Math History Summary By Topic
Spring 2011
Bolded items are more important.
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Numeration/Notation
Numeration
Egypt
• 3200BC – 200
• decimal; hieroglyphic, hieratic numerals
• fractions: unit fractions only
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Old Kingdom (before 2050 BC): Eye of Horus fractions; Middle
Kingdom: hieroglyphic fractions
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Rhind Papyrus, c. 1650 BC, is our most important source for
Egyptian mathematics
Babylonia
• tokens in Mesopotamia, 8000 BC – 2000 BC
• cuneiform, 2000 BC: sexagesimal
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dot for placeholder
• fractions: sexagesimal
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India
• Hindu (Brahmin) numerals, 3rd cent BC
• place value, 8th-9th cent
a
• fractions written b (Muslims added the bar later)
al-Khwarizmi, 800: book on numeration taught Muslim world the Hindu
numeration system
• worked in the House of Wisdom in Baghdad
10th century, decimal fractions in Muslim world
Fibonacci, 1200: taught Europe the Hindu numeration system (Liber Abaci)
Notation
Diophantus, 250: some algebraic notation, didn’t catch on
15-17th centuries, symbols for arithmetic develop
16th century, decimal fractions in Europe (Rudolff, Stevin)
Viéte, 1600: symbolism for algebra
Leibniz, early 18th century: notation for calculus
3
Algebra
Babylonia, 2000 BC: some linear equations;
solving some quadratics by completing the square
Egypt, Rhind Mathematical Papyrus, 1650 BC: some linear
equations
Egyptian, Babylonian mathematics all examples using specific numbers
• no proofs
• no abstractions
Pythagorean theorem led to incommensurables
Zeno’s paradoxes resulted in avoidance of study of infinity
(“horror infiniti”)
• wanted to show that change (specifically, motion) was impossible
Eudoxus, 400 BC: geometric algebra
• essentially algebraic problems recast as geometrical to avoid prob-
lems with irrational numbers
• only objects with same dimension can be equated
• solutions are line segments, not numbers
4
al-Khwarizmi, 800: book on algebra
(“completion and balancing”)
• solutions to all quadratics with at least one positive real root
• classifies them into five types
Khayyam, 1100: geometric solutions to many cubics
Fibonacci, 1225: book on quadratics and problems leading to them (Liber
Quadratorum)
Stifel, 15th century: allows negative coefficients but not neg solns
to equations
Solution of the cubic
• del Ferro, 1500: x3 + px = q
• Tartaglia, 1530: x3 + px2 = q
• Cardano, 1540: general cubic (Cardano’s formulas)
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used complex numbers, but didn’t understand them at all (casus
irreducibilis)
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but they seemed unavoidable
Ferrari, 1548: solution of the quartic
√
−1, a, b ∈ R
Bombelli, 1572: complex numbers can be written a + b −1
Harriott, 1600: negative solutions to equations allowed;
move all terms to one side to solve equations
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Viéte, 1600
• developed algebraic symbolism
• Viéte’s formulas: coefficients of a polynomial are symmetric functions of
the roots
Descartes (Discourse on Method), 1637: invention of analytic
geometry, connecting algebra and geometry
• showed that Eudoxus’ dimensional restriction was unnecessary by
showing that all geometric computations could be considered to result in lengths
• knew that a polynomial of degree n must have n roots (no proof)
Ruffini, 1799: almost proves that the general quintic and higher
degree polynomial cannot be solved by radicals
• invents lots of mathematics to do it, including stuff about permutation
groups
• ignored
Gauss
• 1799: Fundamental Theorem of Algebra
• 1801: modular arithmetic; amounts to much of abelian group theory
Abel, 1824: unsolvability of quintic
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Galois, 1832: solvability of polynomials by radicals linked to
properties of groups of permutations of their roots
• normal subgroups of those groups
Cauchy, 1834: studied permutation groups
• products, order of a perm, cycles, conjugacy, Cauchy’s theorem
W.R. Hamilton, 1843: quaternions, first “artificial” algebraic
system
• development of such systems led to problems with negative numbers
fading away
Kummer, 1844: ideal numbers (generalization of integers)
Cayley, 1849, 1878: abstract groups, group tables
Dedekind, 1871: ideals, prime ideals (in the Gaussian integers)
van Dyck, 1882-3: free groups, generators and relations
Hilbert, 1888: Hilbert Basis Theorem
Burnside, 1897: modern group theory
Fraenkel, 1914: first definition of an abstract ring
Noether, 1920: modern defn of ring; many theorems, esp. in ideal theory
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Number Theory
Before Greeks, just arithmetic
Pythagoreans, from 500 BC
• many results
• used figurative numbers
• all things held in common, including credit for mathematical results
• Pythagorean triples: could generate infinitely many, but not all
• Pythagorean theorem implied existence of incommensurables
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threatened Pythagorean idea that all quantities were ratios of
integers
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Eudoxus’ “geometric algebra” avoided the problem by dealing
only with magnitudes, not with numbers
Euclid, 325 BC: Elements: two chapters on number theory
• formulas for all possible Pythagorean triples, but no proof
Eratosthenes, 200 BC: Sieve of Eratosthenes for finding primes
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Diophantus, 250 (Arithmetica): many number theory problems
Brahmagupta, 7th century: explains negative numbers by
“debt and fortune”
India, 9th century: zero is a number
Fibonacci, 1225 (Liber Quadratorum): proof that Euclid’s formulas
give all Pythagorean triples
Fermat, first half of 17th century
• many problems, theorems, most without proof
• method of infinite descent
• Last Theorem: no solution to x n + yn = zn in integers for n > 2
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not proved until 1995
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attempts to prove it generated much good mathematics
Pascal, first half of 17th century: Pascal’s triangle, connection to
binomial coefficients
• full development of mathematical induction from Maurolico’s first
use
Euler, 18th century
• proofs of many of Fermat’s theorems
• conjectured the quadratic reciprocity law
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Lagrange, second half of 18th century into 19th
• Wilson’s theorem, solution to a Pell’s equation, proofs of many of Fer-
mat’s theorems
Legendre, late 18th into 19th century
• conjectured a form of the quadratic reciprocity law
• contributed to proof of Fermat’s Last Theorem
Gauss
• 1801: Disquisitiones Arithmeticae: modular arithmetic
• proof of the quadratic reciprocity law
• conjectured the prime number theorem
Dirichlet, first half of 19th century
• Dirichlet series, the zeta function
• this is the beginning of analytic number theory
Riemann, 1859: Riemann zeta function, theorems on the
distribution of primes
Dedekind, second half of 19th into 20th century
• algebraic number fields, ideals, zeta function of a number field
• Dedekind cuts construct R from Q
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but require the use of completed infinities
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Hilbert
• 1893: synthesis of algebraic number theory based on Dedekind’s
work
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formed algebraic number theory into a field with its own methods and results
• some of his 23 problems were number-theoretic and were very influential
Hardy/Littlewood/Ramanujan, first half of 20th century: many
results in number theory
11
Geometry
Before the Greeks, some formulas for areas of plane figures,
volumes of solids
Thales, 600 BC: first proofs, a few theorems (Thales’ theorem)
Pythagoreans, from 500 BC
• many theorems (Pythagorean theorem)
• Platonic solids
• believed both that lines were made up of points and that they were
infinitely divisible
Eudoxus, 400 BC: method of exhaustion
• no records left; Archimedes says he invented it
Three big classical problems of geometry
• squaring the circle
• doubling the cube
• trisecting the angle
Hippocrates of Chios, 400 BC: worked on all three of the big
problems
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Euclid
• 325 BC: Elements of Geometry
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axiomatized geometry; all results derived from a few axioms
·
·
five Common Notions: assumptions about quantity, especially equality
five Postulates: specifically geometric assumptions
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first axiomatic system
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Parallel Postulate (P5) controversial from the beginning
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geometric algebra
Archimedes, 3rd century BC
• The Measurement of a Circle
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ratio of circumference to diameter, approximation of π
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method of exhaustion: approximate circumference more and
more closely by polygons the perimeters of which can be calculated
• On the Sphere and the Cylinder
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surface area of sphere, other results
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• The Sand-Reckoner
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shows how to extend the Greek numeration system to describe arbitrarily large numbers
• On Spirals
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spiral of Archimedes
• Quadrature of the Parabola
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by a different use of the method of exhaustion
• The Method of Mechanical Theorems
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think of surfaces as “made up of” lines, volumes of revolution as
“made up of” circles
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discovery technique, not a proof technique, for Archimedes
Apollonius of Perga, 200 BC: Conics
Proclus, 450: our source for much ancient work
• tried to revive Greek geometry (unsuccessfully)
Saccheri
• 1733: tried to prove that adding the negation of the Parallel Postulate
to the other postulates of Euclidean geometry resulted in a contradiction
• Saccheri quadrilaterals
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Pascal, 1639: Mystic Hexagon Theorem
Legendre, 1794: famous geometry text, first to displace Euclid
Gauss
• 1796: construction of regular 17-gon
• 1816–1824: non-Euclidean geometry
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first to develop it
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hyperbolic geometry
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told only a few people at first
• 1827: differential geometry (Theorema Egregium, Gauss-Bonnet Theo-
rem)
Bolyai, 1823: independently developed hyperbolic geometry
Lobachevsky, 1826: independently developed hyperbolic geometry
Riemann, 1854: elliptic geometry
Beltrami, 1868:
• Parallel Postulate is independent of the other four
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proved that geometry of geodesics on the pseudosphere was hyperbolic geometry
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found a map from the plan to the pseudosphere that preserved
angles
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this sent Postulates 1–4 to true statements
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but Postulate 5 is not true on the pseudosphere, so it cannot be
proved from the other four
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• hyperbolic geometry consistent iff Euclidean geometry is so
Klein, 1872: Erlanger Programm: general defn of geometry in
terms of symmetry groups
Hilbert
• 1892: Nullstellensatz (algebraic geometry)
• 1899: first completely rigorous axiomatization of Euclidean geome-
try
Poincaré, 1895: invented algebraic topology
• Poincaré conjecture (surfaces with same fundamental group as Sn are
homeomorphic to Sn)
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finally proved by Perelman, 2003
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Calculus/Analysis
Archimedes
• used method of exhaustion two different ways to approximate ratio of
circumference of circle to diameter and to do quadrature of parabola
• Method of Mechanical Theorems
Napier, 1614: logarithms
Descartes, 1637: analytic geometry, solution of tangent problem
Fermat, first half of 17th century
• independent invention of analytic geometry
p
q
• quadrature of y = x (by ad hoc method)
• method for finding extrema of some curves
Pascal, first half of 17th century: quadrature of sine curve
Cavalieri, first half of 17th century
• quadrature of y = xn for small n
• Cavalieri’s Principle
Wallis, 1655: quadrature of y = xn
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Barrow, mid 17th century: finding tangents using the differential triangle
• explicitly let quantities → 0
Newton
• 1665: General Binomial Theorem
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allows infinite-series expansion of some functions
• 1666: method of fluxions (differential calculus)
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manuscript De Analysi, 1669
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curve generated by moving point
·
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curve is a fluent, velocity of generation is its fluxion
algebraic approach based on binomial theorem
• 1687: Principia (mathematical physics)
Leibniz, 1670s: developed much calculus
• geometric approach
• product rule, Fundamental Theorem of Calculus
• great notation; we use it today
Bernoullis
• Jacob and Johann, late 17th into 18th cent
• development and applications of calculus and DEs
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• Jacob: beginnings of calc of variations
• Jacob: book on probability, left unfinished
Berkeley, 1734: The Analyst: criticism of infinitesimals
• in calculations, people first divided by these (so they can’t be zero)
and then threw them away and treated the results as exact (so the
must be zero)
• both Newton and Leibniz were concerned about them
Taylor, 1715: finite differences, Taylor series, Taylor’s Theorem
Maclaurin
• Maclaurin series
• 1742: Treatise on Fluxions
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convinced English mathematicians that calculus could be founded
on geometry
d’Alembert, mid-18th century
• mechanics, calculus/DEs (esp. PDE—wave equation)
• idea of limit, but too vague to be useful
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thought calculus should somehow be based on limits
• ratio test
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Laplace, late 18th–early 19th cent
• analysis/DEs
• celestial mechanics
• determinants
• full development of probability theory using calculus
Euler, 18th century
• calculus/DEs, esp. infinite series
• complex analysis
• definitions of function—first in terms of formulas, then in terms of
functional dependency
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sine, cosine are functions of a real variable
• invented graph theory for solution of Seven Bridges problem
Lagrange, second half of 18th into 19th cent
• theoretical mechanics (Lagrangian mechanics)
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mechanics as pure mathematics
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special solution to the three-body problem (Lagrange points)
• calculus/DEs (variation of parameters)
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tried to base calculus on infinite series
• calculus of variations
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Legendre, 18th to 19th century: mechanics, elliptic functions
Bolzano (late 18th–19th century): much work on limits
• mostly ignored; Cauchy and Weierstrass had to rediscover it
Gauss, late 18th–19th century
• differential geometry (Theorema Egregium, Gauss-Bonnet Theorem)
• complex plane, 1799
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not the first: Wessel, 1797
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Argand also thought of it, 1806
Fourier, 1822: Fourier series, study of heat
Jacobi, first half of 19th century: elliptic functions, PDEs,
determinants (the Jacobian)
• Abel did similar work on elliptic functions at about the same time
Dirichlet, first half of 19th century: Dirichlet series, the zeta function
Cauchy
• precise defn of limit, derivative, continuity, sum of infinite series
• developed calculus from these; makes infinitesimals unnecessary
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this finally answered Berkeley’s criticism of calculus
• Cauchy criterion for convergence of a sequence
• complex analysis (Cauchy Integral Theorem, etc.)
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Riemann, mid-19th century
• Riemann integral
• elliptic functions
• analytic number theory (Riemann zeta function, the Riemann hy-
pothesis)
Weierstrass, second half of 19th century
• “father of modern analysis”
• complete rigor
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we do and teach analysis in his way
• much real, complex analysis
Poincaré, 19th to early 20th cent
• DEs, dynamical systems, chaos (Poincaré-Bendixson theorem)
• complex analysis
Hilbert, 19th to early 20th cent
• functional analysis (Hilbert spaces)
• mathematical physics
• address in 1900 gave 23 problems which set course for much of 20th
century mathematics
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Function concept
Aristotle, 350 BC: used line segment to indicate duration
Oresme, 1350: perpendicular lines, one for duration, one for a quantity
depending on it
Galileo, 1638: a 1–1 mapping between concentric circles
Leibniz, 1692: “function” : tangent line as function of point on curve
(and other geometric dependencies)
Euler, 18th cent: defn first in terms of algebraic formulæ, later as one
quantity depending on another
Fourier, 1822: function is any relation between quantities
Dirichlet, 1837: pretty modern; like Fourier’s
Frege, late 19th cent: function = set of ordered pairs
Wiener, 1914: fully modern defn
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Infinity
Greeks allowed only “potential infinities,” not completed ones
Zeno’s paradoxes resulted in avoidance of study of infinity
• wanted to show that change (specifically, motion) was impossible
Aristotle, 4th century BC
• allowed only “potential infinities” (processes that never have to stop,
like counting), not “completed infinities” (infinite sets, like N)
Augustine (400) accepted the totality of the natural numbers as a real thing
Aquinas (1250) accepted the infinite divisibility of the line
Gauss (19th century) agreed with no completed infinity
Bolzano (early 19th century): paradoxes of infinite sets (mostly ignored)
Kronecker (19th century) begins constructivism
• mathematical objects exist only if an algorithm can be given to con-
struct them
Cantor, late 19th century: consistent theory of infinite sets
• definition of set, equal cardinality of sets, ordinals
• proved Q countable, R uncountable (Cantor’s diagonal argument)
• proved card(Rn ) = card(R)
• conjectured the well-ordering axiom and the continuum hypothesis
• theory met with much resistance (and some support)
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Matrices
appear in the Nine Chapters, 263
Vandermonde uses idea of determinants, 1772
used but not studied by Gauss, 1801
studied by Cauchy, 1812
• determinant theroems, eigenvalues, diagonalization, but none of these
in general
Jacobi, 1830: determinants
Sylvester: 1850: determinant theorems; 1884: rank-nullity theorem
Cayley, middle of 19th cent: more general theory; inverse of a matrix;
case of Cayley-Hamilton theorem
Frobenius, 1878: general theory
• full proof of C-H theorem, rank, orthogonality, etc.
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Group theory
started with Euler and Gauss, 18th to first part of 19th cent — modular
arithmetic
Lagrange, 1771: studied perms, but didn’t define a product
Ruffini, 1799: proved lots of stuff about perm groups, but was ignored
Cauchy, 1815: groups of perms of roots of polynomials; 1844: groups of
permutations
Galois, 1831: normal subgroups
Cayley, 1849: abstract groups, group tables; 1878: much theory
van Dyck, 1882-3: free groups, generators and relations
Burnside, 1897: modern group theory
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