A Short Account of the History of Mathematics By W. W. Rouse Ball
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A SHORT ACCOUNT OF THE HISTORY OF MATHEMATICS written by W. W. ROUSE BALL, FELLOW OF TRINITY COLLEGE, CAMBRIDGE
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Preface
Table of Contents
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 the Egyptians, Note on ignorance of mathematics shewn by the Chinese
The Ionian and Pythagorean Schools. Circ. 600 b.c.–400 b.c.
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. Philolaus. 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
The Schools of Athens and Cyzicus. Circ. 420–300 b.c.
Authorities, 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. TheaetetusCirc. Aristotle, 384–322 b.c.Circ.Questions on mechanics. Letters used to indicate magnitudesCirc.
The First Alexandrian School. Circ. 300–30 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, 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 statics of solids and fluids, His astronomy, 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, 275–194 b.c., The Sieve of Eratosthenes, The Second Century before Christ, Hypsicles (Euclid, book xiv). Nicomedes. Diocles, Perseus. Zenodorus, Hipparchus, circ. 130 b.c., Foundati on 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
The Second Alexandrian School. Circ. 30 b.c.–641 a.d.
Authorities, The First Century after Christ, Serenus. Menelaus, Nicomachus, Introduction of the arithmetic current in medieval Europe, The Second Century after Christ, Theon of Smyrna. Thymaridas, Ptolemy, died in 168, The Almagest, Ptolemy’s geometry, The Third Century after Christ, Pappus, circ. , The S??a????, 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. Eutocius, 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
The Byzantine School. 641–1453.
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
Systems of Numeration and Primitive Arithmetic.
Authorities, Methods of counting and indicating numbers among primitive races, Use of the abacus or swan-pan for practical calculation, 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
The Rise Of Learning In Western Europe. Circ. 600–1200.
Authorities, Education in the Sixth, Seventh, and Eighth Centuries, The Monastic Schools, Boethius, circ. 475–526, Medieval text-books in geometry and arithmetic, Cassiodorus, 490–566. Isidorus of Seville, 570–636, The Cathedral and Conventual Schools, The Schools of Charles the Great, Alcuin, 735–804, Education in the Ninth and Tenth Centuries, Gerbert (Sylvester II.), died in 1003, Bernelinus, The Early Medieval Universities, Rise during the twelfth century of the earliest universities, Development of the medieval universities, Outline of the course of studies in a medieval university
The Mathematics Of The Arabs.
Authorities, Extent of Mathematics obtained from Greek Sources, The College of Scribes, Extent of Mathematics obtained from the (Aryan) Hindoos, Arya-Bhata, circ. 530, His algebra and trigonometry (in his Aryabhathiya), Brahmagupta, circ. 640, His algebra and geometry (in his Siddhanta), Bhaskara, circ. 1140, The Lilavati or arithmetic; decimal numeration used, The Bija Ganita or algebra, Development of Mathematics in Arabia, Alkarismi or Al-Khwarizm ¯ ¯i, 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
Introduction of Arabian Works into Europe. Circ. 1150–1450.
The Eleventh Century, Moorish Teachers. Geber ibn Aphla. Arzachel, The Twelfth Century, Adelhard of Bath, Ben-Ezra. Gerard. John Hispalensis, The Thirteenth Century, Leonardo of Pisa, circ. 1175–1230, The Liber Abaci, 1202, The introduction of the Arabic numerals into commerce, The introduction of the Arabic numerals into science, The mathematical 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
The Development Of Arithmetic.
Circ. 1300–1637., 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, Simplification of the fundamental processes, Introduction of signs for addition and subtraction, Invention of logarithms, 1614, Use of decimals, 1619
The Mathematics of the Renaissance. Circ. 1450–1637.
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, D¨urer, 1471–1528. Copernicus, 1473–1543, Record, 1510–1558; introduction of symbol for equality, Rudolff, circ. 1525. Riese, 1489–1559, Stifel, 1486–1567, His Arithmetica Integra, 1544, Tartaglia, 1500–1557, His solution of a cubic equation, 1535, His arithmetic, 1556–1560, Cardan, 1501–1576, 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, 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, 1595–1632; development of trigonometry and algebra, Napier, 1550–1617; introduction of logarithms, 1614, Briggs, 1561–1631; 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
The Close 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, D¨urer, 1471–1528. Copernicus, 1473–1543, Record, 1510–1558; introduction of symbol for equality, Rudolff, circ. 1525. Riese, 1489–1559, Stifel, 1486–1567, His Arithmetica Integra, 1544, Tartaglia, 1500–1557, His solution of a cubic equation, 1535, His arithmetic, 1556–1560, Cardan, 1501–1576, 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, 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, 1595–1632; development of trigonometry and algebra, Napier, 1550–1617; introduction of logarithms, 1614, Briggs, 1561–1631; 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
Circ. 1586–1637, Authorities, Development of Mechanics and Experimental Methods, Stevinus, 1548–1620, Commencement of the modern treatment of statics, 1586, Galileo, 1564–1642, Commencement of the science of dynamics, Galileo’s astronomy, Francis Bacon, 1561–1626. Guldinus, 1577–1643, Wright, 1560–1615; construction of maps, Snell, 1591–1626, 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
The History of Modern Mathematics.
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, D¨urer, 1471–1528. Copernicus, 1473–1543, Record, 1510–1558; introduction of symbol for equality, Rudolff, circ. 1525. Riese, 1489–1559, Stifel, 1486–1567, His Arithmetica Integra, 1544, Tartaglia, 1500–1557, His solution of a cubic equation, 1535, His arithmetic, 1556–1560, Cardan, 1501–1576, 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, 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, 1595–1632; development of trigonometry and algebra, Napier, 1550–1617; introduction of logarithms, 1614, Briggs, 1561–1631; 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
Treatment of the subject, Invention of analytical geometry and the method of indivisibles, Invention of the calculus, Development of mechanics, Application of mathematics to physics, Recent development of pure mathematics
History of Mathematics from Descartes to Huygens. Circ. 1635–1675.
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, D¨urer, 1471–1528. Copernicus, 1473–1543, Record, 1510–1558; introduction of symbol for equality, Rudolff, circ. 1525. Riese, 1489–1559, Stifel, 1486–1567, His Arithmetica Integra, 1544, Tartaglia, 1500–1557, His solution of a cubic equation, 1535, His arithmetic, 1556–1560, Cardan, 1501–1576, 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, 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, 1595–1632; development of trigonometry and algebra, Napier, 1550–1617; introduction of logarithms, 1614, Briggs, 1561–1631; 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
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 arithmetical 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 investigations 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, Mersenne; theorem on primes and perfect numbers, Roberval. Van Schooten. Saint-Vincent, Torricelli. Hudde. Fr´enicle, De Laloub`ere. 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
The Life and Works of Newton.
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, D¨urer, 1471–1528. Copernicus, 1473–1543, Record, 1510–1558; introduction of symbol for equality, Rudolff, circ. 1525. Riese, 1489–1559, Stifel, 1486–1567, His Arithmetica Integra, 1544, Tartaglia, 1500–1557, His solution of a cubic equation, 1535, His arithmetic, 1556–1560, Cardan, 1501–1576, 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, 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, 1595–1632; development of trigonometry and algebra, Napier, 1550–1617; introduction of logarithms, 1614, Briggs, 1561–1631; 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
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 on gravitation, 1679, 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 means of infinite series, Appendix on method of fluxions, The invention of fluxions and the infinitesimal calculus, Newton’s death, 1727, List of his works, Newton’s character, Newton’s discoveries
Leibnitz and the Mathematicians of the First Half of the Eighteenth Century.
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, D¨urer, 1471–1528. Copernicus, 1473–1543, Record, 1510–1558; introduction of symbol for equality, Rudolff, circ. 1525. Riese, 1489–1559, Stifel, 1486–1567, His Arithmetica Integra, 1544, Tartaglia, 1500–1557, His solution of a cubic equation, 1535, His arithmetic, 1556–1560, Cardan, 1501–1576, 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, 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, 1595–1632; development of trigonometry and algebra, Napier, 1550–1617; introduction of logarithms, 1614, Briggs, 1561–1631; 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
Authorities, Leibnitz and the Bernoullis, Leibnitz, 1646–1716, 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 his work, James Bernoulli, 1654–1705, John Bernoulli, 1667–1748, The younger Bernouillis, Development of Analysis on the Continent, L’Hospital, 1661–1704, Varignon, 1654–1722. De Montmort. Nicole, Parent. Saurin. De Gua. Cramer, 1704–1752, Riccati, 1676–1754. Fagnano, 1682–1766, Clairaut, 1713–1765, D’Alembert, 1717–1783, Solution of a partial differential equation of the second order, Daniel Bernoulli, 1700–1782, English Mathematicians of the Eighteenth Century, David Gregory, 1661–1708. Halley, 1656–1742, Ditton, 1675–1715, Brook Taylor, 1685–1731, Taylor’s theorem, Taylor’s physical researches, Cotes, 1682–1716, 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
Lagrange, Laplace, and their
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, D¨urer, 1471–1528. Copernicus, 1473–1543, Record, 1510–1558; introduction of symbol for equality, Rudolff, circ. 1525. Riese, 1489–1559, Stifel, 1486–1567, His Arithmetica Integra, 1544, Tartaglia, 1500–1557, His solution of a cubic equation, 1535, His arithmetic, 1556–1560, Cardan, 1501–1576, 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, 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, 1595–1632; development of trigonometry and algebra, Napier, 1550–1617; introduction of logarithms, 1614, Briggs, 1561–1631; 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
Contemporaries. Circ. 1740–1830., Characteristics of the mathematics of the period, Development of Analysis and Mechanics, Euler, 1707–1783, The Introductio in Analysin Infinitorum, 1748, The Institutiones Calculi Differentialis, 1755, The Institutiones Calculi Integralis, 1768–1770, The Anleitung zur Algebra, 1770, Euler’s works on mechanics and astronomy, Lambert, 1728–1777, B´ezout, 1730–1783. Trembley, 1749–1811. Arbogast, 1759–1803, Lagrange, 1736–1813, Memoirs on various subjects, The M´ecanique analytique, 1788, The Th´eorie and Calcul des fonctions, 1797, 1804, The R´esolution des ´equations num´eriques, 1798, Characteristics of Lagrange’s work, Laplace, 1749–1827, Memoirs on astronomy and attractions, 1773–1784, Use of spherical harmonics and the potential, Memoirs on problems in astronomy, 1784–1786, The M´ecanique c´eleste and Exposition du syst`eme du monde, The Nebular Hypothesis, The Meteoric Hypothesis, The Th´eorie analytique des probabilit´es, 1812, The Method of Least Squares, Other researches in pure mathematics and in physics, Characteristics of Laplace’s work, Character of Laplace, Legendre, 1752–1833, His memoirs on attractions, The Th´eorie des nombres, 1798, Law of quadratic reciprocity, The Calcul int´egral and the Fonctions elliptiques, Pfaff, 1765–1825, Creation of Modern Geometry, Monge, 1746–1818, Lazare Carnot, 1753–1823. Poncelet, 1788–1867, Development of Mathematical Physics, Cavendish, 1731–1810, Rumford, 1753–1815. Young, 1773–1829, Dalton, 1766–1844, Fourier, 1768–1830, Sadi Carnot; foundation of thermodynamics, Poisson, 1781–1840, Amp`ere, 1775–1836. Fresnel, 1788–1827. Biot, 1774–1862, Arago, 1786–1853, Introduction of Analysis into England, Ivory, 1765–1842, The Cambridge Analytical School, Woodhouse, 1773–1827, Peacock, 1791–1858. Babbage, 1792–1871. John Herschel, 1792–1871
Mathematics of the Nineteenth Century.
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, D¨urer, 1471–1528. Copernicus, 1473–1543, Record, 1510–1558; introduction of symbol for equality, Rudolff, circ. 1525. Riese, 1489–1559, Stifel, 1486–1567, His Arithmetica Integra, 1544, Tartaglia, 1500–1557, His solution of a cubic equation, 1535, His arithmetic, 1556–1560, Cardan, 1501–1576, 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, 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, 1595–1632; development of trigonometry and algebra, Napier, 1550–1617; introduction of logarithms, 1614, Briggs, 1561–1631; 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
Creation of new branches of mathematics, Difficulty in discussing the mathematics of this century, Account of contemporary work not intended to be exhaustive, Authorities, Gauss, 1777–1855, Investigations in astronomy, Investigations in electricity, The Disquisitiones Arithmeticae, 1801, His other discoveries, Comparison of Lagrange, Laplace, and Gauss, Dirichlet, 1805–1859, Development of the Theory of Numbers, Eisenstein, 1823–1852, Henry Smith, 1826–1883, Kummer, 1810–1893, Notes on other writers on the Theory of Numbers, Development of the Theory of Functions of Multiple Periodicity, Abel, 1802–1829. Abel’s Theorem, Jacobi, 1804–1851, Riemann, 1826–1866, Notes on other writers on Elliptic and Abelian Functions, Weierstrass, 1815–1897, Notes on recent writers on Elliptic and Abelian Functions, The Theory of Functions, Development of Higher Algebra, Cauchy, 1789–1857, Argand, 1768–1822; geometrical interpretation of complex numbers, Sir William Hamilton, 1805–1865; introduction of quaternions, Grassmann, 1809–1877; his non-commutative algebra, 1844, Boole, 1815–1864. De Morgan, 1806–1871, Galois, 1811–1832; theory of discontinuous substitution groups, Cayley, 1821–1895, Sylvester, 1814–1897, Lie, 1842–1889; theory of continuous substitution groups, Hermite, 1822–1901, Notes on other writers on Higher Algebra, Development of Analytical Geometry, Notes on some recent writers on Analytical Geometry, Line Geometry, Analysis. Names of some recent writers on Analysis, Development of Synthetic Geometry, Steiner, 1796–1863, Von Staudt, 1798–1867, Other writers on modern Synthetic Geometry, Development of Non-Euclidean Geometry, Euclid’s Postulate on Parallel Lines, Hyperbolic Geometry. Elliptic Geometry, Congruent Figures, Foundations of Mathematics. Assumptions made in the subject, Kinematics, Development of the Theory of Mechanics, treated Graphically, Development of Theoretical Mechanics, treated Analytically, Notes on recent writers on Mechanics, Development of Theoretical Astronomy, Bessel, 1784–1846, Leverrier, 1811–1877. Adams, 1819–1892, Notes on other writers on Theoretical Astronomy, Recent Developments, Development of Mathematical Physics
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