PREFACE |

TABLE OF CONTENTS |

CHAPTER I. EGYPTIAN AND PHOENICIAN MATHEMATICS. |

The history of mathematics begins with that of the Ionian Greeks |

Greek indebtedness to Egyptians and Phoenicians |

Knowledge of the science of numbers possessed by the Phoenicians |

Knowledge of the science of numbers possessed by the Egyptians |

Knowledge of the science of geometry possessed by Egyptians |

Note on ignorance of mathematics shewn by the Chinese |

First Period. Mathematics under Greek Influence. |

CHAPTER II. THE IONIAN AND PYTHAGOREAN SCHOOLS. |

Authorities |

The Ionian School |

"THALES, 640-550 B.C." |

His geometrical discoveries |

His astronomical teaching |

Anaximander. Anaximenes. Mamercus. Mandryatus |

The Pythagorean School |

"PYTHAGORAS, 569-500 B.C." |

The Pythagorean teaching |

The Pythagorean geometry |

The Pythagorean theory of numbers |

Epicharmus. Hippasus. Phiololaus. Archippus. Lysis |

"ARCHYTAS, circ. 400 B.C." |

His solution of the duplication of a cube |

Theodorus. Timaeus. Bryso |

Other Greek Mathematical Schools in the Fifth Century B.C. |

Oenopides of Chios |

Zeno of Elea. Democritus of Abdera |

CHAPTER III. THE SCHOOLS OF ATHENS AND CYZICUS. |

Authorities |

Mathematical teachers at Athens prior to 420 B.C. |

Anaxogoras. 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 selections |

His two solutions of the duplication of the cube |

Aristaeus. Theaetetus |

"Aristotle, 384-322 B.C." |

Questions on mechanics. Letters used to indicate magnitudes |

CHAPTER IV. THE FIRST ALEXANDRIAN SCHOOL |

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 geo |

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 |

"ARCHIMEDES, 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 statistics of solids and fluids |

His astronomy |

The principles of geometry and that of Archimedes |

"APOLLONIUS, circ. 260-200 B.C." |

His conic sections |

His other works |

His solution of the duplication of a cube |

Contrast between his geometry and that of Archimedes |

"Erathosthenes, 275-194 B.C." |

The Sieve of Eratosthenes |

The Second Century before Christ |

"Hypsicles (Euclid, book XIV). Nicomedes. Diocles" |

Perseus. Zejodorus |

"HIPPARCHUS, circ. 130 B.C." |

Foundation of scientific astronomy |

Foundation of trigonometry |

"HERO of Alexandria, circ. 125 B.C." |

Foundation of scientific engineering and of land-surveying |

Area of a triangle determined in terms of its sides |

Features of Hero's works |

The First Century before Christ |

Theodosius |

Dionysodorus |

End of the First Alexandrian School |

Egypt constituted a Roman province |

CHAPTER V. THE SECOND ALEXANDRIAN SCHOOL |

Authorities |

The First Century after Christ |

Serenus. Menelaus |

Nicomachus |

Introduction of the arithmetic current in medieval Europe |

The Second Century after Christ |

Theon of Smyran. Thymaridas |

"PTOLEMY, died in 168" |

The Almagest |

Ptolemy's astronomy |

Ptolemy's geometry |

The Third Century after Christ |

"Pappus, circ. 280" |

"The Suagwg?, a synopsis of Greek mathematics" |

The Fourth Century after Christ |

Metrodorus. Elementary problems in arithmetic and algebra |

Three stages in the development of algebra |

"DIOPHANTUS, circ. 320 (?)" |

Introduction of syncopated algebra in his Arithmetic |

"The notation, methods, and subject-matter of the work" |

His Porisms |

Subsequent neglect of his discoveries |

Iamblichus |

Theon of Alexandria. Hypatia |

Hostility of the Eastern Church to Greek science |

The Athenian School (in the Fifth Century) |

"Proclus, 412-485. Damascius. Euto |

Roman Mathematics |

Nature and extent of the mathematics read at Rome |

Contrast between the conditions at Rome and at Alexandria |

End of the Second Alexandrian School |

"The capture of Alexandria, and end of the Alexandrian Schools" |

CHAPTER VI. THE BYZANTINE SCHOOL. |

Preservation of works of the great Greek Mathematicians |

Hero of Constantinople. Psellus. Planudes. Barlaam. Argyrus |

Nicholas Rhabdas. Pachymeres. Moschopulus (Magic Squares) |

"Capture of Constantinople, and dispersal of Greek Mathematicians" |

CHAPTER VII. SYSTEMS OF NUMERATION AND PRIMITIVE ARITHMETIC. |

Authorities |

Methods of counting and indicating numbers amoung primitive races |

Use of the abacus or swan-pan for practical calculation |

Methods of representing nu |

The Lilavati or arithmetic ; decimal numeration used |

The Bija Ganita or algebra |

Development of Mathematics in Arabia |

"ALKARISMI or AL-KHWARIZMI, circ. 830" |

His Al-gebr we 'l 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. Albuzjani. Development of trigonometry |

Alhazen. Abd-al-gehl. Development of geometry |

Characteristics of the Arabian School |

CHAPTER X. INTRODUCTION OF ARABIAN WORKS INTO EUROPE. |

The Eleventh Century |

Moorish Teachers. Geber ibn Aphla. Arzachel |

The Twelfth Century |

Adelhard of Bath |

Ben-Ezra. Gerad. John Hispalensis |

The Thirteenth Century |

"LEONARDO OF PISA, circ. 1175-1230" |

"The Liber Abaci, 1202" |

The introduction of the Arabic numerals into commerceThe introduction of the Arabic numerals into science |

The mathematic tournament |

"Frederick II., 1194-1250" |

"JORDANUS, circ. 1220" |

His De Numeris Datis ; syncopated algebra |

Holywood |

"ROGER BACON, 1214-1294" |

Campanus |

The Fourteenth Century |

Bradwardine |

Oresmus |

The reform of the university curriculum |

The Fifteenth Century |

Beldomandi |

CHAPTER XI. THE DEVELOPMENT OF ARITHMETIC. |

Authorities |

The Boethian arithmetic |

Algorism or modern arithmetic |

The Arabic (or Indian) symbols : history of |

"Introduction into Europe by science, commerce, and calendars" |

Improvements introduced in algoristic arithmetic |

(I) Simplification of the fundemental processe |

(ii) Introduction of signs for addition and subtra |

(iii) "Invention of logarithms, 1614" |

(iv) "Use of decimals, 1619" |

CHAPTER XII. THE MATHEMATICS OF THE RENAISSANCE. |

Authorities |

Effect of invention of printing. The renaissance |

Development of Syncopated Algebra and Trigonometry |

"REGIOMONTANUS, 1436-1476" |

His De Triangulis (printed in 1496) |

"Purbach, 1423-1461. Cusa, 1401-1464. Chuquet, circ. 1484" |

Introduction and origin of symbols + and - |

"Pacioli or Lucas di Burgo, circ. 1500" |

"His arithmetic and geometry, 1494" |

"Leonardo da Vinci, 1452-1519" |

"Dürer, 1471-1528. Copernicus, 1473-1543" |

"Record, 1510-1588 ; introduction of symbol for equality" |

"Rudolff, circ. 1525. Riese, 1489-1559" |

"STIFEL, 1486-1567" |

"His Arithmetica Integra, 1544" |

"TARTAGLIA, 1500-1559" |

"His solution of a cubic equation, 1535" |

"His arithmetic, 1556-1560" |

"CARDAN, 1501-1576" |

"Hid 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" |

"Commandino. Peletier. Romanus. Pitiscus. Ramus, 1515-1572" |

"Bombelli, circ. 1570" |

Development of Symbolic Algebra |

"VIETA, 1540-1603" |

"The In Artem ; introduction of symbolic algebra, 1591" |

Vieta's other works |

"Girard, 1590-1633 ; development of trigonometry and algebra" |

"NAPIER, 1550-1617 ; development of trigonometry and algebra" |

"Briggs, 1556-1631 ; calculations of tables of logarithms" |

"HARRIOT, 1560-1621 ; development of analysis in algebra" |

"Oughtred, 1574-1660" |

The Origin of the more Common Symbols in Algebra |

CHAPTER XIII. THE CLOSE OF THE RENAISSANCE. |

Authorities |

Development of Mechanics and Experimental Methods |

"STEVINUS, 1548-1620" |

"Commencement of the modern treatment of statistics, 1586" |

"GALILEO, 1564-1642" |

Commencement of the science of dynamics |

Galileo's astronomy |

"Francis Bacon, 1561-1626" |

Revival of Interest in Pure Geometry |

"KEPLER, 1571-1630" |

"His Paralipomena, 1604 ; principle of continuity" |

"His Stereometria, 1615 ; use of infinitesimals" |

"Kepler's laws of planetary motion, 1609 and 1619" |

"Desargues, 1593-1662" |

His Brouillon project ; use of projective geometry |

Mathematical Knowledge at the Close of the Renaissance |

Third Period. Modern Mathematics |

CHAPTER XIV. THE HISTORY OF MODERN MATHEMATICS. |

Treatment of the subject |

Invention of analytical geometry and the method of indivis |

Invention of the calculus |

Development of mechanics |

Application of mathematics to physics |

Recent development of pure mathematics |

CHAPTER XV. HISTORY OF MATHEMATICS FROM DESCARTES TO HUYGENS. |

Authorities |

"DESCARTES, 1596-1650" |

His views on philosophy |

"His invention of analytical geometry, 1637" |

"His algebra, optics, and theory of vortices" |

"CAVALIERI, 1598-1647" |

The method of indivisibles |

"PASCAL, 1623-1662" |

His geometrical conics |

The arthmetical triangle |

"Foundation of the theory of probabilities, 1654" |

His discussion of the cycloid |

"WALLIS, 1616-1703" |

"The Arithmetica Infinitorum, 1656" |

Law of indices in algebra |

Use of series in quadratures |

"Earliest rectification of curves, 1657" |

Wallis's algebra |

"FERMAT, 1601-1665" |

His investigation on the theory of numbers |

His use in geometry of analysis and of infinitesimals |

"Foundation of the theory of probabilities, 1654" |

"HUYGENS, 1629-1695" |

"The Horologium Oscillatorium, 1673" |

The undulatory theory of light |

Other Mathematicians of this Time |

Bachet |

Marsenne ; theorem on primes and perfect numbers |

Roberval. Van Schooten. Saint-Vincent |

Torricelli. Hudde. Frénicle |

De Laloubère. Mercator. Barrow ; the differential triangle |

Brouncker ; continued fractions |

James Gregory ; distinction between convergent and divergent series |

Sir Christopher Wren |

Hooke. Collins |

Pell. Sluze. Viviani |

Tschirnhausen. De la Hire. Roemer. Rolle. |

CHAPTER XVI. THE LIFE AND WORKS OF NEWTON. |

Authorities |

Newton's school and undergraduate life |

"Investigations in 1665-1666 on fluxions, optics, and gravitation" |

"His views on gravitation, 1666" |

Researches in 1667-1669 |

"Elected Lucasian professor, 1669" |

"Optical lectures and discoveries, 1669-1671" |

"Emission theory of light, 1675" |

"The Leibnitz Letters, 1676" |

"Discoveries and lectures on algebra, 1673-1683" |

"Discoveries and lectures on gravitation, 1684" |

"The Principia, 1685-1686" |

The subject-matter of the Principia |

Publication of the Principia |

Investigations and work from 1686 to 1696 |

"Appointment at the Mint, and removal to London, 1696" |

"Publication of the Optics, 1704" |

Appendix on classification of cubic curves |

Appendix on quadrature by |

The controversy as t |