A Short Account of the History of Mathematics

$18.95

Publication Date: 30th March 2012

This is a new printing, the first inexpensive one, of one of the most honored histories of mathematics of all time. When the last revised edition appeared in 1908, it was hailed by mathematicians and laymen alike, and it remains one of the clearest, most authoritative, and most accurate works in the field. Mathematicians welcomed it as a lucid overview of the development of mathematics down through the centuries. Laymen welcomed it as a work which gave them an opportunity to understand the development of one of the most recondite and difficult of all intellectual endeavors, and the ind... Read More

Format: eBook
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This is a new printing, the first inexpensive one, of one of the most honored histories of mathematics of all time. When the last revised edition appeared in 1908, it was hailed by mathematicians and laymen alike, and it remains one of the clearest, most authoritative, and most accurate works in the field. Mathematicians welcomed it as a lucid overview of the development of mathematics down through the centuries. Laymen welcomed it as a work which gave them an opportunity to understand the development of one of the most recondite and difficult of all intellectual endeavors, and the ind... Read More

Description

This is a new printing, the first inexpensive one, of one of the most honored histories of mathematics of all time. When the last revised edition appeared in 1908, it was hailed by mathematicians and laymen alike, and it remains one of the clearest, most authoritative, and most accurate works in the field. Mathematicians welcomed it as a lucid overview of the development of mathematics down through the centuries. Laymen welcomed it as a work which gave them an opportunity to understand the development of one of the most recondite and difficult of all intellectual endeavors, and the individual contributions of its great men.
In this standard work, Dr. Ball treats hundreds of figures and schools that have been instrumental in the development of mathematics from the Egyptians and Phoenicians to such giants of the 19th century as Grassman, Hermite, Galois, Lie, Riemann, and many others who established modern mathematics as we know it today. This semi-biographical approach gives you a real sense of mathematics as a living science, but where Dr. Ball has found that the biographical approach is not sufficient or suited to presenting a mathematical discovery or development, he does not hesitate to depart from his major scheme and treat the mathematics in detail by itself. Thus, while the book is virtually a pocket encyclopedia of the major figures of mathematics and their discoveries, it is also one of the best possible sources for material on such topics as the problems faced by Greek mathematicians, the contributions of the Arab mathematicians, the development of mathematical symbolism, and the invention of the calculus.
While some background in mathematics is desirable to follow the reference in some of the later sections, most of the book can be read without any more preparation than high school algebra. As a history of mathematics to browse through, or as a convenient reference work, it has never been excelled.


Reprint of the 1908 edition.
Details
  • Price: $18.95
  • Pages: 560
  • Publisher: Dover Publications
  • Imprint: Dover Publications
  • Series: Dover Books on Mathematics
  • Publication Date: 30th March 2012
  • Trim Size: 5.37 x 8.5 in
  • ISBN: 9780486157849
  • Format: eBook
  • BISACs:
    MATHEMATICS / History & Philosophy
Author Bio

H. S. M. Coxeter: Through the Looking Glass
Harold Scott MacDonald Coxeter (1907–2003) is one of the greatest geometers of the last century, or of any century, for that matter. Coxeter was associated with the University of Toronto for sixty years, the author of twelve books regarded as classics in their field, a student of Hermann Weyl in the 1930s, and a colleague of the intriguing Dutch artist and printmaker Maurits Escher in the 1950s.

In the Author's Own Words:
"I'm a Platonist — a follower of Plato — who believes that one didn't invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."

"In our times, geometers are still exploring those new Wonderlands, partly for the sake of their applications to cosmology and other branches of science, but much more for the sheer joy of passing through the looking glass into a land where the familiar lines, planes, triangles, circles, and spheres are seen to behave in strange but precisely determined ways."

"Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about the physical world. The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry."

"Let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused." — H. S. M. Coxeter

Table of Contents
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 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 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. 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"
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 commerce
The 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 subtraction
(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 indivisibles
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 to the

This is a new printing, the first inexpensive one, of one of the most honored histories of mathematics of all time. When the last revised edition appeared in 1908, it was hailed by mathematicians and laymen alike, and it remains one of the clearest, most authoritative, and most accurate works in the field. Mathematicians welcomed it as a lucid overview of the development of mathematics down through the centuries. Laymen welcomed it as a work which gave them an opportunity to understand the development of one of the most recondite and difficult of all intellectual endeavors, and the individual contributions of its great men.
In this standard work, Dr. Ball treats hundreds of figures and schools that have been instrumental in the development of mathematics from the Egyptians and Phoenicians to such giants of the 19th century as Grassman, Hermite, Galois, Lie, Riemann, and many others who established modern mathematics as we know it today. This semi-biographical approach gives you a real sense of mathematics as a living science, but where Dr. Ball has found that the biographical approach is not sufficient or suited to presenting a mathematical discovery or development, he does not hesitate to depart from his major scheme and treat the mathematics in detail by itself. Thus, while the book is virtually a pocket encyclopedia of the major figures of mathematics and their discoveries, it is also one of the best possible sources for material on such topics as the problems faced by Greek mathematicians, the contributions of the Arab mathematicians, the development of mathematical symbolism, and the invention of the calculus.
While some background in mathematics is desirable to follow the reference in some of the later sections, most of the book can be read without any more preparation than high school algebra. As a history of mathematics to browse through, or as a convenient reference work, it has never been excelled.


Reprint of the 1908 edition.
  • Price: $18.95
  • Pages: 560
  • Publisher: Dover Publications
  • Imprint: Dover Publications
  • Series: Dover Books on Mathematics
  • Publication Date: 30th March 2012
  • Trim Size: 5.37 x 8.5 in
  • ISBN: 9780486157849
  • Format: eBook
  • BISACs:
    MATHEMATICS / History & Philosophy

H. S. M. Coxeter: Through the Looking Glass
Harold Scott MacDonald Coxeter (1907–2003) is one of the greatest geometers of the last century, or of any century, for that matter. Coxeter was associated with the University of Toronto for sixty years, the author of twelve books regarded as classics in their field, a student of Hermann Weyl in the 1930s, and a colleague of the intriguing Dutch artist and printmaker Maurits Escher in the 1950s.

In the Author's Own Words:
"I'm a Platonist — a follower of Plato — who believes that one didn't invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."

"In our times, geometers are still exploring those new Wonderlands, partly for the sake of their applications to cosmology and other branches of science, but much more for the sheer joy of passing through the looking glass into a land where the familiar lines, planes, triangles, circles, and spheres are seen to behave in strange but precisely determined ways."

"Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about the physical world. The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry."

"Let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused." — H. S. M. Coxeter

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 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 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. 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"
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 commerce
The 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 subtraction
(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 indivisibles
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 to the