch like this book it is only the leading
men whom I propose to mention.
To David Gregory, Halley, and Ditton I need devote but
few words.
David Gregory. David Gregory, the nephew of the James
Gregory mentioned above on p. 315, born at Aberdeen on
June 24, 1661, and died at Maidenhead on Oct. 10, 1708, was
appointed professor at Edinburgh in 1684, and in 1691 was on
Newton s recommendation elected Savilian professor at Oxford.
His chief works are one on geometry, issued in 1684 ; one on
optics, published in 1695, which contains [p. 98] the earliest
suggestion of the possibility of making an achromatic combina
tion of lenses ; and one on the Newtonian geometry, physics,
and astronomy, issued in 1702.
Halley. Edmund Halley, born in London in 1656, and
died at Greenwich in 1742, was educated at St Paul s School,
London, and Queen s College, Oxford, in 1703 succeeded Wallis
as Savilian professor, and subsequently in 1720 was appointed
astronomer royal in succession to Flamsteed (see above, p. 345)
whose Historia Coelestis Britannica he edited in 1712 (first
and imperfect edition). Halley s name will be recollected for
the generous manner in. which he secured the immediate
publication of Newton s Principia in 1687. Most of his
original work was on astronomy and allied subjects, and lies
outside the limits of this book ; it may be however said that
the work is of excellent quality, and both Lalande and Mairan
speak of it in the highest terms. Halley conjectu rally restored
the eighth and lost book of the conies of Apollonius, and in
1710 brought out a magnificent edition of the whole work:
he also edited the works of Serenus, those of Menelaus, and
some of the minor works of Apollonius. He was in his turn
succeeded at Greenwich as astronomer royal by Bradley*.
* James Bradley, born in Gloucestershire in 1692, and died in 1762,
was the most distinguished astronomer of the first half of the eighteenth
century. Among his more important discoveries were the explanation
of astronomical aberration (1729), the cause of nutation (1748), and his
252
388 DITTON. TAYLOR.
Ditton. Humphry Ditton was born at Salisbury on May 29,
1675, and died in London in 1715 at Christ s Hospital where
he was mathematical master. He does not seem to have paid
much attention to mathematics until he came to London about
1705, and his early death was a distinct loss to English science.
He published in 1706 a text-book on fluxions; this and
another similar work by William Jones which was issued in
1711 occupied in England much the same place that [ Hospital s
treatise did in France; in 1709 Ditton issued an algebra; and
in 1712 a treatise on perspective. He also wrote numerous
papers in the Philosophical Transactions ; he was the earliest
writer to attempt to explain the phenomenon of capillarity on
mathematical principles ; and he invented a method for finding
the longitude which has been since used on various occasions.
Taylor*. Brook Taylor, born at Edmonton on Aug. 18,
1685, and died in London on Dec. 29, 1731, was educated at
St John s College, Cambridge, and was among the most en
thusiastic of Newton s admirers. From the year 1712 onwards
he wrote numerous papers in the Philosophical Transactions in
which, among other things, he discussed the motion of pro
jectiles, the centre of oscillation, and the forms of liquids
raised by capillarity. In. 1719 he resigned the secretaryship
of the Royal Society and abandoned the study of mathematics.
His earliest work, and that by which he is generally known,
is his Methodus Incrementorum Directa et Inversa published
in London, in 1715. This contains [prop. 7] a proof of the
well-known theorem
/ (x + h) =/ (x) + hf (x) + ^f (x) + .,.,
by which any function of a single variable can be expanded
empirical formula for corrections for refraction. It is perhaps not too
much to say that he was the first astronomer who made the art of observ
ing part of a methodical science.
* An account of his life by Sir William Young is prefixed to the
Contemplatio Philosophica : this was printed at London in 1793 for private
circulation and is now extremely rare.
TAYLOR. 389
in powers of it. He does not consider the convergency
of the series, and the proof which involves numerous assump
tions is not worth reproducing. The work also includes
several theorems on interpolation. Taylor was the earliest
writer to deal with theorems on the change of the inde
pendent variable ; he was perhaps the first to realize the
possibility of a calculus of operation, and just as he denotes
the nth differential coefficient of y by y n , so he uses y_ l to
represent the integral of y\ lastly he is usually recognized
as the creator of the theory of finite differences.
The applications of the calculus to various questions given
in the Methodus have hardly received that attention they
deserve. The most important of them is the theory of the
transverse vibrations of strings, a problem which had baffled
previous investigators. In this investigation Taylor shews
that the number of half-vibrations executed in a second is
/DP
"V LN>
where L is the length of the string, N its weight, P the weight
which stretches it, and D the length of a seconds pendulum.
This is correct, but in arriving at it he assumes that every
point of the string will pass through its position of equili
brium at the same instant, a restriction which D Alembert
subsequently shewed to be unnecessary. Taylor also found
the form which the string assumes at any instant. This work
also contains the earliest determination of the differential
equation of the path of a ray of light when traversing a
heterogeneous medium ; and, assuming that the density of the
air depends only on its distance from the earth s surface,
Taylor obtained by means of quadratures the approximate form
of the curve. The form of the catenary and the determination
of the centres of oscillation and percussion are also discussed.
A treatise on perspective, published in 1719, contains the
earliest general enunciation of the principle of vanishing
points ; though the idea of vanishing points for horizontal and
390 COTES.
parallel linns in a picture hung in a vortical plane had been
enunciated by (Juido Ubaldi in his l^rxprMivac, Libri, Pi ,;i,
1600, and by Htevinus in his Sciayraphia, Loyden, 1G08.
Cotes*. Roger Cotes was born near Leicester on July 10,
If>H2, and died at Cambridge on June 5, 1710. Hi; was
educated at Trinity College, Cambridge, of which society he
was a fellow, and in 1700 was elected to the nowly-created
Plumian chair of astronomy in the university of Cambridge.
From 1709 to 1713 his time was mainly occupied in editing
the second edition of the 1 ri itcipia. Tin- remark of Newton
that if only Cotes had lived " we should have learnt some
thing" indicates the opinion of his abilities held by most of
his contemporaries.
Colon s writings were collected and published in 1722
under the titles Ha/nn&nMi Af&nsurci/rufn and Opera MisceL-
latifia. His lectures on hydrostatics were published in 1738.
A largr part <>f I IK- //armonid Mvnxti/t d rn/m is givrn up
to UK; decomposition and integration of rational algebraical
expressions : that part which deals with the theory of partial
fractions was left unfmishe d, but was completed by Domoivro.
Cotes s theorem in trigonometry, which depends on forming the
quadratic factors of , *:"- I, is well known. The proposition that
"if from a fixed point a line bo drawn cutting a curve; in
Qn Qvi . . . ,Q , mid a point / be, taken on tin; line so that the
reciprocal of 1* is the arithmetic mean of the reciprocals of ()Cj t ,
OQ. 29 ... t OQ ut then the locus of 1* will be a straight lino" is also
due to Cotes. The title of the book was derived from the
latter theorem. The Opera MisCGllcMMd contains a paper on
the method for determining the most probable result from a
number of observations : this was the earliest attempt to
frame a theory of errors. It also contains essays on Newton s
Methodux Differentially, on the construction of tables by the
method of differences, on the descent of a body under gravity,
on the cycloidal pendulum, and on projectiles.
* Boo my Hinton/ of tin* Nhnly of Mtttlwrnatics (it CamWdye, Cain
es IHH<), p. HH.
DEMOIVRE. 391
Demoivre. Abraluim Demoivre (more correctly written
as de Moivre) was born at Vitry on May 20, 1667, and
died in London on Nov. 27, 1754. His parents came to
England when he was a boy, and his education and friends
were alike English. His interest in the higher mathematics
is said to have originated in his coming by chance across a
copy of Newton s Princijna. From the eloye on him de
livered in 1754 before the French Academy it would seem
that as a young fellow his work as a teacher of mathe
matics had led him to the house of the Earl of Devonshire at
the instant when Newton, who had asked permission to present
a copy of his work to the earl, was coming out. Faking up
the book, and charmed by the far-reaching conclusions and
the apparent simplicity of the reasoning, Demoivre thought
nothing would be easier than to master the subject, but to his
surprise found that to follow the argument overtaxed his
powers. He however bought a copy, and as he had but little
leisure he tore out the pages in order to carry one or two
of them loose in his pocket so that he could study them in the
intervals of his work as a teacher. Subsequently he joined
the Royal Society, and became intimately connected with
Newton, Haliey, and other mathematicians of the English
school. The manner of his death has a curious interest for
psychologists. Shortly before it, he declared that it was
necessary for him to sleep some ten minutes or a quarter of an
hour longer each day than the preceding one : the day after he
had thus reached a total of something over twenty-three hours
he slept up to the limit of twenty-four hours, and then died in
his sleep.
He is best known for having, together with Lambert,
created that part of trigonometry which deals with imaginary
quantities. Two theorems on this part of the subject aiv .-.till
connected with his name, namely, that which asserts tl.al
sin nx + i cos nx is one of the values of (sin x + icos #)", and
that which gives the various quadratic larlurs of u? n 2px* + 1
His chief works, other than numerous papers in the /V//A/
392 MACLAURIN.
sopkical Transactions, were The Doctrine of Chances published
in 1718, and the Miscellanea Analytica published in 1730. In
the former the theory of recurring series was first given, and
the theory of partial fractions which Cotes s premature death
had left unfinished was completed, while the rule for finding
the probability of a compound event was enunciated. The
latter, besides the trigonometrical propositions mentioned above,
contains some theorems in astronomy but they are treated as
problems in analysis.
Maclaurin.* Colin Maclaurin, who was born at Kilmodan
in Argyllshire in February 1698, and died at York on June 14,
1746, was educated at the university of Glasgow; in 1717,
he was elected, at the early age of nineteen, professor of
mathematics at Aberdeen; and in 1725, he was appointed the
deputy of the mathematical professor at Edinburgh, and ulti
mately succeeded him: there was some difficulty in securing a
stipend for a deputy, and Newton privately wrote offering to
bear the cost so as to enable the university to secure the
services of Maclaurin. Maclaurin took an active part in
opposing the advance of the Young Pretender in 1745 : on the
approach of the Highlanders he fled to York, but the exposure
in the trenches at Edinburgh and the privations he endured in
his escape proved fatal to him.
His chief works are his Geometria Organica, London, 1719;
his De Linearum Geometricarum Proprietatibus, London, 1720;
his Treatise on Fluxions, Edinburgh, 1742; his Algebra,
London, 1748; and his Account of Newton s Discoveries,
London, 1748.
The Geometria Organica is on the extension of a theorem
given by Newton. Newton had shewn that, if two angles
bounded by straight lines turn round their respective summits
so that the point of intersection of two of these lines moves
along a straight line, the other point of intersection will
describe a conic ; and, if the first point move along a conic, the
* A sketch of Maclaurin s life is prefixed to his posthumous account
of Newton s discoveries, London, 1748.
MACLAURIN. 393
second will describe a quartic. Maclaurin gave an analytical
discussion of the general theorem, and shewed how by this
method various curves could be practically traced. This work
contains an elaborate discussion on curves and their pedals,
a branch of geometry which he had created in two papers
published in the Philosophical Transactions for 1718 and 1719.
In the following year, 1720, Maclaurin issued a supplement
which is practically the same as his De Linearum Geometri-
carum Proprietatibus. It is divided into three sections and an
appendix. The first section contains a proof of Cotes s theorem
above alluded to ; and also the analogous theorem (discovered
by himself) that, if a straight line OP^^... drawn through a
fixed point cut a curve of the nth degree in n points
P lt P 3 ,..., and if the tangents at P lt P^,..- cut a fixed line Ox
in points A ly A 2 ,..., then the sum of the reciprocals of the
distances OA lt OA 2y ... is constant for all positions of the
line 0/VPo.... These two theorems are generalizations of
those given by Newton on diameters and asymptotes. Either
is deducible from the other. In the second section these
theorems are applied to conies ; most of the harmonic pro
perties connected with an inscribed quadrilateral are deter
mined ; and in particular the theorem on an inscribed hexagon
which is known by the name of Pascal is deduced. Pascal s
essay was not published till 1779, and the earliest printed
enunciation of his theorem was that given by Maclaurin. In
the third section these theorems are applied to cubic curves.
Amongst other propositions he here shews that, if a quadri
lateral be inscribed in a cubic, and if the points of intersection
of the opposite sides also lie on the curve, then the tangents to
the cubic at any two opposite angles of the quadrilateral will
meet on the curve. The appendix contains some general
theorems. One of these (which includes Pascal s as a
particular case) is that if a polygon be deformed so that while
each of its sides passes through a fixed point, its angles (save
one) describe respectively curves of the ?>ith, wth, />th, degrees,
then shall the remaining angle describe a curve of the degree
394 MACLAURIN.
2mnp . . ; but, if the given points be collinear, the resulting
curve will be only of the degree mnp ____ This essay was re
printed with additions in the Philosophical Transactions for
1735.
The Treatise of Fluxions published in 1742 was the first
logical and systematic exposition of the method of fluxions.
The cause of its publication was an attack by Berkeley on the
principles of the infinitesimal calculus. In it [art. 751, p. 610]
Maclaurin gave a proof of the theorem that
This was obtained in the manner given in many modern text
books by assuming that f (x) can be expanded in a form
like
/ (x) = A + A& + A#? +...,
then on differentiating and putting x = Q in the successive
results, the values of A Q , A 1J ... are obtained: but he did
not investigate the convergency of the series. The result had
been previously given in 1730 by James Stirling in his
Methodus Differentialis [p. 102], and of course is at once
deducible from Taylor s theorem on which the proofs by
Stirling and Maclaurin are admittedly founded. Maclaurin
also here enunciated [art. 350, p. 289] the important theorem
that, if < (x) be positive and decrease as x increases from x = a to
x oo , then the series
< () + </> (a+ 1) + < (a + 2)+ ...
roc
is convergent or divergent as / < (x) dx is finite or infinite.
J a
He also gave the correct theory of maxima and minima, and
rules for finding and discriminating multiple points.
This treatise is however especially valuable for the solu
tions it contains of numerous problems in geometry, statics,
the theory of attractions, and astronomy. To solve these he
reverted to classical methods, and so powerful did these pro
cesses seem, when used by him, that Olairaut after reading the
MACLAURIN. 395
work abandoned analysis, and attacked the problem of the
6gure of the earth again by pure geometry. At a later time
this part of the book was described by Lagrange as the " chef-
d oeuvre de geometric qu on peut comparer a tout ce qu Archi-
mede nous a laisse* de plus beau et de plus ingdnieux."
Maclaurin also determined the attraction of a homogeneous
ellipsoid at an internal point, and gave some theorems on its
attraction at an external point ; in effecting this he introduced
the conception of level surfaces, i.e. surfaces at every point of
which the resultant attraction is perpendicular to the surface.
No further advance in the theory of attractions was made
until Lagrange in 1773 introduced the idea of the potential
(see below, p. 412). Maclaurin also shewed that a spheroid
was a possible form of equilibrium of a mass of homogeneous
liquid rotating about an axis passing through its centre of
mass. Finally he discussed the tides : this part had been
previously published (in 1740) and had received a prize from
the French Academy.
Among Maclaurin s minor works is his Algebra, published
in 1748, and founded on Newton s Universal Arithmetic. It
contains the results of some early papers of Maclaurin ; notably
of two, written in 172G and 1729, on the number of imaginary
roots of an equation, suggested by Newton s theorem (see above,
p. 332); and of one, written in 1729, containing the well-known
rule for finding equal roots by means of the derived equation.
To this a treatise, entitled De Linearum Geomeiricarum Pro-
prietatibus Generalibus, was added as an appendix ; besides
the paper of 1720 above alluded to, it contains some additional
and elegant theorems. Maclaurin also produced in 1728 an
exposition of the Newtonian philosophy, which is incorporated
in the posthumous work printed in 1748. Almost the last
paper he wrote was one printed in the Philosophical Trans
actions for 1743 in which he discussed from a mathematical
point of view the form of a bee s cell.
Maclaurin was succeeded in his chair at Edinburgh by his
pupil Matthew Stewart, born at Roth say in 1717 and died at
396 SIMPSON.
Edinburgh on Jan. 23, 1785, a mathematician of considerable
power, to whom I allude in passing for his theorems on the
problem of three bodies and for his discussion, treated by
transversals and involution, of the properties of the circle and
straight line.
Maclaurin was one of the most able mathematicians of the
eighteenth century, but his influence on the progress of British
mathematics was on the whole unfortunate. By himself
abandoning the use both of analysis and of the infinitesimal
calculus he induced Newton s countrymen to confine them
selves to Newton s methods, and as I remarked before it was
riot until about 1820, when the differential calculus was
introduced into the Cambridge curriculum, that English
mathematicians made any general use of the more powerful
methods of modern analysis.
Simpson*. The last member of the English school whom
I need mention here is Thomas Simpson, who was born in
Leicestershire on Aug. 20, 1710, and died on May 14, 1761.
His father was a weaver and he owed his education to his
own efforts. His mathematical interests were first aroused by
the solar eclipse which took place in 1724, and with the aid
of a fortune- telling pedler he mastered Cocker s Arithmetic and
the elements of algebra. He then gave up his weaving, and
became an usher at a school, and by constant and laborious
efforts improved his mathematical education so that by 1735
he was able to solve several questions involving the infini
tesimal calculus, which had been recently proposed. He next
moved to London, and in 1743 was appointed professor of
mathematics at Woolwich, a post which he continued to occupy
till his death.
The works published by Simpson prove him to have been
a man of extraordinary natural genius and extreme industry.
The most important of them are his Fluxions, 1737 and 1750,
with numerous applications to physics and astronomy ; his
* A life of Simpson, with a bibliography of his writings, by Be vis and
Hutton was published in London in 1764.
SIMPSON. 397
Laws of Chance and his Essays, 1740 ; his theory of Annuities
and Reversions (a branch of mathematics that is due to James
Dodson, 1597 1657, who was a master at Christ s Hospital,
London), with tables of the value of lives, 1742; his Dis
sertations, 1743, in which the figure of the earth, the force
of attraction at the surface of a nearly spherical body, the
theory of the tides, and the law of astronomical refraction
are discussed; his Algebra, 1745; his Geometry, 1747; his
Trigonometry, 1748, in which he introduced the current ab
breviations for the trigonometrical functions ; his Select Exer
cises, 1752, containing the solutions of numerous problems
and a theory of gunnery ; and lastly, his Miscellaneous Tracts,
1754. The last consists of eight memoirs and these contain
his best known investigations. The first three papers are on
various problems in astronomy ; the fourth is on the theory of
mean observations ; the fifth and sixth on problems in fluxions
and algebra ; the seventh contains a general solution of the
isoperimetrical problem ; the eighth contains a discussion of
the third and ninth sections of the Principia, and their appli
cation to the lunar orbit. In this last memoir Simpson
obtained a differential equation for the motion of the apse of
the lunar orbit similar to that arrived at by Clairaut, but
instead of solving it by successive approximations he deduced
a general solution by indeterminate coefficients. The result
agrees with that given by Clairaut. Simpson first solved
this problem in 1747, two years later than the publication of
Clairaut s memoir, but the solution was discovered independently
of Clairaut s researches of which Simpson first heard in 1748.
398
CHAPTER XVIII.
LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
CIRC. 17401830.
THE last chapter contains the history of two separate
schools the continental and the British. In the early years
of the eighteenth century the English school appeared vigorous
and fruitful, but decadence rapidly set in, and after the deaths
of Maclaurin and Simpson no British mathematician appeared
who is at all comparable to the continental mathematicians of
the latter half of the eighteenth century. This fact is partly
explicable by the isolation of the school, partly by its tendency
to rely too exclusively on geometrical and fiuxioiial methods.
Some attention was however given to practical science, but,
except for a few remarks on English physicists, I do not think
it necessary to discuss, English mathematics further, until
about 1820 when analytical methods again came into vogue.
On the continent under the influence of John Bernoulli
the calculus had become an instrument of great analytical power
expressed in an admirable notation and for practical applica
tions it is impossible to over-estimate the value of a good
notation. The subject of mechanics remained however in much
the condition in which Newton had left it, until D Alembert,
in putting Newton s results into the language of the differential
calculus, did something to extend it. Universal gravitation as
enunciated in the Principia was accepted as an established fact,
but the geometrical methods adopted in proving it were diffi
cult to follow or to use in analogous problems ; Maclaurin,
Simpson, and Clairaut may be regarded as the last mathe-
EULER. 399
maticians of distinction who employed them. Lastly the
Newtonian theory of light was generally received as correct.
The leading mathematicians of the era on which we are
now entering are Euler, Lagrange, Laplace, and Legendre.
Briefly we may say that Euler extended, summed up, and com
pleted the work of his predecessors ; while Lagrange with
almost unrivalled skill developed the infinitesimal calculus
and theoretical mechanics into the form in which we now
know them. At the same time Laplace made some additions
to the infinitesimal calculus, and applied that calculus to the
theory of universal gravitation ; he also created a calculus of
probabilities. Legendre invented spherical harmonic analysis
and elliptic integrals, and added to the theory of numbers.
The works of these writers are still standard authorities and
are hardly yet the subject-matter of history. I shall therefore
content myself with a mere sketch of their chief discoveries,
referring anyone who wishes to know more to the works
themselves. Lagrange, Laplace, and Legendre created a
French school of mathematics of which the younger members
are divided into two groups ; one (including Poisson and
Fourier) began to apply mathematical analysis to physics, and
the other (including Monge, Carnot, and Poncelet) created
modern geometry. Strictly speaking some of the great mathe
maticians of recent times, such as Gauss and Abel, were con
temporaries of the mathematicians last named ; but, except for
this remark, I think it convenient to defer any consideration
of them to the next chapter.
The development of analysis and mechanics.
Euler *. Leonliard Euler was bom at Bale on April 15,
* The chief facts in Euler s life are given by Fuss, and a list of
Eulcr s writings is prefixed to his Corrt tpnnth ncc., 2 vols, St Petersburg,
1843. Nicholas Fuss born at Bale in 1755, and died at St Petersburg
in 1826, was a pupil of Daniel Bernoulli, and subsequently was appointed
assistant to Euler : Fuss wrote on spherical conies and on lines of
curvature. No complete edition of Euler s writings has been published,
though the work has been begun twice.
400 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
1707, and died at St Petersburg on Sept. 7, 1783. He was
the son of a Lutheran minister who had settled at Bale, and
was educated in his native town under the direction of John
Bernoulli, with whose sons Daniel and Nicholas he formed a
life-long friendship. When, in 1725, the younger Bernoullis
went to Russia, on the invitation of the empress, they pro
cured a place there for Euler, which in 1733 he exchanged for
the chair of mathematics then vacated by Daniel Bernoulli.
The severity of the climate affected his eyesight, and in 1735
he lost the use of one eye completely. In 1741 he moved to
Berlin at the request, or rather command, of Frederick the
Great ; here he stayed till 1766, when he returned to Russia,
and was succeeded at Berlin by Lagrange. Within two or
three years of his going back to St Petersburg he became
blind ; but in spite of this, and although his house together
with many of his papers were burnt in 1771, he recast and
improved most of his earlier works. He died of apoplexy in
1783. He was married twice.
I think we may sum up Euler s work by saying that he
created analysis, and revised almost all the branches of pure
mathematics which were then known, filling up the details,
adding proofs, and arranging the whole in a consistent form.
Such work is very important, and it is fortunate for science
when it falls into hands as competent as those of Euler.
Euler wrote an immense number of memoirs on all kinds
of mathematical subjects. His chief works, in which many
of the results of earlier memoirs are embodied, are as fol
lows.
In the first place, he wrote in 1748 his Introductio in
Analysin Infinitorum, which was intended to serve as an
introduction to pure analytical mathematics. This is divided
into two parts.
The first part of the Analysis Infinitorum contains the
bulk of the matter which is to be found in modern text-books
on algebra, theory of equations, and trigonometry. In the
algebra he paid particular attention to the expansion of various
EULER. 401
functions in series, and to the summation of given series; and
pointed out explicitly that an infinite series cannot be safely
employed unless it is convergent. In the trigonometry, much
of which is founded on F. C. Mayer s Arithmetic of Sines which
had been published in 1727, Euler developed the idea of John
Bernoulli that the subject was a branch of analysis and not a
mere appendage of astronomy or geometry : he also introduced
(contemporaneously with Simpson) the current abbreviations
for the trigonometrical functions, and shewed that the trigo
nometrical and exponential functions were connected by the
relation cos + i sin = e ie .
Here too [pp. 85, 90, 93] we meet the symbol e used to de
note the base of the Napierian logarithms, namely, the incom
mensurable number 271828..., and the symbol TT used to
denote the incommensurable number 3-14159.... The use of a
single symbol to denote the number 2*71828... seems to be due
to Cotes who denoted it by M. Newton was (as far as I know)
the first to employ the literal exponential notation, and Euler,
using the form a 2 , had taken a as the base of any system of
logarithms : it is probable that the choice of e for a particular
base was determined by its being the vowel consecutive to a.
The use of a single symbol to denote the number 3*14159...
appears to have been introduced by John Bernoulli who repre
sented it by c; Euler in 1734 denoted it by /?, and in a letter
of 1736 (in which he enunciated the theorem that the sum of
the squares of the reciprocals of the natural numbers is ^TT*)
he used the letter c; Chr. Goldbach in 1742 used TT; and after
the publication of Euler s Analysis the symbol TT was generally
employed.
The numbers e and TT would enter into mathematical analysis
from whatever side the subject was approached. The latter
represents among other things the ratio of the circumfer
ence of a circle to its diameter, but it is a mere accident
that that is taken for its definition. De Morgan in the Budget
of Paradoxes tells an anecdote which illustrates how little the
usual definition suggests its real use. He was explaining to
B. 26
402 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES?
an actuary what was the chance that at the end of a given
time a certain proportion of some group of people would be
alive ; and quoted the actuarial formu]a involving TT, which in
answer to a question he explained stood for the ratio of the
circumference of a circle to its diameter. His acquaintance
who had so far listened to the explanation with interest inter
rupted him and explained, " My dear friend, that must be a
delusion ; what can a circle have to do with the number of
people alive at the end of a given time ? "
The second part of the Analysis Infinitorum is on ana
lytical geometry. Euler commenced this part by dividing
curves into algebraical and transcendental, and established a
variety of propositions which are true for all algebraical curves.
He then applied these to the general equation of the second
degree in two dimensions, shewed that it represents the various
conic sections, and deduced most of their properties from the
general equation. He also considered the classification of
cubic, quartic, and other algebraical curves. He next dis
cussed the question as to what surfaces are represented by the
general equation of the second degree in three dimensions, and
how they may be discriminated one from the other : some of
these surfaces had not been previously investigated. In the
course of this analysis he laid down the rules for the transfor
mation of coordinates in space. Here also we find the earliest
attempt to bring the curvature of surfaces within the domain of
mathematics, and the first complete discussion of tortuous curves.
The Analysis Infinitorum was followed in 1755 by the
Institution** Calculi Differentialis to which it was intended as
an introduction. This is the first text-book on the differential
calculus which has any claim to be regarded as complete, and
it may be said that most modern treatises on the subject are
based on it ; at the same time it should be added that the
exposition of the principles of the subject is often prolix and
obscure, and sometimes not altogether accurate.
This series of works was completed by the publication in
three volumes in 1768 to 1770 of the Institutions Calculi
EULER. 403
Integralis in which the results of several of Euler s earlier
memoirs on the same subject and on differential equations are
included. This, like the similar treatise on the differential
calculus, summed up what was then known on the subject,
but many of the theorems were recast and the proofs improved.
The Beta and Gamma* functions were invented by Euler and
are discussed here, but only as illustrations of methods of
reduction and integration. His treatment of elliptic integrals
is superficial ; it was due to a theorem given by John Landeri
a writer who was suggestive rather than powerful in the
Philosophical Transactions for 1755 connecting the arcs of a
hyperbola and an ellipse. Euler s works that form this trilogy
have gone through numerous subsequent editions.
The classic problems on isoperimetrical curves, the brachis-
tochrone in a resisting medium, and the theory of geodesies
(all of which had been suggested by his master John Ber
noulli) had engaged Euler s attention at an early date ; and
in solving them he was led to the calculus of variations. The
general idea of this was laid down in his Curvarum Maximi
Miniimve Proprietate Gaudentium Inventio Nova ac Facilis
published in 1744, but the complete development of the new
calculus was first effected by Lagrange in 1759. The method
used by Lagrange is described in Euler s integral calculus, and
is the same as that given in most modern text-books on the
subject.
In 1770 Euler published the Anleitung zur Algebra in two
volumes. The first volume treats of determinate algebra. This
contains one of the earliest attempts to place the fundamental
processes on a scientific basis : the same subject had attracted
D Alembert a attention. This work also includes the proof of
the binomial theorem for an unrestricted index which is still
known by Euler s name ; the proof is founded on the principle
of the permanence of equivalent forms, but Euler made no
attempt to investigate the convergency of the series : that he
* The history of the Gamma function is given in a monograph by
Brunei in the Mtmoires de la soviet? <les sciences, Bordeaux, 1886.
262
404 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
should have omitted this essential step is the more curious as
he had himself recognized the necessity of considering the
coiivergency of infinite series. The second volume treats of
indeterminate or Diophantine algebra. This contains the
solutions of some of the problems proposed by Fermat, and
which had hitherto remained unsolved. A. French translation
of the algebra, with numerous and valuable additions by
Jjagrange, was brought out in 1794; and a treatise on
arithmetic by Euler was appended to it.
These four works comprise most of what Euler produced in
pure mathematics. He also wrote numerous memoirs on nearly
all the subjects of applied mathematics and mathematical
physics then studied : the chief results in them are as follows.
In the mechanics of a rigid system he determined the
general equations of motion of a body about a fixed point,
which are ordinarily written in the form
and he gave the general equations of motion of a free body,
which are usually presented in the form
(mu) - mv0 3 + mw0 3 X, and h y 3 + A 3 2 = L.
cut dt
He also defended and elaborated the theory of "least action"
which had been propounded by Maupertuis in 1751 in his
Essai de cosmologie [p. 70].
In hydrodynamics Euler established the general equations
of motion, which are commonly expressed in the form
1 dp ^ du du du du
--f- = A-- r --u- r -v -w -j-.
p ax at ax dy dz
At the time of his death he was engaged in writing a treatise
on hydromechanics in which the treatment of the subject would
have been completely recast.
His most important works on astronomy are his Theoria
EULER. 405
Motuum Planetarum et Cometarum, published in 1744; his
Theoria Motus Lunaris, published in 1753; and his Theoria
Motuum Lunae, published in 1772. In these he attacked the
problem of three bodies : he supposed the body considered, e.g.
the moon, to carry three rectangular axes with it in its motion,
the axes moving parallel to themselves, and to these axes all
the motions were referred. This method is not convenient, but
it was from Euler s results that Mayer* constructed the lunar
tables for which his widow in 1770 received .5000, being the
prize offered by the English parliament, and in recognition of
Euler s services a sum of ,300 was voted as an honorarium to
him.
Euler was much interested in optics. In 1746 he discussed
the relative merits of the emission and undulatory theories of
light; he on the whole preferred the latter. In 1770 71
he published his optical researches in three volumes under
the title Dioptrica.
He also wrote an elementary work on physics and the
fundamental principles of mathematical philosophy. This ori
ginated from an invitation he received when he first went to
Berlin to give lessons on physics to the princess of Anhalt-
Dessau. These lectures were published in 1768 1772 in
three volumes under the title Lettres...sur quelques sujets
de physique..., and for half a century remained a standard
treatise on the subject.
Of course Euler s magnificent works were not the only
text-books containing original matter produced at this time.
Amongst numerous writers I would specially single out Daniel
Bernoulli, Simpson, Lambert, Bezout, Trembley, and Arbogast
as having influenced the development of mathematics. To
the two first-mentioned I have already alluded in the last
chapter.
* Johann Tobias Mayer, born in Wiirtemberg in 1723 and died in
1762, was director of the English observatory at Gottingen. Most of his
memoirs, other than his lunar tables, were published in 1775 under the
title Opera Inedlta.
406 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
Lambert*. Johann Heinrich Lambert was born at Miil-
hausen on Aug. 28, 1728, and died at Berlin on Sept. 25, 1777.
He was the son of a small tailor, and had to rely on his own
efforts for his education ; from a clerk in some iron- works, he
got a place in a newspaper office, and subsequently on the
recommendation of the editor he was appointed tutor in a
private family which secured him the use of a good library and
sufficient leisure to use it. In 1759 he settled at Augsburg,
and in 1763 removed to Berlin where he was given a small
pension and finally made editor of the Prussian astronomical
almanack.
Lambert s most important works were one on optics, issued
in 1759, which suggested to Arago the lines of investigation he
subsequently pursued ; a treatise on perspective, published in
1759 (to which in 1768 an appendix giving practical appli
cations was added); and a treatise on comets, printed in 1761,
containing the well-known expression for the area of a focal
sector of a conic in terms of the chord and the bounding
radii. Besides these he communicated numerous papers to
the Berlin Academy. Of these the most important are his
memoir in 1768 on transcendental magnitudes, in which he
proved that TT is incommensurable (the proof is given in Le-
gendre s Geometriej and is there extended to ?r 2 ) : his paper on
trigonometry, read in 1768, in which he developed Demoivre s
theorems on the trigonometry of complex variables, and intro
duced the hyperbolic sine and cosinef denoted by the symbols
sinh x, cosh x : his essay entitled analytical observations, pub
lished in 1771, which is the earliest attempt to form functional
equations by expressing the given properties in the language
* See Lambert nach seinem Leben und Wirken by D. Huber, Bale,
1829. Most of Lambert s memoirs are collected in his Beitraye zum
Gebrauche der Mathematik, published in four volumes, Berlin, 1765
1772.
t These functions are said to have been previously suggested by
F. C. Mayer, see Die Lehre von den Hyperbelfunktionen by S. Giinther,
Halle, 1881, and Beitrdge zur Geschichte der neueren Mathematik, Ans-
bach, 1881.
LAMBERT. B^ZOUT. ARBOGAST. LAGRANGE. 407
of the differential calculus, and then integrating : lastly his
paper on vis viva, published in 1783, in which for the first
time he expressed Newton s second law of motion in the no
tation of the differential calculus.
Of the other mathematicians above mentioned I here add a
few words. Etienne Be*zout, born at Nemours on March 31,
1730, and died on Sept. 27, 1783, besides numerous minor
works, wrote a Theorie generate des Equations algebriques, pub
lished at Paris in 1779, which in particular contained much
new and valuable matter on the theory of elimination and
symmetrical functions of the roots of an equation : he used
determinants in a paper in the Histoire de Vacademie royale,
1764, but did not treat of the general theory. Jean Trembley,
born at Geneva in 1749, and died on Sept. 18, 1811, con
tributed to the development of differential equations, finite
differences, and the calculus of probabilities. Louis Frangois
Antoine Arbogast, born in Alsace on Oct. 4, 1759, and died at
Strassburg, where he was professor, on April 8, 1803, wrote on
series and the derivatives known by his name : he was the first
writer to separate the symbols of operation from those of
quantity.
I do not wish to crowd my pages with an account of those
who have not distinctly advanced the subject, but I have
mentioned the above writers because their names are still well
known. We may however say that the discoveries of Euler
and Lagrange in the subjects which they treated were so com
plete and far-reaching that what their less gifted contempo
raries added is not of sufficient importance to require mention
in a book of this nature.
Lagrange*. Joseph Louis Lagrange^ the greatest mathe
matician of the eighteenth century, was born at Turin on
* Summaries of the life and works of Lagrange are given in the
English Cyclopaedia and the Encyclopaedia Britannica (ninth edition),
of which I have made considerable use : the former contains a biblio
graphy of his writings. Lagrange s works, edited by MM. Serret and
Darboux, are now being published by the French government. Delambre s
account of his life is printed in the first volume.
408 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
Jan. 25, 1736, and died at Paris on April 10, 1813. His
father, who had the charge of the Sardinian military chest, was
of good social position and wealthy, but before his son grew up
he had lost most of his property in speculations, and young
Lagrange had to rely for his position on his own abilities. He
was educated at the college of Turin, but it was not until he
was seventeen that he shewed any taste for mathematics : his
interest in the subject being first excited by a memoir by
Halley (Phil. Trans, vol. xvm. p. 960), across which he came
by accident. Alone and unaided he threw himself into mathe
matical studies, and at the end of a year s incessant toil he
was already an accomplished mathematician, and was made
a lecturer in the artillery school. The first fruit of these
labours was his letter, written when he was still only nineteen,
to Euler in which he solved the isoperimetrical problem which
for more than half a century had been a subject of discussion.
To effect the solution (in which he sought to determine the
form of a function so that a formula in which it entered should
satisfy a certain condition) he enunciated the principles of the
calculus of variations. Euler recognized the generality of the
method adopted, and its superiority to that used by himself ;
and with rare courtesy he withheld a paper he had previously
written, which covered some of the same ground, in order that
the young Italian might have time to complete his work, and
claim the undisputed invention of the new calculus. The
name of this branch of analysis was suggested by Euler.
This memoir at once placed Lagrange in the front rank of
mathematicians then living.
In 1758 Lagrange established with the aid of his pupils
a society, which was subsequently incorporated as the Turin
Academy, and in the five volumes of its transactions, usually
known as the Miscellanea Taurinensia, most of his early
writings are to be found. Many of these are elaborate
works. The first volume contains a memoir on the theory of
the propagation of sound ; in this he indicates a mistake
made by Newton, obtains the general differential equation for
LAGRANGE. 409
the motion, and integrates it for motion in a straight line.
This volume also contains the complete solution of the problem
of a string vibrating transversely ; in this paper he points out
a lack of generality in the solutions previously given by
Taylor, D Alernbert, and Euler, and arrives at the conclusion
that the form of the curve at any time t is given by the
equation y = a sin mx sin nt. The article concludes with a
masterly discussion of echoes, beats, and compound sounds.
Other articles in this volume are on recurring series, proba
bilities, and the calculus of variations.
The second volume contains a long paper embodying the
results of several memoirs in the first volume on the theory
and notation of the calculus of variations ; and he illustrates
its use by deducing the principle of least action, and also by
solutions of various problems in dynamics.
The third volume includes the solution of several dynamical
problems by means of the calculus of variations ; some papers
on the integral calculus; a solution of Fermat s problem
mentioned above, p. 296 (/) ; and the general differential
equations of motion for three bodies moving under their
mutual attractions.
In 1761 Lagrange stood without a rival as the foremost
mathematician living ; but the unceasing labour of the pre
ceding nine years had seriously affected his health, and the
doctors refused to be responsible for his reason or life unless
he would take rest and exercise. Although his health was
temporarily restored his nervous system never quite recovered
its tone, and henceforth he constantly suffered from attacks of
profound melancholy.
The next work he produced was in 1764 on the libratiou
of the moon, and an explanation as to why the same face was
always turned to the earth, a problem which he treated by the
aid of virtual work. His solution is especially interesting as
containing the germ of the idea of generalized equations of
motion, equations which he first formally proved in 1780.
He now started to go on a visit to London, but on the
410 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
way fell ill at Paris. There he was received with the most
marked honour, and it was with regret he left the brilliant
society of that city to return to his provincial life at Turin.
His further stay in Piedmont was however short. In 1766
Euler left Berlin, and Frederick the Great immediately wrote
expressing the wish of " the greatest king in Europe " to
have " the greatest mathematician in Europe " resident at
his court. Lagrange accepted the offer and spent the next
twenty years in Prussia, where he produced not only the
long series of memoirs published in the Berlin and Turin trans
actions but his monumental work, the Mecanique analytique.
His residence at Berlin commenced with an unfortunate mis
take. Finding most of his colleagues married, and assured by
their wives that it was the only way to be happy, he married;
his wife soon died, but the union was not a happy one.
Lagrange was a favourite of the king, who used frequently
to discourse to him on the advantages of perfect regularity of
life. The lesson went home, and thenceforth Lagrange studied
his mind and body as though they were machines, and found
by experiment the exact amount of work which he was able to
do without breaking down. Every night he set himself a
definite task for the next day, and on completing any branch
of a subject he wrote a short analysis to see what points in the
demonstrations or in the subject-matter were capable of im
provement. He always thought out the subject of his papers
before he began to compose them, and usually wrote them
straight off without a single erasure or correction.
His mental activity during these twenty years was amazing.
Not only did he produce his splendid Mecanique analytique,
but he contributed between one and two hundred papers to
the Academies of Berlin, Turin, and Paris. Some of these
are complete treatises, and all without exception are of a
high order of excellence. Except for a short time when he
was ill he produced on an average about one memoir a month.
Of these I note the following as among the most important.
First, his contributions to the fourth and fifth volumes
LAGKANGE. 411
(1766 1773) of the Miscellanea Taurinensia ; of which the
most important was the one in 1771 in which he discussed
how numerous astronomical observations should be combined
so as to give the most probable result. And later, his con
tributions to the first two volumes (1784 1785) of the trans
actions of the Turin Academy ; to the first of which he
contributed a paper on the pressure exerted by fluids in
motion, and to the second an article on integration by infinite
series, and the kind of problems for which it is suitable.
Most of the memoirs sent to Paris were on astronomical
questions, and among these I ought particularly to mention
his memoir on the Jovian system in 1766, his essay on the
problem of three bodies in 1772, his work on the secular
equation of the moon in 1773, and his treatise on cometary
perturbations in 1778. These were all written on subjects
proposed by the French Academy, and in each case the prize
was awarded to him.
The greater number of his papers during this time were
however contributed to the Berlin Academy. Several of
them deal with questions on algebra. In particular I may
mention (i) his discussion of the solution of indeterminate
equations in integers (1770); with special notice of inde
terminate quadratics (1769). (ii) His tract on the theory of
elimination (1770). (lii) His memoirs on a general process for
solving an algebraical equation of any degree (1770 and 1771) ;
this method fails for equations of an order above the fourth,
because it then involves the solution of an equation of higher
dimensions than the one proposed, but it gives all the solutions
of his predecessors as modifications of a single principle. He
found however the complete solution of a binomial equation of
any degree, (iv) Lastly in 1773 he treated of determinants
of the second and third order.
Several of his early papers also deal with questions con
nected with the neglected but singularly fascinating subject of
the theory of numbers. Among these are (i) his proof of
the theorem that every integer which is not a square can
412 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
be expressed as the sum of either two, three, or four integral
squares (1770). (ii) His proof of Wilson s theorem that if n
be a prime, then n 1 + 1 is always a multiple of n (1771). (iii)
His memoirs of 1773, 1775, and 1777, which give the demon
strations of several results enunciated by Fermat, and not
previously proved. (iv) And lastly his method for deter
mining the factors of numbers of the form x z + ay*.
There are also numerous articles on various points of analy
tical geometry. In two of them (in 1792 and 1793) he reduced the
equations of the quadrics (or conicoids) to their canonical forms.
During the years from 1772 to 1785 he contributed a long
series of memoirs which created the science of differential
equations, at any rate as far as partial differential equations
are concerned. I do not think that any previous writer had
done anything beyond considering equations of some particular
form. A large part of these results were collected in the second
edition of Euler s integral calculus which was published in 1794.
His papers on ?nechanics require no separate mention here
as the results arrived at are embodied in the Mecanique
analytique which is described below.
Lastly there are numerous memoirs on problems in
astronomy. Of these the most important are the following,
(i) On the attraction of ellipsoids (1773) : this is founded on
Maclaurin s work, (ii) On the secular equation of the moon
(1773); also noticeable for the earliest introduction of the
idea of the potential. The potential of a body at any point
is the sum of the mass of every element of the body when
divided by its distance from the point. Lagrange shewed
that if the potential of a body at an external point were known,
the attraction in any direction could be at once found. The
theory of the potential was elaborated in a paper sent to
Berlin in 1777. (iii) On the motion of the nodes of a
planet s orbit (1774). (iv) On the stability of the planetary
orbits (1776). (v) Two memoirs in which the method of
determining the orbit of a comet from three observations is
completely worked out (1778 and 1783) : this has not indeed
LAGRANGE. 413
proved practically available, but his system of calculating the
perturbations by means of mechanical quadratures has formed
ti/a basis of most subsequent researches on the subject, (vi) His
determination of the secular and periodic variations of the
elements of the planets (1781 1784): the upper limits assigned
for these agree closely with those obtained later by Leverrier,
and he proceeded as far as the knowledge then possessed of the
masses of the planets permitted, (vii) Three memoirs on the
method of interpolation (1783, 1792, and 1793): the part
of finite differences dealing therewith is now in the same
stage as that in which Lagrange left it.
Over and above these various papers, he composed his great
treatise, the Mecanique analytique. In this he lays down the
law of virtual work, and from that one fundamental principle
by the aid of the calculus of variations deduces the whole
of mechanics, both of solids and fluids. The object of the
book is to shew that the subject is implicitly included in a
single principle, and to give general formulae from which any
particular result can be obtained. The method of generalized
coordinates by which he obtained this result is perhaps the
most brilliant result of his analysis. Instead of following the
motion of each individual part of a material system, as
D Alembert and Euler had done, he shewed that, if we deter
mine its configuration by a sufficient number of variables
whose number is the same as that of the degrees of freedom
possessed by the system, then the kinetic and potential energies
of the system can be expressed in terms of these, and the
differential equations of motion thence deduced by simple
differentiation. For example, in dynamics of a rigid system
he replaces the consideration of the particular problem by
the general equation which is now usually written in the form
Amongst other minor theorems here given I may mention the
proposition that the kinetic energy imparted by given impulses
414 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
to a material system under given constraints is a maximum,
and the principle of least action. All the analysis is *p
elegant that Sir William Rowan Hamilton said the work
could be only described as a scientific poem. It may be
interesting to note that Lagrange remarked that mechanics
was really a branch of pure mathematics analogous to a
geometry of four dimensions, namely, the time and the three
coordinates of the point in space ; and it is said that he prided
himself that from the beginning to the end of the work there
was not a single diagram. At first no printer could be found
who would publish the book ; but Legendre at last persuaded
a Paris firm to undertake it, and it was issued under his
supervision in 1788.
In 1787 Frederick died, and Lagrange, who had found
the climate of Berlin trying, gladly accepted the offer of
Louis XVI. to migrate to Paris. He received similar invita
tions from Spain and Naples. In France he was received with
every mark of distinction, and special apartments in the Louvre
were prepared for his reception. For the first two years of his
residence here he was seized with an attack of melancholy, and
even the printed copy of his Mecanique on which he had
worked for a quarter of a century lay for more than two years
unopened on his desk. Curiosity as to the results of the
French revolution first stirred him out of his lethargy, a
curiosity which soon turned to alarm as the revolution de
veloped. It was about the same time, 1792, that the un
accountable sadness of his life and his timidity moved the
compassion of a young girl who insisted on marrying him, and
proved a devoted wife to whom he became warmly attached.
Although the decree of October, 1793, which ordered all
foreigners to leave France, specially exempted him by name,
he was preparing to escape when he was offered the presidency
of the commission for the reform of weights and measures.
The choice of the units finally selected was largely due to him,
and it was mainly owing to his influence that the decimal
subdivision was accepted by the commission of 1799. The
LAGRANGE. 415
general idea of the decimal system was taken from a work by
Thomas Williams entitled Method .. .for fixing an universal
standard for weights and measures, published in London in
1/88: this almost unknown writer has hardly received the
credit due to his suggestion.
Though Lagrange had determined to escape from France
while there was yet time, he was never in any danger ; and
the different revolutionary governments (and at a later time
Napoleon) loaded him with honours and distinctions. A
striking testimony to the respect in which he was held was
shewn in 1796 when the French commissary in Italy was
ordered to attend in full state on Lagrange s father, and tender
the congratulations of the republic on the achievements of his
son, who "had done honour to all mankind by his genius,
and whom it was the special glory of Piedmont to have
produced."
In 1795 Lagrange was appointed to a mathematical chair
at the newly-established Ecole normale which only enjoyed a
brief existence of four months. His lectures here were quite
elementary and contain nothing of any special importance, but
they were published because the professors had to "pledge
themselves to the representatives of the people and to each
other neither to read nor to repeat from memory," and the
discourses were ordered to be taken down in shorthand in
order to enable the deputies to see how the professors ac
quitted themselves.
On the establishment of the Nicole poly technique in 1797
Lagrange was made a professor; and his lectures there are
described by mathematicians who had the good fortune to be
able to attend them, as almost perfect both in form and matter.
Beginning with the merest elements he led his hearers on
until, almost unknown to themselves, they were themselves
extending the bounds of the subject : above all he impressed
on his pupils the advantage of always using general methods
expressed in a symmetrical notation.
His lectures on the differential calculus form the basis of
416 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
his Theorie des fonctions analytiques which was published in
1797. This work is the extension of an idea contained in L
paper he had sent to the Berlin Memoirs in 1772, and its object *
is to substitute for the differential calculus a group of theorems
based on the development of algebraic functions in series. A
somewhat similar method had been previously used by John
Landen in his Residual Analysis, published in London in 1758.
Lagrange believed that he could thus get rid of those diffi
culties, connected with the use of infinitely large or infinitely
small quantities, which philosophers professed to see in the usual
treatment of the differential calculus. The book is divided
into three parts ; of these the first treats of the general theory
of functions, and gives an algebraic proof of Taylor s theorem,
the validity of which is however open to question ; the second
deals with applications to geometry; and the third with
applications to mechanics. Another treatise on the same
lines was his Lemons sur le calcul des /auctions, issued in 1804.
These works may be considered as the starting-point for the
researches of Cauchy and Jacobi. At a later period Lagrange
reverted to the use of infinitesimals in preference to founding
the differential calculus on a study of algebraic forms : and
in the preface to the second edition of the Mecanique, which
was issued in 1811, he justifies their use and concludes
by saying that "when we have grasped the spirit of the
infinitesimal method, and have verified the exactness of its
results either by the geometrical method of prime and ultimate
ratios or by the analytical method of derived functions, we may
employ infinitely small quantities as a sure and valuable means
of shortening and simplifying our proofs."
His Resolution des equations numeriques, published in 1798,
was also the fruit of his lectures at the Polytechnic. In this
he gives the method of approximating to the real roots of an
equation by means of continued fractions, and enunciates several
other theorems. In a note at the end he shews how Fermat s
theorem that a p-1 1 = (mod p), where p is a prime and a is
prime to p, combined with a, certain suggestion due to Gauss,
LAGRANGE. 417
may be applied to give the complete algebraical solution of any
binomial equation. He also here explains how the equation
whose roots are the squares of the differences of the roots of
the original equation may be used so as to give considerable
information as to the position, and nature of those roots.
The theory of the planetary motions had formed the subject
of some of the most remarkable of Lagrange s Berlin papers.
In 1806 the subject was reopened by Poisson who in a paper
read before the French Academy shewed that Lagrange s
formulae led to certain limits for the stability of the orbits.
Lagrange, who was present, now discussed the whole subject
afresh, and in a memoir communicated to the Academy in
1808 explained how by the variation of arbitrary constants the
periodical and secular inequalities of any system of mutually
interacting bodies could be determined.
In 1810 Lagrange commenced a thorough revision of the
Mecaniqiw analytique, but he was able to complete only about
two- thirds of it before his death.
In appearance he was of medium height, and slightly
formed, with pale blue eyes, and a colourless complexion. In
character he was nervous and timid, he detested controversy,
and to avoid it willingly allowed others to take the credit for
what he had himself done.
Lagrange was above all a student of pure mathematics : he
sought and obtained far-reaching abstract results, and was
content to leave the applications to others. Indeed no in
considerable part of the discoveries of his great contemporary
Laplace consists of the application of the Lagrangian formulae
to the facts of nature ; for example, Laplace s conclusions on
the velocity of sound and the secular acceleration of the moon
are implicitly involved in Lagrange s results. The only difficulty
in understanding Lagrange is that of the subject-matter and the
extreme generality of his processes ; but his analysis is " as
lucid and luminous as it is symmetrical and ingenious." A
recent writer speaking of Lagrange says truly that he took a
prominent part in the advancement of almost every branch of
B. 27
418 LA1MIANGE, LAPLACE, AND THEIR CONTEMPORARIES,
pure mathematics. Like Diophantus and Format he possessed
a special genius for the theory of numbers, and in this subject
he gave solutions of most of the problems which had been pro
posed by Format, and added some theorems of his own. lie
created the calculus of variations. To him too the theory of
differential equations is indebted for its position as a science
rather than a collection of ingenious artifices for the solution
of particular problems. To the calculus of finite differences he
contributed the formula of interpolation which bears his name.
But above all he impressed on mechanics (which it will be
remembered lie considered a part of pure mathematics) that
generality and completeness towards which his labours in
variably tended.
Laplace*. Pierre Simon Laplace was born at Beaumont-en-
Auge in Normandy on March 23, 1749, and died at Paris on
March 5, 1827. He was the son of a small cottager or perhaps
a farm -labourer, and owed his education to the interest excited
in some wealthy neighbours by his abilities and engaging
presence. Very little is known of his early years, for when
he became distinguished he held himself aloof both from his
relatives and from those who had assisted him. A similar
pettiness of character marked many of his actions. It would
seem that from a pupil he became an usher in the school at
Beaumont ; but, having procured a letter of introduction to
D Alembert, he went to Paris to push his fortune. A paper on
the principles of mechanics excited D Alembert s interest, and
on his recommendation a place in the military school was
offered to Laplace.
Secure of a competency, Laplace now threw himself into
original research, and in the next seventeen years, 1771 1787,
he produced much of his original work in astronomy. This
* The following account of Laplace s life and writings is mainly
founded on the articles in the English Cyclopaedia and the Encyclopedia
Rritannica. Laplace s works were published in seven volumes by the
French government in 1843 7; and a new edition with considerable
additional matter was issued at Paris in six volumes, 187884.
LAPLACE. H!)
commenced with a memoir, read before the French Academy
in 1773, in which he shewed that the planetary motions were
stable, and carried the proof as far as the cubes of the eccen
tricities and inclinations. This was followed by several papers
on points in the integral calculus, finite differences, differential
equations, and astronomy.
During the years 1784 1787 he produced some memoirs
of exceptional power. Prominent among these is one read
in 1784, and reprinted in the third volume of the Mecanique
celeste, in which he completely determined the attraction of a
spheroid on a particle outside it. TJiis is memorable for the
introduction into analysis of spherical harmonics or Laplace s
coefficients, and also for the development of the use of the
potential ; a name first given by Green in 1828.
If the coordinates of two points be (r, /i, <o) and (/, pf CD ),
arid if r r, then the reciprocal of the distance between them
can be expanded in powers of r/r 9 and the respective coefficients
are Laplace s coefficients. Their utility arises from the fact that
every function of the coordinates of a point on a sphere can be
expanded in a series of them. It should be stated that the
similar coefficients for space of two dimensions, together with
some of their properties, had been previously given by
Legendre in a paper sent to the French Academy in 1783.
Legendre had good reason to complain of the way in which he
was treated in this matter.
This paper is also remarkable for the development of the
idea of the potential, which was appropriated from Lagrange*
who had used it in his memoirs of 1773, 1777, and 1780.
Laplace shewed that the potential always satisfies the diffe
rential equation
and on this result his subsequent work on attractions was
* See the Bulletin of the New York Mathematical Society, 1HH2,
vol. i., pp. 06 74.
272
420 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
based. The quantity V 2 F has been termed the concentration
of F, and its value at any point indicates the excess of the
value of V there over its mean value in the neighbourhood of
the point. Laplace s equation, or the more general form
V 2 V=-4:7rp, appears in all branches of mathematical physics.
According to some writers this follows at once from the fact
that V 2 is a scalar operator; or the equation may represent
analytically some general law of nature which has not been yet
reduced to words; or possibly (I have sometimes thought) it
might be regarded by a Kantian as the outward sign of one
of the necessary forms through which all phenomena are
perceived.
This memoir was followed by another on planetary in
equalities, which was presented in three sections in 1784, 1785,
and 1786. This deals mainly with the explanation of the
" great inequality" of Jupiter and Saturn. Laplace shewed
by general considerations that the mutual action of two
planets could never largely affect the eccentricities and in
clinations of their orbits; and that the peculiarities of the
Jovian system were due to the near approach to commen-
surability of the mean motions of Jupiter and Saturn : further
developments of these theorems on planetary motion were given
in his two memoirs of 1788 and 1789. It was on these data
that Delambre computed his astronomical tables.
The year 1787 was rendered memorable by Laplace s expla
nation and analysis of the relation between the lunar accelera
tion and the secular changes in the eccentricity of the earth s
orbit : this investigation completed the proof of the stability
of the whole solar system on the assumption that it consists of
a collection of rigid bodies. All the memoirs above alluded
to were presented to the French Academy, and they are
printed in the Memoires presentes par divers savans.
Laplace now set himself the task to write a work which
should "offer a complete solution of the great mechanical
problem presented by the solar system, and bring theory to
coincide so closely with observation that empirical equations
LAPLACE. 421
should no longer find a place in astronomical tables." The
result is embodied in the Exposition du systeme du monde and
the Mecanique celeste.
The former was published in 1796, and gives a general
explanation of the phenomena with a summary of the history
of astronomy, but omits all details. The nebular hypothesis
was here enunciated*. According to this hypothesis the solar
system has been evolved from a globular mass of incandescent
gas rotating round an axis through its centre of mass. As it
cooled, this mass contracted and successive rings broke off
from its outer edge. These rings in their turn cooled, and
finally condensed into the planets, while the sun represents
the central core which is still left. Certain corrections required
by modern science were added by M. Roche, and recently
the theory has been discussed critically by R. Wolf. The
arguments against the hypothesis are summed up in Faye s
Oriyine du monde, Paris, 1884, where an ingenious modi
fication of the hypothesis is proposed, by which the author
attempts to explain the peculiarities of the axial rotation
of Neptune and Uranus, and the retrograde motion of the
satellites of the latter planet. Perhaps modern opinion is
inclined to attribute the separation of the various members
of a planetary system to tidal friction rather than to the
successive separation and condensation of nebulous rings ; but
the subject is one of great difficulty. According to the rule
published by Titius of Wittemberg in 1766 but generally
known as Bode s law, from the fact that attention was called
to it by Johann Elert Bode in 1778 the distances of the
planets from the sun are nearly in the ratio of the numbers
+ 4, 3 + 4, 6 + 4, 12 + 4, <tc., the (n + 2)th term being
(2 n x 3) + 4. It would be an -interesting fact if this could be
deduced from either the nebular or the tidal hypothesis, but so
far as I am aware only one serious attempt to do so has been
made, and the conclusion was that the law was not sufficiently
* On the history of the nebular hypothesis, see The Visible Universe,
by J. E. Gore, London, 1893.
422 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
exact to be more than a convenient means of remembering
the general result. The substance of Laplace s hypothesis had
been published by Kant in 1755 in his Allgemeine Natur-
geschichte, but it is probable that Laplace was not aware of
this. The historical summary procured for its author the
honour of admission to the forty of the French Academy ; it
is commonly esteemed one of the master-pieces of French
literature, though it is not altogether reliable for the later
periods of which it treats.
The full analytical discussion of the solar system is given
in the Mecanique celeste published in five volumes: vols. I. and
n. in 1799; vol. in. in 1802; vol. iv. in 1805; and vol. v. in
1825. An analysis of the contents is given in the English
Cyclopaedia. The first two volumes contain methods for
calculating the motions of the planets, determining their
figures, and resolving tidal problems. The third and fourth
volumes contain the application of these methods, and also
several astronomical tables. The fifth volume is mainly
historical, but it gives as appendices the results of Laplace s
latest researches. Laplace s own investigations embodied in it
are so numerous and valuable that it is regrettable to have to add
that many results are appropriated from writers with scanty
or no acknowledgment, and the conclusions which have been
described as the organized result of a century of patient toil
are generally mentioned as if they were due to Laplace ; and
it is said (for I have not looked into the matter myself) that
the praise which he lavishes on Newton and Clairaut is only
the cloak under which he appropriates the work of other and
less known writers.
The Mecanique celeste is by no means easy reading. Biot,
who assisted Laplace in revising it for the press, says that
Laplace himself was frequently unable to recover the details
in the chain of reasoning, and, if satisfied that the conclusions
were correct, he was content to insert the constantly recurring
formula " II est aise a voir." The best tribute to the excellency
of the work is that it left very little for his successors to add.
LAPLACE. 423
It is not only the translation of the Principia into the language
of the differential calculus, but it also completes parts of which
Newton had been unable to fill in the details. M. Tisserand s
recent work may be considered as a continuation of Laplace s
treatise.
Laplace went in state to beg Napoleon to accept a copy of
his work, and the following account of the interview is well
authenticated, and so characteristic of all the parties concerned
that I quote it in full. Someone had told Napoleon that the
book contained no mention of the name of God ; Napoleon,
who was fond of putting embarrassing questions, received it
with 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, who, though the most
supple of politicians, was as stiff as a martyr on every point of
his philosophy, drew himself up and answered bluntly, "Je
n avais pas besoin de cette hypothese-la. " Napoleon, greatly
amused, told this reply to Lagrange, who exclaimed, " Ah !
c est une belle hypothese ; ^a explique beaucoup de choses."
In 1812 Laplace issued his TJieorie analytiqne des proba-
bilites. The theory is stated to be only common sense ex
pressed in mathematical language. The method of estimating
the ratio of the number of favourable cases to the whole
number of possible cases had been indicated by Laplace
in a paper written in 1779. It consists in treating the
successive values of any function as the coefficients in the
expansion of another function with reference to a different
variable. The latter is therefore called the generating function
of the former. Laplace then shews how by means of interpola
tion these coefficients may be determined from the generating
function. Next he attacks the converse problem, and from the
coefficients he finds the generating function; this is effected by
the solution of an equation in finite differences. The method
is cumbersome, and in consequence of the increased power of
analysis is now rarely used. A summary of Laplace s reason
ing is given in the article on Probability in the Encyclopaedia
Metropolitana.
424 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
This treatise includes an exposition of the method of least
squares, which is a remarkable testimony to Laplace s com
mand over the processes of analysis. The method of least
squares for the combination of numerous observations had
been given empirically by Gauss and Legendre, but the fourth
chapter of this work contains a formal proof of it; on which
the whole of the theory of errors has been since based. This
was effected only by a most intricate analysis specially invented
for the purpose, but the form in which it is presented is so
meagre and unsatisfactory that in spite of the uniform accuracy
of the results it was at one time questioned whether Laplace
had actually gone through the difficult work he so briefly and
often incorrectly indicates.
In 1819 Laplace published a popular account of his work on
probability. This book bears the same relation to the Theorie
des probabilites that the Systeme du monde does to the
Mecanique celeste.
Amongst the minor discoveries of Laplace in pure mathe
matics I may mention his discussion (simultaneously with Yan-
dermonde) of the general theory of determinants in 1772; his
proof that every equation of an even degree must have at least
one real quadratic factor; his reduction of the solution of linear
differential equations to definite integrals ; and his solution of
the linear partial differential equation of the second order. He
was also the first to consider the difficult problems involved in
equations of mixed differences, and to prove that the solution of
an equation in finite differences of the first degree and the
second order might be always obtained in the form of a
continued fraction. Besides these original discoveries he
determined in his theory of probabilities the values of a
number of the more common definite integrals : and in the
same book gave the general proof of the theorem enunciated
by Lagrange for the development of any implicit function in a
series by means of differential coefficients.
In theoretical physics the theory of capillary attraction
is due to Laplace who accepted the idea propounded by
Hauksbee, in the Philosophical Transactions for 1709, that
LAPLACE. 425
the phenomenon was due to a force of attraction which was
insensible at sensible distances. The part which deals with
the action of a solid on a liquid and the mutual action of two
liquids was not worked out thoroughly, but ultimately was
completed by Gauss : Neumann later filled in a few details.
In 1862 Lord Kelvin (Sir William Thomson) shewed that, if
we assume the molecular constitution of matter, the laws of
capillary attraction can be deduced from the Newtonian law of
gravitation. Laplace in 1816 was the first to point out
explicitly why Newton s theory of vibratory motion gave an
incorrect value for the velocity of sound. The actual velocity
is greater than that calculated by Newton in consequence of the
heat developed by the sudden compression of the air which
increases the elasticity and therefore the velocity of the sound
transmitted. Laplace s investigations in practical physics were
confined to those carried on by him jointly with Lavoisier in
the years 1782 to 1784 on the specific heat of various bodies.
Laplace seems to have regarded analysis merely as a means
of attacking physical problems, though the ability with which
he invented the necessary analysis is almost phenomenal. As
long as his results were true he took but little trouble to ex
plain the steps by which he arrived at them ; he never studied
elegance or symmetry in his processes, and it was sufficient
for him if he could by any means solve the particular question
he was discussing. In these respects he stands in marked con
trast to his great contemporary Lagrange.
It would have been well for Laplace s reputation if he had
been content with his scientific work, but above all things he
coveted social fame. The skill and rapidity with which he
managed to change his politics as occasion required would be
amusing if they had not been so servile. As Napoleon s power
increased Laplace abandoned his republican principles (which,
since they had faithfully reflected the opinions of the party in
power, had themselves gone through numerous changes) and
begged the first consul to give him the post of minister of the
interior. Napoleon, who desired the support of men of science,
426 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
accepted the offer; but a little less than six weeks saw the
close of Laplace s political career. Napoleon s memorandum on
the subject is as follows. "Geometre de premier rang, Laplace
ne tarda pas a se montrer administrateur plus que mediocre ;
des son premier travail nous reconnumes que nous nous etions
trompe. Laplace ne saisissait aucune question sous son veri
table point de vue : il cherchait des subtilites partout, n avait
que des idees problematiques, et portait en fin 1 esprit des
infiniment petits jusque dans Tad ministration."
Although Laplace was expelled from office it was desirable to
retain his allegiance. He was accordingly raised to the senate,
and to the third volume of the Mecanique celeste he prefixed a
note that of all the truths therein contained the most precious
to the author was the declaration he thus made of his devotion
towards the peace-maker of Europe. In copies sold after the
restoration this was struck out. In 1814 it was evident that
the empire was falling; Laplace hastened to tender his services
to the Bourbons, and on the restoration was rewarded with the
title of marquis : the contempt that his more honest colleagues
felt for his conduct in the matter may be read in the pages of
Paul Louis Courier. His knowledge was useful on the
numerous scientific commissions on which he served, and
probably accounts for the manner in which his political in
sincerity was overlooked ; but the pettiness of his character
must not make us forget how great were his services to
science.
That Laplace was vain and selfish is not denied by his
warmest admirers ; his conduct to the benefactors of his youth
and his political friends was ungrateful and contemptible ;
while his appropriation of the results of those who were com
paratively unknown seems to be well established and is
absolutely indefensible of those whom he thus treated three
subsequently rose to distinction (Legendre and Fourier in
France and Young in England) and never forgot the injustice
of which they had been the victims. On the other side it may
be said that on some questions he shewed independence of
LAPLACE. LEGENDRE. 427
character, and he never concealed his views on religion,
philosophy, or science however distasteful they might be to
the authorities in power ; it should be also added that towards
the close of his life and especially to the work of his pupils
Laplace was both generous and appreciative, and in one case
suppressed a paper of his own in order that a pupil might have
the sole credit of the discovery.
J7 Legendre. Adrien Marie Legendre was born at Toulouse
[on Sept. 18, 1752, and died at Paris on Jan. 10, 1833. The
leading events of his life are very simple and may be summed
up briefly. He was educated at the Mazarin College in Paris,
appointed professor at the military school in Paris in 1777,
was a member of the Anglo-French commission of 1787 to
connect Greenwich and Paris geodetically ; served on several
of the public commissions from 1792 to 1810; was made a
professor at the Normal school in 1795; and subsequently
held a few minor government appointments. The influence
of Laplace was steadily exerted against his obtaining office
or public recognition, and Legendre who was a timid student
accepted the obscurity to which the hostility of his colleague
condemned him.
Legendre s analysis is of a high order of excellence and is
second only to that produced by Lagrange and Laplace, though
it is not so original. His chief works are his Geometrie, his
Theorie des nombres, his Calcul integral, and his Fonctions
elliptiques. These include the results of his various papers on
these subjects. Besides these he wrote a treatise which gave
the rule for the method of least squares, and two groups of
memoirs, one on the theory of attractions, and the other
on geodetical operations.
The memoirs on attractions are analyzed and discussed in
Tod hunter s History of tJie Theories of Attraction. The earliest
of these memoirs, presented in 1783, was on the attraction
of spheroids. This contains the introduction of Legendre s
coefficients, which are sometimes called circular (or zonal)
harmonics, and which are particular casos of Laplace s co-
428 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
efficients (see above, p. 419); it also includes the solution of a
problem in which the potential is used. The second memoir
was communicated in 1784, and is on the form of equilibrium
of a mass of rotating liquid which is approximately spherical.
The third, written in 1786, is on the attraction of confocal
ellipsoids. The fourth is on the figure which a fluid planet
would assume, and its law of density.
His papers on geodesy are three in number and were
presented to the Academy in 1787 and 1788. The most im
portant result is that by which a spherical triangle may be
treated as plane, provided certain corrections are applied to the
angles. In connection with this subject he paid considerable
attention to geodesies.
The method of least squares was enunciated in his Nouvelles
methodes published in 1806, to which supplements were added
in 1810 and 1820. Gauss independently had arrived at the
same result, had used it in 1795, and published it and the
law of facility in 1809. Laplace was the earliest writer to
give a proof of it : this was in 1812 (see above, p. 424).
Of the other books produced by Legendre, the one most
widely known is his Elements de geometrie which was published
in 1794, and was generally adopted on the continent as a sub
stitute for Euclid. The later editions contain the elements of
trigonometry, and proofs of the irrationality of TT and ?r 2 (see
above, p. 406). An appendix on the difficult question of the
theory of parallel lines was issued in 1803, and is bound up
with most of the subsequent editions.
His Theorie des nombres was published in 1798, and ap
pendices were added in 1816 and 1825 : the third edition,
issued in two volumes in 1830, includes the results of his various
later papers, and still remains a standard work on the subject.
It may be said that he here carried the subject as far as was
possible by the application of ordinary algebra ; but he did not
realize that it might be regarded as a higher arithmetic, and so
form a distinct subject in mathematics.
The law of quadratic reciprocity, which connects any two
LEGENDRE. 429
odd primes is first proved in this book, but the result had been
enunciated in a memoir of 1785. Gauss called the proposition
" the gem of arithmetic," and no less than six separate proofs
are to be found in his works. The theorem is as follows. If
p be a prime and n be prime to p, then we know that the
remainder when n^ p ~ l) is divided by JP is either -t- 1 or-1.
Legendre denoted this remainder by ( ) When the re
mainder is + 1 it is possible to find a square number which
when divided by p leaves a remainder n, that is, n is a
quadratic residue of p ; when the remainder is - 1 there exists
no such square number, and n is a non-residue of p. The
law of quadratic reciprocity is expressed by the theorem that,
if a and b be any odd primes, then
thus, if 6 be a residue of a, then a is also a residue of 6, unless
both of the primes a and b are of the form 4?n -f 3. In other
words, if a and b be odd primes, we know that
a*0-i) = * 1 (mod 6), and W(-D = l( mo d a) ;
but by Legendre s law the two ambiguities will be either
both positive or both negative, unless a and b are both of the
form 4w + 3. Thus, if one odd prime be a non- residue of
another, then the latter will be a non-residue of the former.
Gauss and Kummer have subsequently proved similar laws of
cubic and biquadratic reciprocity ; and an important branch of
the theory of numbers has been based on these researches.
This work also contains the useful theorem by which,
when it is possible, an indeterminate equation of the second
degree can be reduced to the form ax 2 + by* + cz 2 = 0. Legendre
too here discussed the forms of numbers which can be expressed
as the sum of three squares; and he proved [art. 404] that
the number of primes less than n is very approximately
w/(logw- 1-08366).
The Exercices de calcul integral was published in three
430 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
volumes, 1811, 1817, 1826. Of these the third and most of the
first are devoted to elliptic functions ; the bulk of this being
ultimately included in the Fonctions elliptiques. The contents
of the remainder of the treatise are of a miscellaneous
character ; they include integration by series, definite integrals,
and in particular an elaborate discussion of the Beta and the
Gamma functions.
The Traite desfonctions elliptiques was issued in two volumes
in 1825 and 1826, and is the most important of Legendre s
works. A third volume was added a few weeks before his
death, and contains three memoirs on the researches of Abel
and Jacobi. Legendre s investigations had commenced with a
paper written in 1786 on elliptic arcs, but here and in his
other papers he treated the subject merely as a branch of
the integral calculus. Tables of the elliptic integrals were
constructed by him. The modern treatment of the subject is
founded on that of Abel and Jacobi. The superiority of their
methods was at once recognized by Legendre, and almost the
last act of his life was to recommend those discoveries which
he knew would consign his own labours to comparative oblivion.
This may serve to remind us of a fact which I wish to
specially emphasize, namely, that Gauss, Abel, Jacobi, and some
others of the mathematicians alluded to in the next chapter
were contemporaries of the members of the French school.
Pfaff. I may here mention another writer who also made
a special study of the integral calculus. This was Johann
Friederich Pfaff, born at Stuttgart on Dec. 22, 1765, and
died at Halle on April 21, 1825, who was described by
Laplace as the most eminent mathematician in Germany at
the beginning of this century, a description which, had it not
been for Gauss s existence, would have been true enough.
PfafF was the precursor of the German school, which under
Gauss and his followers has largely determined the lines on
which mathematics have developed during this century. He
was an intimate friend of Gauss, and in fact the two mathe
maticians lived together at Helmstadt for the year after Gauss
PFAFF. MONGE. 431
finished his university course in 1798. Pfaflfs chief work
was his (unfinished) Disquisitiones Analyticae on the integral
calculus, published in 1797 ; and his most important memoirs
were either on the calculus or on differential equations : on the
latter subject his paper read before the Berlin Academy in
1814 is still a standard authority.
The creation of modern geometry.
While Euler, Lagraiige, Laplace, and Legendre were per
fecting analysis, the members of another group of French
mathematicians were extending the range of geometry by
methods similar to those previously used by Desargues and
Pascal. The most eminent of those who created modern
synthetic geometry was Poncelet, but the subject is also
associated with the names of Monge and L. Carnot ; its de
velopment in more recent times is largely due to Steiner, von
Staudt, and Cremona (see below, p. 482).
Monge*. Gaspard Monge was born at Beaune on May 10,
1746, and died at Paris on July 28, 1818. He was the son
of a small pedlar, and was educated in the schools of the
Oratorians, in one of which he subsequently became an usher.
A plan of Beaune which he had made fell into the hands
of an officer who recommended the military authorities to
admit him to their training-school at Mezieres. His birth
however precluded his receiving a commission in the army,
but his attendance at an annexe of the school where surveying
and drawing were taught was tolerated, though he was told
that he was not sufficiently well born to be allowed to attempt
problems which required calculation. At last his opportunity
came. A plan of a fortress having to be drawn from the
data supplied by certain observations, he did it by a geo
metrical construction. At first the officer in. charge refused to
o
receive it, because etiquette required that not less than a
* See A xx// hixtorique sur /<> tnirnu.r...dt Montje, by F. P. C. Dupin,
Paris, 1819; also the Notice hi*tin iqiu. .s//r Monge by B. Brisson, Paris,
1818.
432 LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.
certain time should be used in making such drawings, but the
superiority of the method over that then taught was so
obvious that it was accepted; and in 1768 Monge was made
professor, on the understanding that the results of his descrip
tive geometry were to be a military secret confined to officers
above a certain rank.
In 1780 he was appointed to a chair of mathematics in
Paris, and this with several provincial appointments which he
held gave him a comfortable income. The earliest paper of
any special importance which he communicated to the French
Academy was one in 1781 in which he discussed the lines of
curvature drawn on a surface. These had been first considered
by Euler in 1760, and defined as those normal sections whose
curvature was a maximum or a minimum. Monge treated
them as the locus of those points on the surface at which suc
cessive normals intersect, and thus obtained the general differ
ential equation. He applied his results to the central quadrics
in 1795. In 1786 he published his well-known work on statics.
Monge eagerly embraced the doctrines of the revolution,
In 1792 he became minister of the marine, arid assisted the
committee of public safety in utilizing science for the defence
of the republic. When the Terrorists obtained power he was
denounced, and only escaped the guillotine by a hasty flight.
On his return in 1794 he was made a professor at the short
lived Normal school where he gave lectures on descriptive
geometry ; the notes of these were published under the regula
tion above alluded to (see above, p. 415). In 1796 he went to
Italy on the roving commission which was sent with orders to
compel the various Italian towns to offer any pictures, sculpture,
or other works of art that they might possess as a present or in
lieu of contributions to the French republic for removal to Paris.
In 1798 he accepted a mission to Rome, and after executing it
joined Napoleon in Egypt. Thence after the naval and military
victories of England he escaped to France. He was then made
professor at the Polytechnic school, where he gave lectures on
descriptive geometry ; these were published in 1800 in the form
MONGE. CARNOT. 433
of a text-book entitled Geometrie descriptive. This work con
tains propositions on the form and relative position of geometrical
figures deduced by the use of transversals. The theory of per
spective is considered ; this includes the art of representing in
two dimensions geometrical objects which are of three dimen
sions, a problem which Monge usually solved by the aid of two
diagrams, one being the plan and the other the elevation.
Monge also discussed the question as to whether, if in solving
a problem certain subsidiary quantities introduced to facilitate
the solution become imaginary, the validity of the solution is
thereby impaired, and he shewed that the result would not be
affected. On the restoration he was deprived of his offices and
honours, a degradation which preyed on his mind and which
he did not long survive.
Most of his miscellaneous papers are embodied in his
works Application de Valgebre a la geometrie published in 1805,
and Application de Vanalyse a la geometrie, the fourth edition
of which, published in 1819, was revised by him just before
his death. It contains among other results his solution of a
partial differential equation of the second order.
Carnot*. Lazare Nicholas Marguerite Camot, born at
Nolay on May 13, 1753, and died at Magdeburg on Aug. 22,
1823, was educated at Burgundy, and obtained a commission
in the engineer-corps of Conde. Although in the army, he
continued his mathematical studies in which he felt great
interest. His first work, published in 1784, was on machines:
it contains a statement which foreshadows the principle of
energy as applied to a falling weight, and the earliest proof of
the fact that kinetic energy is lost in the collision of bodies.
On the outbreak of the revolution in 1789 he threw himself
into politics. In 1793 he was elected on the committee of
public safety, and the victories of the French army were
largely due to his powers of organization and enforcing disci
pline. He continued to occupy a prominent place in every
* See the eloge by Arago, which, like most obituary notices, is a
panegyric rather than an impartial biography.
B. 28
434 CARNOT. PONCELET.
successive form of government till 1796 when, having opposed
Napoleon s coup detat, he had to fly from France. He took
refuge in Geneva, and there in 1797 issued his La metaphysique
du calcul infinitesimal. In 1802 he assisted Napoleon, but
his sincere republican convictions were inconsistent with the
retention of office. In 1803 he produced his Geometric de
position. This work deals with protective rather than des
criptive geometry, it also contains an elaborate discussion of
the geometrical meaning of negative roots of an algebraical
equation. In 1814 he offered his services to fight for France,
though not for the empire ; and on the restoration he was
exiled.
Poncelet*. Jean Victor Poncelet, born at Metz on July 1,
1788, and died at Paris on Dec. 22, 1867, held a commission
in the French engineers. Having been made a prisoner in the
French retreat from Moscow in 1812 he occupied his enforced
leisure by writing the Traite des proprietes projectiles des
figures, published in 1822, which was long one of the best
known text-books on modern geometry. By means of pro
jection, reciprocation, and homologous figures he established
all the chief properties of conies and quadrics. He also treated
the theory of polygons. His treatise on practical mechanics in
1826, his memoir on water-mills in 1826, and his report on
the English machinery and tools exhibited at the International
exhibition held in London in 1851 deserve mention. He
contributed numerous articles to Crelle s journal. The most
valuable of these deal with the explanation of imaginary
solutions in geometrical problems by the aid of the doctrine of
continuity.
The development of mathematical physics.
It will be noticed that Lagrange, Laplace, and Legendre
mostly occupied themselves with analysis, geometry, and astro-
* See La vie et les ouvrages de Poncelet by Didion and Dupin, Paris,
1869.
CAVENDISH. RUMFORD. 435
nomy. I am inclined to regard Cauchy and the French mathe
maticians of the present day as belonging to a different school
of thought to that considered in this chapter and I place them
amongst modern mathematicians, but I think that Fourier,
Poisson, and the majority of their contemporaries are the lineal
successors of Lagrange and Laplace. If this view be correct, it
would seem that the later members of the French school
devoted themselves mainly to the application of mathematical
analysis to physics. Before considering these mathematicians
I may mention the distinguished English experimental physic
ists who were their contemporaries, and whose merits have only
recently received an adequate recognition. Chief among these
are Cavendish and Young.
Cavendish*. The honourable Henry Cavendish was born at
Nice 011 Oct. 10, 1731, and died in London on Feb. 24, 1810.
His tastes for scientific research and mathematics were formed
at Cambridge, where he resided from 1749 to 1753. Recreated
experimental electricity, and was one of the earliest writers to
treat chemistry as an exact science. T mention him here on
account of his experiment in 1798 to determine the density of
the earth, by estimating its attraction as compared with that
of two given lead balls : the result is that the mean density of
the earth is about five and a half times that of water. This
experiment was carried out in accordance with a suggestion
which had been first made by John Michell, a fellow of Queens
College, Cambridge, who had died before he was able to carry
it into effect.
Rumfordf. Sir Benjamin Thomson, Count Rumford, born
at Concord on March 26, 1753, and died at Auteuil on Aug.
* An account of his life by G. Wilson will be found in the first
volume of the publications of the Cavendish Society, London, 1851. His
Electrical Researches were edited by J. C. Maxwell, and published at
Cambridge in 1879.
f An edition of Rumford s works, edited by George Ellis, accom
panied by a biography was published by the American Academy of
Sciences at Boston in 187*2.
282
436 RUMFORD. YOUNG.
21, 1815, was of English descent and fought on the side of the
loyalists in the American War of secession : on the conclusion
of peace, he settled in England, but subsequently entered the
service of Bavaria where his military and civil powers of
organization proved of great value. At a later period he
again resided in England, and when there founded the Royal
Institution. The majority of his papers were communicated
to the Royal Society of London; of these the most important
is his memoir in which he shewed that heat and work are
mutually convertible.
Young *. Among the most eminent physicists of his time
was Thomas Young, who was born at Milverton on June 13,
1773, and died in London on May 10, 1829. He seems as a
boy to have been somewhat of a prodigy, being well read in
modern languages and literature as well as in science ; he
always kept up his literary tastes and it was he who first
furnished the key to decipher the Egyptian hieroglyphics.
He was destined to be a doctor, and after attending lectures
at Edinburgh and Gottingen entered at Emmanuel College,
Cambridge, from which he took his degree in 1799 ; and to
his stay at the university he attributed much of his future
distinction. His medical career was not particularly suc
cessful, and his favourite maxim that a medical diagnosis is
only a balance of probabilities was not appreciated by his
patients who looked for certainty in return for their fee.
Fortunately his private means were ample. Several papers
contributed to various learned societies from 1798 onwards
prove him to have been a mathematician of considerable
power ; but the researches which have immortalized his name
are those by which he laid down the laws of interference of
waves and of light, and was thus able to suggest the means by
which the chief difficulties in the way of acceptance of the
undulatory theory of light could be overcome.
* His collected works and a memoir on his life were published by
G. Peacock, 4 volumes, London, 1855.
WOLLASTON. D ALTON. FOURIER. 437
Wollaston. Another experimental physicist of the same
time and school was William Hyde Wollaston, who was born
at Dereham on Aug. G, 1766, and died in London on Dec.
22, 1828. He was educated at Gains College, Cambridge,
of which society he was a fellow. Besides his well-known
chemical experiments, he is celebrated for his researches on
experimental optics, and for the improvements which he
effected in astronomical instruments.
Dalton*. Another distinguished writer of the same period
was John Dalton, who was born in Cumberland on Sept. 5,
1766, and died at Manchester on July 27, 1844. Dalton in
vestigated the tension of vapours, and the law of the expansion
of a gas under changes of temperature. He also founded the
atomic theory in chemistry.
It will be gathered from these notes that the English
school of physicists at the beginning of this century were
mostly concerned with the experimental side of the subject.
But in fact no satisfactory theory could be formed without some
similar careful determination of the facts. The most eminent
French physicists of the same time were Fourier, Poisson,
Ampere, and Fresnel. Their method of treating the subject
is more mathematical than that of their English contem
poraries, and the two first named were distinguished for
general mathematical ability.
Fourier t- The first of these French physicists was Jean
Baptiste Joseph Fourier, who was born at Auxerre on March 21,
1768, and died at Paris on May 16, 1830. He was the son of
a tailor, and was educated by the Benedictines. The com
missions in the scientific corps of the army were, as is still the
case in Russia, reserved for those of good birth, and being
thus ineligible he accepted a military lectureship on mathe-
* See the Memoir of Dalton by B. A. Smith, London, 1856; and
W. C. Henry s memoir in the Cavendish Society Transactions, London,
1854.
t An edition of his works, edited by Gaston Darboux, is now being
issued by the French government.
438 FOURIER.
matics. He took a prominent part in his own district in
promoting the Revolution, and was rewarded by an appoint
ment in 1795 in the Normal school, and subsequently by a
chair at the Polytechnic school.
He went with Napoleon on his eastern expedition in 1798,
and was made governor of Lower Egypt. Cut off from France
by the English fleet, he organized the workshops on which the
French army had to rely for their munitions of war. He also
contributed several mathematical papers to the Egyptian In
stitute which Napoleon founded at Cairo with a view of
weakening English influence in the East. After the British
victories and the capitulation of the French under General
Menou in 1801, he returned to France and was made prefect
of Grenoble, and it was while there that he made his experi
ments on the propagation of heat. He moved to Paris in
1816. In 1822 he published his Theorie analytique de la
chaleur, in which he bases his reasoning on Newton s law of
cooling, namely, that the flow of heat between two adjacent
molecules is proportional to the infinitely small difference of
their temperatures. He states that the theory demands that
the temperature of stellar space should be between 50 C. and
60 C., a conclusion which it has been as yet impossible to
prove or disprove. In this work be shews that any function
of a variable, whether continuous or discontinuous, can be
expanded in a series of sines of multiples of the variable ; a
result which is constantly used in modern analysis. Lagrange
had given particular cases of the theorem and had implied that
the method was general, but he had not pursued the subject.
Fourier left an unfinished work on determinate equations
which was edited by Navier, and published in 1831 ; this
contains much original matter, in particular there is a demon
stration of Fourier s theorem on the position of the roots of
an algebraical equation. Lagrange had shewn how the roots
of an algebraical equation might be separated by means of
another equation whose roots were the squares of the differ
ences of the roots of the original equation. Budan, in 1807
SADI CARNOT. POISSON. 439
and 1811, had enunciated the theorem generally known by
the name of Fourier, but the demonstration was clumsy and
not altogether satisfactory. Fourier s proof is the same as
that usually given in text-books on the theory of equations.
The final solution of the problem was given in 1829 by Jacques
Charles Fra^ois Sturm.
Sadi Carnot*. Among Fourier s contemporaries who were
interested in the theory of heat the most eminent was Sadi
Carnot, a son of the eminent geometrician mentioned above.
Sadi Carnot was born at Paris in 1796, and died there of
cholera in August, 1832; he was an officer in the French
army. In 1824 he issued a short work entitled Reflexions sur
la puissance motrice dufeu in which he attempted to determine
in what way heat produced its mechanical effect. He made
the mistake of assuming that heat was material, but his essay
was the commencement of the modern theory of thermo
dynamics.
Poissont. Simeon Denis Poisson, born at Pithiviers on
June 21, 1781, and died at Paris on April 25, 1840, is almost
equally distinguished for his applications of mathematics to
mechanics and to physics. His father had been a common
soldier, and on his retirement was given some small adminis
trative post in his native village : when the revolution broke
out he appears to have assumed the government of the place,
and, being left undisturbed, became a person of some local
importance. The boy was put out to nurse, and he used to
tell how one day his father coming to see him found that the
nurse had gone out on pleasure bent, while she had left him
suspended by a small cord to a nail fixed in the wall. This
she explained was a necessary precaution to prevent him from
* A sketch of his life and an English translation of his Reflexions was
published by R. H. Thurston, London and New York, 1890.
t Memoirs of Poisson will be found in the Encyclopaedia Britannica,
the Transactions of the Royal Astronomical Society, vol. v., and Arago s
Eloges, vol. n.; the latter contains a bibliography of Poisson s papers and
works.
440 POISSON.
perishing under the teeth of the various animals and animal-
cula that roamed on the floor. Poisson used to add that his
gymnastic efforts carried him incessantly from one side to
the other, and it was thus in his tenderest infancy that he
commenced those studies on the pendulum that were to occupy
so large a part of his mature age.
He was educated by his father, and destined much against
his will to be a doctor. His uncle offered to teach him the art;
and began by making him prick the veins of cabbage-leaves
with a lancet. When perfect in this, he was allowed to
put on blisters ; but in almost the first case he did this by
himself, the patient died in a few hours, and though all the
medical practitioners of the place assured him that "the event
was a very common one" he vowed he would have nothing
more to do with the profession. Returning home he found
amongst the official papers sent to his father a copy of the
questions set at the Polytechnic school, and at once found his
career. At the age of seventeen he entered the Polytechnic,
and his abilities excited the interest of Lagrange and
Laplace whose friendship he retained to the end of their
lives. A memoir on finite differences which he wrote when
only eighteen was reported on so favourably by Legendre that
it was ordered to be published in the Recueil des savants etran-
gers. Directly he had finished his course he was made a
lecturer at the school, and he continued through his life to
hold various government scientific posts and professorships.
He was somewhat of a socialist, and remained a rigid republican
till 1815 when, with a view to making another empire im
possible, he joined the legitimists. He took however no active
part in politics, and made the study of mathematics his amuse
ment as well as his business.
His works and memoirs are between three and four hundred
in number. The chief treatises which he wrote were his Traite
de mecanique*, 2 volumes, 1811 and 1833, which was long a
* Among Poisson s contemporaries who studied mechanics and of
whose works he made use I may mention Louis Poinsot, who was born
POISSON. 441
standard work; his Theorie nouvelle de fraction capillaire, 1831;
his Tlieorie mathematique de la chaleur, 1835, to which a supple
ment was added in 1837 ; and his Recherckes sur la probabilite
des jugements, 1837. He had intended if he had lived to write
a work which should cover all mathematical physics and in
which these would have been incorporated.
Of his memoirs on the subject of pure mathematics the
most important are those on definite integrals, and Fourier s
series (these are to be found in the Journal poll/technique from
1813 to 1823, and in the Memoires de Vacademie for 1823),
their application to physical problems constituting one of his
chief claims to distinction ; his essay on the calculus of varia
tions (Memoires de I academie, 1833); and his papers on the
probability of the mean results of observations (Connaissance
des temps, 1827 and following years). Most of his memoirs
were published in the three periodicals here mentioned.
Perhaps the most remarkable of his memoirs in applied
mathematics are those on the theory of electrostatics and
magnetism, which originated a new branch of mathematical
physics : he supposed that the results were due to the
attractions and repulsions of imponderable particles. The
most important of those on physical astronomy are the two
read in 1806 (printed in 1809) on the secular inequalities of
the mean motions of the planets, and on the variation of
arbitrary constants introduced into the solutions of questions
on mechanics ; in these Poisson discusses the question of the
stability of the planetary orbits, which Lagrange had already
proved to the first degree of approximation for the disturbing
forces, and shews that the result can be extended to the third
order of small quantities : these were the memoirs which led
to Lagrange s famous memoir of 1808. Poisson also published
a paper in 1821 on the libration of the moon; and another in
in Paris on Jan. 3, 1777, and died there on Dec. 5, 1859. In his Statique
published in 1803 he treated the subject without any explicit reference
to dynamics: the theory of couples is largely due to him (1806), as also
the motion of a body in space under the action of no forces.
442 AMPERE. FRESNEL. BIOT.
1827 on the motion of the earth about its centre of gravity.
His most important memoirs on the theory of attraction are
one in 1829 on the attraction of spheroids, and another in
1835 on the attraction of a homogeneous ellipsoid: the
substitution of the correct equation involving the potential,
namely, V 2 F= 4?rp ? for Laplace s form of it, V 2 F=0, was
first published in 1813 in the Bulletin des sciences of the
Societe philomatique. Lastly I may mention his memoir in
1825 on the theory of waves.
Ampere *. Andre Marie Ampere was born at Lyons on
January 22, 1775, and died at Marseilles on June 10, 1836.
He was widely read in all branches of learning, and lectured
and wrote on many of them, but after the year 1809, when he
was made professor of analysis at the Polytechnic school in
Paris, he confined himself almost entirely to mathematics and
science. His papers on the connection between electricity and
magnetism were written in 1820. According to his theory,
propounded in 1826, a molecule of matter which can be mag
netized is traversed by a closed electric current, and magnet
ization is produced by any cause which makes the direction of
these currents in the different molecules of the body approach
parallelism.
Fresnel. Augustin Jean Fresnel, born at Broglie on May
10, 1788, and died at Ville-d Avray on July 14, 1827, was a
civil engineer by profession, but he devoted his leisure to the
study of physical optics. The undulatory theory of light which
Hooke, Huygens, and Euler had supported on a priori grounds
had been based ori experiment by the researches of Young.
Fresnel deduced the mathematical consequences of these ex
periments, and explained the phenomena of interference both
of ordinary and polarized light.
Biot. Fresnel s friend and contemporary, Jean Baptiste
Biot, who was born at Paris on April 21, 1774, and died there
in 1862, requires a word or two in passing. Most of his
* See Valson s Etude sur la vie et les ouvrages d* Ampere, Lyons, 1885.
ARAGO. 443
mathematical work was in connection with the subject of
optics and especially the polarization of light. His systematic
works were produced within the years 1805 and 1817: a
selection of his more valuable memoirs was published in Paris
in 1858.
Arago*. Francois Jean Dominique Arago was born at
Estagel in the Pyrenees on Feb. 26, 1786, and died in Paris
on Oct. 2, 1853. He was educated at the Polytechnic school,
Paris, and we gather from his autobiography that however
distinguished were the professors of that institution they were
remarkably incapable of imparting their knowledge or main
taining discipline. In 1804 he was made secretary to the
observatory, and from 1806 to 1809 he was engaged in mea
suring a meridian arc in order to determine the exact length
of a metre. He was then made one of the astronomers at
Paris, given a residence there, and made a professor at the
Polytechnic school, where he enjoyed a marked success as a
lecturer. He subsequently gave popular lectures on astronomy
which were both lucid and accurate, a combination of qualities
which was rarer then than now. He reorganized the national
observatory, the management of which had long been in
efficient, but in doing this he shewed himself dictatorial and
passionate, and the same defects of character revealed them
selves in many of the events of his life. He remained to the
end a consistent republican, and after the coup d etat of 1852
though half blind and dying he resigned his post as astronomer
rather than take the oath of allegiance. It is to the credit of
Napoleon III. that he gave directions that the old man should
be in no way disturbed, and should be left free to say and do
what he liked.
His earliest physical researches were on the pressure of
* Arago s works, which include loges on many of the leading mathe
maticians of the last five or six centuries, have been edited by M. J. A.
Barral and published in fourteen volumes, Paris, 1856 7. An auto
biography is prefixed to the first volume.
444 ARAGO.
steam at different temperatures, and the velocity of sound, 1818
to 1822. His magnetic observations mostly took place from
1823 to 1826. He discovered what has been called rotatory
magnetism, and the fact that most bodies could be magnetized :
these discoveries were completed and explained by Faraday.
He warmly supported Fresnel s optical theories, and the two
philosophers conducted together those experiments on the polar
ization of light which led to the inference that the vibrations
of the luminiferous ether were transverse to the direction of
motion, and that polarization consisted in a resolution of recti
linear motion into components at right angles to each other.
The subsequent invention of the polariscope and discovery of
rotatory polarization are due to Arago. The general idea of
the experimental determination of the velocity of light in
the manner subsequently effected by Fizeau and Foucault
was suggested by him in 1838, but his failing eyesight pre
vented his arranging the details or making the experiments.
It will be noticed that some of the last members of the
French school were alive at a comparatively recent date, but
nearly all their mathematical work was done before the year
1830. They are the direct successors of the French writers
who flourished at the commencement of this century, and
seem to have been out of touch with the great German
mathematicians of the early part of it on whose researches
much of the best work of this century is based ; they are thus
placed here though their writings are in some cases of a later
date than those of Gauss, Abel, Jacobi, and other mathe
maticians of recent times.
The introduction of analysis into England.
The complete isolation of the English school and its
devotion to geometrical methods are the most marked features
in its history during the latter half of the eighteenth century ;
and the absence of any considerable and valuable contribution
IVORY. THE CAMBRIDGE ANALYTICAL SCHOOL. 445
to the advancement of mathematical science was a natural
consequence. One result of this was that the energy of English
men of science was largely devoted to practical physics and
practical astronomy, which were in consequence studied in
Britain perhaps more than elsewhere.
Ivory. Almost the only English mathematician at the
beginning of this century who used analytical methods and
whose work requires mention here is Ivory, to whom the
celebrated theorem in attractions is due. James Ivory was
born in Dundee in 1765, and died at Douglastown on Sept.
21, 1845. After graduating at St Andrews he became the
managing partner in a flax-spinning company in Forfarshire,
but continued to devote most of his leisure to mathematics.
In 1804 he was made professor at the Royal Military College
at Marlow, which is now moved to Sandhurst. He contributed
numerous papers to the Philosophical Transactions, the most
remarkable being those on attractions. In one of these, in
1809, he shewed how the attraction of a homogeneous ellipsoid
on an external point is a multiple of that of another ellipsoid
on an internal point: the latter can be easily obtained. He
criticized Laplace s solution of the method of least squares with
unnecessary bitterness, and in terms which proved his incompe
tence to understand it.
The Cambridge Analytical School. Towards the close of
the last century the more thoughtful members of the Cam
bridge school of mathematics began to recognize that their
isolation from their continental contemporaries was a serious
evil. The earliest attempt in England to explain the notation
and methods of the calculus as used on the continent was due
to Wood house, who stands out as the apostle of the new move
ment. It is doubtful if he could have brought the analytical
methods into vogue by himself ; but his views were enthusi
astically adopted by three undergraduates, Babbage, Peacock,
and Herschel, who succeeded in carrying out the reforms he had
suggested. In a book which will fall into the hands of few but
English readers I may be pardoned for making space for a few
446 WOODHOUSE.
remarks on these four mathematicians*. The original stimulus
came from French sources and I therefore place these remarks
at the close of my account of the French school, but I should
add that the English mathematicians of this century at once
struck out a line independent of their French contemporaries.
Woodhouse. Robert Woodhouse was born at Norwich on
April 28, 1773; was educated at Caius College, Cambridge, of
which society he was subsequently a fellow; was Plumian pro
fessor in the university; and continued to live at Cambridge
till his death on December 23, 1827. His earliest work,
entitled the Principles of Analytical Calculation, was published
at Cambridge in 1803. In this he explained the differential
notation and strongly pressed the employment of it, but he
severely criticized the methods used by continental writers,
and their constant assumption of non- evident principles. This
was followed in 1809 by a trigonometry (plane and spherical),
and in 1810 by a historical treatise on the calculus of variations
and isoperimetrical problems. He next produced an astro
nomy ; the first volume (usually bound in two) on practical
and descriptive astronomy being issued in 1812, the second
volume, containing an account of the treatment of physical
astronomy by Laplace and other continental writers, being
issued in 1818. All these works deal critically with the
scientific foundation of the subjects considered a point which
is not unfrequently neglected in modern text-books.
A man like Woodhouse, of scrupulous honour, universally
respected, a trained logician, and with a caustic wit, was well
fitted to introduce a new system ; and the fact that when he
first called attention to the continental analysis, he exposed
the unsoundness of some of the usual methods of establishing
it more like an opponent than a partizan, was as politic as it
was honest. Woodhouse did not exercise much influence on
the majority of his contemporaries, and the movement might
have died away for the time being, if it had not been for the
advocacy of Peacock, Herschel, and Babbage who formed an
* The following account is condensed from my History of the Study
of Mathematics at Cambridge, Cambridge, 1889.
PEACOCK. BABBAGE. HERSCHEL. 447
Analytical Society, with the object of advocating the general
use in the university of analytical methods and of the diffe
rential notation.
Peacock. George Peacock, who was the most influential of
the early members of the new school, was born at Denton on
April 9, 1791. He was educated at Trinity College, Cam
bridge, of which society he was subsequently a fellow and
tutor. The establishment of the university observatory was
mainly due to his efforts, and in 1836 he was appointed to the
Lowndean professorship of astronomy and geometry. In 1839
he was made dean of Ely, and resided there till his death on
Nov. 8, 1858. Although Peacock s influence on English
mathematicians was considerable he has left but few me
morials of his work; but I may note that his. report on recent
progress in analysis, 1833, commenced those valuable summaries
of scientific progress which enrich many of the annual volumes
of the Transactions of the British Association.
Babbage. Another important member of the Analytical
Society was Charles Babbage, who was born at Totnes on
Dec. 26, 1792; he entered at Trinity College, Cambridge, in
1810; subsequently became Lucasian professor in the univer
sity; and died in London on Oct. 18, 1871. It was he who
gave the name to the Analytical Society, which he stated was
formed to advocate " the principles of pure d-isrn as opposed
to the dW-age of the university." In 1820 the Astronomical
Society was founded mainly through his efforts, and at a later
time, 1830 to 1832, he took a prominent part in the foundation
of the British Association. He will be remembered for his
mathematical memoirs on the calculus of functions, and his
invention of an analytical machine which could not only
perform the ordinary processes of arithmetic but could tabu
late the values of any function and print the results.
Herschel. The third of those who helped to bring ana
lytical methods into general use in England was the son of
Sir William Herschel (1738 1S22), the most illustrious
astronomer of the latter half of the last century and the
creator (it may be fairly said) of stellar astronomy. Sir John
448 HERSCHEL.
Frederick William Herschel was born on March 7, 1792,
educated at St John s College, Cambridge, and died on May
11, 1871. His earliest original work was a paper on Cotes s
theorem, and it was followed by others on mathematical
analysis, but his desire to complete his father s work led ulti
mately to his taking up astronomy. His papers on light and
astronomy contain a clear exposition of the principles which
underlie the mathematical treatment of those subjects.
In 1813 the Analytical Society published a volume of
memoirs, of which the preface and the first paper (on continued
products) are due to Babbage; and three years later they
issued a translation of Lacroix s Traite elementaire du calcul
differential et du calcul integral. In 1817, and again in 1819,
the differential notation was used in the university examina
tions, and after 1820 its use was well established. The
Analytical Society followed up this rapid victory by the issue
in 1820 of two volumes of examples illustrative of the new
method; one by Peacock on the differential and integral
calculus, and the other by Herschel on the calculus of finite
differences. Since then English works on the infinitesimal
calculus have abandoned the exclusive use of the fluxional
notation. It should be noticed in passing that Lagrange and
Laplace, like the majority of other modern writers, employ
both the fluxional and the differential notation ; it was the
exclusive adoption of the former that was so hampering.
Amongst those who materially assisted in extending the
use of the new analysis were William Whewell (1794 1866)
and George Biddell Airy (18011892), both fellows of Trinity
College, Cambridge. The former issued in 1819 a work on
mechanics, and the latter, who was a pupil of Peacock, pub
lished in 1826 his Tracts^ in which the new method was
applied with great success to various physical problems. The
efforts of the society were supplemented by the rapid publica
tion of good text-books in which analysis was freely used.
The employment of analytical methods spread from Cambridge
over the rest of Britain, and by 1830 these methods had come
into general use there.
CHAPTER XIX.
MATHEMATICS OF RECENT TIMES.
IT is evidently impossible for me to discuss adequately the
mathematicians of the age in which we live, especially as I
purposely exclude from this work any detailed reference to
living writers. I make therefore no attempt to give a complete
history of this period, but as a sort of appendix to the preceding
chapters I add a few notes on some of the more striking
features in the history, during this century, of pure mathe
matics (in which I include theoretical dynamics and astronomy) ;
but except for a few allusions I shall not discuss the applica
tion of mathematics to physics. These notes are brief, and in
many cases consist merely of a list of the names of some of
those to whom the development of any branch of the subject
is chiefly due, and an indication of that part of it to which
they have directed most attention. I would refer any one
who wishes for more details to the invaluable catalogue
which has been compiled by the Royal Society of London,
and which contains a list, under the names of the authors, of
all the scientific papers contributed during this century to
journals and learned societies. In only a few cases do I add
any account of the life and works of the mathematicians men
tioned. Even with these limitations it has been very difficult
to put together a connected account of the mathematics of
recent times ; and I wish to repeat explicitly that I do not
suggest, nor do I wish my readers to suppose, that my notes
on a subject give the names of all the chief writers who have
studied it. In fact the quantity of matter produced has been
B. 29
450 MATHEMATICS OF RECENT TIMES.
so enormous that no one can expect to do more than make
himself acquainted with the work in some small department :
as an illustration of this remark I may say that I have reason
to believe that something like 15,000 separate scientific memoirs
are now published every year by the different societies and jour
nals of Europe and America.
Most of the treatises on the history of mathe matics omit
all reference to the work produced during this century. The
chief exceptions with which I am acquainted are a short disser
tation by H. Hank el, entitled Die Entwickelung der Mathematik
in den letzten Jahrhunderten, Tubingen, 1885; the eleventh
and twelfth volumes of Marie s Histoire des sciences in which
are some notes on mathematicians who were born in the last
century; Gerhard t s Geschichte der Mathematik in Deutschland,
Munich, 1877 ; and a Discours on the professors at the Sorbonne
by Ch. Hermite in the Bulletin des sciences mathematiques, 1890,
pp. 6 36. A few histories of the development of particular
subjects have been written such as those by the late Isaac
Todhunter on the theories of attraction and on the calculus of
probabilities while the annual volumes of the British Asso
ciation contain a number of reports on the progress in several
different branches of modern mathematics ; a few similar reports
(and notably one in 1857 by J. Bertrand on the development
of mathematical analysis) have been presented to the French
Academy. I have found these authorities and these reports
useful, but I have derived most assistance in writing this
chapter from the obituary notices in the proceedings of various
learned Societies, foreign as well as British; I am also in
debted to information kindly furnished me by various friends,
and if I do not further dwell on this, it is only that I would
not seem to make them responsible for errors and omissions
which they would have avoided in their own works.
A period of exceptional intellectual activity in any subject
is usually followed by one of comparative stagnation ; and
after the deaths of Lagrange, Laplace, Legendre, and Poisson
GAUSS. 451
the French school, which had occupied so prominent a position
at the beginning of this century, ceased for some years to
produce much new work. Some of the mathematicians whom
I intend first to mention, Gauss, Abel, and Jacobi, were
contemporaries of the later years of the French mathematicians
just named, but their writings appear to me to belong to a
different school, and thus are properly placed at the beginning
of a fresh chapter.
There is no mathematician of this century whose writings
have had a greater effect than those of Gauss ; nor is it on
only one branch of the science that his influence has left a
permanent mark. I cannot therefore commence my account
of the mathematics of recent times better than by describing
very briefly his more important researches.
Gauss*. Karl Friedrich Gauss was born at Brunswick on
April 23, 1777, and died at Gottingen on Feb. 23, 1855. His
father was a bricklayer, and Gauss was indebted for a liberal
education (much against the will of his parents who wished
to profit by his wages as a labourer) to the notice which his
talents procured from the reigning duke. In 1792 he was sent
to the Caroline College, and by 1795 professors and pupils
alike admitted that he knew all that the former could teach
him : it was while there that he investigated the method of
least squares, and proved by induction the law of quadratic
reciprocity. Thence he went to Gottingen, where he studied
under Kastner : many of his discoveries in the theory of num
bers were made while a student here. In 1798 he returned
to Brunswick, where he earned a somewhat precarious liveli
hood by private tuition.
In 1799 Gauss published his demonstration that every
algebraical equation has a root of the form a + bi ; a theorem
of which altogether he gave three distinct proofs. In 1801 this
was followed by his Disquisitioiies Aritkmeticae, which is printed
* Gauss s collected works have been edited by E. J. Schering, and pub
lished by the Royal Society of Gottingen in 7 volumes, 1863 71.
292
452 MATHEMATICS OF RECENT TIMES.
as the first volume of his collected works. The greater part
of it had been sent to the French Academy in the preceding
year, and had been rejected with a sneer which, even if the
book had been as worthless as the referees believed, would
have been unjustifiable ; Gauss was deeply hurt, and his
reluctance to publish his investigations may be partly
attributable to this unfortunate incident.
The next discovery of Gauss was in a totally different
department of mathematics. The absence of any planet in the
space between Mars and Jupiter, where Bode s law would have
led observers to expect one, had been long remarked, but it
was not till 1801 that any one of the numerous group of
minor planets which occupy that space was observed. The
discovery was made by Piazzi of Palermo ; and was the more
interesting as its announcement occurred simultaneously with
a publication by Hegel in which he severely criticized as
tronomers for not paying more attention to philosophy, a
science, said he, which would at once have shewn them that
there could not possibly be more than seven planets, and a
study of which would therefore have prevented an absurd
waste of time in looking for what in the nature of things
could never be found. The new planet was named Ceres, but
it was seen under conditions which appeared to render it almost
impossible to forecast its orbit. The observations were fortu
nately communicated to Gauss ; he calculated its elements,
and his analysis proved him to be the first of theoretical astro
nomers no less than the greatest of " arithmeticians."
The attention excited by these investigations procured for
him in 1807 the offer of a chair at St Petersburg, which he
declined. In the same year he was appointed director of the
Gottingen observatory and professor of astronomy there.
These offices he retained to his death ; and after his ap
pointment he never slept away from his observatory except
on one occasion when he attended a scientific congress at
Berlin. His lectures were singularly lucid and perfect in
form, and it is said that he used here to give the analysis by
GAUSS. 453
which he had arrived at his various results, and which is so
conspicuously absent from his published demonstrations; but
for fear his auditors should lose the thread of his discourse, he
never willingly permitted them to take notes.
I have already mentioned Gauss s publications in 1799,
1801, and 1802. For some years after 1807 his time was
almost wholly occupied by work connected with his observa
tory. In 1809 he published at Hamburg his Theoria Motus
Corporum C oelestium, a treatise which contributed largely to
the improvement of practical astronomy, and introduced the
principle of curvilinear triangulation : and on the same
subject, but connected with observations in general, we have
his memoir Theoma Combinationis Observationum Errombus
Minirnis Obnoxia, with a second part and a supplement.
Somewhat later, he took up the subjects of geodesy, acting
from 1821 to 1848 as scientific adviser to the Danish and
Hanoverian governments for the survey then in progress :
his papers of 1843 and 1866, Ueber Gegenstande der Jioheni
Geodasie, contain his researches on the subject.
Gauss s researches on electricity and magnetism date from
about the year 1830. His first paper on the theory of
magnetism, entitled Intensitas Vis Magneticae Terrestris ad
Mensuram Absolutam Mevocata, was published in. 1833. A
few months afterwards he, together with Weber, invented the
declination instrument and the bitilar magnetometer; and in
the same year they erected at Gottingen a magnetic observa
tory free from iron (as Humboldt and Arago had previously
done on a smaller scale) where they made magnetic observa
tions, and in particular shewed that it was possible and
practicable to send telegraphic signals. In connection with
this observatory Gauss founded the association called the
Magnetische Verein with the object of securing continuous
observations at fixed times. The volumes of their publica
tions, Resultate aus der Beobachtungen des Magnetischen
Vereiiis for 1838 and 1839, contain two important memoirs
by Gauss, one on the general theory of earth-magnetism, and
454 MATHEMATICS OF RECENT TIMES.
the other on the theory of forces attracting according to
the inverse square of the distance. Like Poisson he treated
the phenomena in electrostatics as due to attractions and re
pulsions between imponderable particles. In electrodynamics
he arrived (in 1835) at a result equivalent to that given by
W. E. Weber in 1846, namely, that the attraction between
two electrified particles e and e whose distance apart is r
depends on their relative motion and position according to the
formula
Gauss however held that no hypothesis was satisfactory which
rested on a formula and was not a consequence of a physical
conjecture, and as he could not frame a plausible physical
conjecture he abandoned the subject. Such conjectures were
proposed by Biemann in 1858, and by 0. Neumann and
E. Betti in 1868, but Helmholtz in 1870, 1873, and 1874
shewed that they were untenable. A simpler view which
regards all electric and magnetic phenomena as stresses and
motions of a material elastic medium had been outlined by
Michael Faraday, and was elaborated by James Clerk Maxwell;
the latter, by the use of generalized coordinates, was able to
deduce the consequences, and the agreement with experiment
is close (see below, p. 496). These and other electric theories
were classified and critically discussed in a memoir by J. J.
Thomson in 1885 (see below, p. 497).
Gauss s researches on optics, including systems of lenses,
were published in 1840 in his Dioptrische Untersuchungen.
From this sketch it will be seen that the ground covered
by Gauss s researches was extraordinarily wide. I will now
mention very briefly some of the most important of his
discoveries in pure mathematics.
His most celebrated work in pure mathematics is the
Disquisitiones Arithmeticae which has proved a starting point
for several interesting investigations on the theory of numbers.
This treatise and Legendre s Theorie des nombres remain
GAUSS. 455
standard works on the theory of numbers ; but, just as
in his discussion of elliptic functions Legendre failed to
rise to the conception of a new subject, and confined him
self to regarding their theory as a chapter in the integral
calculus, so he treated the theory of numbers as a chapter in
algebra. Gauss however realized that the theory of discrete
magnitudes or higher arithmetic was of a different kind from
that of continuous magnitudes or algebra, and he introduced
a new notation and new methods of analysis of which
subsequent writers have generally availed themselves. In
particular the Disquisitiones Aritkmeticae introduced the modern
theory of congruences of the first and second orders, and to
this Gauss reduced indeterminate analysis. In it also he
discussed the solution of binomial equations of the form x n = 1 :
this involves the celebrated theorem that the only regular
polygons which can be constructed by elementary geometry
are those of which the number of sides is 2 m (2 n +1), where m
and n are integers and 2 n + 1 is a prime ; a discovery he had
made in 1796. He developed the theory of ternary quadratic
forms involving two indeterminates. He also investigated the
theory of determinants, and it was on Gauss s results that
Jacobi based his researches on that subject.
The theory of functions of double periodicity had its origin
in the discoveries of Abel and Jacobi, which I describe later.
Both these mathematicians arrived at the theta functions,
which play so large a part in the theory of the subject. Gauss
however had independently, and indeed at a far earlier date,
discovered these functions and their chief properties ; having
been led to them by certain integrals which occurred in the
Determinatio Attractionis, to evaluate which he invented the
transformation now associated with the name of Jacobi. Though
Gauss at a later time communicated the fact to Jacobi, he did
not publish his researches ; they occur in a series of note
books of a date not later than 1808, and are included in his
collected works.
Of the remaining memoirs in pure mathematics the most
456 MATHEMATICS OF RECENT TIMES.
remarkable are those on the theory of biquadratic residues
(wherein the notion of complex numbers of the form a + bi
was first introduced into the theory of numbers) in which are
included several tables, and notably one of the number of
the classes of binary quadratic forms; that relating to the
proof of the theorem that every numerical equation has a real
or imaginary root ; that on the summation of series ; that on
hypergeometric series, which contains a discussion of the
Gamma function ; and lastly one on interpolation : his intro
duction of rigorous tests for the convergency of infinite series
is specially noticeable. Finally we have the important memoir
on the conformal representation of one surface upon another,
in which the results given by Lagrange for surfaces of
revolution are generalized for any surfaces.
In the theory of attractions we have a paper on the
attraction of homogeneous ellipsoids ; the already-mentioned
memoir of 1839, Allgemeine Lehrsatze in Beziehung auf die
im verkehrten Verhdltnisse des Quadrats der Entferung, on the
theory of forces attracting according to the inverse square of
the distance ; and the memoir, Determinatio Attractionis, in
which it is shewn that the secular variations, which the
elements of the orbit of a planet experience from the attraction
of another planet which disturbs it, are the same as if the
mass of the disturbing planet were distributed over its orbit into
an elliptic ring in such a manner that equal masses of the ring
would correspond to arcs of the orbit described in equal times.
The great masters of modern analysis are Lagrange, Laplace,
and Gauss, who were contemporaries. It is interesting to note
the marked contrast in their styles. Lagrange is perfect both
in form and matter, he is careful to explain his procedure,
and though his arguments are general they are easy to follow.
Laplace on the other hand explains nothing, is absolutely
indifferent to style, and, if satisfied that his results are correct,
is content to leave them either with no proof or with a faulty
one. Gauss is as exact and elegant as Lagrange, but even
more difficult to follow than Laplace, for he removes every
.DIRICHLET. 457
trace of the analysis by which he reached his results, and
studies to give a proof which while rigorous shall be as concise
and synthetical as possible.
Dirichlet*. One of Gauss s pupils to whom I may here allude
is Lejeune Dirichlet, who is generally known for his exposition
of the discoveries of Jacobi (who was his father-in-law) and of
Gauss, rather than for his own original investigations, valuable
though some of these are. Peter Gustav Lejeune Dirichlet was
born at Diiren on Feb. 13, 1805, and died at Gottingen on
May 5, 1859. He held successively professorships at Breslau
and Berlin, and on Gauss s death in 1855 was appointed to
succeed him as professor of the higher mathematics at Gottin-
gen. He intended to finish Gauss s incomplete works, for
which he was admirably fitted, but his early death prevented
this ; he produced however several memoirs which have
considerably facilitated the comprehension of some of Gauss s
more abstruse methods. Of Dirichlet s original work the most
celebrated is that on the determination of means with applica
tions to the distribution of prime numbers.
The researches of Gauss on the theory of numbers were
continued or supplemented by Jacobi (see below, p. 465) who
first proved the law of cubic reciprocity ; discussed the theory
of residues ; and, in his Canon Aritlimeticus, gave a table of
residues of prime roots.
Eisenstein. This subject was next taken up by Ferdinand
Gotthold Eisenstein, a professor at the university of Berlin,
who was born at Berlin on April 16, 1823, and died there on
Oct. 11, 1852. The theory of numbers may be divided into
two main divisions, namely, the theory of congruences and
the theory of forms. The solution of the problem of the
representation of numbers by binary quadratic forms is one of
* His works are being produced in two volumes, vol. i., by
L. Kronecker, Berlin, 1889. His lectures on the theory of numbers were
edited by B. Dedekind, third edition, Brunswick, 187981: his investi
gations on the theory of the potential have been edited by F. Grube, second
edition, Leipzig, 1887. There is a note on some of his researches by C.
W. Borchardt in Crelle s Journal, vol. LVII., 1859, pp. 9192.
458 EISENSTEIN.
the great achievements of Gauss, and the fundamental principles
upon which the treatment of such questions rest were given
by him in the Disquisitiones Arithmeticae. Gauss there added
some results relating to ternary quadratic forms, but the general
extension from two to three indeterminates was the work of
Eisenstein, who, in his memoir Neue T/ieoreme der hoheren
Arithmetik, denned the ordinal and generic characters of ternary
quadratic forms of an uneven determinant ; and, in the case
of definite forms, assigned the weight of any order or genus ;
but he did not consider forms of an even determinant, nor
give any demonstrations of his work.
Eisenstein also considered the theorems relating to the
possibility of representing a number as a sum of squares, and
shewed that the general theorem was limited to eight squares.
The solutions in the cases of two, four, and six squares may be
obtained by means of elliptic functions, but the cases in which
the number of squares is uneven involve special processes
peculiar to the theory of numbers. Eisenstein gave the solu
tion in the case of three squares. He also left a statement
of the solution he had obtained in the case of five squares*;
but his results were published without proofs, and apply
only to numbers which are not divisible by a square.
Among Eisenstein s other investigations I single out for
special mention the remarkable rule he enunciated by means
of which it is possible to distinguish whether a given series
represents an algebraical or a transcendental function.
Henry Smith t One of the most original and powerful
mathematicians of the school founded by Gauss was Henry
Smith. Henry John Stephen Smith was born in London
on Nov. 2, 1826, and died at Oxford on Feb. 9, 1883. He
* Crelle s Journal, vol. xxxv., 1847, p. 368.
t Smith s collected mathematical works, edited by Dr Glaisher of
Trinity College, Cambridge, will be shortly issued by the university of
Oxford. The following account is extracted from the obituary notice by
Dr Glaisher in the monthly notices of the Astronomical Society, 1884,
pp. 138149.
HENRY SMITH. 459
was educated at Rugby, and at Balliol College, Oxford, of
which latter society he was a fellow; and in 1861 he was
elected Savilian professor of geometry at Oxford, where he
resided till his death.
The subject in connection with which Smith s name will
be always specially remembered is the theory of numbers, and
to this he devoted the years from 1854 to 1864. The results
of his historical researches were given in his report published in
parts in the Transactions of the British Association from 1859
to 1865 ; this report contains an account of what had been done
on the subject to that time together with some additional mat
ter. The chief outcome of his own original work on the sub
ject is included in two memoirs printed in the Philosophical
Transactions for 1861 and 1867 ; the first being on linear
indeterminate equations and congruences, and the second
on the orders and genera of ternary quadratic forms. In the
latter memoir demonstrations of Eisenstein s results and their
extension to ternary quadratic forms of an even determinant
were supplied, and a complete classification of ternary
quadratic forms was given.
Smith, however, did not confine himself to the case of three
indeterminates, but succeeded in establishing the principles on
which the extension to the general case of n indeterminates
depends, and obtained the general formulae ; thus effecting the
greatest advance made in the subject since the publication of
Gauss s work. In the account of his methods and results which
appeared in the Proceedings of the Royal Society*, Smith re
marked that the theorems relating to the representation of
numbers by four squares and other simple quadratic forms, are
deducible by a uniform method from the principles there indi
cated, as also are the theorems relating to the representation of
numbers by six and eight squares. He then proceeded to say
that as the series of theorems relating to the representation of
numbers by sums of squares ceases, for the reason assigned by
Eisenstein, when the number of squares surpasses eight, it was
* See vol. xiii., 1864, pp. 199203, and vol. xvi., 1868, pp. 197208.
460 MATHEMATICS OF RECENT TIMES.
desirable to complete it. The results for even squares were
known. The principal theorems relating to the case of five
squares had been given by Eisenstein, but he had considered
only those numbers which are not divisible by a square,
and he had not considered the case of seven squares. Smith
here completed the enunciation of the theorems for the case of
five squares, and added the corresponding theorems for the case
of seven squares.
This paper was the occasion of a dramatic incident in the
history of mathematics. Fourteen years later, in ignorance of
Smith s work, the demonstration and completion of Eisenstein s
theorems for five squares were set by the French Academy as
the subject of their "Grand prix des sciences mathematiques."
Smith wrote out the demonstration of his general theorems so
far as was required to prove the results in the special case of
five squares, and only a month after his death, in March 1883,
the prize was awarded to him, another prize being also awarded
to H. Minkowski of Bonn. No episode could bring out in a
more striking light the extent of Smith s researches than that
a question of which he had given the solution in 1867 as a
corollary from general formulae which governed the whole
class of investigations to which it belonged should have been
regarded by the French Academy as one whose solution was of
such difficulty and importance as to be worthy of their great
prize. It has been also a matter of comment that they should
have known so little of contemporary English and German
researches on the subject as to be unaware that the result
of the problem they were proposing was then lying in their
own library.
Among Smith s other investigations I may specially mention
his geometrical memoir Sur quelques problemes cubiques el
biquadratiques, for which in 1868 he was awarded the Steiner
prize of the Berlin Academy. In a paper which he contributed
to the Atti of the Accademia dei Lincei for 1877 he established
a very remarkable analytical relation connecting the modular
equation of order n and the theory of binary quadratic forms
THE THEORY OF NUMBERS. 461
belonging to the positive determinant n. In this paper the
modular curve is represented analytically by a curve in such a
manner as to present an actual geometrical image of the
complete systems of the reduced quadratic forms belonging to
the determinant, and a geometrical interpretation is given to
the ideas of " class/ " equivalence," and " reduced form." He
was also the author of important papers in which he succeeded
in extending to complex quadratic forms many of Gauss s
investigations relating to real quadratic forms. He was led
by his researches on the theory of numbers to the theory of
elliptic functions, and the results he arrived at, especially on
the theory of the theta and omega functions, are of importance.
The Theory of Numbers, as treated to-day, may be said to
originate with Gauss. I have already mentioned very briefly
the subject of the subsequent investigations of Jacobi, Dirich-
let, Eisenstein, and Henry Smith.
Among other mathematicians who have written on it I
may allude to the following.
Riemann (see below, p. 468), who investigated the dis
tribution of primes.
James Joseph Sylvester, Savilian professor in the university
of Oxford, born in London on Sept. 3, 1814 (see below, pp. 462,
478, 482), who also has written on the distribution of primes.
Cauchy (see below, p. 473), who in particular discussed the
expression of quadratic binomials.
Joseph LiouviUe, the editor from 1836 to 1874 of the well-
known journal, who was born at St Omer on March 24, 1809,
and died in 1882 (see below, p. 470), most of whose numerous
investigations dealt with the representation of numbers by
special forms.
Ernest Edward Kummer, born at Sorau on Jan. 29, 1810,
and until recently professor at Berlin (see below, p. 477),
to whom we owe the conception of the so-called ideal primes,
which are required in the treatment of complexes, and which
he applied to the problem of Fermat s equation ; and whose
462 THE THEORY OF NUMBERS.
paper on hypergeometric series may rank with that by Gauss
as a classical memoir on the subject.
Leopold Kronecker, professor in Berlin, born at Liegnitz on
Dec. 7, 1823, and died at Berlin on Dec. 29, 1891, most of
whose investigations on this branch of mathematics were on
ternary and quadratic forms : on his investigations generally
see the Bulletin of the New York Mathematical Society, vol. I.,
pp. 173184; see also below, p. 469.
Charles Hermite, professor in Paris, born in Lorraine on
Dec. 24, 1822 (see below, pp. 469, 470, 471, 478), who wrote
on ternary forms.
Julius Wilhelm Richard Dedekind, born at Brunswick on
Oct. 6, 1831, whose more important researches, given in an ap
pendix to his edition of Dirichlet s writings, are on ideal primes :
see also below, p. 493.
Patnutij Tchebycheff, formerly professor at the university
of St Petersburg, born in Russia in 1821, who has written on
the number of primes between given limits : a problem also
considered by Legendre, Dirichlet, and Riemann.
And James Whitbread Lee Glaisher, fellow and tutor of
Trinity College, Cambridge, born at Lewisham on Nov. 5, 1848
(see below, p. 470), from whose numerous papers I may single
out those relating to prime numbers ; those on functions of a
number which are formed from its (real or complex) divisors ;
and those on the possible divisors of numbers of a given form.
Finally I may mention that the problem of the partition of
numbers, to which Euler paid considerable attention, has in
recent times attracted the attention of Arthur Cayley, Sadlerian
professor in the university of Cambridge, born in Richmond,
Surrey, on Aug. 16, 1821 (see below, pp. 469, 478, 481), of
Sylvester (see pp. 461, 478, 482), and of Percy Alexander
Macmahon, professor at Woolwich and a major in the English
artillery, born at Malta on Sept. 26, 1854 (see below, p. 479).
Interest in problems connected with the theory of numbers
seems recently to have flagged, and possibly it may be found
hereafter that the subject is approached better on other lines.
ABEL. 463
The theory of functions of double and multiple periodicity
is another subject to which much attention has been paid
during this century. I have already mentioned that as early
as 1808 Gauss had discovered the theta functions and their chief
properties, but his investigations remained for many years
concealed in his note-books ; and it was to the researches
made between 1820 and 1830 by Abel and Jacobi that
the modern development of the subject is due. Their treat
ment of it has completely superseded that used by Legendre,
and they are justly reckoned as the creators of this branch of
mathematics.
Abel*. Niels Henrick Abel was born at Findoe in Norway
on Aug. 5, 1802, and died at Arendal on April 6, 1829, at the
age of twenty-six. His memoirs on elliptic functions which
were originally published in Crellds Journal treat the subject
from the point of view of the theory of equations and algebraic
forms, a treatment to which his researches naturally led him.
The important and very general result known as Abel s theorem,
which was subsequently applied by Riemann to the theory of
transcendental functions, was sent to the French Academy in
1828, but (mainly through the action of Cauchy) was not
published for several years. The name of Abelian function has
been given to the higher transcendents of multiple periodicity
which were first discussed by Abel. He criticized the use of
infinite series, but I do not know that the results lead to any
definite rules for testing convergency. As illustrating his
fertility of ideas I may in passing notice his celebrated demon
stration that it is impossible to solve a quiutic equation by
means of radicals ; this theorem was the more important since
it definitely limited a field of mathematics which had pre
viously attracted numerous writers. I should add that this
theorem had been enunciated as early as 1798 by Paolo
* The life of Abel by C. A. Bjerknes was published at Stockholm in
1880 Two editions of Abel s works have been published, of which the
last, edited by Sylow and Lie and issued at Christian ia in two volumes
in 1881, is the more complete.
464 JACOBI.
Ruffini, an Italian physician practising at Modena ; but I
believe that the proof he gave was deficient in generality.
Jacob! *. Carl Gustav Jacob Jacobi, born of Jewish parents
at Potsdam on Dec. 10, 1804, and died at Berlin on Feb. 18,
1851, was educated at the university of Berlin where he ob
tained the degree of doctor of philosophy in 1825. In 1827 he
became extraordinary professor of mathematics at Konigsberg,
and in 1829 was promoted to be an ordinary prof essor ; this
chair he occupied till 1842, when the Prussian government
gave him a pension, and he moved to Berlin where he con
tinued to live till his death in 1851.
Jacobi s most celebrated investigations are those on elliptic
functions, the modern notation in which is due to him, and the
theory of which he established simultaneously with Abel but
independently of him. These are given in his treatise Funda-
menta Nova Theoriae Functionum Elliptwarum, Konigsberg,
1829, and in some later papers in Crelle s Journal. The
correspondence between Legendre and Jacobi on elliptic func
tions has been reprinted in the first volume of Jacobi s collected
works. Jacobi, like Abel, recognized that elliptic functions
were not merely a group of theorems on integration, but that
they were types of a new kind of function, namely, one of
double periodicity; hence he paid particular attention to the
theory of the theta function. The following passage! in which
he explains this view is sufficiently interesting to deserve textual
reproduction: "E quo, cum universam, quae fingi potest, am-
plectatur periodicitatem analyticam elucet, functioiies ellipticas
non aliis adnumerari debere transcendentibus, quae quibusdam
gaudent elegantiis, fortasse pluribus illas aut maioribus, sed
speciem quandam iis inesse perfecti et absoluti."
* See C. J. Gerhardt s Geschichte der Mathematik in Deutschland,
Munich, 1877. Jacobi s collected works were edited by Dirichlet, 3
volumes, Berlin, 1846 71, and accompanied by a biography, 1852; a
new edition, under the supervision of C. W. Borchardt and K. Weierstrass,
was issued at Berlin in 7 volumes, 1881 1891.
t His collected works, vol. i., 1881, p. 87.
JACOBI. RIEMANN. 465
Among Jacobi s other investigations I may specially single
out his papers on determinants, which did a great deal to bring
them into general use ; and particularly his invention of the
Jacobian, that is, of the functional determinant formed by the
n a partial differential coefficients of the first order of n given
functions of n independent variables. I ought also to mention
his papers on Abelian transcendents; his investigations on the
theory of numbers (see above, p. 457) ; his important work
on the theory of partial differentia] equations; his development
of the calculus of variations ; and his numerous memoirs on
the planetary theory and other particular dynamical problems,
in the course of which also he extended the theory of differential
equations : most of the results of the researches last named are
included in his Vorlesungen uber Dynamik, edited by Clebsch,
Berlin, 1866.
Riemaim*. Georg Friederich Bernhard Riemann was born
at Breselenz on Sept. 17, 1826, and died at Selasca on July 20,
1866. He studied at Gottingen under Gauss, and subsequently
at Berlin under Jacobi, Dirichlet, Steiner, and Eisenstein, all
of whom were professors there at the same time. His earliest
paper, written in 1850, was on algebraic functions of a complex
variable, and on it the recent investigations of Schwarz,
Klein, and Poincare are largely based : to these I refer very
briefly below (see p. 470). In 1854 Riemann wrote his cele
brated memoir on the hypotheses on which geometry is
founded. This was succeeded by memoirs on elliptic functions
and the theory of numbers; he also wrote on physical subjects.
The question of the truth of the assumptions usually made
in our geometry had been considered by J. Saccheri as long
ago as 1733, and in more recent times had been discussed by
Nicolai Ivanowitsch Lobatschewsky (professor at Kasan, born
at Nijnii-Novgorod in 1793, and died at Kasan on Feb. 12,
* Riemann s collected works, edited by H. Weber and prefaced by an
account of his life by Dedekind, were published at Leipzig, second edition,
1892. Another short biography of Riemann has been written by E. J.
Schering, Gottingen, 18G7.
B. 30
466 MATHEMATICS OF RECENT TIMES.
1856) in 1826 and again in 1840, by Gauss in 1831 and in
1846, and by Johann Bolyai (born at Klausenburg in 1802 and
died at Maros-Vasarhely in 1860) in 1832 in the appendix to
the first volume of his father s Tentamen, but Rieniann s memoir
of 1854 attracted general attention to the subject of hyper-
geometry, and the theory has been since extended and simpli
fied by various writers, notably by Eugenio Beltrami (professor
at Pavia, born at Cremona in 1835), and by Hermann Ludwig
Ferdinand von Helmholtz (professor at Berlin, born at Potsdam
on Aug. 31, 1821)*. The subject is so technical that I confine
myself to a bare sketch of the argument from which the idea
is derived.
That a space of two dimensions should have the geometrical
properties with which we are made familiar in the study of
elementary geometry, it is necessary that it should be possible
at any place to construct a figure congruent to a given figure ;
and this is so only if the product of the principal radii of
curvature at every point of the space or surface be constant.
There are three species of surfaces which possess this property :
namely, (i) spherical surfaces, where the product is positive ;
(ii) plane surfaces (which lead to Euclidean geometry), where
it is zero ; and (iii) what Beltrami has called pseudo-spherical
surfaces, where it is negative. Moreover, if any surface be
bent without dilation or contraction, the measure of curvature
remains unaltered. Thus these three species of surfaces are types
of three kinds on which congruent figures can be constructed.
For instance a plane can be rolled into a cone, and the system
of geometry on a conical surface is similar to that on a plane.
These kinds of space of two dimensions are distinguished
one from the other by a simple test. Through a point of
spherical space no geodetic line a geodetic line being defined
as the shortest distance between two points can be drawn
* For references see my Mathematical Recreations and Problems,
chap. x. A historical summary of the treatment of non-Euclidean
geometry is given in J. Frischauf s Elements der absoluten Geometrie,
Leipzig, 1876.
HYPERGEOMETRY. 467
parallel to a given geodetic line. Through a point of Euclidean
or plane space one and only one geodetic line (i.e. a straight
line) can be drawn parallel to a given geodetic line. Through
a point of pseudo-spherical space more than one geodetic line
can be drawn parallel to a given goedetic line, but all these
lines form a pencil whose vertical angle is constant.
It may be thought that we have a demonstration that our
space is plane, since through a given point we can draw only
one straight line parallel to a given straight line. This is not
so, for it is conceivable that our means of observation do riot
permit us to say with absolute accuracy whether two lines are
parallel ; hence we cannot use this as a means to tell whether
our space is plane or not. A better test can be deduced from
the proposition that in any two-dimensional space of uniform
curvature the sum of the angles of a triangle, if it differ from
two right angles, will differ by a quantity proportional to the
area of the triangle. Hence it may happen possibly that,
although for triangles such as we can measure the difference
is imperceptible, yet for triangles which are millions of times
bigger there would be a sensible difference.
If space be spherical or pseudo-spherical, its extent is finite ;
if it be plane, its extent is infinite. In regard to pseudo-
spherical space, I should add that its extent may be infinite, if
it be constructed in space of four dimensions.
In the preceding sketch of the foundations of non-Euclidean
geometry I have assumed tacitly that the measure of a distance
remains the same everywhere. Klein has shewn that, if this
be not the case and if the law of the measurement of distance be
properly chosen, we can obtain three systems of plane geometry
analogous to the three systems mentioned above. These are
called respectively elliptic, parabolic, and hyperbolic geometries.
The above refers only to hyper-space of two dimensions.
Naturally there arises the question whether there are different
kinds of hyper-space of three or more dimensions. Riemann
shewed that there are three kinds of hyper-space of three
dimensions having properties analogous to the three kinds of
468 ELLIPTIC AND ABELIAN FUNCTIONS.
hyper-space of two dimensions already discussed. These are
differentiated by the test whether at every point no geodetical
surfaces, or one geodetical surface, or a fasciculus of geodetical
surfaces can be drawn parallel to a given surface : a geodetical
surface being defined as such that every geodetic line joining
two points on it lies wholly on the surface.
I return now to Riemann s other investigations. In mul
tiply periodic functions, it is hardly too much to say that he,
in his memoir in Borchardt s Journal for 1857, did for the
Abel i an functions what Abel had done for the elliptic func
tions, and it is this perhaps that will constitute one of his
chief claims to future distinction.
In the theory of numbers, Riemann s short tract of eight
pages on the number of primes which lie between two given
numbers affords a striking instance of his analytical powers.
Legendre had previously shewn (see above, p. 429) that
the number of primes less than n is very approximately
n/(log n - 1-08366); but Riemann went further, and this tract
and a memoir by Tchebycheff contain nearly all that has been
done yet in connection with a problem of so obvious a charac
ter that it has suggested itself to all who have considered the
theory of numbers, and yet which overtaxed the powers even
of Lagrange and Gauss.
Among others than those already named I may mention the
following who have written on Elliptic and Abelian functions.
Johann Georg Rosenhain, professor in Konigsberg, born
there on June 10, 1816, and died in 1887, who wrote (in 1844)
on the hyperelliptic (double theta) function and functions of
two variables with four periods.
Adolphe Gopel, born at Rostok in September, 1812 and
died at Berlin in March, 1847, who discussed hyperelliptic
functions: see C reliefs Journal, vol. xxxv., 1847, pp. 313 318.
Karl Weierstrass, professor in Berlin, born at Ostendfelde
on Oct. 31, 1815, whose earlier researches related to the theta
functions, which he treated under a modified form in which
ELLIPTIC AND ABELIAX FUNCTIONS. 469
they are expressible in powers of the modulus : at a later
period he developed a method for treating all elliptic functions
in a symmetrical manner a process to which he was natur
ally led by his researches on the general theory of functions
(see below, pp. 471, 482); in this theory the theta functions are
independent of the form of their space boundaries.
Leopold Kronecker (see above, p. 462), who wrote on
elliptic functions.
Francesco Brioschi of Rome (see below, p. 478), who wrote
on elliptic and hyperelliptic functions.
Henry Smith (see above, p. 461), who discussed the trans
formation theory, the theta and omega functions, and certain
functions of the modulus.
Cayley (see pp. 462, 478, 481), who was the first to
work out (in 1845) the theory of doubly infinite products
and determine their periodicity, and who has written at length
on the connection between the researches of Legendre and
Jacobi ; his later writings have dealt mainly with the theory
of transformation and the modular equation : Cayley s collected
works are now being issued by the university of Cambridge.
The researches of Uermite (see pp. 462, 470, 471, 478) are
mostly concerned with the transformation theory, the higher
development of the theta functions, and the connection between
the methods and results of Weierstrass and Jacobi.
The transformation of the double theta function has been
also considered by Leo Konigsberger, professor at Heidelberg,
born in Prussia in 1837; see his lectures, published at Leipzig
in 1874.
The investigations of Georges Henri Halphen, an officer in
the French army, born at Rouen on Oct. 30, 1844 and died at
Paris on May 21, 1889, are largely founded on Weierstrass s
work : a sketch of Halphen s life and works is given in
Liouville s Journal for 1889, pp. 345 359, and in the Comptes
Rendus, 1890, vol. ex, pp. 489497; see also below, pp. 481,
482.
Felix Christian Klein, born in 1849 and now professor in
470 THE THEORY OF FUNCTIONS.
Gottingen (see below, pp. 470 1, 479), has written on Abelian
functions, elliptic modular functions, and hyperelliptic functions.
Filially H. A. Schwarz, formerly of Gottingen and now of
Berlin, born in 1845 (see below pp. 470, 482), H. Weber, of
Marburg, M. Nother of Erlangen (see below, p. 481), W. Stahl
of Aix-la-Chapelle, F. G. Frobenius, now of Berlin and formerly
of Zurich (see below, p. 482), and Glaisher (see above, p. 462)
have written on various branches of the theory, and Dr Glaisher
has in particular developed the theory of the zeta function.
The text-book by Briot and Bouquet contains a clear
account of elliptic functions as it exists at present, developed
from the point of view of the complex variable. Albert Briot
was born at St Hippolyte in 1817, occupied a chair at the
Sorbonne in Paris, and died in 1882 : Jean Claude Bouquet
was born in 1819, and died in Paris in 1885.
The consideration of algebraical, trigonometrical, elliptic,
hyperelliptic, and other special kinds of functions paved the
way for a theory of functions, which promises to prove a most
important and far-reaching branch of mathematics. To a
large extent this is the work of living mathematicians, and
therefore outside the limits of this chapter. I will content
myself by referring to the following writers.
First I may mention Cauchy (see below, p. 473) who gave
the general elementary theory of functions, and Liouville (see
above, p. 461), who wrote chiefly on doubly periodic functions :
their investigations were extended and connected in the work
by Briot and Bouquet, and have been further developed by
Hermite (see pp. 462, 469, 471, 478).
Next I may refer to the researches on the theory of
algebraic functions which have their origin in Riemanris paper
of 1850 (see above, p. 465).
Schwarz (see above, p. 470) has established accurately
certain theorems of which the proofs given by Riemann were
open to objection.
Klein (see pp. 469 70, 479) has connected Riemann s
MATHEMATICS OF RECENT TIMES. 471
theory of functions with the theory of groups, and has written
on automorphic functions.
Henri Poincare, professor in Paris, born at Nancy in 1854
(see below, pp. 482, 492), has also written on automorphic
functions, and on the general theory with special applications
to differential equations.
Finally I may refer to the work of Weierstrass and Mittag-
Leffler.
Of these, Karl Weierstrass (see pp. 469, 482) has created
a large part of the modern theory of functions, and in particu
lar has constructed the theory of uniform analytical functions.
And Magnus Gustaf Mittag-Leffler, born at Stockholm,
1846, and now professor there, has greatly developed the
theory of analytical functions ; a subject on which Hermite
(see pp. 462, 469, 470, 478) has also written.
In connection with these researches Paul Emile Appell,
professor in Paris, born at Strassburg in 1858, C. Emile
Picard of Paris, and fidouard G our sat of Paris have written
on special branches of the theory.
As text-books I may mention Dr. Forsyth s Theory of
Functions of a Complex Variable, Cambridge, 1893 ; and Carl
Neumann s Vorlesungen uber Riemann s Theorie der AbeVschen
Integrate , second edition, Leipzig, 1884.
The theory of numbers may be considered as a higher
arithmetic, and the theory of elliptic and Abelian functions as
a higher trigonometry. The theory of higher algebra (including
the theory of equations) has also attracted considerable attention,
and was a favourite subject of study of the three mathematicians,
Cauchy, Hamilton, and De Morgan whom I propose to
mention next though the interests of these writers were by
no means limited to this subject.
Cauchy*. The first of these mathematicians is the best
* See La vie et les travaux de Cauchy by L. Valson, 2 volumes, Paris,
1868. A complete edition of his works is now being issued by the French
government.
472 MATHEMATICS OF RECENT TIMES.
representative of the French school of analysis in this century.
Augustin Louis Cauchy, who was born at Paris on Aug. 21,
1789, and died at Sceaux on May 25, 1857, was educated
at the Polytechnic school, the nursery of so many French
mathematicians of that time, and adopted the profession of
a civil engineer. His earliest mathematical paper was
one on polyhedra in 1811. Legendre thought so highly of it
that he asked Cauchy to attempt the solution of an analogous
problem which had baffled previous investigators, and his
advice was justified by the success of Cauchy in 1812. Memoirs
on analysis and the theory of numbers presented in 1813,
1814, and 1815 shewed that his ability was not confined to
geometry alone : in one of these papers he generalized some
results which had been established by Gauss and Legendre ;
in another of them he gave a theorem on the number of values
which an algebraical function can assume when the literal
constants it contains are interchanged. It was the latter
theorem that enabled Abel to shew that in general an algebraic
equation of a degree higher than the fourth cannot be solved
by the use of purely algebraical expressions.
To Cauchy and Gauss we owe the scientific treatment of
series which have an infinite number of terms, and the former
established general rules for investigating the convergency and
divergency of such series. It is only a few works of an earlier
date that contain any discussion as to the limitations of the
series employed. It is said that Laplace, who was present
when Cauchy read his first paper on the subject, was so im
pressed by the illustrations of the danger of employing such
series without a rigorous investigation of their convergency
that he put on one side the work on which he was then
engaged and denied himself to all visitors, in order to see
if any of the demonstrations given in the earlier volumes of
the Mecanique celeste were invalid ; and he was fortunate
enough to find that no material errors had been thus introduced.
The treatment of series and of the fundamental conceptions
of the calculus in most of the text books then current was
CAUCHY. 473
based on Euler s works, and to any one trained to accurate
habits of thought was not free from objection. It is one of
the chief merits of Cauchy that he placed those subjects on a
logical foundation.
On the restoration in 1816 the French Academy was
purged, and, in spite of the indignation and scorn of French
scientific society, Cauchy accepted a seat which was procured
for him by the expulsion of Monge. He was also at the same
time made professor at the Polytechnic; and his lectures there
on algebraic analysis, the calculus, and the theory of curves
were published as text-books. On the revolution in 1830 he
went into exile, and was first appointed professor at Turin,
whence he soon moved to Prague to undertake the education
of the Comte de Chambord. He returned to France in 1837 ;
and in 1848, and again in 1851, by special dispensation of the
emperor was allowed to occupy a chair of mathematics without
taking the oath of allegiance.
His activity was prodigious, and from 1830 to 1859 he
published in the transactions of the Academy or the Comptes
Rendus over 600 original memoirs and about 150 reports.
In most of them the feverish haste with which they were
thrown off is too visible ; and many are marred by obscurity,
repetition of old results, and blunders.
Among the more important of his researches are the
discussion of tests for the convergency of series; the determina
tion of the number of real and imaginary roots of any algebraic
equation ; his method of calculating these roots approximately ;
his theory of the symmetric functions of the coefficients of
equations of any degree ; his ci priori valuation of a quantity
IMS than the least difference between the roots of an equation ;
and his papers on determinants in 1841 which did a great deal
to bring them into general use. Cauchy also did something to
reduce the art of determining definite integrals to a science, but
this branch of the integral calculus still remains without much
system or method. The rule for finding the principal values
of integrals was enunciated by him ; and the calculus of resi-
474 CAUCHY. ARGAND. SIR WILLIAM HAMILTON.
dues was his invention. His proof of Taylor s theorem seems
to have originated from a discussion of the double periodicity
of elliptic functions. The means of shewing a connection
between different branches of a subject by giving imaginary
values to independent variables is largely due to him. He
also gave a direct analytical method for determining planetary
inequalities of long period ; and to physics he contributed a
memoir on the quantity of light reflected from the surfaces
of metals, as well as other papers on optics.
Argand. I may mention here the name of Jean Robert
Argand who was born at Geneva on July 22, 1768 and died
circ. 1825. In his Essai, issued in 1806, he gave a geo
metrical representation of a complex number, and applied
it to shew that every algebraic equation has a root : this was
prior to the memoirs of Gauss and Cauchy on the same subject,
but the essay did not attract much attention when it was first
published. An earlier demonstration that /v /( 1) indicates
perpendicularity, due to Buee, was published in the Philo
sophical Transactions for 1806, and the idea was foreshadowed
in a memoir by H. Kuhn in the Transactions [pp. 170 223]
for 1750 of the St Petersburg Academy.
Hamilton*. In the opinion of some writers, the theory of
quaternions will be ultimately esteemed one of the great
discoveries of this century : that discovery is due to Sir
William Rowan Hamilton, who was born of Scotch parents in
Dublin on Aug. 4, 1805, and died there on Sept. 2, 1865.
His education, which was carried on at home, seems to have
been singularly discursive : under the influence of an uncle
who was a good linguist he first devoted himself to linguistic
studies ; by the time he was seven he could read Latin, Greek,
French, and German with facility ; and when thirteen he was
able to boast that he was familiar with as many languages as
he had lived years. It was about this time that he came
* See the life of Hamilton (with a bibliography of his writings) by R.
P. Graves, three volumes, Dublin, 188289 : the leading facts are given
in an article in the North British Revieiv for 186G.
SIB W1LLIAU HAMLTOX. 475
across a copy of Newton s CWwno/ Arithmetic; this was his
introduction to modern analysis, and he soon mastered the
elements of analytical geometry and the calculus. He next
read the /ViiicyiX and the f oar Tolmnes then iiiiMMnii
of Laplace s M^mm^mt nffejfc In the latter he detected a
mistake, and his paper on the subject, written in 1823, placed
him at once in the front rank of mathematicians. In the
following year he entered at Trinity College, Dublin: his
university career is unique, for the chair of astronomy be
coming vacant in 1827. while he was yet an undergraduate,
he was asked by the electors to Hnai for it, and was elected
unanimously, it being understood that he should be left free
to pursue his own line of study.
His earliest paper, wriifcem in 1 823, was em eptice and was
published in 1828 under the title of a Theory of System* of
Bmyt, to which two supplements, written in 1831 and 1832,
wove afterwards added ; in the latter of these the phenomenon
of *mfo$ refraction is predicted. This was followed by a
paper in 1827 on the principle of Varying Attim 9 and in 1834
and 1835 by memoirs on a General Method** Dynamic*: the
subject of theoretical dynamics bong piopetlj treated as a
branch of pure mathematics. His luoUum on Quatenno**
were published in 1852. Amongst his other papers, I may
specially mention one on the form of the solution of the gumel
elgfiUriir equation of the fifth degree, which confirmed the
conclusion arrived at by Abel that it cannot be expressed in
terms of the more elementary operations and functions : one
on fluctuating functions ; one on the hodograph ; and lastly
one on the numerical solution of differential equations. His
Mlematf* o/Quatemum* were issued in 1866 : of this a compe
tent authority says that the methods of analysis there given
shew as great an advance over those of analytical geometry, as
the lillm ihneiMl over those of Euclidean geometry. In more
| recent times the subject has been further developed by Tait
(aee below, p. 497).
Hamilton was painfully fastidious on what he published,
476 GRASSMANN. BE MORGAN.
and he left an immense collection of manuscript books which
are now in the library of Trinity College, Dublin, and some of
which it is to be hoped will be ultimately printed.
Grassmann. The idea of non-commutative algebras and
of quaternions seems to have occurred to Grassmann at about
the same time as to Hamilton. Hermann Gunther Grast-
mann, was born in Stettin on April 15, 1809, and died there
in 1877. He was professor at the gymnasium at Stettin.
His researches on non-commutative algebras are contained in
his Ausdehnungslehre, first published in 1844 and enlarged in
1862. The scientific treatment of the fundamental principles
of algebra initiated by Hamilton and Grassman, was con
tinued by De Morgan and Boole, and subsequently was further
developed by H. Hankel in his work on complexes, 1867,
and by G. Cantor in his memoirs on the theory of irrationals,
1871 ; the discussion is however so technical that I am unable
to do more than allude to it. Grassmann also investigated
the properties of homaloidal hyper- space.
De Morgan*. Augustus De Morgan, born in Madura
(Madras) in June, 1806 and died in London on March 18,
1871, was educated at Trinity College, Cambridge, but in
the then state of the law was (as a Unitarian) ineligible to
a fellowship. In 1828 he became professor at the newly-
established university of London, which is the same institution
as that now known as University College. There (except for
five years from 1831 to 1835) he taught continuously till 1867,
and through his works and pupils exercised a wide influence
on English mathematicians of the present day. The London
Mathematical Society was largely his creation, and he took a
prominent part in the proceedings of the Royal Astronomical
Society.
He was perhaps more deeply read in the philosophy and
history of mathematics than any of his contemporaries, but the
results are given in scattered articles which well deserve col-
* His life was written by his widow, S. E. De Morgan, London, 1882.
HIGHER ALGEBRA. 477
lection and republication. A list of these is given in his life,
and I have made considerable use of some of them in this book.
The best known of his works are the memoirs on the founda
tion of algebra, Cambridge Philosophical Transactions, vols. vin.
and ix.; his treatise on the differential calculus published in
1842, a work of great ability and noticeable for the rigorous
treatment of infinite series ; and his articles on the calculus
of functions and on the theory of probabilities in the Encyclo
paedia Metropolitana. The article on the calculus of functions
contains an investigation of the principles of symbolic reason
ing, but the applications deal with the solution of functional
equations rather than with the general theory of functions :
the article on the theory of probabilities gives a clear analysis
of the mathematics of the subject to the time at which it was
written.
Besides those above named, I may mention the following
who have written on the subjects of Higher Algebra, the Theory
of Forms, and the Theory of Equations.
Josef Ludwig Raabe who in 1832 discussed tests for the
convergency of series ; a subject also discussed later by Joseph
Louis Francois Bertrand, secretary of the French Academy,
born in Paris in 1822 (see below, pp. 482, 488), Rummer
(see above, p. 461), Ulisse Dini of Pisa, and A. Pringsheim
of Munich ; on the researches of the above writers see the
Bulletin of the New York Mathematical Society, October,
1892, pp. 110.
George Boole, born at Lincoln on Nov. 2, 1815, and died at
Cork on Dec. 8, 1864, who invented a system of non-commuta
tive algebra, and from whose memoirs on linear transformations
part of the theory of covariants has developed.
Evariste Galois, one of the most original and powerful
mathematicians of this century, born at Paris on Oct. 26,
1811, and killed in a duel on May 30, 1832, at the early
age of 20, whose writings are mainly concerned with the
theory of equations and substitution groups : on his in-
478 HIGHER ALGEBRA.
vestigations, see Liouville s Journal for 1846, vol. xi.,
pp. 381 444 ; and the American Journal of Mathematics for
1891, vol. xiii., pp. 109142.
Carl Wilhelm Borchardt, professor in Berlin, born there
on Feb. 22, 1817, and died there in 1880, who in particular
discussed generating functions in the theory of equations, and
arithmetic-geometric means: a collected edition of his works,
edited by G. Hettner, was issued at Berlin in 1888.
Cayley (see pp. 462, 469, 481), whose ten classical memoirs
on quantics (binary and ternary forms) and researches on non-
commutative algebras, especially on matrices, will be found in
the collected edition of his works.
Sylvester (see pp. 461, 462, 482), from among whose
numerous memoirs I may in particular single out those
on canonical forms, on the theory of contravariants, reci-
procants (i.e., differential invariants), on the theory of equa
tions, and that on Newton s rule; to which I may add that
he has created the language and notation of considerable parts
of the subjects on which he has written.
Camille Jordan, who has written on the theory of substi
tutions in general and with special applications to differential
equations.
Sir George Gabriel Stokes, Lucasian professor in the uni
versity of Cambridge, born near Sligo on Aug. 13, 1819, who
has written on the critical values of the sums of periodic series,
and on the summation of series (Cambridge Philosophical
Transactions, 1847, vol. viii., pp. 533 583) ; see also below,
pp. 492, 496.
Eugen Netto, of Strassburg, who has written on substitutions.
Hermite (see above, pp. 462, 469, 470, 471), who has in
particular discussed the theory of associated covariants in binary
quantics, the theory of ternary quantics, and who has applied
elliptic functions to the solution of the quintic equation.
Enrico Betti of Pisa who died in 1892, and Brioschi (see
above, p. 469), both of whom discussed binary quantics.
Siegfried Heinrich Aronhold, born at Angerburg on July 16,
HIGHER ALGEBRA. 479
1819, who developed symbolic methods, especially in connection
with ternary quantics ; this was done concurrently with but
independently of Cayley s work on the same subject.
Paid Gordan, professor at Erlangen, who has discussed
the theory of forms, and shewn that there are only a finite
number of concomitants of quantics : an edition of his work on
invariants (determinants and binary forms) edited by G. Ker-
schensteiner was issued at Leipzig in three volumes 1885, 1887,
1893.
Rudolph Frederick Alfred Clebsch, born at Konigsberg in
1833, died at Gottingen, where he was professor, in 1872,
who also independently investigated the theory of binary forms
in some papers collected and published in 1871 : an account of
his life and works is printed in the MatJiematische Annalen,
1873, vol. vi., pp. 197202, and 1874, vol. VIL, pp. 155:
see also below, pp. 481, 493.
Macmahon (see above, p. 462), who has written on the
connection of symmetric functions, the derivation of invariants
and covariants from elementary algebra, and the concomitants
of binary forms.
Sophus Lie, professor at Leipzig (see below, p. 482), who
has written on groups of continuous substitutions, differential
invariants, and complexes of lines.
Klein (see above, pp. 469 70, 470 1), who has investigated
the problem of discontinuous substitutions and polyhedral groups.
And lastly Andrew Russell Forsyth, fellow and lecturer of
Trinity College, Cambridge, born at Glasgow on June 18, 1858,
who has developed the theory of invariants of differential
equations, ternariants, and quaternariants.
No account of contemporary writings on this subject would
be complete without a reference to the admirable text-books
produced by George Salmon, provost of Trinity College,
Dublin, born in 1819, in his Higher Algebra, and by Joseph
Alfred Serret, professor at the Sorbonne, born at Paris on
Aug. 30, 1819, and died in 1885, in his Cours cTAlgebre
superieure, in which the chief discoveries of their respective
480 ANALYTICAL GEOMETRY.
authors are embodied. An admirable historical summary of
the theory of the complex variable is given in the Vorlesungen
uber die complexen Zahlen, Leipzig, 1867 by H. Hankel,
professor in Tubingen, born at Halle in 1839, and died at
Schramberg in 1873.
Before mentioning the creators of modern synthetic
geometry it will be convenient to call attention to two other
divisions of pure mathematics which have been greatly
developed in recent years, but any sketch of the results
arrived at or of the methods by which they have been attained
would be so closely connected with the work of living mathe
maticians that I shall do little more than mention the names
of the subjects.
Analytical Geometry has been studied by a host of modern
writers, but I do not propose to describe their investigations,
and I shall content myself by merely mentioning the names of
the following mathematicians.
James Booth, born in the county Leitrim on Aug. 25, 1806
and died in Buckinghamshire on April 15, 1878 was one of
the earliest writers in this century to devote himself to the
development of analytical geometry ; his chief results are
embodied in his work entitled A Treatise on some new Geo
metrical Methods.
The researches of James MacCullagh, professor in Dublin,
born near Strabane in 1809 and died in Dublin on Oct. 24,
1846, which include some valuable discoveries on the theory of
quadrics, will be found in his collected works edited by Jellett
and Haughton, Dublin, 1880 : see also below, p. 496.
Julius PI ticker, professor (after 1836) in Bonn, born at
Elberfeld on July 16, 1801, and died at Bonn on May 22,
1868, devoted himself chiefly to the study of algebraic curves,
of a geometry in which the line is the element in space, and
the theory of congrueness and complexes; his equations con
necting the singularities of curves are well known : in 1847 he
exchanged his chair for one of physics, and his subsequent
ANALYSIS, 481
Researches were on spectra and magnetism. An account of
his works was published by Clebsch, Gesellschaft der Wissen-
schaften, Gottingen, 1872, vol. xvi.
The majority of the memoirs on analytical geometry by
Cayley (see pp. 462, 469, 478) and by Henry Smith (see
above, p. 460) deal with the theory of curves and surfaces ; the
most remarkable" of those of Ludwig Otto Hesse, born at
Konigsberg on April 22, 1811, and died at Munich, where he
was professor, in 1874, are on the plane geometry of curves
(see the notice of these by F. C. Klein); of those of Jean Gaston
Darboux, professor in Paris, born at Mines in 1842, on the
geometry of surfaces ; and of those of Halphen (see pp. 469,
482) on the singularities of surfaces and on tortuous curves.
The singularities of curves and surfaces have also been con
sidered by Hieronymus Georg Zeuthen, professor at Copenhagen,
born in 1839, and by Hermann Cdsar Hannibal Schubert, pro
fessor at Hamburg, born at Potsdam in 1848: the lectures of
the latter have been published by F. Lindemann, two volumes,
Leipzig, 1875, 1891. Nother (see above, p. 470) has discussed
the theory of tortuous curves. And Clebsch (see pp. 479, 493)
has applied Abel s theorem to geometry.
Among more recent text-books are Clebsch s Vorlesuny //
iiSer Geometric, edited by F. Lindemann ; and Salmon s Conic
Sections, Geometry of Three Dimensions, and Higher Pln<
Curves; in which the chief discoveries of these writers ait
embodied.
Finally I may allude to the extension of the subject-matter
of analytical geometry by the introduction of the ideas of
| space of n dimensions in the writings of Grassmann (see above,
p. 476) in 1844 and 186-, liif.inann (see above, p. ;
Cayley (see above, pp. 462, 469, 478, 481), and others.
Among those who have extended the range of
(including the calculus and differential equations) or whom
it is difficult to place in any of the preceding categories
are the following, whom I place in alphabetical order.
B. 31
482 MATHEMATICS OF RECENT
Appell (see above, p. 471). Bertrand (see pp. 450, 477, 488).
Boole (see above, p. 477). Cauchy (see above, p. 473).
Darboux (see above, p. 481). Forsyth (see above, p. 479),
who has written on Pfaff s problem, and is also the author
of the standard English treatise on differential equations.
Frobenius (see above, p. 470). Lazarus Fuchs, professor at
Berlin, born in Prussia in 1833. Halphen (see above, pp. 469,
481). Jacobi (see above, p. 464). Jordan (see above, p. 478).
Konigsberger (see above, p. 469). Sophie Koivalevski, pro
fessor at Stockholm, born on Dec. 27, 1853, and died Feb.
18, 1891 ; see the Bulletin des sciences mathematiques, vol. xv.,
pp. 212 220. Lie (see above, p. 479). Poincare (see pp.
471, 492). Riemann (see above, p. 465) who wrote on the
theory of partial differential equations. Schwarz (see above,
p. 470). Sylvester (see above, pp. 461, 462, 478). And
Weierstrass (see above, pp. 468 9, 471) who has developed
the calculus of variations.
The writers I have mentioned above mostly concerned
themselves with analysis. I will next describe some of the
more important works produced in this century on synthetic
geometry*.
Modern synthetic geometry may be said to have had its
origin in the works of Monge in 1800, Carnot in 1803, and
Poncelet in 1822, but these only dimly foreshadowed the great
extension it was to receive in Germany, of which Steiner and.
von Staudt are perhaps the best known exponents.
Steinerf. Jacob Steiner, "the greatest geometrician since
* The Aperpu historique sur V origins et le developpement des methodes
en geometric by M. Chasles, Paris, second edition, 1875, and the Die
synthetische Geometric im Alterthum und in der Neuzeit by Th. Keye,
Strassburg, 1886, contain interesting summaries of the history of geometry,
but Chasles s work is written from an exclusively French point of
view.
f Steiner s collected works, edited by Weierstrass, were issued in two
volumes, Berlin, 188182. A sketch of his life is contained in the Erin-
nerung an Steiner by C. F. Geiser, Schaffhausen, 1874.
SYNTHETIC GEOMETRY. 483
the time of Apollonius," was born at Utzensdorf on March 18,
1796, and died at Bern on April 1, 1863. His father was a
peasant, and the boy had no opportunity to learn reading and
writing till the age of fourteen. He subsequently went to
Heidelberg and thence to Berlin, supporting himself by giving
lessons. His Systematise?!* Entivickelunyen was published in
1832, and at once made his reputation: it contains a full
discussion of the principle of duality, and of the projective
and homographic relations of rows, pencils, <fec., based on
metrical properties. By the influence of Crelle, Jacobi, and
the von Humboldts, who were impressed by the power of this
work, a chair of geometry was created for Steiner at Berlin,
and he continued to occupy it till his death. The most im
portant of his other researches are contained in papers which
appeared originally in Crelle s Journal, and are embodied
in his tiynthetische Geometric, vol. I. edited by C. F. Geiser,
vol. ii. by H. Schroeter : these relate chiefly to properties of
algebraic curves and surfaces, pedals and roulettes, and maxima
and minima ; the discussion is purely geometrical. Steiner s
works may be considered as the classical authority on recent
synthetic geometry.
Von Staudt. A system of pure geometry, quite distinct
from that expounded by Steiner, was proposed by Karl
Georg Christian von Staudt, born at Rothenburg on Jan.
24, 1798, and died in 1867, who held the chair of mathe
matics at Erlangen. In his Geometric der Laye, published in
1847, he constructed a system of geometry built up without
any reference to number or magnitude, but, in spite of its
abstract form, he succeeded by means of it alone in establishing
the non-metrical projective properties of figures, discussed
imaginary points, lines, and planes, and even obtained a geo
metrical definition of a number : these views were further
elaborated in his Beitrdge zur Geometric der Lage, 1856 1860.
This geometry is curious and brilliant, and has been used by
Culmann as the basis of his graphical statics.
Among other works on pure geometry I may refer to the
312
484 GRAPHICS.
Introduzione ad una teoria yeometrica delle curve piane, 1862,
and its continuation Preliminari di una teoria geometrica delle
superficie by Luigi Cremona, of the Polytechnic School at Rome.
As usual text-books I may mention M. Chasles s Traite de
geometrie superieure, 1852 ; J. Steiner s Vorlesungen iiber syn-
thetische Geometric, 1867; L. Cremona s Elements de geometrie
protective, translated into English by C. Leudesdorf, Oxford,
1885 ; and Th. Reye s Geometrie der Lage 3 volumes, third
edition.
I shall conclude the chapter with a few notes more
or less discursive on branches of mathematics of a less
abstract character and concerned with problems that occur in
nature.
Closely connected with the subject of modern geometry is
the science of graphics in which rules are laid down for solving
various problems by the aid of the drawing-board: the modes
of calculation which are permissible are considered in modern
protective geometry. This method of attacking questions has
been hitherto applied chiefly to problems in mechanics,
elasticity, and electricity; it is especially useful in engineering,
and in that subject an average draughtsman ought to be able
to obtain approximate solutions of most of the equations,
differential or otherwise, with which he is likely to be
concerned, which will not involve errors greater than would
have to be allowed for in any case in consequence of our im
perfect knowledge of the structure of the materials employed.
The theory may be said to have originated with Poncelet s
work, but I believe that it is only within the last twenty
years that systematic expositions of it have been published.
Among the best known of such works I may mention the
Graphische Statik, by C. Culmann, Zurich, 1875, recently edited
by W. Ritter; the Lezioni di statica grafica, by A. Favaro,
Padua, 1877 (French translation annotated by P. Terrier in
2 volumes, 1879 85) ; the Calcolo grafico, by L. Cremona,
Milan, 1879 (English translation by T. H. Beare, Oxford,
CULMANN. CLIFFORD. 485
1889), which is largely founded on Mobius s work; La statique
graphique, by M. Le vy, Paris, 4 volumes, 1886 88; and La
statica grafica, by C. Sairotti, Milan, 1888.
The general character of these books will be sufficiently
illustrated by the following note on the contents of Culmann s
work. Culmann commences with a description of the geo
metrical representation of the four fundamental processes of
addition, subtraction, multiplication, and division; and pro
ceeds to evolution and involution, the latter being effected by
the use of equiangular spiral. He next shews how the quan
tities considered such as volumes, moments, and moments of
inertia may be represented by straight lines ; thence deduces
the laws for combining forces, couples, &c. ; and then explains
the construction and use of the ellipse and ellipsoid of inertia,
the neutral axis, and the kern ; the remaining and larger part
of the book is devoted to shewing how geometrical drawings,
made on these principles, give the solutions of many practical
problems connected with arches, bridges, frameworks, earth
pressure on walls and tunnels, &c.
The subject has been treated during the last twenty years
by numerous writers especially in Italy and Germany, and
applied to a large number of problems. But as I stated at
the beginning of this chapter that I should as far as possible
avoid discussion of the works of living authors I content
o
myself with a bare mention of the subject.
Clifford*. I may however add here a brief note on Clifford,
who was one of the earliest English mathematicians of the latter
half of this century to advocate the use of graphical and geo
metrical methods in preference to analysis. William Kitigdon
Cliford, born at Exeter on May 4, 1845, and died at Madeira
on March 3, 1879, was educated at Trinity College, Cambridge,
of which society he was a fellow. In 1871 he was appointed
professor of applied mathematics at University College, London,
* For further details of Clifford s life and work see the authorities
quoted in the article on him in the Dictionary of National Biography,
vol. xi.
486 THEORETICAL MECHANICS.
a post which he retained till his death. His remarkable felicity
of illustration and power of seizing analogies made him one of
the most brilliant expounders of mathematical principles. His
health failed in 1876, when the writer of this book undertook
his work for a few months ; Clifford then went to Algeria and
returned at the end of the year, but only to break down again
in 1878. His most important works are his Theory of
Biquaternions, On the Classification of Loci (unfinished), and
The Theory of Graphs (unfinished) : his Canonical Dissection
of a Riemann s Surface, and the Elements of Dynamic also
contain much interesting matter.
I next turn to the question of mechanics treated analytically.
The knowledge of mathematical mechanics of solids attained
by the great mathematicians of the last century may be said
to be summed up in the admirable Mecanique analytique by
Lagrange and Traite de mecanique by Poisson, and the appli
cation of the results to astronomy is illustrated by Laplace s
Mecanique celeste. These works have been already described.
The mechanics of fluids is more difficult than that of solids
and the theory is less advanced.
Theoretical Statics, especially the theory of the potential
and attractions has received considerable attention from the
mathematicians of this century.
I have already mentioned (see above, p. 412) that the
introduction of the idea of the potential is due to Lagrange,
and it occurs in a memoir of a date as early as 1773. The
idea was at once grasped by Laplace who, in his memoir of
1784, used it freely and to whom the credit of the invention was
formerly, somewhat unjustly, attributed. In the same memoir
Laplace also extended to space of three dimensions the idea of
circular harmonic analysis which had been introduced by
Legendre in 1783.
Green*. George Green was one of the earliest writers of
* A collected edition of Green s works was published at Cambridge
in 1871.
GREEN. MOEBIUS. 487
this century who investigated further the properties of the
potential. Green was born near Nottingham in 1793 in a
humble condition in life, and died at Cambridge in 1841.
Although self-educated he contrived to get access to various
mathematical books, and in 1827 wrote a paper on the po
tential in which the term was first introduced proved its
chief properties, and applied the results to electricity and
magnetism. This contains the important theorem now known
by his name. This remarkable paper was seen by some neigh
bours who were able to appreciate the power shewn in it : it
was published by subscription in 1828, but does not seem to
have attracted much attention at first. Similar results were
independently established, in 1839 by Gauss to whom their
general dissemination was due.
In 1832 and 1833 Green presented papers to the Cam
bridge Philosophical Society on the equilibrium of fluids and
on attractions in space of n dimensions, and in the latter year
his memoir on the motion of a fluid agitated by the vibrations
of a solid ellipsoid was read before the Royal Society of Edin
burgh. In 1833 he entered at Cains College, Cambridge, and
was subsequently elected to a fellowship. He then threw
himself into original work, and produced in 1837 his paper on
the motion of waves in a canal, and on the reflection and
refraction of sound and light. In the latter the geometrical
laws of sound and light are deduced by the principle of energy
from the undulatory theory, the phenomenon of total reflexion
is explained physically, and certain properties of the vibrating
medium are deduced. He also discussed the propagation of
light in any crystalline medium.
Of Gauss s work on attractions I have already spoken (see
above, p. 456). The theory of level surfaces and lines of force
is largely due to Chasles who also determined the attraction of
an ellipsoid at any external point. I ought not to leave
the subject of theoretical statics without mentioning Mobius.
August Ferdinand Mobius, professor at Leipzig, who was born
at Schulpforta on Nov. 17, 1790, and died on Sept. 26, 1 E
488 THEORETICAL DYNAMICS AND ASTRONOMY.
was one of the best known of Gauss s pupils; he published
his Barycentrisches Calcul in 1826 : his collected works were
published at Leipzig in four volumes, 1885 7. Among living
writers I may allude to Sir Robert Stawell Ball, Lowndean
professor in the university of Cambridge, born in Dublin
on July 1, 1840, who issued his Theory of Screws in
1876.
Theoretical Dynamics has been studied by most of the
writers above mentioned. In addition to these I may repeat
that the principle of " Varying Action " was elaborated by Sir
William Hamilton in 1827. and the " Hamiltonian equations"
were given in 1835; and I may call attention to Bertrand s
work on dynamics. The use of generalized coordinates, intro
duced by Lagrange (see above, p. 409), has become the custo
mary means of attacking dynamical (as well as many physical)
problems. The standard English text-book on the dynamics of
rigid bodies is that by Dr Routh.
On the mechanics of fluids, liquids, and gases, apart from
the physical theories on which they rest, I propose to say
nothing, except to refer to the memoirs of Green, Sir George
Stokes, Lord Kelvin (better known as Sir William Thomson),
and von Helmholtz. The fascinating but difficult theory of
vortex rings is due to the two writers last-mentioned. One
problem in it has been also considered by J. J. Thomson, but
it is a subject which is as yet rather beyond our powers of
analysis. The subject of sound may be treated in connection
with hydrodynamics, but on this I would refer the reader who
wishes for further information to the work published at Cam
bridge in 1877 by Lord Rayleigh, recently Cavendish professor
in the university of Cambridge.
Theoretical Astronomy is included in, or at any rate closely
connected with, theoretical dynamics. Among those who in
this century have devoted themselves to the study of theoreti
cal astronomy the name of Gauss is one of the most prominent;
to his work I have already alluded.
BESSEL. LEVERRIER. 489
Bessel*. The best known of Gauss s contemporaries was
Friedrich Wilkelm Bessel, who was born at Minden on
July 22, 1784, and died at Konigsberg on March 17, 1846.
Bessel commenced his life as a clerk on board ship, but in
1806 he became an assistant in the observatory at Lilienthal,
and was thence in 1801 promoted to be director of the new
Prussian observatory at Konigsberg where he continued to
live during the remainder of his life. Bessel introduced into
pure mathematics those functions which are now called by
his name, this was in 1824 though their use is indicated in a
memoir seven years earlier ; but his most notable achievements
were the reduction (given in his Fundamenta Astronomiae,
Konigsberg, 1818) of the Greenwich observations by Bradley
of 3,222 stars, and his determination of the annual parallax
of 61 Cygni. Bradley s observations have been recently
reduced again by Dr A. Auwers of Berlin.
Leverriert. Among the astronomical events of this century
the discovery of the planet Neptune by Leverrier and Adams
is one of the most striking. Urbain Jean Joseph Leverrier,
the son of a petty Government employe in Normandy, was
born at St L6 on March 11, 1811, and died at Paris on
Sept. 23, 1877. He was educated at the Polytechnic school,
and in 1837 was appointed as lecturer on astronomy there.
His earliest researches in astronomy were communicated to the
Academy in 1839 : in these he calculated within much narrower
limits than Laplace had done the extent within which the incli
nations and eccentricities of the planetary orbits vary. The
independent discovery in 1846 by Leverrier and Adams of the
planet Neptune by means of the disturbance it produced on
* See pp. 36 53 of A. M. Clerke s History of Astronomy, Edinburgh,
1887. Bessel s collected works and correspondence have been edited by
R. Engelmann and published in four volumes at Leipzig, 1875 82.
t For further details of his life see Bertrand s eloye in vol. XLI. of the
M&moires de Vacademie ; and for an account of his work see Adams s
address in vol. xxxvi. of the Monthly Notices of the Royal Astronomical
Society.
490 ADAMS.
the orbit of Uranus attracted general attention to physical
astronomy, and strengthened the opinion as to the universality
of gravity. In 1855 Leverrier succeeded Arago as director
of the Paris observatory, and reorganized it in accordance
with the requirements of modern astronomy. He now set
himself the task of discussing the theoretical investigations
of the planetary motions and of revising all tables which
involved them. He lived just long enough to sign the last
proof-sheet of this work.
Adams*. The co-discoverer of Neptune was John Couch
Adams, who was born in Cornwall on June 5, 1819, educated
at St. John s College, Cambridge, subsequently appointed
Lowndean professor in the University, and director of the
Observatory, and who died at Cambridge on Jan. 21, 1892.
There are three important problems which are specially
associated with the name of Adams. The first of these is his
discovery of the planet Neptune from the perturbations it
produced on the orbit of Uranus : in point of time this was
slightly earlier than Leverrier s investigation.
The second memoir to which I referred was on the secular
acceleration of the moon s mean motion (Philosophical Trans
actions, 1855, vol. CXLIII., p. 377). Laplace had calculated
this on the hypothesis that it was caused by the eccentricity of
the earth s orbit, and had obtained a result which agreed
substantially with the value deduced from a comparison of the
records of ancient and modern eclipses. Adams shewed that
certain terms in an expression had been neglected, and that
if they were taken into account the result was only about
one-half that found by Laplace. The correctness of the
calculations of Adams was denied by Plana, Pontecoulant, and
other continental astronomers, but Delaunay in France and
Cayley in England verified the work.
The third investigation connected with the name of
* A sketch of his life was given in Nature, Oct. 14, 1866, and in The
Observatory, April, 1892, pp. 173189 : his collected works will be issued
shortly at Cambridge.
THEORETICAL ASTRONOMY. 491
Adams, is his determination of the orbit of the Leonids or
shooting stars which were especially conspicuous in November,
1866, and whose period is about thirty-three years. Newton,
of Yale, had shewn that there were only five possible orbits.
Adams calculated the disturbance which would be produced by
the planets on the motion of the node of the orbit of a swarm
of meteors in each of these cases, and found that this dis
turbance agreed with observation for one of the possible orbits,
but for none of the others. Hence the orbit was known
(Monthly Notices of the Royal Astronomical Society, April,
1867, p. 247).
Other well-known astronomers of this century are Giovanni
Antonio Ainadeo Plana, born at Voghera on Nov. 8, 1781, and
died at Turin on Jan. 20, 1864, whose work on the motion
of the moon was published in 1832.
Philip Gustave Doulcet, Count Pontecoulant, born in 1795
and died at Pontecoulant on July 21, 1871.
Charles Eugene Delaunay, born at Lusigny on April 9,
1816, and drowned off Cherbourg on Aug. 3, 1872, whose
work on the lunar theory indicates the best method yet sug
gested for the analytical investigations of the whole problem,
and whose (incomplete) lunar tables are among the astronomical
achievements of this century.
And Peter Andrew Hansen, born in Schleswig on Dec. 8,
O
1795, and died at Gotha where he was head of the observatory
on March 28, 1874, who compiled the lunar tables published
in London in 1857, and elaborated the most delicate methods
yet known for the determination of lunar and planetary pertur
bations; for an account of his numerous memoirs see the
Transactions of the Royal Society of London for 1876 77.
Among living mathematicians 1 may mention the following
names.
Felix Tisserand of Paris, born in 1845, whose Mecanique
celeste forms a worthy pendant to Laplace s work of the same
title.
492 MATHEMATICAL ASTRONOMY AND PHYSICS.
George William Hill, born in New York in 1838, and until
recently on the staff of the American Ephemeris, who (in 1884)
determined the inequalities of the moon s motion due to the
non-spherical figure of the earth an investigation which
completed Delaunay s lunar theory : Hill has also dealt with
the secular motion of the moon s perigee and the motion of a
planet s perigee under certain conditions ; and has written on
the analytical theory of the motion of Jupiter and Saturn,
with a view to the preparation of tables of their positions
at any given time.
Simon Newcomb, born in Nova Scotia on March 12, 1835,
superintendent of the American Ephemeris, who re-examined
the Greenwich observations from the earliest times, applied
the results to the lunar theory, and revised Hansen s tables.
George Howard Darwin, of Trinity College, Cambridge,
born in Kent in 1845, and now Plumian professor in the
university of Cambridge, who has written on the effect of
tides on viscous spheroids, the development of planetary
systems by means of tidal friction, the mechanics of meteoric
swarms, &c.
Perhaps also I may here mention Poincare (see above,
pp. 471, 482), who has discussed the difficult problem of
three bodies, and the form assumed by a mass of fluid under
its own attraction.
Within the last half century the results of spectrum
analysis have been applied to determine the constitution, and
directions of motions of the heavenly bodies to and from the
earth. The early history of spectrum analysis will be always
associated with the names of Gustav Robert Kirclihoff (see
below, p. 495), of A. J. Angstrom, of Upsala, and of Sir George
Stokes of Cambridge (see pp. 478, 496), but it pertains to
optics rather than to astronomy.
Within the last few years the range of astronomy has
been still further extended by the art of photography. To
what new developments this may lead it is as yet impossible
to say.
MATHEMATICAL PHYSICS. 493
Mathematical Physics. An account of the history of mathe
matics in this century would not be other than misleading
if there were no reference to the application of it to numerous
problems in heat, elasticity, light, electricity, and other physical
subjects. The history of mathematical physics is however so
extensive that I could not pretend to do it justice even were its
consideration properly included in a history of mathematics :
moreover, it is so closely 
