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Math History SummarySpring 2011

More or less chronological

Before Greek Mathematics

0.1 Africa

Lebombo bone, 35000 BC

Ishango bone, 19000 BC

0.2 Mesopotamia

tokens, 8000 BC–4000 BC — evolved into cuneiform

0.3 Egypt

hieroglyphics, hieratic numerals, 3200 BC – 200

fractions

• “Eye of Horus” fractions, Old Kingdom (before 2050 BC)

• hieroglyphic fractions after 2050 BC

Moscow papyrus, 1850 BC

• volume of a truncated pyramid

Rhind Mathematical Papyrus (RMP), 1650 BC

• arithmetic

• some linear equations (ax = b by false position)

• some areas of plane figures

• approximate squaring of circle

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0.4 Babylonia

cuneiform, 2000 BC

• sexagesimal

• used a space, then a placeholder dot as a zero

some linear equations

some quadratic equations by 2000 BC (by completing the square)

Plimpton 322, 1800 BC: table of Pythagorean triples

0.5 China

numeration system fully developed by 1400 BC

rod numerals by 200 BC

0.6 India

Baudhayana Sulbasutra, 800 BC

• approx squaring of circle

• approx for√

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• Pyth Thm for squares

Hindu (Brahmi) numerals, 3rd cent BC

place value, 8th-9th cent

0.7 Greece

Attic numeration, 500 BC

Ionic numeration, 400 BC

used Babylonian sexagesimal fractions but with their own notation

0.8 Italy

Roman numerals, 2nd cent BC

A few common fractions, mostly expressed as twelfths

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0.9 Remarks

All mathematics before the Greeks was concrete and by example

Ancient Greece

Thales, 600 BC: first proofs

Pythagoras, 500 BC

• founder of Pythagorean brotherhood

• number religion: “All is number”

• believed in value of contemplative life

Pythagoreans, from 500 BC

• number theory — many results

• Pythagorean triples

◦ Pythagoras could generate an infinite class of them◦ so could Plato

• lines (i) made up of discrete points; (ii) infinitely divisible

• used figurative numbers, probably arrangement of pebbles

• Pythagorean theorem

• incommensurables

• planets held in place by crystal spheres

• music theory

Zeno of Elea, 450 BC

• Zeno’s paradoxes: space and time are not discrete, motion is impossible

◦ big problem for Pythagoreans

Eudoxus of Cnidus, 400 BC

• geometric algebra

◦ based on theory of proportion that doesn’t use definite magnitudes◦ requires dimension-matching (only compare lengths with lengths, etc.)

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◦ avoids the problem of incommensurables—and also any solution

• method of exhaustion

Plato, 400 BC

• philosopher

• Academy (school of philosophy at Athens) — most talented mathematicians hungout there

• probably set down geometric construction rules

◦ straightedge and compass only, to keep geometry pure (i.e., about ideal objects)

three big geometry problems

• squaring the circle

• doubling the cube

• trisecting the angle

Hippocrates of Chios, 400 BC

• book on geometry

◦ propositions in logical chains◦ first squarable region (lune) not bounded by line segments

Alexander the Great, 325 BC

• conquered the then-known world, spread Greek culture everywhere

• beginning of Hellenistic period

Euclid, 325 BC

• Elements of Geometry

◦ axiomatized geometry; all results derived from a few axioms◦ first axiomatic system◦ Parallel Postulate (P5) controversial from the beginning◦ geometric algebra◦ number theory◦ formulas for all Pyth triples (without proof)

Archimedes, 3rd cent BC

• The Measurement of a Circle

◦ ratio of circumference to diameter, approximation of π

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• On the Sphere and the Cylinder

◦ surface area of sphere, other results

• The Sand-Reckoner

◦ shows how to extend the Greek numeration system to describe arbitrarily largenumbers

• On Spirals

◦ spiral of Archimedes

• Quadrature of the Parabola

◦ by a different use of the method of exhaustion

• The Method of Mechanical Theorems

◦ think of surfaces as “made up of” lines, volumes of revolution as “made up of”circles

◦ discovery technique, not a proof technique, for Archimedes

Hipparchus of Rhodes, 2nd cent BC: table of chords (like sines)

Apollonius of Perga, 200 BC: geometer, author of Conics

Eratosthenes, 200 BC

• prime number sieve

• good estimate of diameter of Earth

Diophantus, 250

• number theory (Arithmetica, unknown to Europe until 15th cent)

◦ problems solved ad hoc, no general methods

• some algebraic notation, but didn’t catch on

Proclus, 450

• geometer

• tried to revive Greek geometry (unsuccessfully)

• our source for much ancient work

India and China up to 900

0.10 China

Nine Chapters on the Mathematical Art, 263

• compilation of work from 200 BC with commentary by Liu Hui

• many mathematical techniques, all in practical settings

◦ arithmetic◦ areas of plane figures◦ volumes of solids◦ ratio and proportion◦ percentages◦ arithmetic and geometric progression◦ linear equations◦ systems of linear equations solved by Gaussian elimination◦ Pythagorean theorem (“Gougu rule”)◦ similar triangles◦ extracting square roots

0.11 India

Aryabhata, 500

• table of sines

Brahmagupta, 628

• explained negative numbers using debt and fortune

fractions written as one number over another (Muslims added the bar later)

Hindu development of place value, 8th-9th cent

zero as a number, 9th cent (Hindu)

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Medieval Muslims

al-Khwarizmi, 800

• taught Arabic world about Hindu numeration system

• book on algebra

◦ complete solution of quadratics with at least one positive real root

· classifies them into five types

House of Wisdom, 800–1300

• in Baghdad

• for translation of Persian works, then Greek

decimal fractions, 10th cent

Omar Khayyam, 1100

• great poet and mathematician

• reduced all cubics to 15 types, gave soln techniques

◦ did not know about complex roots, rejected neg roots◦ solutions are geometric, do not yield numbers

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Early Europe

Ptolemy, 200: table of chords (like Hipparchus’)

Charlemagne, 800

• establishment of monastery schools, uniform curriculum

◦ not much math taught beyond arithmetic

Reconquista retakes Toledo, 11th cent: Arabic versions of Greek books available to theWest

• some originals come back with Crusaders, 1100–1200

Fibonacci, 1200

• Liber Abaci

◦ taught Europe the Hindu numeration system◦ negative numbers for money

• Liber Quadratorum

◦ on quadratics and related problems◦ proof that Euclid’s formulas give all Pyth triples

great universities founded, 12th-13th cent

14th century awful: famine, plague, war, peasant revolts, collapse of authority

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Fifteenth Century Europe

originals of Greek works come to the West with scholars fleeing the Turkish conquest ofConstantinople (1453)

negative coefficients used by Chuquet, but work lost for centuries

arithmetic symbols develop, 15th-17th cent

Gutenberg, 1439: books become widely available

Maurolico, 1450:

one proof by mathematical induction

• didn’t understand it in general or realize its importance

governments stabilize

Renaissance in Italy; humanism

Regiomontanus makes trig a subject in its own right (1464)

Pacioli’s book (1494): arithmetic, some algebra; Problem of Points

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Sixteenth Century Europe

negative coefficients used by Stifel, 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)

◦ used complex numbers, but didn’t understand them at all (casus irreducibilis)

Copernicus, 1543: heliocentric solar system

solution of the quartic: Ferrari, 1548

Bombelli, 1572: complex numbers can be written a + b√−1, a,b ∈R

decimal fractions in Europe (Rudolff, Stevin, 1585)

Renaissance all over Europe

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Seventeenth Century Europe

negative solutions to equations allowed by Harriott, 1600

Viéte, 1600

• developed algebraic symbolism

• Viéte’s formulas: coefficients of a polynomial are symmetric functions of the roots

Gilbert

• Magnetism, 1600

◦ first completely experimentally based science book

Brahe, 1600

Kepler, 1600

• planets’ orbits are elliptical with Sun at one focus

• quantitative descriptions of orbits

Napier, 1614: logarithms

• idea is to match an arithmetic progression with a logarithmic one

Francis Bacon

• Novum Organum, 1620: only knowledge derived from experiment should be accepted

Descartes

• Discourse on Method, 1637

◦ beginning of modern philosophy◦ invention of analytic geometry◦ solution of “tangent problem”◦ knew that a polynomial of degree n must have n roots (no proof)◦ clockwork universe, started by God

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Mersenne, first half of century: salons, huge correspondence kept people in touch withothers’ work

Fermat, first half of century

• lawyer and magistrate; did math in spare time, did not publish

• quadrature of y = xpq by ad hoc method using rectangles

• method for finding extrema of some curves

• independent invention of analytic geometry

• number theory — many results, most without proof

◦ method of infinite descent◦ Last Theorem: no solution to xn + yn = zn in integers for n > 2

· not proved until 1995· attempts to prove it generated much good mathematics

• probability (with Pascal)

Pascal, first half of century

• number theory

◦ Pascal’s triangle, binomial coefficients◦ full development of mathematical induction

• geometry

◦ Mystic Hexagon theorem◦ many others, mostly lost

• probability (with Fermat)

• scientific work

• Pascaline, first adding machine

• Christian apologetics (Pensées)

Cavalieri, 1635

• quadrature of y = xn for small n

• Cavalieri’s Principle

◦ if two areas have equal cross-sections everywhere, they are equal; similarly forvolumes

Huygens, second half of century

• book on probability widely read

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• mathematical theory of harmonic motion

• wave theory of light, much other scientific work

Wallis, 1655: quadrature of xn, n ∈N (Arithmetica Infinitorum)

• big influence on Newton

Barrow

• finding tangents using the differential triangle

• explicitly let quantities→ 0

Newton

• physicist, mathematician

• General Binomial Theorem, 1665

◦ allows infinite-series expansion of some functions

• theory of fluxions (differential calculus), 1666

◦ manuscript De Analysi, 1669◦ curve generated by moving point

· curve is a fluent, velocity of generation is its fluxion

• universal gravitation, 1666

• corpuscular theory of light, 1666

• reflecting telescope, 1668

• Principia Mathematica, 1686-7

◦ theory and applications of universal gravitation◦ explanation of many phenomena from a few principles + mathematics and

logic

· elliptical orbits of planets and comets, tides, motion of bodies in fluids, etc.

◦ first really comprehensive (convincing) physical theory

• became Warden, then Master, of the Mint

• personally, very touchy

◦ hypochondriac, emotionally unstable◦ oversensitive to criticism, jealous of scientific priority◦ Arianism and devotion to alchemy made him secretive

Leibniz, second half of 17th into 18th cent

• philosopher and mathematician

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• developed much calculus

◦ geometric approach◦ product rule, Fundamental Theorem◦ great notation; we use it today

• long dispute with Newton over priority

• calculating machine, improvement on Pascal’s

• tried to develop a symbolic language in which all ideas could be expressed

◦ and rules for combining them

Jacob and Johann Bernoulli, late 17th into 18th cent

• development and applications of calculus and DEs

• Jacob: beginnings of calc of variations

• Jacob: book on probability, left unfinished

l’Hôpital, 1696: famous calculus text

• contained a bunch of Johann Bernoulli’s results without proper attribution

Ruffini almost proves that the general quintic and higher degree polynomial cannot besolved by radicals, 1799

• invents lots of mathematics to do it

• ignored

Eighteenth Century Europe

Saccheri

• tried to prove that adding the negation of the Parallel Postulate to the other postu-lates of Euclidean geometry resulted in a contradiction, 1733

◦ Saccheri quadrilaterals

Berkeley, 1734 (The Analyst): criticism of infinitesimals

• both Newton and Leibniz were concerned about these

DeMoivre, first half of century

• DeMoivre’s formula ((cos x + i sin x)n = cos(nx) + i sin(nx))

• probability

◦ uses normal dist to approx binomial; way early on this◦ conjectures Central Limit Theorem (CLT)

Taylor (early part of century)

• finite differences

• model of vibrating string

• Taylor series, Taylor’s Theorem

Maclaurin

• Maclaurin series

• Treatise on Fluxions, 1742

◦ convinced English mathematicians that calculus could be founded on geometry

d’Alembert

• 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

• leading philosophe and contributor to the Encyclopédie

Laplace, late 18th–early 19th cent

• celestial mechanics

• analysis/DEs

• determinants

• full development of probability theory using calculus

◦ full proof that least squares is best when errors are normally distributed◦ first proof of a CLT

Euler

• almost 900 mathematical publications

• calculus/DEs, esp. infinite series

• complex analysis

• graph theory

• number theory

◦ proved many of Fermat’s theorems

• sine, cosine are functions of a real variable

Lagrange, second half of 18th into 19th cent

• theoretical mechanics (Lagrangian mechanics)

◦ mechanics as pure mathematics◦ special solution to the three-body problem (Lagrange points)

• calculus/DEs (variation of parameters)

◦ tried to base calculus on infinite series

• calculus of variations

• number theory

Legendre

• mechanics

• number theory

• elliptic functions

• independently invented least squares (Gauss had it first)

• famous geometry text, first to displace Euclid

Nineteenth Century Europe

Gauss, first half of century

• mathematician, physicist

• construction of regular 17-gon

• least squares

◦ proof in restricted case

• Fundamental Theorem of Algebra

◦ four proofs, last one completely correct

• number theory

◦ modular arithmetic◦ quadratic reciprocity law

· Euler stated the law, Legendre gave an incorrect proof of it◦ conjecture of the prime number theorem

• differential geometry (Theorema Egregium, Gauss-Bonnet Theorem)

• complex plane, 1799

◦ not the first: Wessel, 1797◦ Argand also thought of it, 1806

• non-Euclidean geometry

◦ first to develop it◦ hyperbolic geometry◦ told only a few people at first

• orbit of Ceres

• magnetism (with Weber)

Fourier

• Fourier series, study of heat, 1822

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Bolyai

• independently developed hyperbolic geometry, 1823

Lobachevsky

• independently developed hyperbolic geometry, 1826

Poisson, early part of century

• celestial mechanics

• probability

Galois, early part of century

• relates solvability of polynomials by radicals to groups of symmetries of their roots

• proves that a polynomial is solvable iff its group is so

Dirichlet, first half of cent: number theory

• Dirichlet series, the zeta function

Abel, early part of century

• proof that quintic is unsolvable by radicals, 1824

• elliptic functions

Jacobi, first half of century

• elliptic functions, PDEs, determinants (the Jacobian)

Cauchy, first half of century

• precise defn of limit, derivative, continuity, sum of infinite series

◦ developed calculus from these; makes infinitesimals unnecessary

• Cauchy criterion for convergence of a sequence

• permutation groups (Cauchy’s theorem)

• complex analysis (Cauchy Integral Theorem, etc.)

W.R. Hamilton, first half of century

• mathematical physics (Hamiltonian mechanics)

• first “artificial” algebraic system (quaternions)

Riemann, middle of cent

• Riemann integral

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• elliptic geometry (non-Euclidean)

• elliptic functions

• analytic number theory (Riemann zeta function, the Riemann hypothesis)

Chebyshev, second half of cent

• number theory

• probability

◦ Chebyshev’s inequality◦ general forms of LLN and CLT

Weierstrass, second half of century

• “father of modern analysis”

◦ complete rigor◦ we do and teach analysis in his way

• much real, complex analysis

Kovalevsky, late 19th century

• DEs

• first female math professor

Cayley, 18th to early 19th cent

• permutations, matrices

• abstract groups

Klein, late 19th to early 20th cent

• geometry

◦ Erlanger Programm, 1872: general defn of geometry in terms of symmetrygroups

• number theory, function theory

Poincaré, 19th to early 20th cent

• DEs, dynamical systems, chaos (Poincaré-Bendixson theorem)

• complex analysis

• algebraic topology

◦ Poincaré conjecture (surfaces with same fundamental group as Sn are homeo-morphic to Sn)

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· finally proved by Perelman, 2003

Hilbert, 19th to early 20th cent

• algebraic geometry (Hilbert Basis Theorem, 1890; Nullstellensatz, 1892)

• algebraic number theory: organized the field

• complete and correct axiomatization of Euclidean geometry, 1899

• 23 problems, 1900 (outstanding problems that set the course of much of 20th centmathematics)

• functional analysis (Hilbert spaces)

• mathematical physics

• foundations (Hilbert’s program)

problem of negative numbers faded away with development of formal systems

all three great Greek geometry problems solved (all negatively)

Dedekind, 1872: resolution of magnitude-number problem dating from Greek times(Dedekind cuts)

Twentieth Century

Noether, early part of century

• ring theory, 1920s

• modern defn of ring

• many theorems, esp. in ideal theory

• mathematical physics (Noether’s Theorem)

Lebesgue, first half of cent

• Lebesgue measure, the Lebesgue integral

G.H. Hardy, first half of cent

• analytic number theory

• collaboration with Littlewood, Ramanujan

Ramanujan, early part of cent

• number theory (asymptotic formula for number of partitions, with Hardy; manyothers)

• infinite series

Hausdorff, first half of century

• algebra (Hausdorff Maximal Principle, 1914)

• topology (Hausdorff spaces, defn of top space in terms of neighborhoods, Hausdorffdimension)

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Stuff Described by Topic

Function concept

Aristotle, 350 BC used line segment to indicate durationOresme, 1350 perpendicular lines, one for duration, one for a quantity depending on itGalileo, 1638 a 1–1 mapping between concentric circlesLeibniz, 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 anotherFourier, 1822 function is any relation between quantitiesDirichlet, 1837 pretty modern; like Fourier’sFrege, late 19th cent function = set of ordered pairsWiener, 1914 fully modern defn

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 ingeneral

• Jacobi, 1830: determinants

• Sylvester: 1850: determinant theorems; 1884: rank-nullity theorem

• Cayley, middle of 19th cent: more general theory; inverse of a matrix; case ofCayley-Hamilton theorem

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• Frobenius, 1878: general theory

◦ full proof of C-H theorem, rank, orthogonality, etc.

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 is ignored

• Cauchy, 1815: groups of perms of roots of polynomials; 1844: groups of permuta-tions

• 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

Point-set topology

• Cantor, 1872–90: limit point, closed set, closure, density (all in R)

• Fréchet, 1904–5: metric spaces

• Hausdorff, 1914: defn of topological space in terms of neighborhoods; Hausdorffdimension

• Banach, 1922, 1932: Banach spaces

• Kuratowski, 1920: modern defn of top space (Kuratowski closure operation +Hausdorff’s neighborhoods)

• Alexandroff & Urysohn, 1929: compactness defined by open cover

• Bing, Nagata, Smirnov, 1950s: independent solutions of the metrization problem