Introduction to the History of Mathematics

Fall 2020 - R. L. Herman

Table of Contents

Early Civilizations - Babylonia, Egypt, China, India, Islamic

Beyond Numerals - Decimals, Logarithms, Symbolic Algebra

Italian Mathematics - 16th Century

The Rise of Calculus - 17th Century

Exploiting Calculus - 18th Century

The Birth of Rigor - 19th Century

The Modern Era - 20th Century

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A Civiliation Timeline

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Early Civilizations

ˆ Egypt - 3100 BCE

ˆ Mesopotamia (Babylonia,
Sumer) - 2100 BCE
ˆ China 1600 BCE
ˆ India 1200 BCE
Arithmetic, Geometry,
No proofs
Problems were practical or
recreational

Figure 1: Babylonian tablet - Base 60

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Greek Civilization

ˆ Deductive Reasoning
ˆ Definitions, Axioms
ˆ Propositions via logic
ˆ Geometry, Trigonometry,
Astronomy, Numbers, Conics
ˆ Thales 624-546 BCE
ˆ Pythagoras 6th Century BCE
ˆ Euclid 4th Century BCE
Elements
ˆ Archimedes 3rd Century BCE
ˆ Appolonius 2nd Century BCE
ˆ Heron 10-70 CE
Diophantus 200-284 CE, Figure 2: Euclid
Hypatia 400 CE
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Chinese and Hindu - 700-1200 CE

ˆ Chinese Mathematics 1300 BCE -

1800 CE
ˆ Pythagorean Thm
ˆ π estimates
ˆ Volumes, Applications
ˆ Pascal’s Triangle
ˆ Chinese Remainder Thm
ˆ Indian Mathematics 1200 BCE,
mostly 500-1200 CE
ˆ Geometry
ˆ Trigonometry
ˆ Power sieries
ˆ Astronomy
ˆ π estimate
Figure 3: Liu-Hong and
ˆ Number system, 0
Aryabhatta
ˆ Pell’s Equation
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Middle Eastern Mathematics - 700-1200 CE

In Europe - Dark Ages - 400-1200 CE

ˆ Middle East - Islam 7th Century

ˆ Translations of Greek
Mathematics into Arabic
ˆ Arabic Numerals by 1000 CE
ˆ Algebra 825 CE - al-Khwarizmi
- called al-Jabr
ˆ Omar Khayyam (1048-1131) -
geometric solution of cubic
Figure 4: al-Khwarizmi

Around 10th Century - Middle Eastern Mathematics brought to Spain.

It takes 300 years to accept Arabic numerals. - Fibonacci - 1202
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Beyond Numerals

ˆ Fractions 4000 years ago

ˆ Sexagesimal (base 60) into
17th century
ˆ Decimal (base 10)
ˆ al-Uqlidisi - (920-980)
ˆ al-Kashi (1380-1429)
ˆ Simon Stevin (1548-1620)
ˆ Logarithms
ˆ John Napier (1550-1617)
used a stange base
ˆ Henry Briggs (1561-1630)
Base 10 Tables
ˆ 54 square roots of 10 leading
to 30 decimal places Figure 5: Brigg’s Tables
ˆ Tables to 14 decimal places
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Italian Mathematics - Polynomial Equations, Complex Numbers

ˆ Fibonacci (Leonardo of Pisa)

(1170-1250) Liber Abaci
ˆ Equation Solving contests
ˆ Solutions of cubic and quartic
ˆ Depressed cubic
del Ferro (1465-1526)
ˆ Cubic and quartic equations
Tartaglia (1500-1557)
Cardano (1501-1576)
Ars Magna
Ferrari (1522-1565)
ˆ Bombelli (1526-1572) Figure 6: Cardano and Tartaglia
ˆ Viète (1540-1603)
Adriaan van Roomen Problem
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Unification of Geometry and Algebra

ˆ Symbolic Algebra
ˆ Rhetorical until 15th century
ˆ Syncopated/abbrev. - 1500
ˆ Symbolic algebra developed
16-17th century
ˆ Unification
ˆ Oresme (1320-1382) -
P1
Velocity-time graphs, n
ˆ Descartes (1596-1650)
ˆ Rep. curves by equations
ˆ Coordinate systems -
published The Method
ˆ Made use of variables which
can vary continuously - lines.
ˆ Fermat (1607-1665)
ˆ Rep. equations by curves Figure 7: Fermat and Descartes

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The Rise of Calculus

ˆ Archimedes - 3rd century BCE

ˆ Kepler (1571-1630)
ˆ Cavalieri (1598-1647)
ˆ Fermat (1607-1665)
ˆ Wallis (1616-1673)
ˆ Pascal (1623-1662)
ˆ Barrow (1630-1677)
ˆ Wren (1632-1723)
ˆ Gregory (1638-1675)
ˆ Newton (1642-1726)
ˆ Principia 1687
ˆ Leibniz (1646-1716) Figure 8: Archimedes, Cavalieri,
Z
d Wallis, Gregory, Newton, and Leibniz
ˆ Notation ,
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The Infinitesimal

ˆ Hippasus 500 BCE

ˆ Pythagorean
√
ˆ 2 irrational
ˆ Introduction of Infinitesimals
ˆ Cavilieiri and Torricelli
ˆ Stevin, Wallis, Harriot
ˆ Critics
ˆ Jesuits in Italy
ˆ George Berkeley (1685-1753)
The Analyst, - A Discourse
Addressed to an Infidel
Mathematician, 1734
ˆ infinitesimals undermine
mathematics and rationality
ˆ Augustin-Louis Cauchy - 1821 Figure 9: Berkeley’s The Analyst
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Exploiting Calculus

ˆ Bernoulli Family
ˆ Euler (1707-1783)
ˆ Laplace (1749-1827)
ˆ Neptune discovered using math
- 1846 Nikolas
(1623-1708)

Nicolaus Jacob Johann Hieronymus

(1662-1716) (1654-1705) (1667-1748) (1669-1760)

Nicolaus I Nicolaus II Johann II Daniel

(1687-1759) (1695-1726) (1710-1790) (1700-1782)

Johann III Daniel II Nicolaus IIII Jakob II

(1744-1807) (1759-1789) (1759-1789) (1759-1789)

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From Geometry to Topology

ˆ Bruneschelli and Perspective

ˆ Projective Geometry
ˆ Euler’s geometry without
distance
ˆ Birth of Topology
ˆ How smoke rings led to Knot
Theory.

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The Birth of Rigor - 19th Century

Non-Euclidean Geometry
ˆ Parallel Postulate
ˆ Hyperbolic Geometry
ˆ Nikolai Lobachevsky
(1792-1856)
ˆ Johann Bolyai (1802-1860) Figure 11: Gauss, Lobachevsky, Bolyai
ˆ Johann Carl Friedrich Gauss
(1777-1855)
ˆ Elliptic Geometry
ˆ Georg Friedrich Bernhard
Riemann (1826-1866)
ˆ Prince of Mathematicians
ˆ By 1870’s doubted Euclid
Figure 12: Different Geometries

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19th Century Group Theory

ˆ Joseph-Louis Lagrange
(1736-1813)
ˆ Johann Carl Friedrich Gauss
(1777-1855)
ˆ Augustin Cauchy (1789-1857)
ˆ Niels Henrik Abel (1802-1829)
ˆ Évariste Galois (1811-1832)
ˆ Arthur Cayley (1821-1895)
ˆ Camille Jordan (1838-1922)

Figure 13: Abel and Galois

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19th Century Analysis and Set Theory

ˆ Jean-Baptiste Joseph Fourier

(1768-1830)
ˆ Johann Carl Friedrich Gauss
(1777-1855)
ˆ Augustin Cauchy (1789-1857)
ˆ Karl Weierstrass (1815-1897)
ˆ George Boole (1815-1864)
ˆ Georg Friedrich Bernhard Riemann
(1826-1866)
ˆ Richard Dedekind (1831-1916)
ˆ Georg Ferdinand Ludwig Philipp
Cantor (1845-1918) Figure 14: Gauss and Riemann
ˆ Founder of set theory
ˆ Defined infinite sets
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19th Century Number Theory

ˆ Marie-Sophie Germain (1776-1831)

ˆ Johann Carl Friedrich Gauss (1777-1855)
Disquisitiones Arithmeticae - 1801
ˆ Adrien-Marie Legendre (1752-1833) and
Peter Gustav Lejeune Dirichlet (1805-1859) prove
Fermat’s Last Theorem for n = 5 in 1825
ˆ Dirichlet, n = 14 in 1832.
ˆ Riemann hypothesis, distribution of primes - 1832.
ˆ Charles Jean de la Vallée-Poussin and Jacques
Hadamard - Prime Number Theorem. 1896
ˆ Hermann Minkowski: Geometry of Numbers, Figure 15: Sophie
1896. Germain, Adrien-Marie
Legendre
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The Modern Era

Figure 16: Hilbert, Gödel, Uhlenbeck, Ramanujan, Wiles, Mirzakhani,

Shannon, Russell, Noether

ˆ Chronology of 20th Century Mathematicians

ˆ Greatest Mathematicians born between 1860 and 1975
ˆ Pictures of Famous 20th Century Mathematicians
ˆ The Story of Math Website

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