# ΠΙΝΑΚΕΣ ΙΣΤΟΡΙΑΣ ΜΑΘΗΜΑΤΙΚΩΝ

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A Mathematical Chronology

http://www-history.mcs.st-andrews.ac.uk/

A Mathematical Chronology

Main Index Chronology index Biographies index

About 30000BCPalaeolithic peoples in central Europe and France record numbers on bones.

About 25000BCEarly geometric designs used.

About 5000BCA decimal number system is in use in Egypt.

About 4000BCBabylonian and Egyptian calendars in use.

About 3400BCThe first symbols for numbers, simple straight lines, are used in Egypt.

About 3000BCThe abacus is developed in the Middle East and in areas around the Mediterranean. A somewhat different type of abacus is used in China.

About 3000BCHieroglyphic numerals in use in Egypt. (See this History Topic.)

About 3000BCBabylonians 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 2770BCEgyptian calendar used.

About 2000BCHarappans adopt a uniform decimal system of weights and measures.

About 1950BCBabylonians solve quadratic equations.

About 1900BCThe Moscow papyrus (also called the Golenishev papyrus) is written. It gives details of Egyptian geometry. (See this History Topic.)

About 1850BCBabylonians know Pythagoras's Theorem. (See this History Topic.)

About 1800BCBabylonians use multiplication tables.

About 1750BCThe 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 1700BCThe 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 1360BCA decimal number system with no zero starts to be used in China.

About 1000BCChinese use counting boards for calculation.

About 800BCBaudhayana is the author of one of the earliest of the Indian Sulbasutras. (See this History Topic.)

About 750BCManava writes a Sulbasutra. (See this History Topic.)

About 600BCApastamba writes the most interesting Indian Sulbasutra from a mathematical point of view. (See this History Topic.)

575BCThales brings Babylonian mathematical knowledge to Greece. He uses geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore.

About 540BCCounting rods used in China.

530BCPythagoras of Samos moves to Croton in Italy and teaches mathematics, geometry, music, and reincarnation.

About 500BCThe Babylonian sexagesimal number system is used to record and predict the positions of the Sun, Moon and planets. (See this History Topic.)

About 500BCPanini's work on Sanskrit grammar is the forerunner of the modern formal language theory.

About 465BCHippasus writes of a "sphere of 12 pentagons", which must refer to a dodecahedron.

About 450BCGreeks begin to use written numerals. (See this History Topic.)

About 450BCZeno of Elea presents his paradoxes.

About 440BCHippocrates of Chios writes the Elements which is the first compilation of the elements of geometry.

About 430BCHippias of Elis invents the quadratrix which may have been used by him for trisecting an angle and squaring the circle.

About 425BCTheodorus of Cyrene shows that certain square roots are irrational. This had been shown earlier but it is not known by whom.

About 400BCBabylonians 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.)

387BCPlato founds his Academy in Athens

About 375BCArchytas 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 360BCEudoxus of Cnidus develops the theory of proportion, and the method of exhaustion.

About 340BCAristaeus writes Five Books concerning Conic Sections.

About 330BCAutolycus of Pitane writes On the Moving Sphere which studies the geometry of the sphere. It is written as an astronomy text.

About 320BCEudemus of Rhodes writes the History of Geometry.

About 300BCEuclid gives a systematic development of geometry in his Stoicheion (The Elements). He also gives the laws of reflection in Catoptrics.

About 290BCAristarchus 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 290BCThe Chinese classic Chou pei suan ching is written.

About 250BCIn On the Sphere and the Cylinder, Archimedes gives the formulae for calculating the volume of a sphere and a cylinder. In Measurement of the Circle he gives an approximation of the value of with a method which will allow improved approximations. In Floating Bodies he presents what is now called "Archimedes' principle" and begins the study of hydrostatics. He writes works on two- and three-dimensional geometry, studying circles, spheres and spirals. His ideas are far ahead of his contemporaries and include applications of an early form of integration.

About 235BCEratosthenes of Cyrene estimates the Earth's circumference with remarkable accuracy finding a value which is about 15% too big.

About 230BCNicomedes writes his treatise On conchoid lines which contain his discovery of the curve known as the "Conchoid of Nicomedes".

About 225BCApollonius of Perga writes Conics in which he introduces the terms "parabola", "ellipse" and "hyperbola".

About 230BCEratosthenes of Cyrene develops his sieve method for finding all prime numbers. (See this History Topic.)

About 200BCDiocles writes On burning mirrors, a collection of sixteen propositions in geometry mostly proving results on conics.

About 190BCChinese mathematicians use powers of 10 to express magnitudes.

127BCHipparchus 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 150BCHypsicles writes On the Ascension of Stars. In this work he is the first to divide the Zodiac into 360 degrees.

About 100BCChinese mathematicians are the first to introduce negative numbers.

About 1ADChinese mathematician Liu Hsin uses decimal fractions.

About 20Geminus writes a number of astronomy texts and The Theory of Mathematics. He tries to prove the parallel postulate. (See this History Topic.)

About 50Chinese mathematician Sun-tzi presents the first known example of an indeterminate equation.

About 60Heron of Alexandria writes Metrica (Measurements). It contains formulas for calculating areas and volumes.

About 90The Chinese invent magic squares.

About 90Nicomachus of Gerasa writes Arithmetike eisagoge (Introduction to Arithmetic) which is the first work to treat arithmetic as a separate topic from geometry.

About 100The classical Chinese mathematics text Jiuzhang Suanshu (Nine Chapters on the Mathematical Art) begins to be assembled.

About 110Menelaus of Alexandria writes Sphaerica which deals with spherical triangles and their application to astronomy.

About 150Ptolemy 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 250The Maya civilization of Central America uses an almost place-value number system to base 20. (See this History Topic.)

250Diophantus of Alexandria writes Arithmetica, a study of number theory problems in which only rational numbers are allowed as solutions.

263By using a regular polygon with 192 sides Liu Hui calculates the value of as 3.14159 which is correct to five decimal places. (See this History Topic.)

301Iamblichus writes on astrology and mysticism. His Life of Pythagoras is a fascinating account.

340Pappus of Alexandria writes Synagoge (Collections) which is a guide to Greek geometry.

390Theon of Alexandria produces a version of Euclid's Elements (with textual changes and some additions) on which almost all subsequent editions are based.

About 400Hypatia 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.

450Proclus, a mathematician and Neo-Platonist, is one of the last philosophers at Plato's Academy at Athens.

About 460Tsu Ch'ung Chi gives the approximation 355/113 to which is correct to 6 decimal places. (See this History Topic.)

499Aryabhata I calculates to be 3.1416. He produces his Aryabhatiya, a treatise on quadratic equations, the value of , and other scientific problems.

About 500Metrodorus assembles the Greek Anthology consisting of 46 mathematical problems.

510Eutocius of Ascalon writes commentaries on Archimedes' work.

510Boethius writes geometry and arithmetic texts which are widely used for a long time.

About 530Eutocius writes commentaries on the works of Archimedes and Apollonius.

532Anthemius of Tralles, a mathematician of note, is the architect for the Hagia Sophia at Constantinople. (See this History Topic.)

534Chinese mathematics is introduced into Japan.

575Varahamihira produces Pancasiddhantika (The Five Astronomical Canons). He makes important contributions to trigonometry.

594Decimal notation is used for numbers in India. This is the system on which our current notation is based. (See this History Topic.)

628Brahmagupta writes Brahmasphutasiddanta (The Opening of the Universe), a work on astronomy; on mathematics. He uses zero and negative numbers, gives methods to solve quadratic equations, sum series, and compute square roots.

About 700Mathematicians in the Mayan civilization introduce a symbol for zero into their number system. (See this History Topic.)

729Hsing introduces a new calendar into China, correcting many errors in earlier calendars.

732Qutan Zhuan accuses Hsing of copying an Indian calendar in producing his own. However Hsing's Chinese calendar is far more accurate than the Indian one.

About 775Alcuin of York writes elementary texts on arithmetic, geometry and astronomy.

About 790Chinese begin to use finite difference methods.

About 810House of Wisdom set up in Baghdad. There Greek and Indian mathematical and astronomy works are translated into Arabic.

About 810Al-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 850Thabit 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 850Thabit 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.

850Mahavira writes Ganita Sara Samgraha. It consists of nine chapters and includes all mathematical knowledge of mid-ninth century India.

900Sridhara 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 900Abu 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.

920Al-Battani writes Kitab al-Zij a major work on astronomy in 57 chapters. It contains advances in trigonometry.

950Gerbert of Aurillac (later Pope Sylvester II) reintroduces the abacus into Europe. He uses Indian/Arabic numerals without having a zero.

About 960Al-Uqlidisi writes Kitab al-fusul fi al-hisab al-Hindi which is the earliest surviving book that presents the Hindu system.

About 970Abu'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.

976Codex 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 1000Ibn 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 1010Al-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 1020Ibn 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.

1040Ahmad 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 1050Hermann 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.

1072Al-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.

1093Shen 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.

1130Jabir 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 1140Bhaskara II (sometimes known as Bhaskaracharya) writes Lilavati (The Beautiful) on arithmetic and geometry, and Bijaganita (Seed Arithmetic), on algebra.

1142Adelard of Bath produces two or three translations of Euclid's Elements from Arabic.

1144Gherard of Cremona begins translating Arabic works (and Arabic translations of Greek works) into Latin.

1149Al-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.

1150Arabic 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 1200Chinese start to use a symbol for zero. (See this History Topic.)

1202Fibonacci 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".

1225Fibonacci 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 1225Jordanus Nemorarius writes on astronomy. In mathematics he uses letters in an early form of algebraic notation.

About 1230John of Holywood (sometimes called Johannes de Sacrobosco) writes on arithmetic, astronomy and calendar reform.

1247Ch'in Chiu-Shao writes Mathematical Treatise in Nine Sections. It contains simultaneous integer congruences and the Chinese Remainder Theorem. It considers indeterminate equations, Horner's method, areas of geometrical figures and linear simultaneous equations.

1248Li Yeh writes a book which contains negative numbers, denoted by putting a diagonal stroke through the last digit.

About 1260Campanus 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.

1275Yang 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.

1303Chu Shih-Chieh writes Szu-yuen Yu-chien (The Precious Mirror of the Four Elements), which contains a number of methods for solving equations up to degree 14. He also defines what is now called Pascal's triangle and shows how to sum certain sequences.

1321Levi ben Gerson (sometimes known as Gersonides) writes Book of Numbers dealing with arithmetical operations, permutations and combinations.

1328Bradwardine writes De proportionibus velocitatum in motibus which is an early work on kinematics using algebra.

1335Richard of Wallingford writes Quadripartitum de sinibus demonstratis, the first original Latin treatise on trigonometry.

1336Mathematics becomes a compulsory subject for a degree at the University of Paris.

1342Levi 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.

1343Jean de Meurs writes Quadripartitum numerorum (Four-fold Division of Numbers), a treatise on mathematics, mechanics, and music.

1343Levi ben Gerson (Gersonides) writes De harmonicis numeris (Concerning the Harmony of Numbers), which is a commentary on the first five books of Euclid.

1364Nicole 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.

1382Nicole 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.

1400Madhava of Sangamagramma proves a number of results about infinite sums giving Taylor expansions of trigonometric functions. He uses these to find an approximation for correct to 11 decimal places.

1411Al-Kashi writes Compendium of the Science of Astronomy.

1424Al-Kashi writes Treatise on the Circumference giving a remarkably good approximation to in both sexagesimal and decimal forms.

1427Al-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.

1434Alberti studies the representation of 3-dimensional objects and writes the first general treatise Della Pictura on the laws of perspective.

1437Ulugh 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.

1450Nicholas 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 1470Chuquet writes Triparty en la science des nombres, the earliest French algebra book.

1472Peurbach 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.

1474Regiomontanus publishes his Ephemerides, astronomical tables for the years 1475 to 1506 AD, and proposes a method for calculating longitude by using the moon.

1475Regiomontanus publishes De triangulis planis et sphaericis (Concerning Plane and Spherical Triangles), which studies spherical trigonometry to apply it to astronomy.

1482Campanus of Novara's edition of Euclid's Elements becomes the first mathematics book to be printed.

1489Widman writes an arithmetic book in German which contains the first appearance of + and - signs.

1494Pacioli 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.

1514Vander Hoecke uses the + and - signs.

1515Del Ferro discovers a formula to solve cubic equations. (See this History Topic.)

1522Tunstall publishes De arte supputandi libri quattuor (On the Art of Computation), an arithmetic book based on Pacioli's Summa.

1525Rudolff introduces a symbol resembling for square roots in his Die Coss the first German algebra book. He understands that x0 = 1.

1525Drer publishes Unterweisung der Messung mit dem Zirkel und Richtscheit, the first mathematics book published in German. It is a work on geometric constructions.

1533Frisius publishes a method for accurate surveying using trigonometry. He is the first to propose the triangulation method.

1535Tartaglia solves the cubic equation independently of del Ferro. (See this History Topic.)

1536Hudalrichus Regius finds the fifth perfect number. The number 212(213 - 1) = 33550336 is the first perfect number to be discovered since ancient times. (See this History Topic.)

1540Ferrari discovers a formula to solve quartic equations. (See this History Topic.)

1541Rheticus publishes his trigonometric tables and the trigonometrical parts of Copernicus's work.

1543Copernicus 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.

1544Stifel publishes Arithmetica integra which contains binomial coefficients and the notation +, -, .

1545Cardan 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.)

1550Ries 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.

1551Recorde translates and abridges the ancient Greek mathematician Euclid's Elements as The Pathewaie to Knowledge.

1555J Scheybl gives the sixth perfect number 216(217 - 1) = 8589869056 but his work remains unknown until 1977. (See this History Topic.)

1557Recorde publishes The Whetstone of Witte which introduces = (the equals sign) into mathematics. He uses the symbol "bicause noe 2 thynges can be moare equalle".

1563Cardan writes his book Liber de Ludo Aleae on games of chance but it would not be published until 1663.

1571Vite 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.

1572Bombelli publishes the first three parts of his Algebra. He is the first to gives the rules for calculating with complex numbers.

1575Maurolico publishes Arithmeticorum libri duo which contains examples of inductive proofs.

1585Stevin publishes De Thiende in which he presents an elementary and thorough account of decimal fractions.

1586Stevin publishes De Beghinselen der Weeghconst containing the theorem of the triangle of forces.

1590Cataldi uses continued fractions in finding square roots.

1591Vite 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.

1593Van Roomen calculates to 16 decimal places. (See this History Topic.)

1595Pitiscus becomes the first to employ the term trigonometry in a printed publication.

1595Clavius writes Novi calendarii romani apologia justifying calendar reforms.

1603Cataldi finds the sixth and seventh perfect numbers, 216(217 - 1) =8589869056 and 218(219 - 1) = 137438691328.

1603Accademia dei Lincei founded in Rome.

1606Snell 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.

1609Kepler publishes Astronomia nova (New Astronomy). The work contains Kepler's first and second law on elliptical orbits, but only verified for the planet Mars.

1610Galileo 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.

1612Bachet 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.

1613Cataldi publishes Trattato del modo brevissimo di trovar la radice quadra delli numeri in which he finds square roots using continued fractions.

1614Napier publishes his work on logarithms in Mirifici logarithmorum canonis descriptio (Description of the Marvellous Rule of Logarithms).

1615Kepler 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.

1615Mersenne encourages mathematicians to study the cycloid. (See this Famous curve.)

1617Snell publishes his technique of trigonometrical triangulation which improves the accuracy of cartographic measurements.

1617Briggs publishes Logarithmorum chilias prima (Logarithms of Numbers from 1 to 1,000) which introduces logarithms to the base 10.

1617Napier 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.

1620Brgi publishes Arithmetische und geometrische progress-tabulen which contains his version of logarithms discovered independently of Napier.

1620Gunter makes a mechanical device, Gunter's scale, to multiply numbers based on logarithms using a single scale and a pair of dividers.

1620Guldin gives Guldin's Centroid Theorem which was already known to Pappus.

1621Bachet publishes his Latin translation of Diophantus's Greek text Arithmetica.

1623Schickard 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.

1624Briggs 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.

1626Albert 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.

1629Fermat works on maxima and minima. This work is an early contribution to the differential calculus.

1630Oughtred invents an early form of circular slide rule. It uses two Gunter rulers.

1630Mydorge works on optics and geometry. He gives an extremely accurate measurement of the latitude of Paris.

1631Harriot'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.

1631Oughtred publishes Clavis Mathematicae which includes a description of Hindu-Arabic notation and decimal fractions. It has a considerable section on algebra.

1634Roberval finds the area under the cycloid curve. (See this Famous curve.)

1635Descartes discovers Euler's theorem for polyhedra, V - E + F = 2.

1635Cavalieri 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.

1636Fermat discovers the pair of amicable numbers 17296, 18416 which were known to Thabit ibn Qurra 800 years earlier.

1637Descartes publishes La Gomtrie which describes his application of algebra to geometry.

1639Desargues 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).

1640Pascal publishes Essay pour les coniques (Essay on Conic Sections).

1641Wilkins publishes on codes and ciphers.

1642Pascal builds a calculating machine to help his father with tax calculations. It performs only additions.

1644Torricelli 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.

1647Fermat claims to have proved a theorem, but leaves no details of his proof since the margin in which he writes it is too small. Later known as Fermat's last theorem, it states that the equation xn + yn = zn has no non-zero solutions for x, y and z when n > 2. This theorem is finally proved to be true by Wiles in 1994. (See this History Topic.)

1647Cavalieri publishes Exercitationes geometricae sex (Six Geometrical Exercises) which contains in print for the first time the integral from 0 to a of xn.

1648Wilkins publishes Mathematical Magic giving an account of mechanical devices.

1648Abraham Bosse publishes a work containing Desargues' famous "perspective theorem" - that when two triangles are in perspective the meets of corresponding sides are collinear.

1649Van Schooten publishes the first Latin version of Descartes' La gomtrie.

1649De Beaune writes Notes brives which contains the many results on "Cartesian geometry", in particular giving the now familiar equations for hyperbolas, parabolas and ellipses.

1650De 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.

1651Nicolaus 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).

1653Pascal publishes Treatise on the Arithmetical Triangle on "Pascal's triangle". It had been studied by many earlier mathematicians.

1654Fermat and Pascal begin to work out the laws that govern chance and probability in five letters which they exchange during the summer.

1654Pascal 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.

1655Brouncker gives a continued fraction expansion of 4/. He also computes the quadrature of the hyperbola, a result he will publish three years later.

1656Wallis publishes Arithmetica infinitorum which uses interpolation methods to evaluate integrals.

1656Huygens patents the first pendulum clock.

1657Huygens 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.

1657Neile becomes the first to find the arc length of an algebraic curve when he rectified the cubical parabola. (See this Famous curve.)

1657Frenicle de Bessy publishes Solutio duorm problematum ... which gives solutions to some of Fermat's number theory challenges.

1658Wren finds the length of an arc of the cycloid. (See this Famous curve.)

1659Rahn publishes Teutsche algebra which contains (the division sign) probably invented by Pell.

1660De 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.)

1660Hooke discovers Hooke's law of elasticity.

1660Viviani measures the velocity of sound. He determines the tangent to a cycloid. (See this Famous curve.)

1661Van 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.

1662The Royal Society of London is founded. Brouncker becomes its first President. (See this Article.)

1662Graunt and Petty publish Natural and Political Observations made upon the Bills of Mortality. It is one of the first statistics books.

1663Barrow becomes the first Lucasian Professor of Mathematics at the University of Cambridge in England. (See this Article.)

1665Newton discovers the binomial theorem and begins work on the differential calculus.

1666The Acadmie des Sciences in Paris is founded.

1667James Gregory publishes Vera circuli et hyperbolae quadratura which lays down exact foundations for the infinitesimal geometry.

1668James Gregory publishes Geometriae pars universalis which is the first attempt to write a calculus textbook.

1668Pell gives a table of factors of all integers up to 100000.

1669Wren publishes his result that a hyperboloid of revolution is a ruled surface.

1669Barrow resigns the Lucasian Chair of Mathematics at Cambridge University to allow his pupil Newton to be appointed.

1669Wallis publishes his Mechanica (Mechanics) which is a detailed mathematical study of mechanics.

1670Barrow publishes Lectiones Geometricae which contains his important work on tangents which forms the starting point of Newton's work on the calculus.

1671De Witt publishes A Treatise on Life Annuities. It contains the idea of mathematical expectation.

1671James Gregory discovers Taylor's Theorem and writes to Collins telling him of his discovery. His series expansion for arctan(x) gives a series for /4.

1672Mengoli publishes The Problem of Squaring the Circle which studies infinite series and gives an infinite product expansion for /2.

1672Mohr publishes Euclides danicus in which he shows that all Euclidean constructions can be carried out with compasses alone.

1673Leibniz demonstrates his incomplete calculating machine to the Royal Society. It can multiply, divide and extract roots.

1673Huygens 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.

1675La Hire publishes Sectiones conicae which is a major work on conic sections.

1675Leibniz uses the modern notation for an integral for the first time.

1676Leibniz discovers the differentials of basic functions independently of Newton.

1677Leibniz discovers the rules for differentiating products, quotients, and the function of a function.

1678Giovanni Ceva publishes De lineis rectis containing "Ceva's theorem".

1678Cocker'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.

1679Leibniz introduces binary arithmetic. It was not published until 1701.

1680Cassini 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.)

1682Tschirnhaus studies catacaustic curves, being the envelope of light rays emitted from a point source after reflection from a given curve.

1683Seki Kowa publishes a treatise that first introduces determinants. He considers integer solutions of ax - by = 1 where a, b are integers.

1684Leibniz 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.

1685Wallis 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.

1685Kochanski gives an approximate method to find the length of the circumference of a circle.

1687Newton 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.

1690Jacob Bernoulli uses the word "integral" for the first time to refer to the area under a curve.

1690Rolle publishes Trait d'algbre on the theory of equations.

1691Jacob Bernoulli invents polar coordinates, a method of describing the location of points in space using angles and distances.

1691Rolle publishes Mthods pour rsoudre les galits which contains Rolle's theorem. His proof uses a method due to Hudde.

1692Leibniz introduces the term "coordinate".

1693Halley 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.

1694Johann Bernoulli discovers "L'Hpital's rule".

1696Johann Bernoulli poses the problem of the brachristochrone and challenges others to solve it. Johann Bernoulli, Jacob Bernoulli and Leibniz all solve it.

1702David Gregory publishes Astronomiae physicae et geometricae elementa which is a popular account of Newton's theories.

1706Jones introduces the Greek letter to represent the ratio of the circumference of a circle to its diameter in his Synopsis palmariorum matheseos (A New Introduction to Mathematics).

1707Newton publishes Arithmetica universalis (General Arithmetic) which contains a collection of his results in algebra.

1707De Moivre uses trigonometric functions to represent complex numbers in the form r(cos z + i sin x).

1708La Hire calculates the length of the cardioid. (See this Famous curve.)

1710Arbuthnot 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.

1711Giovanni Ceva publishes De Re Nummeraria (Concerning Money Matters) which is one of the first works in mathematical economics.

1713Jacob 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.

1715Brook 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.

1717Johann Bernoulli declares that the principle of virtual displacement is applicable to all cases of equilibrium.

1718Jacob Bernoulli's work on the calculus of variations is published after his death.

1718De 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.

1719Brook 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.

1722The 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.

1724Jacapo 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.

1724Academy of Sciences is founded in St Petersburg.

1727Euler 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.

1728Grandi 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.

1730De Moivre gives further theorems concerning his trigonometric representation of complex numbers. He gives Stirling's formula.

1731Clairaut publishes Recherches sur les courbes double coubure on skew curves.

1733De 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.

1733In 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.

1734Berkeley 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.

1735Euler introduces the notation f(x).

1736Euler solves the topographical problem known as the "Knigsberg bridges problem". He proves mathematically that it is impossible to design a walk which crosses each of the seven bridges exactly once.

1736Euler publishes Mechanica which is the first mechanics textbook which is based on differential equations.

1737Simpson 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.

1738Daniel 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.

1739D'Alembert publishes Mmoire sur le calcul intgral (Memoir on Integral Calculus).

1740Simpson publishes Treatise on the Nature and Laws of Chance. Much of this probability treatise is based on the work of de Moivre.

1740Maclaurin is awarded the Grand Prix of the Acadmie des Sciences for his work on gravitational theory to explain the tides.

1742Maclaurin 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.

1742Goldbach 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.

1743D'Alembert publishes Trait 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.

1744D'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.

1746D'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.)

1747D'Alembert uses partial differential equations to study the winds in Rflexion sur la cause gnrale des vents (Reflection on the General Cause of Winds) which receives the prize of the Prussian Academy.

1748Agnesi 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.)

1748Euler publishes Analysis Infinitorum (Analysis of the Infinite) which is an introduction to mathematical analysis. He defines a function and says that mathematical analysis is the study of functions. This work bases the calculus on the theory of elementary functions rather than on geometric curves, as had been done previously. The famous formula ei = -1 appears for the first time in this text.

About 1750D'Alembert studies the "three-body problem" and applies calculus to celestial mechanics. Euler, Lagrange and Laplace also work on the three-body problem.

1750Cramer publishes Introduction l'analyse des lignes courbes algbraique. 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.

1750Giulio 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.

1751Euler publishes his theory of logarithms of complex numbers.

1752D'Alembert discovers the Cauchy-Riemann equations while investigating hydrodynamics.

1752Euler states his theorem V - E + F = 2 for polyhedra.

1753Simson notes that in the Fibonacci sequence the ratio between adjacent numbers approaches the golden ratio.

1754Lagrange makes important discoveries on the tautochrone which would contribute substantially to the new subject of the calculus of variations.

1755Euler publishes Institutiones calculi differentialis which begins with a study of the calculus of finite differences.

1757Lagrange is a founding member of a mathematical society in Italy that will eventually become the Turin Academy of Sciences.

1758The appearance of "Halley's comet" on 25 December confirms Halley's predictions 15 years after his death.

1759Aepinus 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.

1761Lambert proves that is irrational. He publishes a more general result in 1768.

1763Monge begins the study of descriptive geometry.

1764Bayes 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".

1765Euler publishes Theory of the Motions of Rigid Bodies which lays the foundation of analytical mechanics.

1766Lambert 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.

1767D'Alembert calls the problems to elementary geometry caused by failure to prove the parallel postulate "the scandal of elementary geometry".

1768Lambert publishes his result that is irrational.

1769Euler publishes the first volume of his three volume work Dioptics.

1769Euler 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.

1770Lagrange proves that any integer can be written as the sum of four squares.

1770Lagrange publishes Rflexions sur la rsolution algbrique des quations 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.

1770Euler publishes his textbook Algebra.

1771Lagrange 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.

1774Buffon uses a mathematical and scientific approach to calculate that the age of the Earth is about 75000 years.

1777Euler introduces the symbol i to represent the square root of -1 in a manuscript which will not appear in print until 1794.

1777Buffon carries out his probability experiment calculating by throwing sticks over his shoulder onto a tiled floor and counting the number of times the sticks fell across the lines between the tiles.

1779Bzout publishes Thorie gnrale des quation algbraiques on the theory of equations. The work includes a result now known as a result known as "Bzout's theorem".

1780Lagrange wins the Grand Prix of the Acadmie des Sciences in Paris for his work on perturbations of the orbits of comets by the planets.

1781Coulomb's major work on friction Thorie des machines simples wins him the Grand Prix from the Acadmie des Sciences.

1781William Herschel discovers the planet Uranus.

1783Royal Society of Edinburgh is founded. (See this Article.)

1784Legendre introduces his "Legendre polynomials" in his work Recherches sur la figure des plantes on celestial mechanics.

1785Condorcet publishes Essai sur l'application de l'analyse la probabilit des dcisions rendues la pluralit 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.

1785Lagrange states the law of quadratic reciprocity but his proof is incorrect.

1785Condorcet 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.

1785Lagrange begins work on elliptic functions and elliptic integrals.

1788Lagrange publishes Mcanique 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.

1792De Prony begins a major task of producing the Cadastre. This consisted of logarithmic and trigonometric tables given to between 14 and 29 decimal places.

1794Legendre publishes Elments de gomtrie, 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.

1796Laplace 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.

1796Gauss gives the first correct proof of the law of quadratic reciprocity.

1797Lagrange publishes Thorie 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.

1797Wessel 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.

1797Mascheroni proves in Geometria del compasso that all Euclidean constructions can be made with compasses alone and so a ruler in not required.

1797Lazare Carnot publishes Rflexions sur la mtaphysique du calcul infinitsimal 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.

1799Gauss 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.)

1799Laplace publishes the first volume of five-volume Trait de mcanique cleste (Celestial Mechanics). It applies calculus to study the orbits of celestial bodies and examines the stability of the Solar System.

1799Monge publishes Gomtrie descriptive which describes orthographic projection, the graphical method used in modern mechanical drawing.

1799Ruffini 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.

1800Lacroix completes publication of his three volume textbook Trait de Calcul differntiel et intgral.

1801Gauss 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.

1801The 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.

1801Gauss proves Fermat's conjecture that every number can be written as the sum of three triangular numbers.

1803Lazare Carnot publishes Gomtrie de position in which sensed magnitudes are first used systematically in geometry.

1804Bessel publishes a paper on the orbit of Halley's comet using data from Harriot's observations 200 years earlier.

1806Argand introduces the Argand diagram as a way of representing a complex number geometrically in the plane.

1806Legendre develops the method of least squares to find best approximations to a set of observed data.

1807Fourier 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.

1808Germain makes an important contribution to Fermat's last theorem. This is named "Germain's theorem" by Legendre.

1809Poinsot discovers two new regular polyhedra.

1809Gauss 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).

1810Gergonne publishes the first volume of his new mathematics journal Annales de mathmatique pures et appliques which became known as Annales de Gergonne.

1811Poisson publishes Trait de mcanique (Treatise on Mechanics). It includes Poisson's work on the applications of mathematics to topics such as electricity, magnetism and mechanics.

1812Laplace publishes the two volumes of Thorie Analytique des probabilits (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.

1814Argand gives a beautiful proof (with some gaps) of the fundamental theorem of algebra. (See this History Topic.)

1814Barlow produces Barlow's Tables which give factors, squares, cubes, square roots, reciprocals and hyperbolic logs of all numbers from 1 to 10000.

1815Peter Roget (the author of Roget's Thesaurus) invents the "log-log" slide rule.

1815Pfaff publishes important work on what are now called "Pfaffian forms".

1816Peacock, Herschel and Babbage are the leaders of the Analytical Society at Cambridge which publishes an English translation of Lacroix's textbook Trait de Calcul differntiel et intgral.

1817Bessel 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.

1817Bolzano 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.

1818Inspired by the work of Laplace, Adrain publishes Investigation of the figure of the Earth and of the gravity in different latitudes.

1819Horner 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.

1820Brianchon publishes Recherches sur la determination d'une hyperbole equilatre, au moyen de quatres conditions donnes which contains a statement and proof of the nine point circle theorem.

1821Navier gives the well known "Navier-Stokes equations" for an incompressible fluid.

1821Cauchy 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.

1822Poncelet develops the principles of projective geometry in Trait des proprits 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.

1822Fourier's prize winning essay of 1811 is published as Thorie 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.

1822Feuerbach publishes his discoveries on the nine point circle of a triangle.

1823Jnos 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.)

1823Babbage 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.

1824Sadi Carnot publishes Rflexions sur la puissance motrice du feu et sur les machines propres dvelopper 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.

1824Abel 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.

1824Bessel develops "Bessel functions" further while undertaking a study of planetary perturbations.

1824Steiner develops synthetic geometry. He publishes his theories on the topic in 1832.

1825Gompertz 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.

1826Ampre 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.

1826Crelle begins publication of his Journal fr die reine und angewandte Mathematik which will become known as Crelle's Journal. The first volume contains several papers by Abel.

1826Poncelet'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.

1827Jacobi writes a letter to Legendre detailing his discoveries on elliptic functions. Abel was independently working on elliptic functions at this time.

1827Mbius 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.

1827Feuerbach writes a paper which, independently of Mbius, introduces homogeneous coordinates.

1828Gauss 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.

1828Green 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.

1828Abel begins a study of doubly periodic elliptic functions.

1828Plcker publishes Analytisch-geometrische which develops the "Plcker abridged notation". He, independently of Mbius and Feuerbach one year earlier, discovers homogeneous coordinates.

1829Galois submits his first work on the algebraic solution of equations to the Acadmie des Sciences in Paris.

1829Lobachevsky 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 1830Babbage creates the first accurate actuarial tables for use in insurance calculations.

1830Poisson introduces "Poisson's ratio" in elasticity which involves stresses and strains on materials.

1830Peacock publishes his Treatise on Algebra which attempts to give algebra a logical treatment comparable to Euclid's Elements.

1831Mbius publishes ber eine besondere Art von Umkehrung der Reihen which introduces the "Mbius function" and the "Mbius inversion formula".

1831Cauchy gives power series expansions of analytic functions of a complex variable.

1832Steiner 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.

1832Jnos Bolyai's work on non-Euclidean geometry is published as an appendix to an essay by Farkas Bolyai, his father. (See this History Topic.)

1833Legendre points out the flaws in 12 "proofs" of the parallel postulate. (See this History Topic.)

1834Hamilton 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.

1835Quetelet publishes Sur l'homme et le dveloppement de ses facults (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.

1835Coriolis publishes Sur les quations du mouvement relatif des systmes 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.

1836Ostrogradski rediscovers Green's theorem.

1836Liouville founds a mathematics journal Journal de Mathmatiques Pures et Appliques. This journal, sometimes known as Journal de Liouville, did much to advance mathematics in France throughout the 19th century.

1836Poncelet publishes Cours de mcanique applique aux machines (A Course in Mechanics Applied to Machines). It is the first to propose the use of mathematics in machine design.

1837Poisson publishes Recherches sur la probabilit 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.

1837The Cambridge and Dublin Mathematical Journals begins publication.

1837Dirichlet gives a general definition of a function.

1837Liouville discusses integral equations and gives the "Sturm-Liouville theory" which is used in solving such equations.

1837Wantzel proves that the classical problems of duplicating a cube and trisecting an angle could not be solved with ruler and compass.

1838Bessel measures the parallax of the star 61 Cygni, the first star for which this is calculated.

1838Cournot publishes Recherches sur les principes mathmatiques de la thorie des richesses in which he discusses mathematical economics, in particular supply- and demand-functions.

1838De Morgan invents the term "mathematical induction" and makes the method precise.

1839Lam proves Fermat's Last Theorem for n = 7. (See this History Topic.)

1840Cauchy publishes the first volume of the four volume work Exercises d'analyse et de physique mathematique.

1841Gauss 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.

1841Jacobi writes a long memoir De determinantibus functionalibus devoted to the functional determinant now called the Jacobian.

1841Quetelet establishes the Belgium Central Statistical Bureau.

1842Hesse introduces the "Hessian determinant" in a paper which investigates cubic and quadratic curves.

1842Stokes begins his research on fluids and publishes On the steady motion of incompressible fluids.

1843Cayley 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.

1843Hamilton discovers quaternions, which generalise complex numbers to four dimensions.

1843Liouville announces to the Acadmie 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.

1843Kummer invents "ideal complex numbers" in his study of unique factorisation. This leads to the development of ring theory.

1844Liouville finds the first transcendental numbers - numbers that cannot be expressed as the roots of an algebraic equation with rational coefficients.

1844Grassmann 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.

1845Cayley publishes Theory of Linear Transformations in which he examines the composition of linear transformations.

1845While examining permutation groups Cauchy proves a fundamental theorem of group theory which became known as "Cauchy's theorem". (See this History Topic.)

1846Liouville publishes Galois' papers on the solution of algebraic equations in Liouville's Journal.

1846Maxwell writes his first paper at the age of 14: On the description of oval curves, and those having a plurality of foci.

1847Boole 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.

1847De Morgan proposes two laws of set theory that are now known as "de Morgan's laws".

1847Von Staudt publishes Geometrie der Lage. It is the first work to completely free projective geometry from any metrical basis.

1848Thomson (Lord Kelvin) proposes the absolute temperature scale now named after him.

1849Hermite applies Cauchy's residue techniques to doubly periodic functions.

1850Chebyshev publishes On Primary Numbers in which he proves new results in the theory of prime numbers. He proves Bertrand's conjecture there is always at least one prime between n and 2n for n > 1.

1850In his paper On a New Class of Theorems Sylvester first uses the word "matrix". (See this History Topic.)

1851Bolzano's book Paradoxien des Undendlichen (Paradoxes of the Infinite) is published three years after his death. It introduces his ideas about infinite sets.

1851Liouville 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.

1851Riemann's doctoral thesis contains ideas of exceptional importance, for example "Riemann surfaces" and their properties.

1852Sylvester establishes the theory of algebraic invariants.

1852Francis Guthrie poses the Four Colour Conjecture to De Morgan. (See this History Topic.)

1852Chasles publishes Trait de gomtrie which discusses cross ratio, pencils and involutions, all notions which he introduced.

1853Hamilton publishes Lectures on Quaternions.

1853Shanks gives to 707 places (in 1944 it was discovered that Shanks was wrong from the 528th place).

1854Riemann 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 ber 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".

1854Boole 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.

1854Cayley 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.)

1855Maxwell 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.

1856Weierstrass publishes his theory of inversion of hyperelliptic integrals in Theorie der Abelschen Functionen which appeared in Crelle's Journal.

1857Riemann 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.

1858Cayley 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.

1858Mbius describes a strip of paper that has only one side and only one edge. Now known as the "Mbius 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.

1858Dedekind 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.

1859Mannheim invents the first modern slide rule that has a "cursor" or "indicator".

1859Riemann 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.

1860Delaunay publishes the first volume of La Thorie 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.

1861Weierstrass discovers a continuous curve that is not differentiable any point.

1862Maxwell proposes that light is an electromagnetic phenomenon.

1862Jevons reads General Mathematical Theory of Political Economy to the British Association.

1862Listing publishes Der Census raumlicher Complexe oder Verallgemeinerung des Euler'schen Satzes von den Polyedern which discusses extensions of "Euler's formula".

1863Weierstrass gives a proof in his lecture course that the complex numbers are the only commutative algebraic extension of the real numbers.

1864Bertrand publishes Treatise on Differential and Integral Calculus.

1864London Mathematical Society founded. (See this Article.)

1864Benjamin 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.

1865Plcker makes further advances in geometry when he defines a four dimensional space in which straight lines rather than points are the basic elements.

1866Hamilton'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.

1867Moscow Mathematical Society is founded.

1868Beltrami publishes Essay on an Interpretation of Non-Euclidean Geometry which gives a concrete model for the non-euclidean geometry of Lobachevsky and Bolyai.

1869Lueroth discovers the "Lueroth quartic".

1870Benjamin Peirce publishes Linear Associative Algebras at his own expense.

1871Betti publishes a memoir on topology which contains the "Betti numbers".

1872Dedekind publishes his formal construction of real numbers and gives a rigorous definition of an integer.

1872Heine publishes a paper which contains the theorem now known as the "Heine-Borel theorem".

1872Socit Mathmatique de France is founded.

1872Mray publishes Nouveau prcis d'analyse infinitsimale which aims to present the theory of functions of a complex variable using power series.

1872Sylow publishes Thormes sur les groupes de substitutions which contains the famous three "Sylow theorems" about finite groups. He proves them for permutation groups.

1872Klein 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.

1873Maxwell publishes Electricity and Magnetism. This work contains the four partial differential equations, now known as "Maxwell's equations".

1873Hermite publishes Sur la fonction exponentielle (On the Exponential Function) in which he proves that e is a transcendental number.

1873Gibbs publishes two important papers on diagrams in thermodynamics.

1873Brocard produces his work on the triangle.

1874Cantor 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.

1876Gibbs publishes On the Equilibrium of Heterogeneous Substances which represents a major application of mathematics to chemistry.

1877Cantor 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.

1878Sylvester founds the American Journal of Mathematics.

1879Kempe published his false proof of the Four Colour Theorem. (See this History Topic.)

1879Lexis publishes On the theory of the stability of statistical series which begins the study of time series.

1879Kharkov Mathematical Society is founded.

1880Poincar publishes important results on automorphic functions.

1881Venn introduces his "Venn diagrams" which become a useful tools in set theory.

1881Gibbs 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.

1882Lindemann proves that is transcendental. This proves that it is impossible to construct a square with the same area as a given circle using a ruler and compass. The classic mathematical problem of squaring the circle dates back to ancient Greece and had proved a driving force for mathematical ideas through many centuries.

1882Mittag-Leffler founds the journal Acta Mathematica.

1883Reynolds 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.

1883Poincar publishes a paper which initiates the study of the theory of analytic functions of several complex variables.

1883The Edinburgh Mathematical Society is founded. (See this Article.)

1884Volterra begins his study of integral equations.

1884Frege publishes The Foundations of Arithmetic.

1884Hlder discovers the "Hlder inequality".

1884Mittag-Leffler publishes Sur la reprsentation analytique fes fonctions monognes uniformes d'une variable indpendante which gives his theorem on the construction of a meromorphic function with prescribed poles and singular parts.

1884Frobenius proves Sylow's theorems for abstract groups.

1884Ricci-Curbastro begins work on the absolute differential calculus.

1884Circolo Matematico di Palermo is founded.

1885Weierstrass shows that a continuous function on a finite subinterval of the real line can be uniformly approximated arbitrarily closely by a polynomial.

1885Edgeworth publishes Methods of Statistics which presents an exposition of the application and interpretation of significance tests for the comparison of means.

1886Reynolds formulates a theory of lubrication

1886Peano proves that if f(x, y) is continuous then the first order differential equation dy/dx = f(x, y) has a solution.

1887Levi-Civita publishes a paper developing the calculus of tensors.

1888Dedekind 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".

1888Galton introduces the notion of correlation.

1888Engel 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.

1889Peano publishes Arithmetices principia, nova methodo exposita (The Principles of Arithmetic) which gives the Peano axioms defining the natural numbers in terms of sets.

1889FitzGerald suggests what is now called the FitzGerald-Lorentz contraction to explain the "Michelson-Morley experiment".

1890Peano discovers a space filling curve.

1890St Petersburg Mathematical Society is founded.

1890Heawood 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.)

1891Fedorov and Schnflies independently classify crystallographic space groups showing that there are 230 of them.

1892Poincar publishes the first of three volumes of Les Mthodes nouvelles de la mcanique cleste (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.

1893Pearson 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.

1894Poincar begins work on algebraic topology.

1894Borel introduces "Borel measure".

1894Cartan, in his doctoral dissertation, classifies all finite dimensional simple Lie algebras over the complex numbers.

1895Poincar 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.

1895Cantor publishes the first of two major surveys on transfinite arithmetic.

1895Heinrich Weber publishes his famous text Lehrbuch der Algebra (Lectures on Algebra).

1896The prime number theorem is proved independently by Hadamard and de la Valle-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.

1896Cesro publishes Lezione di geometria intrinseca in which he formulates intrinsic geometry.

1896Frobenius introduces group characters.

1897Hensel invents the p-adic numbers.

1897Burali-Forti is the first to discover of a set theory paradox.

1897Burnside publishes The Theory of Groups of Finite Order.

1897Frobenius begins the study of the representation theory of groups.

1898Frobenius introduces the notion of induced representations and the "Frobenius Reciprocity Theorem".

1898Hadamard's work on geodesics on surfaces of negative curvature lays the foundations of symbolic dynamics.

1899Hilbert publishes Grundlagen der Geometrie (Foundations of Geometry) putting geometry in a formal axiomatic setting.

1899Lyapunov devises methods which provide ways of determining the stability of sets of ordinary differential equations.

1900Hilbert 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.

1900Goursat begins publication of Cours d'analyse mathematique which introduces many new analysis concepts.

1900Fredholm develops his theory of integral equations in Sur une nouvelle mthode pour la rsolution du problme de Dirichlet.

1900Fejr publishes a fundamental summation theorem for Fourier series.

1900Levi-Civita and Ricci-Curbastro publish Mthodes 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.

1901Russell discovers "Russell's paradox" which illustrates in a simple fashion the problems inherent in naive set theory.

1901Planck proposes quantum theory. (See this History Topic.)

1901The Runge-Kutta method for numerically solving ordinary differential equations is proposed.

1901Lebesgue formulates the theory of measure.

1901Dickson publishes Linear groups with an exposition of the Galois field theory.

1902Lebesgue gives the definition of the "Lebesgue integral".

1902Beppo Levi states the axiom of choice for the first time.

1902Gibbs publishes Elementary Principles of Statistical Mechanics which is a beautiful account putting the foundations of statistical mechanics on a firm foundation.

1903Castelnuovo publishes Geometria analitica e proiettiva his most important work in algebraic geometry.

1904Poincar gives a lecture in which he proposes a theory of relativity to explain the "Michelson and Morley experiment". (See this History Topic.)

1904Zermelo uses the axiom of choice to prove that every set can be well ordered.

1904Lorentz introduces the "Lorentz transformations". (See this History Topic.)

1904Poincar proposes the Poincar Conjecture, namely that any closed 3-dimensional manifold which is homotopy equivalent to the 3-sphere must be the 3-sphere.

1905Einstein publishes the special theory of relativity. (See this History Topic.)

1905Lasker proves the decomposition theorem for ideals into primary ideals in a polynomial ring.

1906Frchet, in his dissertation, investigated functionals on a metric space and formulated the abstract notion of compactness.

1906Markov studies random processes that are subsequently known as "Markov chains".

1906Bateman applies Laplace transforms to integral equations.

1906Koch 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.

1907Frchet discovers an inte