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Palaeolithic peoples in central Europe and France record numbers on bones.

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.

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 1400BC

About this date a decimal number system with no zero starts to be used in China. (See this History Topic.)

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.

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 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 π 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 230BC

Eratosthenes of Cyrene develops his sieve method for finding all prime numbers. (See this History Topic.)

About 225BC

Apollonius of Perga writes *Conics* in which he introduces the terms "parabola", "ellipse" and "hyperbola".

About 200BC

Diocles writes *On burning mirrors*, a collection of sixteen propositions in geometry mostly proving results on conics.

About 200BC

Possible earliest date for the classic Chinese work *Jiuzhang suanshu* or *Nine Chapters on the Mathematical Art*. (See this History Topic.)

About 180BC

Date of earliest Chinese document *Suanshu shu* (A Book on Arithmetic). (See this History Topic.)

About 150BC

Hypsicles writes *On the Ascension of Stars*. In this work he is the first to divide the Zodiac into 360 degrees.

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 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 60

Heron of Alexandria writes *Metrica* (*Measurements*). It contains formulas for calculating areas and volumes.

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 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 π 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

Zu Chongzhi gives the approximation ^{355}/_{113} to π which is correct to 6 decimal places. (See this History Topic.)

499

Aryabhata I calculates π to be 3.1416. He produces his *Aryabhatiya*, a treatise on quadratic equations, the value of π, 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.

644

Li Chunfeng starts to assemble the Chinese Ten Mathematical Classics. (See this History Topic.)

About 700

Mathematicians in the Mayan civilization introduce a symbol for zero into their number system. (See this History Topic.)

About 775

Alcuin of York writes elementary texts on arithmetic, geometry and astronomy.

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

Qin Jinshao 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

Zhu Shijie 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 π 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 π 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 *x*^{0} = 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 2^{12}(2^{13} - 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 2^{16}(2^{17} - 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 π 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, 2^{16}(2^{17} - 1) =8589869056 and 2^{18}(2^{19} - 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 *x*^{n} + *y*^{n} = *z*^{n} 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 *x*^{n}.

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}/_{π} . 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 ^{π}/_{4}.

1672

Mengoli publishes *The Problem of Squaring the Circle* which studies infinite series and gives an infinite product expansion for π/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 π 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 *e*^{πi} = -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 π 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 π 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 π 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

Legendre 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 2*n* 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 π 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 π 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 parties respectivement congruentes*.

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 of 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 *a*^{q} 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 (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 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 Mac Lane 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 Mac Lane 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 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 mathématiques 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 27^{5} + 84^{5} + 110^{5} + 133^{5} = 144^{5}.

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, forme, hasard et dimension* which describes 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 *x*^{n} + *y*^{n} = *z*^{n}. (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 95800^{4} + 217519^{4} + 414560^{4} = 422481^{4}.

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 *x*^{p} + *y*^{q} = *z*^{r} 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: 2^{6972593} -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.

JOC/EFR August 2001
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