Front matter
A SHOET ACCOUNT
OF THE
HISTORY OF MATHEMATICS
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 . 8
Knowledge of the science of geometry possessed by the Egyptians 5
Note on ignorance of mathematics shewn by the Chinese . . 9 jFtrst ^ertoK. Jlatfjematics untrer ffireefe Influence.
This period begins with the teaching of Thales, circ. 600 B.C., and ends with the capture of Alexandria by the Mohammedans in or about 641 a.d. The characteristic feature of this period is the development of geometry.
Chapter II. The Ionian and Pythagorean Schools. Circ. 600 b.c.-400 b.c. The Ionian School .......... 14
His geometrical discoveries ...... 15
His astronomical teaching ...... 17
Anaximander. Anaximenes. Mamercus. Mandryatus . . 18
h
The Pythagorean School
Pythagoras, 569-500 b.c
The Pythagorean teaching
The Pythagorean geometry
The Pythagorean theory of numbers Epicharmus. Hippasus. Philolaus. Archippus. Archytas, circ. 400 B.C.
His solution of the duplication of a cube Theodorus. Timaeus. Bryso Other Greek Mathematical Schools in the Fifth Cen
Oenopides of Chios
Zeno of Elea. Democritus of Abdera .
Lysis ury B.C.
PAGE
Chaptek III. The Schools of Athens and Cyzicus. CiEC. 420-300 B.C.
Mathematical teachers at Athens prior to 420 b.c. .
Anaxagoras. The Sophists. Hippias (The quadratrix)
Antipho
Three problems in which these schools were specially interested Hippocrates of Chios, circ. 420 B.c
Letters used to describe geometrical diagrams . __^.:^^==^ Introduction in geometry of the method of reduction
The quadrature of certain lunes ....
The problem of the duplication of the cube Plato, 429-348 b.c
Introduction in geometry of the method of analysis .
Theorem on the duplication of the cube .
EuDOXus, 408-355 b.c
Theorems on the golden section ....
Introduction of the method of exhaustions
Pupils of Plato and Eudoxus .
Menaechmus, circ. 340 b.c
Discussion of the conic sections ....
His two solutions of the duplication of the cube
Aristaeus. Theaetetus
Aristotle, 384-322 B.C. .
Questions on mechanics. Letters vised to indicate magnitudes
Chapter IV. The First Alexandrian School. CiRC. 300-30 B.C. PAGE
Foundation of Alexandria .51
The Third Century before Christ 52
Euclid's Elements ........ 53
The Elements as a text-book of geometry .... 54
The Elements as a text-book of the theory of numbers . 57
Euclid's other works 60"
Aristarchus, circ. 310-250 b.c 62
Alethod of determining the distance of the sun . . 62
Conon. Dositheus. Zeuxippus. Nicoteles ..... 64
Akchimedes, 287-212 b.c 64
His works on plane geometry ...... 67
His works on geometry of three dimensions ... 70
His two papers on arithmetic, and the "cattle problem " . 71 His works on the statics of solids and fluids . . .73
His astronomy ......... 76
The principles of geometry assumed by Archimedes . . 76
Apollonius, circ. 260-200 b.c 77
His conic sections ........ 77
His other works ........ 80
His solution of the duplication of the cube ... 81
Contrast between his geometry and that of Archimedes . 82
Eratosthenes, 275-194 b.c 83
The Sieve of Eratosthenes 83
The Second Century before Christ 84
Hypsicles (Euclid, book xiv). Nicomedes. Diodes ... 85
Perseus. Zenodorus 86
HiPPARCHUS, circ. 130 b.c 86
Foundation of scientific astronomy ..... 87
Foundation of trigonometry ...... 88
Hero of Alexandria, circ. 125 b.c 88
Foundation of scientific engineering and of land-surveying 88
Area of a triangle determined in terms of its sides . . 89
Features of Hero's works 91 xii TABLE OF CONTENTS
The First Century lefore Christ Theodosius .....
Dionysodorus
End of the First Alexandrian School Egypt constituted a Roman province
PAGE
91 91 92 92 92
Chapter V. The Second Alexandrian School. The First Century after Christ ....... 94
Serenus. Menelaus 94
Mcomachus 94
Introduction of the arithmetic current in medieval Europe 95
The Second Century after Christ .95
Theon of Smyrna. Thymaridas ....... 95
Ptolemy, died in 168 96
The Almagest 96
Ptolemy's astronomy ....... 96
Ptolemy's geometry ........ 98
The Third Century after Christ 99
Pappus, circ. 280 99
The livva'ywyr], a synopsis of Greek mathematics . . 99
The Fourth Century after Christ 101
v,r3»»jyietrodorus. Elementary problems in arithmetic and algebra . 102
^^ Three stages in the development of algebra 103
DiOPHANTUP, circ. 320 (?) 103
Introduction of syncopated algebra in his Arithinetic . 104
The notation, methods, and subject-matter of the work . 104
His Porisms 110
Subsequent neglect of his discoveries .... 110
lamblichus 110
Theon of Alexandria. Hypatia Ill
Hostility of the Eastern Church to Greek science . . . .111 The Athenian School {in the Fifth Century) . . . . .111
Proclus, 412-485. Damascius. Eutocius 112
TABLE OF CONTENTS xiii
PAQE
Roman Mathematics . . . . . . . . .113
Nature and extent of the mathematics read at Rome . . , 113
Contrast between the conditions at Rome and at Alexandria . . 114
End of the Secoiid Alexandrian School . . . . . ,115
The capture of Alexandria, and end of the Alexandrian Schools . 115
Chaptee VL The Byzantine School. 641-1453.
Preservation of works of the great Greek Mathematicians . .116 Hero of Constantinople. Psellus. Planudes. Barlaam. Argyrus 117 Nicholas Rhabdas. Pachymeres. Moschopulus (Magic Squares) . 118 Capture of Constantinople, and dispersal of Greek Mathematicians 120
Chaptek VTI. Systems op Numeration and Primitive Arithmetic.
Methods of counting and indicating numbers among primitive races 121 Use of the abacus or swan- pan for practical calculation . . . 123
126 126 127 127 128
Methods of representing numbers in writing .
The Roman and Attic symbols for numbers .
The Alexandrian (or later Greek) symbols for numbers
Greek arithmetic .......
Adoption of the Arabic system of notation among civilized races xiv TABLE OF CONTENTS
This period begins about the sixth century, and may be said to end with the invention of analytical geometry and of the infinitesimal calculus. The characteristic feature of this period is the creation or development of modem arithmetic, algebra, and trigonometry.
Chapter VIII. The Rise of Learning in Western Europe. CiRC. 600-1200.
PAOE
JEdUfCation in the Sixth, Seventh, and Eighth Centuries . . .131
The Monastic Schools 131
Boethius, circ. 475-526 132
Medieval text-books in geometry and arithmetic . . 133 Cassiodorus, 490-566. Isidorus of Seville, 570-636 . . .133
The Cathedral and Conventual Schools 134
The Schools of Charles the Great . . . . . . .134
Education in the Ninth and Tenth Centuries .... 136
Gerbert (Sylvester IL), died in 1003 136
Bernelinus 139
The Early Medieval Universities 139
Rise during the twelfth century of the earliest universities . .139
Development of the medieval universities 140
Outline of the course of studies in a medieval university . . 141
Chapter IX. The Mathematics of the Arabs.
Extent of Mathematics obtained from Greek Sources . . .144 The College of Scribes 145
TABLE OF CONTENTS xv
Extent of 3Iathematics obtained from the {Aryan) Hindoos Akya-Bhata, circ. 530
His algebra and trigonometry (in his Aryahhathiya) Brahmagupta, circ. 640
His algebra and geometry (in his Siddhanta) . Bhaskara, circ. 1140
The Lilavati or arithmetic ; decimal numeration nsed
The Bija Ganita or algebra .....
Develojyment of Mathematics in Arabia ..... Alkarismi or Al-Khwarizmi, circ. 830 ....
His Al-gebr ice' I mukabala .....
His solution of a quadratic equation
Introduction of Arabic or Indian system of numeration Tabit ibn Korra, 836-901 ; solution of a cubic equation Alkayami. Alkarki. Development of algebra Albategni. Albnzjani. Development of trigonometry . Alhazen. Abd-al-gehl. Development of geometry .^-^y Characteristics of the Arabian School .....
PAGE
Chapter X. Introduction of Arabian Works into Europe. Circ. 1150-1450.
The Eleventh Century 165
Moorish Teachers. Geber ibn Aphla. Arzachel . . . .165
The Twelfth Century 165
Adelhard of Bath .165
Ben-Ezra. Gerard. John Hispalensis . . . .. • .166 The Thirteenth Century . . . . . . . . .167
Leonardo of Pisa, circ. 1175-1230 167
The Liber Abaci, 1202 167
The introduction of the Arabic numerals into commerce . 168 The introduction of the Arabic numerals into science . 168
The mathematical tournament 169 xvi TABLE OF CONTENTS
PAGE
JoRDANUS, circ. 1220 171
His Be Numeris Datis ; syncopated algebra . . . 172
Holywood 174
Campanus 177
The Fourteenth Century . . . . . . . .177
Bradwardine 177
Oresmus . . . . 178
The reform of the university curriculum ..... 179
The Fifteenth Century 180
Beldomandi 180
Chapter XI. The Development of Arithmetic. Circ. 1300-1637.
The Boethian arithmetic . . 182
Algorism or modern arithmetic ....... 183
The Arabic (or Indian) symbols : history of 184
Introduction into Europe by science, commerce, and calendars . 186
Improvements introduced in algoristic arithmetic . . .188
(i) Simplification of the fundamental processes . . .188
(ii) Introduction of signs for addition and subtraction . .194
(iii) Invention of logarithms, 1614 195
(iv) Use of decimals, 1619 197
Chapter XII. The Mathematics of the Renaissance. Circ. 1450-1637.
Authorities .
Effect of invention of printing. The renaissance . Develofment of Syncopated Algebra and Trigonometry
Regiomontanus, 1436-1476
His De Triangulis (printed in 1496) .
Purbach, 1423-1461. Cusa, 1401-1464. Chuquet, circ. Introduction and origin of symbols + and- .
TABLE OF CONTENTS xvii
Pacioli or Lucas di Burgo, circ. 1500 ..... His arithmetic and geometry, 1494 ....
Record, 1510-1588 ; introduction of symbol for equality
His Arithmetica Integra, 1544 .....
His solution of a cubic equation, 1535
His arithmetic, 1556-1560
His Ars Magna, 1545 ; the third work printed on algebra His solution of a cubic equation ....
Ferrari, 1522-1565 ; solution of a biquadratic equation .
Rheticus, 1514-1576. Maurolycus. Borrel. Xylander
Commandiho. Peletier. Romanus. Pitiscus. Ramus, 1515-157
Bombelli, circ. 1570
Development of SymloUc A Igebra
The In Artem ; introduction of symbolic algebra, 1591 Vieta's other works
Girard, 1590-1633 ; development of trigonometry and algebra
Napier, 1550-1617 ; introduction of logarithms, 1614 .
Briggs, 1556-1631 ; calculations of tables of logarithms .
Harriot, 1560-1621 ; development of analysis in algebra
The Origin of the more Common Symbols in Algebra
PAGE
Chaptee XIII. The Close of the Rejjaissance. Circ. 1586-1637.
Development of Mechanics and Experimental Methods . . . 244
Commencement of the modern treatment of statics, 1586 . 245
XVlll
Commencement of the science of dynamics Galileo's astronomy ....
Wright, 1560-1615 ; construction of maps
Revival of Interest in Pure Geometry
His Paralipomena, 1604 ; principle of continuity His Stereometria, 1615 ; use of infinitesimals . Kepler's laws of planetary motion, 1609 and 1619
His Brouillon project ; use of projective geometry
Mathematical Knowledge at the Close of the Renaissance .
PAGE
This period begins with the invention of analytical geometry and the infinitesimal calculus. The mathematics is far more complex than that produced in either of the preceding periods : but it may be generally described as characterized by the development of analysis, and its application to the phenomena of nature.
Chapter XIV. The History of Modern Mathematics.
Treatment of the subject 263
Invention of analytical geometry and the method of indivisibles . 264
Invention of the calculus . 265
Development of mechanics 265
Application of mathematics to physics 266
Recent development of pure mathematics 267
TABLE OF CONTENTS xix
Chapter XV. History of Mathematics from Descartes TO HUYGENS. CiRC. 1635-1675.
PAGE
His views on philosophy . . . . . . ,271
His invention of analytical geometry, 1637 . . . 272
His algebra, optics, and theory of vortices . . . 276
The method of indivisibles ...... 279
His geometrical conies 284
The arithmetical triangle 284
Foundation of the theory of probabilities, 1654 . . 285
His discussion of the cycloid 287
The Arithmetica Infinitorum, 1656 ..... 289
Law of indices in algebra . 289
Use of series in quadratures 290
Earliest rectification of curves, 1657 291
Wallis's algebra 292
His investigations on tlie theory of numbers . . . 294
His use in geometry of analysis and of infinitesimals . 298
Foundation of the theory of probabilities, 1654 . . 300
The Horoloyium Oscillatorium, 1673 .... 302
The undulatory theory of light ..... 303
Other Mathematicians of this Time 305
Bachet 305
Marsenne ; theorem on primes and perfect numbers . . . 306
Roberval. Van Schooten. Saint- Vincent 307
Torricelli. Hudde. Frenicle 308
De Laloubere. Mercator. Barrow ; the difi"erential triangle . 309
Brouncker ; continued fractions ....... 312
James Gregory ; distinction between convergent and divergent scries 313
Sir Christopher Wren . . .314
Hooke. Collins 315
Pell. Sluze. Viviani 316
Tschirnhausen. De la Hire. Roemer. Rolle .... 317
XX
Chapter XVI. The Life and Woeks of Newton.
PAGE
Newton's school and undergraduate life ..... 320
Investigations in 1665-1666 on fluxions, optics, and gravitation . 321
His views on gravitation, 1666 ...... 321
Researches in 1667-1669 323
Elected Lucasian professor, 1669 ....... 324
Optical lectures and discoveries, 1669-1671 324
Emission theory of light, 1675 326
The Leibnitz Letters, 1676 327
Discoveries on gravitation, 1679 330
Discoveries and lectures on algebra, 1673-1683 . . . . 330
Discoveries and lectures on gravitation, 1684 ..... 333
The subject-matter of the Principia ..... 335
Publication of the Principia 337
Investigations and work from 1686 to 1696 ..... 338
Appointment at the Mint, and removal to London, 1696 . . 339
Publication of the Optics, 1704 339
Appendix on classification of cubic curves . . . 340
Appendix on quadrature by means of infinite series . . 341
Appendix on method of fluxions 343
The invention of fluxions and the infinitesimal calculus . . . 347
Newton's death, 1727 348
List of his works . . . . , . . . . . 348
Newton's character ......... 349
Newton's discoveries 351
Chapter XVII. Leibnitz and the Mathematicians OF the FrRST Half of the Eighteenth Century.
Authorities ..........
Leibnitz and the Bernoullis .......
His system of philosophy, and services to literature The controversy as to the origin of the calculus His memoirs on the infinitesimal calculus His papers on various mechanical problems Characteristics of liis work
XXI
James Bernoulli, 1654-1705
John Bernoulli, 1667-1748
The younger Bernouillis
Development of Analysis on the Continent
Varignon, 1654-1722. De Montmort. Nicole
Parent. Saurin. De Gua. Cramer, 1704-1752
Solution of a partial differential equation of the second
Daniel Bernoulli, 1700-1782
English Mathematicians of the Eighteenth Century . David Gregory, 1661-1708. Halley, 1656-1742 .
Brook Taylor, 1685-1731
Taylor's theorem
Taylor's physical researches
Demoivre, 1667-1754 ; development of trigonometry Maclaurin, 1698-1746
His geometrical discoveries
The Treatise of Fluxions ....
His propositions on attractions .
Stewart, 1717-1785. Thomas Simpson, 1710-1761 order
Chapter XVIII. Lagrange, Laplace, and their Contemporaries. Circ. 1740-1830.
Characteristics of the mathematics of the period
Development of Analysis and Mechanics . . . .
The Introductio in Analysin Infinitorum, 1748 The Institutiones Calculi Differ entialis, 1755 . The Institutiones Calculi Ditegralis, 1768-1770 The Anleitung zur Algebra, 1770 Euler's works on mechanics and astronomy
PAGE
Memoirs on various subjects ...... 403
The Mdcanique analytique, 1788 . . , . . 406
The Th6orie and Calcul des fondions, 1797, 1804 . . 410
The Risolution des Equations numeriques, 1798 . . . 410
Characteristics of Lagrange's work ..... 411
Memoirs on astronomy and attractions, 1773-1784 . . 413
Use of spherical harmonics and the potential . . . 413
Memoirs on problems in astronomy, 1784-1786 . . . 414
The M4canique celeste and Exposition du systeme du mmide 415
The Nebular Hypothesis 415
The Meteoric Hypothesis ....... 415
The TMorie analytique des iJrohahilites, 1812 . . , 418
The Method of Least Squares 418
Other researches in pure mathematics and in physics . 419
Characteristics of Laplace's work ..... 420 Character of Laplace . . . . . . .421
His memoirs on attractions 422
The TMorie des nombres, 1798 423
Law of quadratic reciprocity 423
The Calcul integral and the Fonctions elliptiqucs . . 424
Creation of Modern Geometry . . ... . . . . 425
Lazare Carnot, 1753-1823. Poncelet, 1788-1867 . . . .428 DevelopTnent of Mathematical Physics . . . . . .429
Sadi Carnot ; foundation of thermodynamics 433
Introduction of Analysis into Erigland 438
The Cambridge Analytical School 439
TABLE OF CONTENTS xxiii
Chaptee XIX. Mathematics of the Nineteenth Century.
PAGE
Creation of new branches of mathematics ..... 444
Difficulty in discussing the mathematics of this century . . 445
Account of contemporary work not intended to be exhaustive . 445
Investigations in astronomy -. 448
Investigations in electricity ...... 449
The Disquisitiones Arithmeticae, 1801 .... 452
His other discoveries ....... 453
. Comparison of Lagrange, Laplace, and Gauss . . . .454
Development of the Theory of Numbers 455
Notes on other writers on the Theory of Numbers . . . .459
Development of the Theory of Functions of Multiple Periodicity . 461
Abel, 1802-1829. Abel's Theorem 461
Notes on other writers on Elliptic and Abelian Functions . . 465
Notes on recent writers on Elliptic and Abelian Functions . . 467
The Theory of Functions 467
Development of Higher Algebra ....... 468
Argand, 1768-1822 ; geometrical interpretation of complex numbers 471
Sir William Hamilton, 1805-1865 ; introduction of quaternions . 472
Grassmann, 1809-1877 ; his non-commutative algebra, 1884 . 473
Galois, 1811-1832 ; theory of discontinuous substitution groups . 475
Lie, 1842-1889 ; theory of continuous substitution groups . . 477
Notes on other writers on Higher Algebra 479
Development of Analytical Geometry ...... 480
Notes on some recent writers on Analytical Geometry . . . 481
Line Geometry 482
Analysis. Names of some recent writers on Analysis . . . 482
XXIV the subject
Development of Synthetic Geometry .
Other writers on modern Synthetic Geometry
Development of Non-Euclidean Geometry
Euclid's Postulate on Paraller Lines .
Hyperbolic Geometry. Elliptic Geometry Congruent Figures ....
Foundations of Mathematics. Assumptions made
Kinematics .........
Development of the Theory of Jlechanics, treated Graphically
Development of Theoretical Mechanics, treated Analytically
Notes on recent writers on Mechanics ....
Development of Theoretical Astronomy . . . .
Notes on other writers on Theoretical Astronomy .
Recent Developments
Development of Mathematical Physics . .
PAGE
Index .
Press Notices
499 523
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