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| Cronologia della matematica |
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About 25000BC
Early geometric designs used.
About 5000BC
A decimal number system is in use in Egypt.
About 4000BC
Babylonian and Egyptian calendars in use.
About 3400BC
The first symbols for numbers, simple straight lines, are used in Egypt.
About 3000BC
The abacus is developed in the Middle East and in areas around
the Mediterranean. A somewhat different type of abacus is used in China.
About 3000BC
Hieroglyphic numerals in use in Egypt. (See this History
Topic.)
About 3000BC
Babylonians begin to use a sexagesimal number system for recording financial transactions.
It is a place-value system without a zero place value. (See this History
Topic.)
About 2770BC
Egyptian calendar used.
About 2000BC
Harappans adopt a uniform decimal system of weights and measures.
About 1950BC
Babylonians solve quadratic equations.
About 1900BC
The Moscow papyrus (also called the Golenishev papyrus) is written. It
gives details of Egyptian geometry. (See this History
Topic.)
About 1850BC
Babylonians know Pythagoras's
Theorem. (See this History
Topic.)
About 1800BC
Babylonians use multiplication tables.
About 1750BC
The Babylonians solve linear and quadratic algebraic equations, compile
tables of square and cube roots. They use Pythagoras's
theorem and use mathematics to extend knowledge of astronomy. (See this
History
Topic.)
About 1700BC
The Rhind papyrus (sometimes called the Ahmes
papyrus) is written. It shows that Egyptian mathematics has developed
many techniques to solve problems. Multiplication is based on repeated
doubling, and division uses successive halving. (See this History
Topic.)
About 1360BC
A decimal number system with no zero starts to be used in China.
About 1000BC
Chinese use counting boards for calculation.
About 800BC
Baudhayana
is the author of one of the earliest of the Indian Sulbasutras. (See this
History
Topic.)
About 750BC
Manava
writes a Sulbasutra. (See this History
Topic.)
About 600BC
Apastamba
writes the most interesting Indian Sulbasutra from a mathematical point
of view. (See this History
Topic.)
575BC
Thales
brings Babylonian mathematical knowledge to Greece. He uses geometry to
solve problems such as calculating the height of pyramids and the distance
of ships from the shore.
About 540BC
Counting rods used in China.
530BC
Pythagoras
of Samos moves to Croton in Italy and teaches mathematics, geometry,
music, and reincarnation.
About 500BC
The Babylonian sexagesimal number system is used to record and predict
the positions of the Sun, Moon and planets. (See this History
Topic.)
About 500BC
Panini's
work on Sanskrit grammar is the forerunner of the modern formal language
theory.
About 465BC
Hippasus writes of a "sphere of 12 pentagons", which must refer to a dodecahedron.
About 450BC
Greeks begin to use written numerals. (See this History
Topic.)
About 450BC
Zeno
of Elea presents his paradoxes.
About 440BC
Hippocrates
of Chios writes the Elements which is the first compilation
of the elements of geometry.
About 430BC
Hippias
of Elis invents the quadratrix which may have been used by him for
trisecting an angle and squaring the circle.
About 425BC
Theodorus
of Cyrene shows that certain square roots are irrational. This had been shown earlier but it is not known
by whom.
About 400BC
Babylonians use a symbol to indicate an empty place in their numbers recorded
in cuneiform writing. There is no indication that this was in any way
thought of as a number. (See this History
Topic.)
387BC
Plato
founds his Academy in Athens
About 375BC
Archytas
of Tarentum develops mechanics. He studies the "classical problem"
of doubling the cube and applies
mathematical theory to music. He also constructs the first automaton.
About 360BC
Eudoxus
of Cnidus develops the theory of proportion, and the method of exhaustion.
About 340BC
Aristaeus
writes Five Books concerning Conic Sections.
About 330BC
Autolycus
of Pitane writes On the Moving Sphere which studies the geometry
of the sphere. It is written as an astronomy text.
About 320BC
Eudemus
of Rhodes writes the History of Geometry.
About 300BC
Euclid
gives a systematic development of geometry in his Stoicheion (The
Elements). He also gives the laws of reflection in Catoptrics.
About 290BC
Aristarchus
of Samos uses a geometric method to calculate the distance of the
Sun and the Moon from Earth. He also proposes that the Earth orbits the
Sun.
About 290BC
The Chinese classic Chou pei suan ching is written.
About 250BC
In On the Sphere and the Cylinder, Archimedes
gives the formulae for calculating the volume of a sphere and a cylinder.
In Measurement of the Circle he gives an approximation of the value
of 
with a method which will allow
improved approximations. In Floating Bodies he presents what is
now called "Archimedes' principle" and begins the study of hydrostatics.
He writes works on two- and three-dimensional geometry, studying circles,
spheres and spirals. His ideas are far ahead of his contemporaries and
include applications of an early form of integration.
About 235BC
Eratosthenes
of Cyrene estimates the Earth's circumference with remarkable accuracy
finding a value which is about 15% too big.
About 230BC
Nicomedes
writes his treatise On conchoid lines which contain his discovery
of the curve known as the "Conchoid of Nicomedes".
About 225BC
Apollonius
of Perga writes Conics in which he introduces the terms "parabola", "ellipse" and "hyperbola".
About 230BC
Eratosthenes
of Cyrene develops his sieve method for finding all prime numbers. (See this History
Topic.)
About 200BC
Diocles
writes On burning mirrors, a collection of sixteen propositions
in geometry mostly proving results on conics.
About 190BC
Chinese mathematicians use powers of 10 to express magnitudes.
127BC
Hipparchus
discovers the precession of the equinoxes and calculates the length of
the year to within 6.5 minutes of the correct value. His astronomical
work uses an early form of trigonometry.
About 150BC
Hypsicles
writes On the Ascension of Stars. In this work he is the first
to divide the Zodiac into 360 degrees.
About 100BC
Chinese mathematicians are the first to introduce negative numbers.
About 1AD
Chinese mathematician Liu Hsin uses decimal fractions.
About 20
Geminus
writes a number of astronomy texts and The Theory of Mathematics.
He tries to prove the parallel postulate. (See
this History
Topic.)
About 50
Chinese mathematician Sun-tzi presents the first known example of an indeterminate
equation.
About 60
Heron
of Alexandria writes Metrica (Measurements). It contains
formulas for calculating areas and volumes.
About 90
The Chinese invent magic squares.
About 90
Nicomachus
of Gerasa writes Arithmetike eisagoge (Introduction to Arithmetic)
which is the first work to treat arithmetic as a separate topic from geometry.
About 100
The classical Chinese mathematics text Jiuzhang Suanshu (Nine
Chapters on the Mathematical Art) begins to be assembled.
About 110
Menelaus
of Alexandria writes Sphaerica which deals with spherical triangles
and their application to astronomy.
About 150
Ptolemy
produces many important geometrical results with applications in astronomy.
His version of astronomy will be the accepted one for well over one thousand
years.
About 250
The Maya civilization of Central America uses an almost place-value number
system to base 20. (See this History
Topic.)
250
Diophantus
of Alexandria writes Arithmetica, a study of number theory
problems in which only rational numbers are allowed as solutions.
263
By using a regular polygon with 192 sides Liu Hui calculates the value
of as 3.14159 which is correct to five
decimal places. (See this History
Topic.)
301
Iamblichus writes on astrology and mysticism. His Life
of Pythagoras is a fascinating account.
340
Pappus
of Alexandria writes Synagoge (Collections) which is
a guide to Greek geometry.
390
Theon
of Alexandria produces a version of Euclid's
Elements (with textual changes and some additions) on which almost
all subsequent editions are based.
About 400
Hypatia
writes commentaries on Diophantus
and Apollonius.
She is the first recorded female mathematician and she distinguishes herself
with remarkable scholarship. She becomes head of the Neo-Platonist school
at Alexandria.
450
Proclus,
a mathematician and Neo-Platonist, is one of the last philosophers at
Plato's Academy at Athens.
About 460
Tsu
Ch'ung Chi gives the approximation 355/113 to
which is correct to 6 decimal
places. (See this History
Topic.)
499
Aryabhata
I calculates
to be 3.1416. He produces his Aryabhatiya, a treatise on quadratic
equations, the value of ,
and other scientific problems.
About 500
Metrodorus assembles the Greek Anthology consisting of 46 mathematical
problems.
510
Eutocius
of Ascalon writes commentaries on Archimedes'
work.
510
Boethius
writes geometry and arithmetic texts which are widely used for a long
time.
About 530
Eutocius
writes commentaries on the works of Archimedes
and Apollonius.
532
Anthemius
of Tralles, a mathematician of note, is the architect for the Hagia
Sophia at Constantinople. (See this History
Topic.)
534
Chinese mathematics is introduced into Japan.
575
Varahamihira
produces Pancasiddhantika (The Five Astronomical Canons).
He makes important contributions to trigonometry.
594
Decimal notation is used for numbers in India. This is the system on which
our current notation is based. (See this History
Topic.)
628
Brahmagupta
writes Brahmasphutasiddanta (The Opening of the Universe),
a work on astronomy; on mathematics. He uses zero and negative numbers,
gives methods to solve quadratic equations, sum series, and compute square
roots.
About 700
Mathematicians in the Mayan civilization introduce a symbol for zero into
their number system. (See this History
Topic.)
729
Hsing introduces a new calendar into China, correcting many errors in
earlier calendars.
732
Qutan Zhuan accuses Hsing of copying an Indian calendar in producing his
own. However Hsing's Chinese calendar is far more accurate than the Indian
one.
About 775
Alcuin
of York writes elementary texts on arithmetic, geometry and astronomy.
About 790
Chinese begin to use finite difference methods.
About 810
House of Wisdom set up in Baghdad. There Greek and Indian mathematical
and astronomy works are translated into Arabic.
About 810
Al-Khwarizmi
writes important works on arithmetic, algebra, geography, and astronomy.
In particular Hisab al-jabr w'al-muqabala (Calculation by Completion
and Balancing), gives us the word "algebra", from "al-jabr". From
al-Khwarizmi's name, as a consequence of his arithmetic book, comes the
word "algorithm".
About 850
Thabit
ibn Qurra makes important mathematical discoveries such as the extension
of the concept of number to (positive) real numbers, integral calculus,
theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry.
About 850
Thabit
ibn Qurra writes Book on the determination of amicable numbers
which contains general methods to construct amicable numbers. He knows
the pair of amicable numbers 17296, 18416.
850
Mahavira
writes Ganita Sara Samgraha. It consists of nine chapters and includes
all mathematical knowledge of mid-ninth century India.
900
Sridhara
writes the Trisatika (sometimes called the Patiganitasara)
and the Patiganita. In these he solves quadratic equations, sums
series, studies combinations, and gives methods of finding the areas of
polygons.
About 900
Abu
Kamil writes Book on algebra which studies applications of
algebra to geometrical problems. It will be the book on which Fibonacci
will base his works.
920
Al-Battani
writes Kitab al-Zij a major work on astronomy in 57 chapters. It
contains advances in trigonometry.
950
Gerbert of Aurillac (later Pope Sylvester II) reintroduces the abacus
into Europe. He uses Indian/Arabic numerals without having a zero.
About 960
Al-Uqlidisi
writes Kitab al-fusul fi al-hisab al-Hindi which is the earliest
surviving book that presents the Hindu system.
About 970
Abu'l-Wafa
invents the wall quadrant for the accurate measurement of the declination of stars in the sky. He writes important books
on arithmetic and geometric constructions. He introduces the tangent function
and produces improved methods of calculating trigonometric tables.
976
Codex Vigilanus copied in Spain. Contains the first evidence of
decimal numbers in Europe.
About 990
Al-Karaji
writes Al-Fakhri in Baghdad which develops algebra. He gives Pascal's triangle.
About 1000
Ibn
al-Haytham (often called Alhazen) writes works on optics, including
a theory of light and a theory of vision, astronomy, and mathematics,
including geometry and number theory. He gives Alhazen's problem: Given
a light source and a spherical mirror, find the point on the mirror were
the light will be reflected to the eye of an observer.
About 1010
Al-Biruni
writes on many scientific topics. His work on mathematics covers arithmetic,
summation of series, combinatorial analysis, the rule of three, irrational
numbers, ratio theory, algebraic definitions, method of solving algebraic
equations, geometry, Archimedes' theorems, trisection of the angle and
other problems which cannot be solved with ruler and compass alone,
conic sections, stereometry, stereographic projection,
trigonometry, the sine theorem in the plane, and solving spherical triangles.
About 1020
Ibn
Sina (usually called Avicenna) writes on philosophy, medicine, psychology,
geology, mathematics, astronomy, and logic. His important mathematical
work Kitab al-Shifa' (The Book of Healing) divides mathematics
into four major topics, geometry, astronomy, arithmetic, and music.
1040
Ahmad
al-Nasawi writes al-Muqni'fi al-Hisab al-Hindi which studies
four different number systems. He explains the operations of arithmetic,
particularly taking square and cube roots in each system.
About 1050
Hermann
of Reichenau (sometimes called Hermann the Lame or Hermann Contractus)
writes treatises on the abacus and the astrolabe. He introduces into Europe the astrolabe, a portable
sundial and a quadrant with a cursor.
1072
Al-Khayyami
(usually known as Omar Khayyam) writes Treatise on Demonstration of
Problems of Algebra which contains a complete classification of cubic
equations with geometric solutions found by means of intersecting conic
sections. He measures the length of the year to be 365.24219858156 days,
a remarkably accurate result.
1093
Shen
Kua writes Meng ch'i pi t'an (Dream Pool Essays), which
is a work on mathematics, astronomy, cartography, optics and medicine.
It contains the earliest mention of a magnetic compass.
1130
Jabir
ibn Aflah writes works on mathematics which, although not as good
as many other Arabic works, are important since they will be translated
into Latin and become available to European mathematicians.
About 1140
Bhaskara
II (sometimes known as Bhaskaracharya) writes Lilavati (The
Beautiful) on arithmetic and geometry, and Bijaganita (Seed
Arithmetic), on algebra.
1142
Adelard
of Bath produces two or three translations of Euclid's
Elements from Arabic.
1144
Gherard
of Cremona begins translating Arabic works (and Arabic translations
of Greek works) into Latin.
1149
Al-Samawal
writes al-Bahir fi'l-jabr (The brilliant in algebra). He
develops algebra with polynomials using negative powers and zero. He solves
quadratic equations, sums the squares of the first n natural numbers,
and looks at combinatorial problems.
1150
Arabic numerals are introduced into Europe with Gherard
of Cremona's translation of Ptolemy's
Almagest. The name of the "sine" function comes from this translation.
About 1200
Chinese start to use a symbol for zero. (See this History
Topic.)
1202
Fibonacci
writes Liber abaci (The Book of the Abacus), which sets
out the arithmetic and algebra he had learnt in Arab countries. It also
introduces the famous sequence of numbers now called the "Fibonacci sequence".
1225
Fibonacci
writes Liber quadratorum (The Book of the Square), his most
impressive work. It is the first major European advance in number theory
since the work of Diophantus
a thousand years earlier.
About 1225
Jordanus
Nemorarius writes on astronomy. In mathematics he uses letters in
an early form of algebraic notation.
About 1230
John
of Holywood (sometimes called Johannes de Sacrobosco) writes on arithmetic,
astronomy and calendar reform.
1247
Ch'in
Chiu-Shao writes Mathematical Treatise in Nine Sections. It
contains simultaneous integer congruences and the Chinese Remainder Theorem.
It considers indeterminate equations, Horner's
method, areas of geometrical figures and linear simultaneous equations.
1248
Li Yeh writes a book which contains negative numbers, denoted by putting
a diagonal stroke through the last digit.
About 1260
Campanus
of Novara, chaplain to Pope Urban IV, writes on astronomy and publishes
a Latin edition of Euclid's
Elements which became the standard Euclid for the next 200 years.
1275
Yang
Hui writes Cheng Chu Tong Bian Ben Mo (Alpha and omega of
variations on multiplication and division). It uses decimal fractions
(in the modern form) and gives the first account of Pascal's triangle.
1303
Chu
Shih-Chieh writes Szu-yuen Yu-chien (The Precious Mirror
of the Four Elements), which contains a number of methods for solving
equations up to degree 14. He also defines what is now called Pascal's triangle and shows
how to sum certain sequences.
1321
Levi
ben Gerson (sometimes known as Gersonides) writes Book of Numbers
dealing with arithmetical operations, permutations and combinations.
1328
Bradwardine
writes De proportionibus velocitatum in motibus which is an early
work on kinematics using algebra.
1335
Richard of Wallingford writes Quadripartitum de sinibus demonstratis,
the first original Latin treatise on trigonometry.
1336
Mathematics becomes a compulsory subject for a degree at the University
of Paris.
1342
Levi
ben Gerson (Gersonides) writes De sinibus, chordis et arcubus
(Concerning Sines, Chords and Arcs), a treatise on trigonometry
which gives a proof of the sine theorem for plane triangles and gives
five figure sine tables.
1343
Jean de Meurs writes Quadripartitum numerorum (Four-fold Division
of Numbers), a treatise on mathematics, mechanics, and music.
1343
Levi
ben Gerson (Gersonides) writes De harmonicis numeris (Concerning
the Harmony of Numbers), which is a commentary on the first five books
of Euclid.
1364
Nicole
d'Oresme writes Latitudes of Forms, an early work on coordinate
systems which may have influence Descartes.
Another work by Oresme contains the first use of a fractional exponent.
1382
Nicole
d'Oresme publishes Le Livre du ciel et du monde (The Book
of Heaven and Earth). It is a compilation of treatises on mathematics,
mechanics, and related areas. Oresme opposed the theory of a stationary
Earth.
1400
Madhava
of Sangamagramma proves a number of results about infinite sums giving
Taylor
expansions of trigonometric functions. He uses these to find an approximation
for correct to
11 decimal places.
1411
Al-Kashi
writes Compendium of the Science of Astronomy.
1424
Al-Kashi
writes Treatise on the Circumference giving a remarkably good approximation
to in both sexagesimal and decimal forms.
1427
Al-Kashi
completes The Key to Arithmetic containing work of great depth
on decimal fractions. It applies arithmetical and algebraic methods to
the solution of various problems, including several geometric ones and
is one of the best textbooks in the whole of medieval literature.
1434
Alberti
studies the representation of 3-dimensional objects and writes the first
general treatise Della Pictura on the laws of perspective.
1437
Ulugh
Beg publishes his star catalogue Zij-i Sultani. It contains
trigonometric tables correct to eight decimal places based on Ulugh Beg's
calculation of the sine of one degree which he calculated correctly to
16 decimal places.
1450
Nicholas
of Cusa studies geometry and logic. He contributes to the study of
infinity, studying the infinitely large and the infinitely small. He looks
at the circle as the limit of regular polygons.
About 1470
Chuquet
writes Triparty en la science des nombres, the earliest French
algebra book.
1472
Peurbach
publishes Theoricae Novae Planetarum (New Theory of the Planets).
He uses Ptolemy's
epicycle theory of the planets but believes they are controlled
by the sun.
1474
Regiomontanus
publishes his Ephemerides, astronomical tables for the years 1475
to 1506 AD, and proposes a method for calculating longitude by using the
moon.
1475
Regiomontanus
publishes De triangulis planis et sphaericis (Concerning Plane
and Spherical Triangles), which studies spherical trigonometry to
apply it to astronomy.
1482
Campanus
of Novara's edition of Euclid's
Elements becomes the first mathematics book to be printed.
1489
Widman
writes an arithmetic book in German which contains the first appearance
of + and - signs.
1494
Pacioli
publishes Summa de arithmetica, geometria, proportioni et proportionalita
which is a review of the whole of mathematics covering arithmetic, trigonometry,
algebra, tables of moneys, weights and measures, games of chance, double-entry
book-keeping and a summary of Euclid's
geometry.
1514
Vander Hoecke uses the + and - signs.
1515
Del
Ferro discovers a formula to solve cubic equations. (See this History
Topic.)
1522
Tunstall
publishes De arte supputandi libri quattuor (On the Art of Computation),
an arithmetic book based on Pacioli's
Summa.
1525
Rudolff
introduces a symbol resembling 
for square roots in his Die Coss
the first German algebra book. He understands that x0
= 1.
1525
Dürer
publishes Unterweisung der Messung mit dem Zirkel und Richtscheit,
the first mathematics book published in German. It is a work on geometric
constructions.
1533
Frisius
publishes a method for accurate surveying using trigonometry. He is the
first to propose the triangulation method.
1535
Tartaglia
solves the cubic equation independently of del
Ferro. (See this History
Topic.)
1536
Hudalrichus Regius finds the fifth perfect number. The number 212(213
- 1) = 33550336 is the first perfect number to be discovered since ancient
times. (See this History
Topic.)
1540
Ferrari
discovers a formula to solve quartic equations. (See this
History
Topic.)
1541
Rheticus
publishes his trigonometric tables and the trigonometrical parts of Copernicus's
work.
1543
Copernicus
publishes De revolutionibus orbium coelestium (On the revolutions
of the heavenly spheres). It gives a full account of the Copernican
theory, namely that the Sun (not the Earth) is at rest in the centre of
the Universe.
1544
Stifel
publishes Arithmetica integra which contains binomial coefficients and
the notation +, -, .
1545
Cardan
publishes Ars Magna giving the formula that will solve any cubic equation based on Tartaglia's
work and the formula for quartics discovered by Ferrari.
(See this History
Topic.)
1550
Ries
publishes his famous arithmetic book Rechenung nach der lenge, auff
den Linihen vnd Feder. It taught arithmetic both by the old abacus method and the new Indian method.
1551
Recorde
translates and abridges the ancient Greek mathematician Euclid's
Elements as The Pathewaie to Knowledge.
1555
J Scheybl gives the sixth perfect number 216(217 - 1) = 8589869056
but his work remains unknown until 1977. (See this History
Topic.)
1557
Recorde
publishes The Whetstone of Witte which introduces = (the equals
sign) into mathematics. He uses the symbol "bicause noe 2 thynges can
be moare equalle".
1563
Cardan
writes his book Liber de Ludo Aleae on games of chance but it would
not be published until 1663.
1571
Viète
begins publishing the Canon Mathematicus which he intends as a
mathematical introduction to his astronomy treatise. It covers trigonometry,
containing trigonometric tables and the theory behind their construction.
1572
Bombelli
publishes the first three parts of his Algebra. He is the first
to gives the rules for calculating with complex numbers.
1575
Maurolico
publishes Arithmeticorum libri duo which contains examples of inductive
proofs.
1585
Stevin
publishes De Thiende in which he presents an elementary and thorough
account of decimal fractions.
1586
Stevin
publishes De Beghinselen der Weeghconst containing the theorem
of the triangle of forces.
1590
Cataldi
uses continued fractions in finding
square roots.
1591
Viète
writes In artem analyticam isagoge (Introduction to the analytical
art), using letters as symbols for quantities, both known and unknown.
He uses vowels for the unknowns and consonants for known quantities. Descartes,
later, introduces the use of letters x, y ... at the end
of the alphabet for unknowns.
1593
Van
Roomen calculates
to 16 decimal places. (See this History
Topic.)
1595
Pitiscus
becomes the first to employ the term trigonometry in a printed publication.
1595
Clavius
writes Novi calendarii romani apologia justifying calendar reforms.
1603
Cataldi
finds the sixth and seventh perfect numbers, 216(217 - 1) =8589869056
and 218(219 - 1) = 137438691328.
1603
Accademia dei Lincei founded in Rome.
1606
Snell
makes the first attempt to measure a degree of the meridian arc on the
Earth's surface, and so determine the size of the Earth. He publishes
Hypomnemata mathematica (Mathematical Memoranda) which is
a Latin translation of Stevin's
work on mechanics.
1609
Kepler
publishes Astronomia nova (New Astronomy). The work contains
Kepler's first and second law on elliptical orbits, but only verified
for the planet Mars.
1610
Galileo
publishes Sidereus Nuncius (Message from the stars) which
describes the astronomical discoveries he has made with his telescopes.
Harriot
also observes the moons of Jupiter but does not publish his work.
1612
Bachet
publishes a work on mathematical puzzles and tricks which will form the
basis for almost all later books on mathematical recreations. He devises
a method of constructing magic squares.
1613
Cataldi
publishes Trattato del modo brevissimo di trovar la radice quadra delli
numeri in which he finds square roots using continued fractions.
1614
Napier
publishes his work on logarithms in Mirifici logarithmorum canonis
descriptio (Description of the Marvellous Rule of Logarithms).
1615
Kepler
publishes Nova stereometria doliorum vinarorum (Solid Geometry
of a Wine Barrel), an investigation of the capacity of casks, surface
areas, and conic sections. He first had the idea at his marriage celebrations
in 1613. His methods are early uses of the calculus.
1615
Mersenne
encourages mathematicians to study the cycloid. (See this Famous
curve.)
1617
Snell
publishes his technique of trigonometrical triangulation which improves
the accuracy of cartographic measurements.
1617
Briggs
publishes Logarithmorum chilias prima (Logarithms of Numbers
from 1 to 1,000) which introduces logarithms to the base 10.
1617
Napier
invents Napier's bones, consisting of numbered sticks, as a mechanical
calculator. He explains their function in Rabdologiae (Study
of Divining Rods) published in the year of his death.
1620
Bürgi
publishes Arithmetische und geometrische progress-tabulen which
contains his version of logarithms discovered independently of Napier.
1620
Gunter
makes a mechanical device, Gunter's scale, to multiply numbers
based on logarithms using a single scale and a pair of dividers.
1620
Guldin
gives Guldin's Centroid Theorem which was already known to Pappus.
1621
Bachet
publishes his Latin translation of Diophantus's
Greek text Arithmetica.
1623
Schickard
makes a "mechanical clock", a wooden calculating machine that add and
subtract and aid with multiplication and division. He writes to Kepler
suggesting using mechanical means to calculate ephemeredes.
1624
Briggs
publishes Arithmetica logarithmica (The Arithmetic of Logarithms)
which introduces the terms "mantissa" and "characteristic". It gives the
logarithms of the natural numbers from 1 to 20,000 and 90,000 to 100,000
computed to 14 decimal places as well as tables of the sine function to
15 decimal places, and the tangent and secant functions to 10 decimal
places.
1626
Albert
Girard publishes a treatise on trigonometry containing the first use
of the abbreviations sin, cos, tan. He also gives formulas for the area
of a spherical triangle.
1629
Fermat
works on maxima and minima. This work is an early contribution to the
differential calculus.
1630
Oughtred
invents an early form of circular slide rule. It uses two Gunter
rulers.
1630
Mydorge
works on optics and geometry. He gives an extremely accurate measurement
of the latitude of Paris.
1631
Harriot's
contributions are published ten years after his death in Artis analyticae
praxis (Practice of the Analytic Art). The book introduces
the symbols > and < for "greater than" and "less than" but these
symbols are due to the editors of the work and not Harriot himself. His
work on algebra is very impressive but the editors of the book do not
present it well.
1631
Oughtred
publishes Clavis Mathematicae which includes a description of Hindu-Arabic
notation and decimal fractions. It has a considerable section on algebra.
1634
Roberval
finds the area under the cycloid curve. (See this Famous
curve.)
1635
Descartes
discovers Euler's
theorem for polyhedra, V - E + F = 2.
1635
Cavalieri
presents his development of Archimedes'
method of exhaustion in his Geometria indivisibilis continuorum nova.
The method incorporates Kepler's
theory of infinitesimally small geometric quantities.
1636
Fermat
discovers the pair of amicable numbers 17296, 18416
which were known to Thabit
ibn Qurra 800 years earlier.
1637
Descartes
publishes La Géométrie which describes his application of algebra
to geometry.
1639
Desargues
begins the study of projective geometry, which
considers what happens to shapes when they are projected on to a non-parallel
plane. He describes his ideas in Brouillon project d'une atteinte aux
evenemens des rencontres du Cone avec un Plan (Rough draft for
an essay on the results of taking plane sections of a cone).
1640
Pascal
publishes Essay pour les coniques (Essay on Conic Sections).
1641
Wilkins
publishes on codes and ciphers.
1642
Pascal
builds a calculating machine to help his father with tax calculations.
It performs only additions.
1644
Torricelli
publishes Opera geometrica which contains his results on projectiles.
He investigates the point which minimises the sum of its distances from
the vertices of a triangle.
1647
Fermat
claims to have proved a theorem, but leaves no details of his proof since
the margin in which he writes it is too small. Later known as Fermat's last theorem, it
states that the equation xn + yn
= zn has no non-zero solutions for x,
y and z when n > 2. This theorem is finally proved
to be true by Wiles
in 1994. (See this History
Topic.)
1647
Cavalieri
publishes Exercitationes geometricae sex (Six Geometrical Exercises)
which contains in print for the first time the integral from 0 to a
of xn.
1648
Wilkins
publishes Mathematical Magic giving an account of mechanical devices.
1648
Abraham Bosse publishes a work containing Desargues'
famous "perspective theorem" - that when two triangles are in perspective
the meets of corresponding sides are collinear.
1649
Van
Schooten publishes the first Latin version of Descartes'
La géométrie.
1649
De
Beaune writes Notes brièves which contains the many results
on "Cartesian geometry", in particular giving the now familiar equations
for hyperbolas, parabolas and ellipses.
1650
De
Witt completes writing Elementa curvarum linearum. It is the
first systematic development of the analytic geometry of the straight
line and conic. It is not published, however, until 1661 when it appears
as an appendix to van
Schooten's major work.
1651
Nicolaus
Mercator publishes three works on trigonometry and astronomy, Trigonometria
sphaericorum logarithmica, Cosmographia and Astronomica
sphaerica. He gives the well known series expansion of log(1 + x).
1653
Pascal
publishes Treatise on the Arithmetical Triangle on "Pascal's triangle". It had
been studied by many earlier mathematicians.
1654
Fermat
and Pascal
begin to work out the laws that govern chance and probability in five letters
which they exchange during the summer.
1654
Pascal
publishes his Treatise on the Equilibrium of Liquids on hydrostatics.
He recognizes that force is transmitted equally in all directions through
a fluid, and gives Pascal's law of pressure.
1655
Brouncker
gives a continued fraction expansion
of 4/. He also computes the quadrature of the hyperbola, a result he will publish three years later.
1656
Wallis
publishes Arithmetica infinitorum which uses interpolation methods
to evaluate integrals.
1656
Huygens
patents the first pendulum clock.
1657
Huygens
publishes De ratiociniis in ludi aleae (On Reasoning in Games
of Chance). It is the first published work on probability theory, outlining
for the first time the concept called mathematical expectation based on
the ideas in the letters of Fermat
and Pascal
from 1654.
1657
Neile
becomes the first to find the arc length of an algebraic curve when he
rectified the cubical parabola. (See this Famous curve.)
1657
Frenicle
de Bessy publishes Solutio duorm problematum ... which gives
solutions to some of Fermat's
number theory challenges.
1658
Wren
finds the length of an arc of the cycloid. (See this Famous
curve.)
1659
Rahn
publishes Teutsche algebra which contains 
(the division sign) probably invented
by Pell.
1660
De
Sluze discusses spirals, points of inflection and the finding of geometric
means in his works. He studies curves which Pascal
names the "pearls of Sluze". (See this Famous curve.)
1660
Hooke
discovers Hooke's law of elasticity.
1660
Viviani
measures the velocity of sound. He determines the tangent to a cycloid.
(See this Famous
curve.)
1661
Van
Schooten publishes the second and final volume of Geometria a Renato
Des Cartes. This work establishes analytic geometry as a major mathematical
topic. The book also contains appendices by three of his disciples, de
Witt, Hudde,
and Heuraet.
1662
The Royal Society of London is founded. Brouncker
becomes its first President. (See this Article.)
1662
Graunt and Petty publish Natural and Political Observations made upon
the Bills of Mortality. It is one of the first statistics books.
1663
Barrow
becomes the first Lucasian Professor of Mathematics at the University
of Cambridge in England. (See this Article.)
1665
Newton
discovers the binomial theorem and begins
work on the differential calculus.
1666
The Académie des Sciences in Paris is founded.
1667
James
Gregory publishes Vera circuli et hyperbolae quadratura which
lays down exact foundations for the infinitesimal geometry.
1668
James
Gregory publishes Geometriae pars universalis which is the
first attempt to write a calculus textbook.
1668
Pell
gives a table of factors of all integers up to 100000.
1669
Wren
publishes his result that a hyperboloid of revolution is a ruled surface.
1669
Barrow
resigns the Lucasian Chair of Mathematics at Cambridge University to allow
his pupil Newton
to be appointed.
1669
Wallis
publishes his Mechanica (Mechanics) which is a detailed
mathematical study of mechanics.
1670
Barrow
publishes Lectiones Geometricae which contains his important work
on tangents which forms the starting point of Newton's
work on the calculus.
1671
De
Witt publishes A Treatise on Life Annuities. It contains the
idea of mathematical expectation.
1671
James
Gregory discovers Taylor's
Theorem and writes to Collins
telling him of his discovery. His series expansion for arctan(x)
gives a series for /4.
1672
Mengoli
publishes The Problem of Squaring the Circle which studies infinite
series and gives an infinite product expansion for /2.
1672
Mohr
publishes Euclides danicus in which he shows that all Euclidean
constructions can be carried out with compasses alone.
1673
Leibniz
demonstrates his incomplete calculating machine to the Royal Society.
It can multiply, divide and extract roots.
1673
Huygens
publishes Horologium Oscillatorium sive de motu pendulorum. As
well as work on the pendulum he investigates evolutes and involutes of curves and finds the evolutes of the cycloid
and of the parabola.
1675
La
Hire publishes Sectiones conicae which is a major work on conic sections.
1675
Leibniz
uses the modern notation for an integral for the first time.
1676
Leibniz
discovers the differentials of basic functions independently of Newton.
1677
Leibniz
discovers the rules for differentiating products, quotients, and the function
of a function.
1678
Giovanni
Ceva publishes De lineis rectis containing "Ceva's theorem".
1678
Cocker's
Arithmetic is published two years after Cocker's death. It would
run to more than 100 editions over a period of about 100 years.
1679
Leibniz
introduces binary arithmetic. It was not published until 1701.
1680
Cassini
studies the "Cassinian curve" which is the locus of a point the product
of whose distances from two fixed foci is constant. (See this Famous
curve.)
1682
Tschirnhaus
studies catacaustic curves, being the envelope of light rays emitted from
a point source after reflection from a given curve.
1683
Seki
Kowa publishes a treatise that first introduces determinants. He considers integer solutions of ax
- by = 1 where a, b are integers.
1684
Leibniz
publishes details of his differential calculus in Nova Methodus pro
Maximis et Minimis, itemque Tangentibus. In contains the familiar
d notation, and the rules for computing the derivatives of powers,
products and quotients.
1685
Wallis
publishes De Algebra Tractatus (Treatise of Algebra) which
contains the first published account of Newton's
binomial theorem. It made
Harriot's
remarkable contributions known.
1685
Kochanski gives an approximate method to find the length of the circumference
of a circle.
1687
Newton
publishes The Principia or Philosophiae naturalis principia
mathematica (The Mathematical Principles of Natural Philosophy).
In this work, recognised as the greatest scientific book ever written,
Newton presents his theories of motion, gravity, and mechanics. His theories
explain the eccentric orbits of comets, the tides and their variations,
the precession of the Earth's axis, and motion of the Moon.
1690
Jacob
Bernoulli uses the word "integral" for the first time to refer to
the area under a curve.
1690
Rolle
publishes Traité d'algèbre on the theory of equations.
1691
Jacob
Bernoulli invents polar coordinates, a method of describing the location
of points in space using angles and distances.
1691
Rolle
publishes Méthods pour résoudre les égalités which contains Rolle's
theorem. His proof uses a method due to Hudde.
1692
Leibniz
introduces the term "coordinate".
1693
Halley
publishes his mortality tables for the city of Breslau (now Wroclaw) in
Poland. His attempts to relate mortality and age in a population and proves
highly influential in the future production of actuarial tables in life
insurance.
1694
Johann
Bernoulli discovers "L'Hôpital's rule".
1696
Johann
Bernoulli poses the problem of the brachristochrone and challenges
others to solve it. Johann
Bernoulli, Jacob
Bernoulli and Leibniz
all solve it.
1702
David
Gregory publishes Astronomiae physicae et geometricae elementa
which is a popular account of Newton's
theories.
1706
Jones
introduces the Greek letter to represent the ratio of the circumference
of a circle to its diameter in his Synopsis palmariorum matheseos
(A New Introduction to Mathematics).
1707
Newton
publishes Arithmetica universalis (General Arithmetic) which
contains a collection of his results in algebra.
1707
De
Moivre uses trigonometric functions to represent complex numbers in
the form r(cos z + i sin x).
1708
La
Hire calculates the length of the cardioid. (See this Famous
curve.)
1710
Arbuthnot
publishes an important statistics paper in the Royal Society which discusses
the slight excess of male births over female births. This paper is the
first application of probability to social statistics.
1711
Giovanni
Ceva publishes De Re Nummeraria (Concerning Money Matters)
which is one of the first works in mathematical economics.
1713
Jacob
Bernoulli's book Ars conjectandi (The Art of Conjecture)
is an important work on probability. It contains
the Bernoulli numbers which appear
in a discussion of the exponential series.
1715
Brook
Taylor publishes Methodus incrementorum directa et inversa
(Direct and Indirect Methods of Incrementation), an important contribution
to the calculus. The book discusses singular solutions to differential
equations, a change of variables formula, and a way of relating the derivative
of a function to the derivative of the inverse function. There is also
a discussion on vibrating strings.
1717
Johann
Bernoulli declares that the principle of virtual displacement is applicable
to all cases of equilibrium.
1718
Jacob
Bernoulli's work on the calculus of variations is
published after his death.
1718
De
Moivre publishes The Doctrine of Chances. The definition of
statistical independence appears in this book together with many problems
with dice and other games. He also investigated mortality statistics and
the foundation of the theory of annuities.
1719
Brook
Taylor publishes New principles of linear perspective. The
first edition appeared four years earlier under the title Linear perspective.
The work gives the first general treatment of vanishing points.
1722
The work unfinished by Cotes
on his death is published as Harmonia mensurarum. It deals with
integration of rational functions. It contains a thorough treatment of
the calculus applied to logarithmic and circular functions.
1724
Jacapo
Riccati studies the Riccati differential equation in a paper.
He gives solutions for certain special cases to the equation which was
first studied by Jacob
Bernoulli.
1724
Academy of Sciences is founded in St Petersburg.
1727
Euler
is appointed to St Petersburg. He introduces the symbol e for the
base of natural logarithms in a manuscript entitled Meditation upon
Experiments made recently on firing of Cannon. The manuscript was
not published until 1862.
1728
Grandi
publishes Flora geometrica (Geometrical Flowers). He gives
a geometrical definition of curves which resemble petals and leaves of
flowers. For example the rhodonea curves are so called since they
look like roses while the clelie curve is named after the Countess
Clelia Borromeo to whom he dedicated his book.
1730
De
Moivre gives further theorems concerning his trigonometric representation
of complex numbers. He gives Stirling's
formula.
1731
Clairaut
publishes Recherches sur les courbes à double coubure on skew curves.
1733
De
Moivre first describes the normal distribution curve, or law of errors,
in Approximatio ad summam terminorum binomii (a+b)n
in seriem expansi. Gauss,
in 1820, also investigated the normal distribution.
1733
In Euclides ab Omni Naevo Vindicatus Saccheri
does important early work on non-euclidean geometry, although he considers it an attempt
to prove the parallel postulate of Euclid.
1734
Berkeley
publishes The analyst: or a discourse addressed to an infidel mathematician.
He argues that although the calculus led to true results its foundations
were no more secure than those of religion.
1735
Euler
introduces the notation f(x).
1736
Euler
solves the topographical problem known as the "Königsberg bridges problem".
He proves mathematically that it is impossible to design a walk which
crosses each of the seven bridges exactly once.
1736
Euler
publishes Mechanica which is the first mechanics textbook which
is based on differential equations.
1737
Simpson
publishes his Treatise on Fluxions written as a textbook for his
private students. In the book he uses infinite series to find the definite
integrals of functions.
1738
Daniel
Bernoulli publishes Hydrodynamica (Hydrodynamics). It
gives for the first time the correct analysis of water flowing from a
hole in a container and discusses pumps and other machines to raise water.
He also gives, in Chapter 10, the basis of the kinetic theory of gases.
1739
D'Alembert
publishes Mémoire sur le calcul intégral (Memoir on Integral
Calculus).
1740
Simpson
publishes Treatise on the Nature and Laws of Chance. Much of this
probability treatise is based
on the work of de
Moivre.
1740
Maclaurin
is awarded the Grand Prix of the Académie des Sciences for his work on
gravitational theory to explain the tides.
1742
Maclaurin
publishes Treatise on Fluxions which aims to provide a rigorous
foundation for the calculus by appealing to the methods of Greek geometry.
It is the first systematic exposition of Newton's
methods written in reply to Berkeley's
attack on the calculus for its lack of rigorous foundations.
1742
Goldbach
conjectures, in a letter to Euler,
that every even number 
4 can be written as the sum of two primes. It is not yet known whether
Goldbach's conjecture is
true.
1743
D'Alembert
publishes Traité de dynamique (Treatise on Dynamics). In
this celebrated work he states his principle that the internal actions
and reactions of a system of rigid bodies in motion are in equilibrium.
1744
D'Alembert
publishes Traite de l'equilibre et du mouvement des fluides (Treatise
on Equilibrium and on Movement of Fluids). He applies his principle
to the equilibrium and motion of fluids.
1746
D'Alembert
further develops the theory of complex numbers in making the first serious
attempt to prove the fundamental theorem of algebra.
(See this History
Topic.)
1747
D'Alembert
uses partial differential equations
to study the winds in Réflexion sur la cause générale des vents
(Reflection on the General Cause of Winds) which receives the prize
of the Prussian Academy.
1748
Agnesi
writes Instituzioni analitiche ad uso della giovent italiana which
is an Italian teaching text on the differential calculus. The book contains
many examples which were carefully selected to illustrate the ideas. There
is an investigation of a curve that becomes known as "the witch of Agnesi".
(See this Famous
curve.)
1748
Euler
publishes Analysis Infinitorum (Analysis of the Infinite)
which is an introduction to mathematical analysis. He defines a function
and says that mathematical analysis is the study of functions. This work
bases the calculus on the theory of elementary functions rather than on
geometric curves, as had been done previously. The famous formula ei = -1 appears for the
first time in this text.
About 1750
D'Alembert
studies the "three-body problem" and applies
calculus to celestial mechanics. Euler,
Lagrange
and Laplace
also work on the three-body problem.
1750
Cramer
publishes Introduction à l'analyse des lignes courbes algébraique.
The work investigates curves. The third chapter looks at a classification
of curves and it is in this chapter that the now famous "Cramer's rule"
is given.
1750
Giulio
Fagnano publishes much of his previous work in Produzioni matematiche.
It contains remarkable properties of the lemniscate and the duplication
formula for integrals. This latter result led Euler
to prove the addition formula for elliptic integrals.
1751
Euler
publishes his theory of logarithms of complex numbers.
1752
D'Alembert
discovers the Cauchy-Riemann
equations while investigating hydrodynamics.
1752
Euler
states his theorem V - E + F = 2 for polyhedra.
1753
Simson
notes that in the Fibonacci sequence the ratio
between adjacent numbers approaches the golden ratio.
1754
Lagrange
makes important discoveries on the tautochrone which would contribute substantially to the
new subject of the calculus of variations.
1755
Euler
publishes Institutiones calculi differentialis which begins with
a study of the calculus of finite differences.
1757
Lagrange
is a founding member of a mathematical society in Italy that will eventually
become the Turin Academy of Sciences.
1758
The appearance of "Halley's comet" on 25 December confirms Halley's
predictions 15 years after his death.
1759
Aepinus
publishes Tentamen theoriae electriciatis et magnetismi (An
Attempt at a Theory of Electricity and Magnetism). It is the first
work to develop a mathematical theory of electricity and magnetism.
1761
Lambert
proves that is
irrational. He publishes a more general result in 1768.
1763
Monge
begins the study of descriptive geometry.
1764
Bayes
publishes An Essay Towards Solving a Problem in the Doctrine of Chances
which gives Bayes theory of probability. The work contains
the important "Bayes' theorem".
1765
Euler
publishes Theory of the Motions of Rigid Bodies which lays the
foundation of analytical mechanics.
1766
Lambert
writes Theorie der Parallellinien which is a study of the parallel postulate. By assuming
that the parallel postulate is false, he manages to deduce a large number
of results about non-euclidean geometry.
1767
D'Alembert
calls the problems to elementary geometry caused by failure to prove the
parallel postulate "the scandal
of elementary geometry".
1768
Lambert
publishes his result that
is irrational.
1769
Euler
publishes the first volume of his three volume work Dioptics.
1769
Euler
makes Euler's Conjecture, namely
that it is impossible to exhibit three fourth powers whose sum is a fourth
power, four fifth powers whose sum is a fifth power, and similarly for
higher powers.
1770
Lagrange
proves that any integer can be written as the sum of four squares.
1770
Lagrange
publishes Réflexions sur la résolution algébrique des équations
which makes a fundamental investigation of why equations of degrees up
to four can be solved by radicals. The paper
is the first to consider the roots of a equation as abstract quantities
rather than numbers. He studies permutations of the roots and this work leads to group theory.
1770
Euler
publishes his textbook Algebra.
1771
Lagrange
proves Wilson's
theorem (first stated without proof by Waring)
that n is prime if and only if (n - 1)! + 1 is divisible
by n.
1774
Buffon
uses a mathematical and scientific approach to calculate that the age
of the Earth is about 75000 years.
1777
Euler
introduces the symbol i to represent the square root of -1 in a
manuscript which will not appear in print until 1794.
1777
Buffon
carries out his probability experiment calculating
by throwing sticks
over his shoulder onto a tiled floor and counting the number of times
the sticks fell across the lines between the tiles.
1779
Bézout
publishes Théorie générale des équation algébraiques on the theory
of equations. The work includes a result now known as a result known as
"Bézout's theorem".
1780
Lagrange
wins the Grand Prix of the Académie des Sciences in Paris for his work
on perturbations of the orbits of comets by the planets.
1781
Coulomb's
major work on friction Théorie des machines simples wins him the
Grand Prix from the Académie des Sciences.
1781
William Herschel discovers the planet Uranus.
1783
Royal Society of Edinburgh is founded. (See this Article.)
1784
Legendre
introduces his "Legendre polynomials" in his work Recherches sur la
figure des planètes on celestial mechanics.
1785
Condorcet
publishes Essai sur l'application de l'analyse à la probabilité des
décisions rendues à la pluralité des voix (Essay on the Application
of the Analysis to the Probability of Majority Decisions). It is a
major advance in the study of probability in the social
sciences.
1785
Lagrange
states the law of quadratic reciprocity but
his proof is incorrect.
1785
Condorcet
publishes Essay on the Application of Analysis to the Probability of
Majority Decisions which is an extremely important work in the development
of the theory of probability.
1785
Lagrange
begins work on elliptic functions and elliptic
integrals.
1788
Lagrange
publishes Mécanique analytique (Analytical Mechanics). It
summarises all the work done in the field of mechanics since the time
of Newton
and is notable for its use of the theory of differential equations. With
this work Lagrange transforms mechanics into a branch of mathematical
analysis.
1792
De
Prony begins a major task of producing the Cadastre. This consisted
of logarithmic and trigonometric tables given to between 14 and 29 decimal
places.
1794
Legendre
publishes Eléments de géométrie, an account of geometry which would
be a leading text for 100 years. It will replace Euclid's Elements
as a textbook in most of Europe and, in succeeding translations, in the
United States. It becomes the prototype of later geometry texts.
1796
Laplace
presents his famous nebular hypothesis in Exposition du systeme du
monde which views the solar system as originating from the contracting
and cooling of a large, flattened, and slowly rotating cloud of incandescent
gas.
1796
Gauss
gives the first correct proof of the law of quadratic reciprocity.
1797
Lagrange
publishes Théorie des fonctions analytiques (Theory of Analytical
Functions). It is the first treatise on the theory of functions of
a real variable. It uses modern notation like dy/dx
for derivatives.
1797
Wessel
presents a paper on the vector representation of complex numbers which
is published in Danish in 1799. The idea first appears in a report he
wrote in 1787.
1797
Mascheroni
proves in Geometria del compasso that all Euclidean constructions can
be made with compasses alone and so a ruler in not required.
1797
Lazare
Carnot publishes Réflexions sur la métaphysique du calcul infinitésimal
in which he treats zero and infinity as limits. He also considers that
infinitely small quantities are real objects, being representable as differences
between limits.
1799
Gauss
proves the fundamental theorem of algebra
and notes that earlier proofs, such as by d'Alembert
in 1746, could easily be corrected. (See this History
Topic.)
1799
Laplace
publishes the first volume of five-volume Traité de mécanique céleste
(Celestial Mechanics). It applies calculus to study the orbits
of celestial bodies and examines the stability of the Solar System.
1799
Monge
publishes Géométrie descriptive which describes orthographic projection,
the graphical method used in modern mechanical drawing.
1799
Ruffini
publishes the first proof that algebraic equations of degree greater than
four cannot be solved by radicals. It was
largely ignored as were the further proofs he would publish in 1803, 1808
and 1813.
1800
Lacroix
completes publication of his three volume textbook Traité de Calcul
differéntiel et intégral.
1801
Gauss
publishes Disquisitiones Arithmeticae (Discourses on Arithmetic).
It contains seven sections, the first six of which are devoted to number
theory and the last to the construction of a regular 17-gon by ruler and compasses.
1801
The minor planet Ceres is discovered but then lost. Gauss
computes its orbit from the few observations that had been made leading
to Ceres being rediscovered in almost exactly the position predicted by
Gauss.
1801
Gauss
proves Fermat's
conjecture that every number can be written as the sum of three triangular
numbers.
1803
Lazare
Carnot publishes Géométrie de position in which sensed magnitudes
are first used systematically in geometry.
1804
Bessel
publishes a paper on the orbit of Halley's
comet using data from Harriot's
observations 200 years earlier.
1806
Argand
introduces the Argand diagram as a way of representing a complex number
geometrically in the plane.
1806
Legendre
develops the method of least squares to find best approximations to a
set of observed data.
1807
Fourier
discovers his method of representing
continuous functions by the sum of a series of trigonometric functions
and uses the method in his paper On the Propagation of Heat in Solid
Bodies which he submits to the Paris Academy.
1808
Germain
makes an important contribution to Fermat's last theorem. This
is named "Germain's theorem" by Legendre.
1809
Poinsot
discovers two new regular polyhedra.
1809
Gauss
describes the least-squares method which he uses to find orbits of celestial
bodies in Theoria motus corporum coelestium in sectionibus conicis
Solem ambientium (Theory of the Movement of Heavenly Bodies).
1810
Gergonne
publishes the first volume of his new mathematics journal Annales de
mathématique pures et appliquées which became known as Annales
de Gergonne.
1811
Poisson
publishes Traité de mécanique (Treatise on Mechanics). It
includes Poisson's work on the applications of mathematics to topics such
as electricity, magnetism and mechanics.
1812
Laplace
publishes the two volumes of Théorie Analytique des probabilités
(Analytical Theory of Probabilities). The first book studies generating
functions and also approximations to various expressions occurring in
probability theory. The second
book contains Laplace's definition of probability, Bayes's rule, and mathematical
expectation.
1814
Argand
gives a beautiful proof (with some gaps) of the fundamental theorem of algebra.
(See this History
Topic.)
1814
Barlow
produces Barlow's Tables which give factors, squares, cubes, square roots,
reciprocals and hyperbolic logs of all numbers from 1 to 10000.
1815
Peter Roget (the author of Roget's Thesaurus) invents the "log-log" slide
rule.
1815
Pfaff
publishes important work on what are now called "Pfaffian forms".
1816
Peacock,
Herschel
and Babbage
are the leaders of the Analytical Society at Cambridge which publishes
an English translation of Lacroix's
textbook Traité de Calcul differéntiel et intégral.
1817
Bessel
discovers a class of integral functions, now called "Bessel functions",
in his study of a problem of Kepler
to determine the motion of three bodies moving under
mutual gravitation.
1817
Bolzano
publishes Rein analytischer Beweis (Pure Analytical Proof)
which contain an attempt to free calculus from the concept of the infinitesimal.
He defines continuous functions without the use of infinitesimals. The work contains the Bolzano-Weierstrass
theorem.
1818
Inspired by the work of Laplace,
Adrain
publishes Investigation of the figure of the Earth and of the gravity
in different latitudes.
1819
Horner
submits a paper giving "Horner's method" for solving algebraic equations
to the Royal Society and was published in the same year in the Philosophical
Transactions of the Royal Society.
1820
Brianchon
publishes Recherches sur la determination d'une hyperbole equilatère,
au moyen de quatres conditions données which contains a statement
and proof of the nine point circle theorem.
1821
Navier
gives the well known "Navier-Stokes equations" for an incompressible fluid.
1821
Cauchy
publishes Cours d'analyse (A Course in Analysis), which
sets mathematical analysis on a formal footing for the first time. Designed
for students at the Ecole Polytechnique it was concerned with developing
the basic theorems of the calculus as rigorously as possible.
1822
Poncelet
develops the principles of projective geometry in Traité
des propriétés projectives des figures (Treatise on the Projective
Properties of Figures). This work contains fundamental ideas of projective
geometry such as the cross-ratio, perspective, involution and the circular
points at infinity.
1822
Fourier's
prize winning essay of 1811 is published as Théorie analytique de la
chaleur (Analytical Theory of Heat). It makes widely available
the techniques of Fourier analysis, which will
have widespread applications in mathematics and throughout science.
1822
Feuerbach
publishes his discoveries on the nine point circle of a triangle.
1823
János
Bolyai completes preparation of a treatise on a complete system of
non-Euclidean geometry. When Bolyai discovers that Gauss
had anticipated much of his work, but not published anything, he delays
publication. (See this History
Topic.)
1823
Babbage
begins construction of a large "difference engine" which is able to calculate
logarithms and trigonometric functions. He was using the experience gained
from his small "difference engine" which he constructed between 1819 and
1822.
1824
Sadi
Carnot publishes Réflexions sur la puissance motrice du feu et
sur les machines propres à développer cette puissance (Thoughts
on the Motive Power of Fire, and on Machines Suitable for Developing that
Power). A book on steam engines, it will be of fundamental importance
in thermodynamics. The "Carnot cycle" which forms the basis of the second
law of thermodynamics also appears in the book.
1824
Abel
proves that polynomial equations of degree greater than four cannot be
solved by radicals. He publishes
it at his own expense as a six page pamphlet.
1824
Bessel
develops "Bessel functions" further while undertaking a study of planetary
perturbations.
1824
Steiner
develops synthetic geometry. He publishes his theories on the topic in
1832.
1825
Gompertz
gives "Gompertz's Law of Mortality" which shows that the mortality rate
increases in a geometric progression so when death rates are plotted on
a logarithmic scale, a straight line known as the "Gompertz function"
is obtained.
1826
Ampère
publishes Memoir on the Mathematical Theory of Electrodynamic Phenomena,
Uniquely Deduced from Experience. It contains a mathematical derivation
of the electrodynamic force law and describes four experiments. It lays
the foundation for electromagnetic theory.
1826
Crelle
begins publication of his Journal für die reine und angewandte Mathematik
which will become known as Crelle's Journal. The first volume contains
several papers by Abel.
1826
Poncelet's
work on the pole and polar lines associated with conics lead him to discover
the principle of duality. Gergonne,
who introduced the word polar, discovers independently the principle of
duality.
1827
Jacobi
writes a letter to Legendre
detailing his discoveries on elliptic functions. Abel
was independently working on elliptic functions at this time.
1827
Möbius
publishes Der barycentrische Calkul on analytical geometry. It
becomes a classic and includes many of his results on projective and affine geometry.
In it he introduces homogeneous coordinates and also discusses geometric
transformations, in particular projective transformations.
1827
Feuerbach
writes a paper which, independently of Möbius,
introduces homogeneous coordinates.
1828
Gauss
introduces differential geometry and
publishes Disquisitiones generales circa superficies. This paper
arises from his geodesic interests, but it contains such geometrical ideas
as "Gaussian curvature". The paper also includes Gauss's famous theorema
egregrium.
1828
Green
publishes Essay on the Application of Mathematical Analysis to the
Theory of Electricity and Magnets, in which he applies mathematics
to the properties of electric and magnetic fields. He introduces the term
potential, develops properties of the potential function and applies them
to electricity and magnetism. The formula connecting surface and volume
integrals, now known as "Green's theorem", appears for the first time
in the work, as does the "Green's function" which would be extensively
used in the solution of partial differential equations.
1828
Abel
begins a study of doubly periodic elliptic functions.
1828
Plücker
publishes Analytisch-geometrische which develops the "Plücker abridged
notation". He, independently of Möbius
and Feuerbach
one year earlier, discovers homogeneous coordinates.
1829
Galois
submits his first work on the algebraic solution of equations to the Académie
des Sciences in Paris.
1829
Lobachevsky
develops non-euclidean geometry, in particular hyperbolic geometry,
and his first account of the subject is published in the Kazan Messenger.
When it was submitted for publication in the St Petersburg Academy of
Sciences Ostrogradski
rejects it. (See this History
Topic.)
About 1830
Babbage
creates the first accurate actuarial tables for use in insurance calculations.
1830
Poisson
introduces "Poisson's ratio" in elasticity which involves stresses and
strains on materials.
1830
Peacock
publishes his Treatise on Algebra which attempts to give algebra
a logical treatment comparable to Euclid's
Elements.
1831
Möbius
publishes Über eine besondere Art von Umkehrung der Reihen which
introduces the "Möbius function" and the "Möbius inversion formula".
1831
Cauchy
gives power series expansions of analytic functions of a complex variable.
1832
Steiner
publishes Systematische Entwicklungen ... (Systematic Development
of the Dependency of Geometrical Forms on One Another) which gives
a treatment of projective geometry based
on metric considerations.
1832
János
Bolyai's work on non-Euclidean geometry is published as an appendix to an
essay by Farkas
Bolyai, his father. (See this History
Topic.)
1833
Legendre
points out the flaws in 12 "proofs" of the parallel postulate. (See
this History
Topic.)
1834
Hamilton
uses algebra in treating dynamics in On a General Method in Dynamics.
This paper gives the first statement of the characteristic function applied
to dynamics.
1835
Quetelet
publishes Sur l'homme et le développement de ses facultés (A
treatise on Man and the Development of his Faculties). He presents
his conception of the "average man" as the central value about which measurements
of a human trait are grouped according to the normal curve.
1835
Coriolis
publishes Sur les équations du mouvement relatif des systèmes de corps.
He introduces the "Coriolis force" and shows that the laws of motion can
be used in a rotating frame of reference if an extra force called the
"Coriolis acceleration" is added to the equations of motion. In the same
year Coriolis publishes a work on a mathematical theory of billiards.
1836
Ostrogradski
rediscovers Green's
theorem.
1836
Liouville
founds a mathematics journal Journal de Mathématiques Pures et Appliquées.
This journal, sometimes known as Journal de Liouville, did much
to advance mathematics in France throughout the 19th century.
1836
Poncelet
publishes Cours de mécanique appliquée aux machines (A Course
in Mechanics Applied to Machines). It is the first to propose the
use of mathematics in machine design.
1837
Poisson
publishes Recherches sur la probabilité des jugements (Researches
on the Probabilities of Opinions). In this work he establishes the
rules of probability, gives "Poisson's
law of large numbers" and describes the "Poisson distribution" for a discrete
random variable which is a limiting case of the binomial distribution.
1837
The Cambridge and Dublin Mathematical Journals begins publication.
1837
Dirichlet
gives a general definition of a function.
1837
Liouville
discusses integral equations and gives
the "Sturm-Liouville theory" which is used in solving such equations.
1837
Wantzel
proves that the classical problems of duplicating a cube and trisecting an angle could
not be solved with ruler and compass.
1838
Bessel
measures the parallax of the star 61 Cygni, the first star for which this
is calculated.
1838
Cournot
publishes Recherches sur les principes mathématiques de la théorie
des richesses in which he discusses mathematical economics, in particular
supply- and demand-functions.
1838
De
Morgan invents the term "mathematical induction" and makes the method
precise.
1839
Lamé
proves Fermat's Last Theorem for
n = 7. (See this History
Topic.)
1840
Cauchy
publishes the first volume of the four volume work Exercises d'analyse
et de physique mathematique.
1841
Gauss
publishes a treatise on optics in which he gives a formulae for calculating
the position and size of the image formed by a lens with a given focal
length.
1841
Jacobi
writes a long memoir De determinantibus functionalibus devoted
to the functional determinant now called the Jacobian.
1841
Quetelet
establishes the Belgium Central Statistical Bureau.
1842
Hesse
introduces the "Hessian determinant" in a paper which investigates cubic
and quadratic curves.
1842
Stokes
begins his research on fluids and publishes On the steady motion of
incompressible fluids.
1843
Cayley
is the first person to investigate "geometry of n dimensions" which
occurs in the title of his paper of that year. He uses determinants as the major tool.
1843
Hamilton
discovers quaternions, which generalise complex numbers to four dimensions.
1843
Liouville
announces to the Académie des Sciences in Paris that he had found deep
results in Galois's
unpublished work and promises to publish Galois's papers together with
his own commentary.
1843
Kummer
invents "ideal complex numbers" in his study of unique factorisation.
This leads to the development of ring theory.
1844
Liouville
finds the first transcendental numbers - numbers that cannot be expressed
as the roots of an algebraic equation with rational coefficients.
1844
Grassmann
publishes Die lineale Ausdehnundslehre, ein neuer Zweig der Mathematik
in which he develops the idea of an algebra in which the symbols representing
geometric entities such as points, lines and planes, are manipulated using
specific rules.
1845
Cayley
publishes Theory of Linear Transformations in which he examines
the composition of linear transformations.
1845
While examining permutation groups Cauchy
proves a fundamental theorem of group theory which became known as "Cauchy's theorem".
(See this History
Topic.)
1846
Liouville
publishes Galois'
papers on the solution of algebraic equations in Liouville's Journal.
1846
Maxwell
writes his first paper at the age of 14: On the description of oval
curves, and those having a plurality of foci.
1847
Boole
publishes The Mathematical Analysis of Logic, in which he shows
that the rules of logic can be treated mathematically rather than metaphysically.
Boole's work lays the foundation of computer logic.
1847
De
Morgan proposes two laws of set theory that are now known as "de Morgan's
laws".
1847
Von
Staudt publishes Geometrie der Lage. It is the first work to
completely free projective geometry from
any metrical basis.
1848
Thomson
(Lord Kelvin) proposes the absolute temperature scale now named after
him.
1849
Hermite
applies Cauchy's
residue techniques to doubly periodic functions.
1850
Chebyshev
publishes On Primary Numbers in which he proves new results in
the theory of prime numbers. He proves Bertrand's
conjecture there is always at least one prime between n and 2n
for n > 1.
1850
In his paper On a New Class of Theorems Sylvester
first uses the word "matrix". (See this History
Topic.)
1851
Bolzano's
book Paradoxien des Undendlichen (Paradoxes of the Infinite)
is published three years after his death. It introduces his ideas about
infinite sets.
1851
Liouville
publishes a second work on the existence of specific transcendental numbers which are now known as "Liouville
numbers". In particular he gave the example 0.1100010000000000000000010000...
where there is a 1 in place n! and 0 elsewhere.
1851
Riemann's
doctoral thesis contains ideas of exceptional importance, for example
"Riemann surfaces" and their properties.
1852
Sylvester
establishes the theory of algebraic invariants.
1852
Francis Guthrie poses the Four Colour Conjecture to
De
Morgan. (See this History
Topic.)
1852
Chasles
publishes Traité de géométrie which discusses cross ratio, pencils
and involutions, all notions which he introduced.
1853
Hamilton
publishes Lectures on Quaternions.
1853
Shanks
gives to 707 places
(in 1944 it was discovered that Shanks was wrong from the 528th place).
1854
Riemann
completes his Habilitation. In his dissertation he studied the representability
of functions by trigonometric series. He gives the conditions for a function
to have an integral, what we now call the condition of "Riemann integrability".
In his lecture Über die Hypothesen welche der Geometrie zu Grunde liegen
(On the hypotheses that lie at the foundations of geometry), delivered
on 10 June 1854 he defines an n-dimensional space and gives a definition
of what today is called a "Riemannian space".
1854
Boole
publishes The Laws of Thought on Which are founded the Mathematical
Theories of Logic and Probabilities. He reduces logic to algebra and
this algebra of logic is now known as Boolean algebra.
1854
Cayley
makes an important advance in group theory when he makes the first attempt, which is
not completely successful, to define an abstract group. (See this History
Topic.)
1855
Maxwell
publishes On Faraday's lines of force showing that a few relatively
simple mathematical equations could express the behaviour of electric
and magnetic fields and their interrelation.
1856
Weierstrass
publishes his theory of inversion of hyperelliptic integrals in Theorie
der Abelschen Functionen which appeared in Crelle's Journal.
1857
Riemann
publishes Theory of abelian functions. It develops further the
idea of Riemann surfaces and their topological properties, examines multi-valued
functions as single valued over a special "Riemann surface", and solves
general inversion problems special cases of which had been solved by Abel
and Jacobi.
1858
Cayley
gives an abstract definition of a matrix, a term introduced by Sylvester
in 1850, and in A Memoir on the Theory of Matrices he studies its
properties.
1858
Möbius
describes a strip of paper that has only one side and only one edge. Now
known as the "Möbius strip", it has the surprising property that it remains
in one piece when cut down the middle. Listing
makes the same discovery in the same year.
1858
Dedekind
discovers a rigorous method to define irrational numbers with "Dedekind
cuts". The idea comes to him while he is thinking how to teach differential
and integral calculus.
1859
Mannheim
invents the first modern slide rule that has a "cursor" or "indicator".
1859
Riemann
makes a conjecture about the zeta function which involves prime numbers. It is still
not known whether Riemann's hypothesis is true
in general although it is known to be true in millions of cases. It is
perhaps the most famous unsolved problem in mathematics in the 21st century.
1860
Delaunay
publishes the first volume of La Théorie du mouvement de la lune
which is the result of 20 years work. Delaunay solves the three-body problem by giving
the longitude, latitude and parallax of the Moon as infinite series.
1861
Weierstrass
discovers a continuous curve that is not differentiable any point.
1862
Maxwell
proposes that light is an electromagnetic phenomenon.
1862
Jevons
reads General Mathematical Theory of Political Economy to the British
Association.
1862
Listing
publishes Der Census raumlicher Complexe oder Verallgemeinerung des
Euler'schen Satzes von den Polyedern which discusses extensions of
"Euler's formula".
1863
Weierstrass
gives a proof in his lecture course that the complex numbers are the only
commutative algebraic extension of the real numbers.
1864
Bertrand
publishes Treatise on Differential and Integral Calculus.
1864
London Mathematical Society founded. (See this Article.)
1864
Benjamin
Peirce presents his work on Linear Associative Algebras to the American
Academy. It classifies all complex associative algebras of dimension less
than seven using the, now familiar, tools of idempotent and nilpotent
elements.
1865
Plücker
makes further advances in geometry when he defines a four dimensional
space in which straight lines rather than points are the basic elements.
1866
Hamilton's
Elements of Quaternions is unfinished on his death but the 800
page work which took seven years to write is published posthumously by
his son.
1867
Moscow Mathematical Society is founded.
1868
Beltrami
publishes Essay on an Interpretation of Non-Euclidean Geometry
which gives a concrete model for the non-euclidean geometry of Lobachevsky
and Bolyai.
1869
Lueroth
discovers the "Lueroth quartic".
1870
Benjamin
Peirce publishes Linear Associative Algebras at his own expense.
1871
Betti
publishes a memoir on topology which contains the "Betti numbers".
1872
Dedekind
publishes his formal construction of real numbers and gives a rigorous
definition of an integer.
1872
Heine
publishes a paper which contains the theorem now known as the "Heine-Borel
theorem".
1872
Société Mathématique de France is founded.
1872
Méray
publishes Nouveau précis d'analyse infinitésimale which aims to
present the theory of functions of a complex variable using power series.
1872
Sylow
publishes Théorèmes sur les groupes de substitutions which contains
the famous three "Sylow theorems" about finite groups. He proves them
for permutation groups.
1872
Klein
gives his inaugural address at Erlanger. He defines geometry as the study
of the properties of a space that are invariant under a given group of
transformations. This became known as the "Erlanger programm" and profoundly
influences mathematical development.
1873
Maxwell
publishes Electricity and Magnetism. This work contains the four
partial differential equations,
now known as "Maxwell's equations".
1873
Hermite
publishes Sur la fonction exponentielle (On the Exponential
Function) in which he proves that e is a transcendental number.
1873
Gibbs
publishes two important papers on diagrams in thermodynamics.
1873
Brocard
produces his work on the triangle.
1874
Cantor
publishes his first paper on set theory. He rigorously describes the notion
of infinity. He shows that infinities come in different sizes. He proves
the controversial result that almost all numbers are transcendental.
1876
Gibbs
publishes On the Equilibrium of Heterogeneous Substances which
represents a major application of mathematics to chemistry.
1877
Cantor
is surprised at his own discovery that there is a one-one correspondence
between points on the interval [0, 1] and points in a square.
1878
Sylvester
founds the American Journal of Mathematics.
1879
Kempe
published his false proof of the Four Colour Theorem. (See
this History
Topic.)
1879
Lexis
publishes On the theory of the stability of statistical series
which begins the study of time series.
1879
Kharkov Mathematical Society is founded.
1880
Poincaré
publishes important results on automorphic functions.
1881
Venn
introduces his "Venn diagrams" which become a useful tools in set theory.
1881
Gibbs
develops vector analysis in a pamphlet written for the use of his own
students. The methods will be important in Maxwell's
mathematical analysis of electromagnetic waves.
1882
Lindemann
proves that is
transcendental. This proves that it is impossible to construct
a square with the same area as a given circle using a ruler and compass. The classic
mathematical problem of squaring the circle dates
back to ancient Greece and had proved a driving force for mathematical
ideas through many centuries.
1882
Mittag-Leffler
founds the journal Acta Mathematica.
1883
Reynolds
publishes An experimental investigation of the circumstances which
determine whether the motion of water in parallel channels shall be direct
or sinuous and of the law of resistance in parallel channels. The
"Reynolds number" (as it is now called) used in modelling fluid flow appears
in this work.
1883
Poincaré
publishes a paper which initiates the study of the theory of analytic
functions of several complex variables.
1883
The Edinburgh Mathematical Society is founded. (See this Article.)
1884
Volterra
begins his study of integral equations.
1884
Frege
publishes The Foundations of Arithmetic.
1884
Hölder
discovers the "Hölder inequality".
1884
Mittag-Leffler
publishes Sur la représentation analytique fes fonctions monogènes
uniformes d'une variable indépendante which gives his theorem on the
construction of a meromorphic function with prescribed poles and singular
parts.
1884
Frobenius
proves Sylow's
theorems for abstract groups.
1884
Ricci-Curbastro
begins work on the absolute differential calculus.
1884
Circolo Matematico di Palermo is founded.
1885
Weierstrass
shows that a continuous function on a finite subinterval of the real line
can be uniformly approximated arbitrarily closely by a polynomial.
1885
Edgeworth
publishes Methods of Statistics which presents an exposition of
the application and interpretation of significance tests for the comparison
of means.
1886
Reynolds
formulates a theory of lubrication
1886
Peano
proves that if f(x, y) is continuous then the first
order differential equation dy/dx
= f(x, y) has a solution.
1887
Levi-Civita
publishes a paper developing the calculus of tensors.
1888
Dedekind
publishes Was sind und was sollen die Zahlen? (The Nature and
Meaning of Numbers). He puts arithmetic on a rigorous foundation giving
what were later known as the "Peano axioms".
1888
Galton
introduces the notion of correlation.
1888
Engel
and Lie
publish the first of three volumes of Theorie der Transformationsgruppen
(Theory of Transformation Groups) which is a major work on continuous
groups of transformations.
1889
Peano
publishes Arithmetices principia, nova methodo exposita (The
Principles of Arithmetic) which gives the Peano axioms defining the
natural numbers in terms of sets.
1889
FitzGerald
suggests what is now called the FitzGerald-Lorentz
contraction to explain the "Michelson-Morley experiment".
1890
Peano
discovers a space filling curve.
1890
St Petersburg Mathematical Society is founded.
1890
Heawood
publishes Map colour theorems in which he points out the error
in Kempe's
proof of the Four Colour Theorem. He proves
that five colours suffice. (See this History
Topic.)
1891
Fedorov and Schönflies
independently classify crystallographic space groups showing that there
are 230 of them.
1892
Poincaré
publishes the first of three volumes of Les Méthodes nouvelles de la
mécanique céleste (New Methods in Celestial Mechanics). He
aims to completely characterise all motions of mechanical systems, invoking
an analogy with fluid flow. He also shows that series expansions previously
used in studying the three-body problem, for example
by Delaunay,
were convergent, but not in general uniformly convergent. This puts in
doubt the stability proofs of the solar system given by Lagrange
and Laplace.
1893
Pearson
publishes the first in a series of 18 papers, written over the next 18
years, which introduce a number of fundamental concepts to the study of
statistics. These papers contain contributions to regression analysis,
the correlation coefficient and includes the chi-square test of statistical
significance.
1894
Poincaré
begins work on algebraic topology.
1894
Borel
introduces "Borel measure".
1894
Cartan,
in his doctoral dissertation, classifies all finite dimensional simple
Lie algebras over the complex numbers.
1895
Poincaré
publishes Analysis situs his first work on topology which gives an early systematic treatment of the
topic. He is the originator of algebraic topology publishing
six papers on the topic. He introduces fundamental groups.
1895
Cantor
publishes the first of two major surveys on transfinite arithmetic.
1895
Heinrich
Weber publishes his famous text Lehrbuch der Algebra (Lectures
on Algebra).
1896
The prime number theorem is proved
independently by Hadamard
and de
la Vallée-Poussin. This theorem gives an estimate of the number of
primes there are up to a given number, showing that the number of primes
less than n tends to infinity as n/log n.
1896
Cesàro
publishes Lezione di geometria intrinseca in which he formulates
intrinsic geometry.
1896
Frobenius
introduces group characters.
1897
Hensel
invents the p-adic numbers.
1897
Burali-Forti
is the first to discover of a set theory paradox.
1897
Burnside
publishes The Theory of Groups of Finite Order.
1897
Frobenius
begins the study of the representation theory of
groups.
1898
Frobenius
introduces the notion of induced representations and the "Frobenius Reciprocity
Theorem".
1898
Hadamard's
work on geodesics on surfaces of negative curvature lays the foundations
of symbolic dynamics.
1899
Hilbert
publishes Grundlagen der Geometrie (Foundations of Geometry)
putting geometry in a formal axiomatic setting.
1899
Lyapunov
devises methods which provide ways of determining the stability of sets
of ordinary differential equations.
1900
Hilbert
poses 23 problems at the Second International Congress of Mathematicians
in Paris as a challenge for the 20th century. The problems include the
continuum hypothesis, the
well ordering of the real numbers, Goldbach's conjecture, the
transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension
of "Dirichlet's principle" and many more. Many of the problems were solved
during the 20th century, and each time one of the problems was solved
it was a major event for mathematics.
1900
Goursat
begins publication of Cours d'analyse mathematique which introduces
many new analysis concepts.
1900
Fredholm
develops his theory of integral equations in Sur
une nouvelle méthode pour la résolution du problème de Dirichlet.
1900
Fejér
publishes a fundamental summation theorem for Fourier series.
1900
Levi-Civita
and Ricci-Curbastro
publish Méthodes de calcul differential absolu et leures applications
in which they set up the theory of tensors in the form that will be used in the general theory
of relativity 15 years later.
1901
Russell
discovers "Russell's paradox" which illustrates in a simple fashion the
problems inherent in naive set theory.
1901
Planck
proposes quantum theory. (See this
History
Topic.)
1901
The Runge-Kutta
method for numerically solving ordinary differential equations is proposed.
1901
Lebesgue
formulates the theory of measure.
1901
Dickson
publishes Linear groups with an exposition of the Galois field theory.
1902
Lebesgue
gives the definition of the "Lebesgue integral".
1902
Beppo Levi states the axiom of choice for the first
time.
1902
Gibbs
publishes Elementary Principles of Statistical Mechanics which
is a beautiful account putting the foundations of statistical mechanics on
a firm foundation.
1903
Castelnuovo
publishes Geometria analitica e proiettiva his most important work
in algebraic geometry.
1904
Poincaré
gives a lecture in which he proposes a theory of relativity to explain
the "Michelson and Morley experiment". (See this History
Topic.)
1904
Zermelo
uses the axiom of choice to prove
that every set can be well ordered.
1904
Lorentz
introduces the "Lorentz transformations". (See this History
Topic.)
1904
Poincaré
proposes the Poincaré Conjecture, namely
that any closed 3-dimensional manifold which is homotopy equivalent to
the 3-sphere must be the 3-sphere.
1905
Einstein
publishes the special theory of relativity. (See this History
Topic.)
1905
Lasker
proves the decomposition theorem for ideals into primary ideals in a polynomial
ring.
1906
Fréchet,
in his dissertation, investigated functionals on a metric space and formulated
the abstract notion of compactness.
1906
Markov
studies random processes that are subsequently known as "Markov chains".
1906
Bateman
applies Laplace transforms to integral equations.
1906
Koch
publishes Une methode geometrique elementaire pour l'etude de certaines
questions de la theorie des courbes plane which contains the "Koch
curve". It is a continuous curve which is of infinite length and nowhere
differentiable.
1907
Fréchet
discovers an integral representation theorem for functionals on the space
of "quadratic Lebesgue integrable functions". A similar result was discovered
independently by Riesz.
1907
Einstein
publishes his principle of equivalence, in which says that gravitational
acceleration is indistinguishable from acceleration caused by mechanical
forces. It is a key ingredient of general relativity. (See this History
Topic.)
1907
Heegaard
and Dehn
publish Analysis Situs which marks the beginnings of combinatorial
topology.
1907
Brouwer's
doctoral thesis on the foundations of mathematics attacked the logical
foundations of mathematics and marks the beginning of the Intuitionist School.
1907
Dehn
formulates the word problem and the isomorphism problem for group presentations.
1907
Riesz
proves the theorem now called the "Riesz-Fischer theorem" concerning Fourier analysis on Hilbert space.
1908
Gosset
introduces "Student's t-test" to handle small samples.
1908
Hardy
and Weinberg present a law describing how the proportions of dominant
and recessive genetic traits would be propagated in a large population.
This establishes the mathematical basis for population genetics.
1908
Zermelo
publishes Untersuchungen über die Grundlagen der Mengenlehre (Investigations
on the Foundations of Set Theory). He bases set theory on seven axioms
: Axiom of extensionality, Axiom of elementary sets, Axiom of separation,
Power set axiom, Union axiom, Axiom of choice and Axiom
of infinity. This aims to overcome the difficulties with set theory encountered
by Cantor.
1908
Poincaré
publishes Science et méthode (Science and Method), perhaps
his most famous popular work.
1909
Carmichael investigates pseudoprimes.
1909
Edmund
Landau gives the first systematic presentation of analytic number theory.
1910
Russell
and Whitehead
publish the first volume of Principia Mathematica. They attempt
to put the whole of mathematics on a logical foundation. They were able
to provide detailed derivations of many major theorems in set theory,
finite and transfinite arithmetic, and elementary measure theory. The
third and final volume will appear three years later, while a fourth volume
on geometry was planned but never completed.
1910
Steinitz
gives the first abstract definition of a field in Algebraische Theorie
der Körper.
1911
Sergi
Bernstein introduces the "Bernstein polynomials" in giving a constructive
proof of Weierstrass's
theorem of 1885.
1912
Denjoy
introduces the "Denjoy integral".
1913
Hardy
receives a letter from Ramanujan.
He brings Ramanujan to Cambridge and they go on to write five remarkable
number theory papers together.
1913
Weyl
publishes Die Idee der Riemannschen Flache which brings together
analysis, geometry and topology.
1914
Hausdorff
publishes Grundzüge der Mengenlehre in which he creates a theory
of topological and metric spaces.
1914
Bieberbach
introduces the "Bieberbach polynomials" which approximate a function that
conformally maps a given
simply-connected domain onto a disc.
1914
Harald
Bohr and Edmund
Landau prove their theorem on the distribution of zeros of the zeta function.
1915
Einstein
submits a paper giving a definitive version of the general theory of relativity.
(See this History
Topic.)
1916
Bieberbach
formulates the Bieberbach Conjecture.
1916
Macaulay
publishes The algebraic theory of modular systems which studies
ideals in polynomial rings. It contains many ideas which today occur in
the theory of "Grobner bases".
1916
Sierpinski
gives the first example of an absolutely normal number, that is a number
whose digits occur with equal frequency in whichever base it is written.
1917
Kakeya poses his problem on minimising areas.
1919
Russell
publishes Introduction to Mathematical Philosophy which had been
largely written while he was in prison for anti-war activities.
1919
Hausdorff
introduces the notion of "Hausdorff dimension", which is a real number
lying between the topological dimension of an object and 3. It is used
to study objects such as Koch's
curve.
1920
Takagi
publishes his fundamental paper on class field theory.
1920
Hasse
discovers the "local-global" principle.
1920
Siegel's
dissertation is important in the theory of Diophantine approximations.
1920
Fundamenta Mathematica is founded by Sierpinski
and Mazurkiewicz.
1921
Keynes
publishes his Treatise on Probability in which he argues that probability is a logical
relation and so it is objective. A statement involving probability relations
has a truth-value independent of people's opinions. This is to have a
profound effect on statistics as well as economics.
1921
Fisher
introduces the concept of likelihood into statistics.
1921
Borel
publishes the first in a series of papers on game theory and becomes the first to define games of strategy.
1921
Emmy
Noether publishes Idealtheorie in Ringbereichen which is of
fundamental importance in the development of modern abstract algebra.
1922
Richardson
publishes Weather Prediction by Numerical Process. He is the first
to apply mathematics, in particular the method of finite differences,
to predicting the weather. The calculations are prohibitive by hand calculation
and only the development of computers will make his idea a reality.
1922
Banach
is awarded his habilitation for a thesis on measure theory. He begins his work on a development of
normed vector spaces.
1922
Fraenkel
attempts to put set theory into an axiomatic setting.
1922
Chebotaryov
proves the density theorem on primes in an arithmetical progression.
1922
Fejér
and Riesz
publish an important work on conformal mappings.
1922
Kolmogorov
constructs a summable function which diverges almost everywhere.
1923
Study
publishes important work on real and complex algebras of low dimension.
1924
Alexander
introduces the now famous "Alexander horned sphere".
1925
Fisher
publishes Statistical Methods for Research Workers. He gives experimental
and statistical methods which can be used in biology.
1925
Whitehead
publishes Science and the Modern World. It results from a series
of lectures given in the United States and serves as an introduction to
his later metaphysics. He considers the growth, success, and impact of
"scientific materialism" which is the notion that nature is merely matter
and energy.
1925
Besicovitch
solves "Kakeya's problem" on minimising areas.
1925
Krull
proves the "Krull-Schmidt theorem" for decomposing abelian groups of operators.
1926
Reidemeister
publishes an important book on knot theory: Knoten und gruppen.
1926
Artin
and Schreier
publish a paper on ordering formally real fields and real closed fields.
1926
Banach
and Tarski
publish the "Banach-Tarski paradox" in a joint paper in Fundamenta
Mathematicae: Sur la decomposition des ensembles de points en partiens
respectivement congruent.
1927
Emmy
Noether, Helmut
Hasse and Richard
Brauer work on non-commutative algebras.
1927
Artin
publishes his reciprocity law in Beweis des allgemeinen Reziprozitätsgesetzes.
1928
Von
Mises publishes Probability, Statistics and Truth.
1928
Von
Neumann proves the minimax theorem in game theory.
1928
Hopf
introduces homology groups.
1929
Gelfond
makes his Conjecture about the linear
independence of algebraic numbers over the rational numbers.
1930
Van
der Waerden's famous work Modern Algebra is published. This
two volume work presents the algebra developed by Emmy
Noether, Hilbert,
Dedekind
and Artin.
1930
Hurewicz
proves his embedding theorem for separable metric spaces into compact
spaces.
1930
Kuratowski
proves his theorem on planar graphs.
1931
G
D Birkhoff proves the general ergodic theorem. This will transform
the Maxwell-Boltzmann
kinetic theory of gases into a rigorous principle through the use of Lebesgue
measure.
1931
Gödel
publishes Über formal unentscheidbare Sätze der Principia Mathematica
und verwandter Systeme (On Formally Undecidable Propositions in
Principia Mathematica and Related Systems). He proves fundamental
results about axiomatic systems showing in any axiomatic mathematical
system there are propositions that cannot be proved or disproved within
the axioms of the system. In particular the consistency of the axioms
cannot be proved.
1931
Von
Mises introduces the idea of a sample space into probability theory.
1931
Borsuk
publishes his theory or retracts in metric differential geometry.
1932
Haar
introduces the "Haar measure" on groups.
1932
Hall
publishes A contribution to the theory of groups of prime power order.
1932
Magnus
proves that the word problem is true for one relator groups.
1932
Von
Neumann publishes Grundlagen der Quantenmechanik on quantum mechanics. (See this
History
Topic.)
1933
Kolmogorov
publishes Foundations of the Theory of Probability which presents
an axiomatic treatment of probability.
1934
Gelfond
and Schneider solve "Hilbert's Seventh problem" independently. They proved
that aq is transcendental when a is algebraic (
0 or 1) and q is an irrational algebraic number.
1934
Leray
shows the existence of weak solutions to the Navier-Stokes
equations.
1934
Zorn
establishes "Zorn's lemma" so named by (probably) by Tukey.
It is equivalent to the axiom of choice.
1935
Church
invents "lambda calculus" which today is an invaluable tool for computer
scientists.
1936
Turing
publishes On Computable Numbers, with an application to the Entscheidungsproblem
which describes a theoretical machine, now known as the "Turing machine".
It becomes a major ingredient in the theory of computability.
1936
Church
publishes An unsolvable problem in elementary number theory. "Church's
Theorem", which shows there is no decision procedure for arithmetic, is
contained in this work.
1937
Vinogradov
publishes Some theorems concerning the theory of prime numbers
in which he proves that every sufficiently large odd integer can be expressed
as the sum of three odd primes. This is a major contribution to the solution
of the Goldbach conjecture.
1938
Kolmogorov
publishes Analytic Methods in Probability Theory which lays the
foundations of the theory of Markov
random processes.
1939
Douglas
gives a complete solution to the Plateau problem, proving
the existence of a surface of minimal area bounded by a contour.
1939
Abraham
Albert publishes Structure of Algebras.
1940
Baer
introduces the concept of an injective module, then begins studying group
actions in geometry.
1940
Aleksandrov
introduces exact sequences.
1941
Linnik
introduces the large sieve method in number theory.
1941
Abraham
Albert starts work on nonassociative algebras.
1942
Steenrod
publishes a paper in which "Steenrod squares" are introduced for the first
time.
1942
Eilenberg
and MacLane
publish a paper which introduces "Hom" and "Ext" for the first time.
1943
Marshall
Hall publishes on projective planes.
1943
Naimark
proves the "Gelfand-Naimark theorem" on self-adjoint algebras of operators
in Hilbert space.
1944
Von
Neumann and Morgenstern publish Theory of Games and Economic Behaviour.
The theory of games is used in the study of economics.
1944
Artin
studies rings with the minimum condition, now called "Artinian rings".
1945
Eilenberg
and MacLane
introduce the terms "category" and "natural transformation".
1946
Weil
publishes Foundations of Algebraic Geometry.
1947
George
Dantzig introduces the simplex method of optimisation.
1948
Norbert
Wiener publishes Cybernetics: or, Control and Communication in
the Animal and the Machine. The term "cybernetics" is due to Wiener.
The book details work done on the theory of information control, particularly
applied to as computers.
1948
Shannon
invents information theory and applies mathematical methods to study errors
in transmitted information. This becomes of vital importance in computer
science and communications.
1948
Schwartz
publishes Généralisation de la notion de fonction, de dérivation, de
transformation de Fourier et applications mathématique et physiques
which is his first important publication on the theory of distributions.
1949
Mauchly
and John
Eckert build the Binary Automatic Computer (BINAC). One of the major
advances of this machine is that data is stored on magnetic tape rather
than on punched cards.
1949
Selberg
and Erdös
find an elementary proof of the prime number theorem that
makes no use of complex function theory.
1950
Carnap publishes Logical Foundations of Probability.
1950
Hamming
publishes a fundamental paper on error-detecting and error-correcting
codes.
1950
Hodge
puts forward the "Hodge Conjecture" on projective algebraic varieties.
1951
Serre
uses spectral sequences to the study of the relations between the homology
groups of fibre, total space and base space in a fibration. This enables
him to discover fundamental connections between the homology groups and homotopy groups of a space and to prove important results
on the homotopy groups of spheres.
1952
Hörmander
begins working on the theory of partial differential equations.
Ten years later he will receive a Fields Medal for this work.
1954
Serre
is awarded a Fields Medal for his work on spectral sequences and his
work developing complex variable theory in terms of sheaves.
1954
Kolmogorov
publishes his second paper on the theory of dynamical systems. This marks
the beginning of KAM-theory, which is named after Kolmogorov, Arnold and
Moser.
1955
Cartan
and Eilenberg
develop homological algebra which allows powerful algebraic methods and
topological methods to be related.
1955
Novikov
proves the insolubility of the word problem for groups.
1955
Taniyama
poses his conjecture on elliptic curves which will play a major role in
the proof of Fermat's Last Theorem.
1956
Milnor
publishes On manifolds homeomorphic to the 7-sphere which opens
up the new field of differential topology.
1957
Kolmogorov
solves "Hilbert's Thirteenth Problem" on continuous functions of three
variables which cannot be represented by continuous functions of two variables.
1958
Thom
is awarded a Fields Medal for his work on topology, in particular on
characteristic classes, cobordism theory and the "Thom transversality
theorem".
1959
Boone
proves that many decision problems for groups are insoluble.
1959
Marshall
Hall publishes his famous text Theory of Groups.
1960
M Suzuki discovers new infinite families of finite simple groups.
1961
Edward Lorenz discovers a simple mathematical system with chaotic behaviour.
It leads to the new mathematics of chaos theory which is widely applicable.
1961
Smale
proves the higher dimensional Poincaré conjecture for n
> 4, namely that any closed n-dimensional manifold which is
homotopy equivalent to the n-sphere must be the n-sphere.
1962
Jacobson
publishes his classic text Lie algebras.
1962
Sobolev
publishes Applications of Functional Analysis in Mathematical Physics.
1963
John
Thompson and Feit publish Solvability of Groups of Odd Order
which proves that all nonabelian finite simple groups are of even order.
Their paper requires 250 pages to prove the theorem.
1963
Cohen
proves the independence of the axiom of choice and of the
continuum hypothesis.
1964
Hironaka
solves a major problem concerning the resolution of singularities on an
algebraic variety.
1965
Sergi
Novikov's work on differential topology, in
particular in calculating stable homotopy groups and classifying smooth simply-connected
manifolds, leads him to make the "Novikov Conjecture".
1965
Bombieri
uses his improved large sieve method to prove what is now called "Bombieri's
mean value theorem", which concerns the distribution of primes in arithmetic
progressions.
1965
Tukey
and Cooley publish a paper introducing the "Fast Fourier Transform" algorithm.
1965
Selten
publishes important work on distinguishing between reasonable and unreasonable
decisions in predicting the outcome of games. It will lead to the award
of a Nobel Prize in 1994.
1966
Grothendieck
receives a Fields Medal for his work on geometry, number theory, topology
and complex analysis. His theory of schemes allows certain of Weil's
number theory conjectures to be solved. His theory of topoi is highly
relevant to mathematical logic, he had given an algebraic proof of the
Riemann-Roch
theorem, and provided an algebraic definition of the fundamental group of a curve.
1966
Lander and Parkin use a computer to find a counterexample to Euler's Conjecture. They
find 275 + 845 + 1105 + 1335
= 1445.
1966
Alan
Baker proves "Gelfond's Conjecture" about the linear independence
of algebraic numbers over the rational numbers.
1967
Atiyah
publishes K-theory which details his work on K-theory and
the index theorem which led to the award of a Fields Medal in 1966.
1968
Novikov
and Adian jointly publish a proof that the Burnside group B(d, n) is infinite
for every d > 1 and every n > 4380.
1969
Conway
publishes details of his discovery of new sporadic finite simple groups.
1970
Alan
Baker is awarded a Fields Medal for his work on Diophantine equations.
1970
Matiyasevich shows that "Hilbert's tenth problem" is unsolvable, namely
that there is no general method for determining when polynomial equations
have a solution in whole numbers.
1971
Stephen Cook formulates the P versus NP problem regarding
polynomial time algorithms.
1972
Thom
publishes Structural Stability and Morphogenesis which explains
catastrophe theory. The theory examines situations in which gradually
changing forces lead to so-called catastrophes, or abrupt changes, and
has important applications in biology and optics.
1972
Quillen
formulates higher algebraic K-theory, a new tool that uses geometric
and topological methods and ideas to formulate and solve major problems
in algebra, particularly ring theory and module theory.
1973
Deligne
proves the three "Weil conjectures".
1973
Chen Jingrun shows that every sufficiently large even integer is the sum
of a prime and a number with at most two prime factors. It makes a major
contribution to the Goldbach Conjecture.
1974
Mumford
is awarded a Fields Medal for his work on algebraic varieties.
1975
Feigenbaum
discovers a new constant, approximately 4.669201660910..., which is related
to period-doubling bifurcations and plays an important part in chaos theory.
1975
Mandelbrot
publishes Les objets fractals, forn, hasard et dimension which
describe the theory of fractals.
1976
Lakatos
work Proofs and Refutations is published as a book two years after
his death. First published in four parts in 1963-64 the work gives Lakatos's
account of how mathematics develops.
1976
Thurston
is awarded the Oswald Veblen Geometry Prize of the American Mathematical
Society for his work on foliations.
1976
Appel and Haken show that the Four Colour Conjecture is
true using 1200 hours of computer time to examine around 1500 configurations.
(See this History
Topic.)
1977
Adleman, Rivest, and Shamir introduce public-key codes, a system for passing
secret messages using large primes and a key which can be published.
1978
Fefferman
is awarded a Fields Medal for his work on partial differential equations,
Fourier analysis, in particular
convergence, multipliers, divergence, singular integrals and "Hardy spaces".
1978
Mori
proves the "Hartshorne conjecture", that projective spaces are the only
smooth complete algebraic varieties with ample tangent bundles.
1979
Connes
publishes work on non-commutative integration theory.
1980
The classification of finite simple groups is complete.
1982
Mandelbrot
publishes The fractal geometry of nature which develops his theory
of fractal geometry more fully than his work of 1975.
1982
Freedman
proves that any closed 4-dimensional manifold which is homotopy equivalent
to the 4-sphere must be the 4-sphere. This proves a further case of the
higher dimensional Poincaré conjecture following
Smale's
work in 1961.
1982
Shing-Tung
Yau is awarded a Fields Medal for his contributions to partial differential equations,
to the "Calabi conjecture" in algebraic geometry, to the
positive mass conjecture of general relativity theory, and to real and
complex Monge-Ampère
equations.
1983
Donaldson
publishes Self-dual connections and the topology of smooth 4-manifolds
which leads to totally new ideas concerning the geometry of 4-manifolds.
1983
Faltings
proves the "Mordell conjecture". He makes a major contribution to Fermat's Last Theorem showing
that for every n there are at most a finite number of coprime integers
x, y, z satisfying xn +
yn = zn. (See this History
Topic.)
1984
Louis de Brange solves the Bieberbach Conjecture.
1984
Vaughan
Jones discovers a new polynomial invariant for knots and links in 3-space.
1984
Witten
publishes Supersymmetry and Morse theory containing ideas that
have become of central importance in the study of differential geometry.
1986
Margulis
proves the "Oppenheim conjecture" on the values of indefinite irrational
quadratic forms at integer points.
1987
Zelmanov
proves an important conjecture about when an infinite dimensional Lie algebra is nilpotent.
1988
Langlands
is the first recipient of the National Academy of Sciences Award in Mathematics.
He receives it for "extraordinary vision that has brought the theory of
group representations into
a revolutionary new relationship with the theory of automorphic forms
and number theory."
1988
Elkies finds a counterexample to Euler's Conjecture with n
= 4, namely 958004 + 2175194 + 4145604
= 4224814.
1989
Bourgain,
using analytic and probabilistic methods, solves the L(p)
problem which had been a longstanding one in "Banach space" theory and
harmonic analysis.
1990
Drinfeld
is awarded a Fields Medal at the International Congress of Mathematicians
in Kyoto, Japan for his work on quantum groups and for his work in number
theory.
1991
Zelmanov
solves the restricted Burnside problem for groups.
1991
Quidong Wang finds infinite series solutions to the n-body problem (with
minor exceptions).
1993
Menasco and Thistlethwaite prove the knot theory conjecture known as "Tait's
Second Conjecture", namely that any two reduced alternating diagrams of
the same prime knot are related by a sequence of twists.
1994
Wiles
proves Fermat's Last Theorem. (See
this History
Topic.)
1994
Connes
publishes a major text on noncommutative geometry.
1994
Lions
is awarded a Fields Medal for his work on the theory of nonlinear partial differential equations.
1994
Yoccoz
is awarded a Fields Medal for his work on dynamical systems.
1994
Krystyna Kuperberg solves the "Seifert Conjecture" about the topology
of dynamical systems.
1995
A large prize is offered by banker Andrew Beal for a solution to the Beal
Conjecture: the equation xp + yq
= zr has no solutions for p, q,
r > 2 and coprime integers x, y, z.
1997
Wiles
is awarded the Wolfskehl Prize for solving Fermat's last theorem.
1998
Borcherds is awarded a Fields Medal for his work in automorphic forms and mathematical
physics; Gowers receives one for his work in functional analysis and combinatorics;
Kontsevich receives one for his work in algebraic geometry, algebraic topology, and mathematical
physics; and McMullen receives one for his work on holomorphic dynamics
and geometry of 3-dimensional manifolds.
1998
Thomas Hales proves Kepler's problem on sphere
packing.
1999
The Great Internet Mersenne Prime Search project finds the 38th Mersenne
prime: 26972593 -1.
1999
Conrad and Taylor prove the "Taniyama-Shimura conjecture". Wiles
proved a special case in 1993 on his way to giving a proof of Fermat's Last Theorem.
2000
At a meeting of the American Mathematical Society in Los Angeles "Mathematical
Challenges of the 21st Century" were proposed. Unlike "Hilbert's problems"
from 100 years earlier, these were given by a team of 30 leading mathematicians
of whom eight were Fields Medal winners.