Rahn publishes Teutsche algebra which contains
(the division sign) probably invented by Pell.
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About 25000BC
Early geometric designs used.
About 5000BC
A decimal number system is in use in Egypt.
About 4000BC
Babylonian and Egyptian calendars in use.
About 3400BC
The first symbols for numbers, simple straight lines, are used in Egypt.
About 3000BC
The abacus is developed in the Middle East and in areas around the Mediterranean.
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 x0 = 1.
1525
Dürer publishes Unterweisung der Messung mit dem Zirkel und Richtscheit, the first mathematics book published in German. It is a work on geometric constructions.
1533
Frisius publishes a method for accurate surveying using trigonometry. He is the first to propose the triangulation method.
1535
Tartaglia solves the cubic equation independently of del Ferro. (See this History Topic.)
1536
Hudalrichus Regius finds the fifth perfect number. The number 212(213 - 1) = 33550336 is the first perfect number to be discovered since ancient times. (See this History Topic.)
1540
Ferrari discovers a formula to solve quartic equations. (See this History Topic.)
1541
Rheticus publishes his trigonometric tables and the trigonometrical parts of Copernicus's work.
1543
Copernicus publishes De revolutionibus orbium coelestium (On the revolutions of the heavenly spheres). It gives a full account of the Copernican theory, namely that the Sun (not the Earth) is at rest in the centre of the Universe.
1544
Stifel publishes Arithmetica integra which contains binomial coefficients and the notation +, -, √.
1545
Cardan publishes Ars Magna giving the formula that will solve any cubic equation based on Tartaglia's work and the formula for quartics discovered by Ferrari. (See this History Topic.)
1550
Ries publishes his famous arithmetic book Rechenung nach der lenge, auff den Linihen vnd Feder. It taught arithmetic both by the old abacus method and the new Indian method.
1551
Recorde translates and abridges the ancient Greek mathematician Euclid's Elements as The Pathewaie to Knowledge.
1555
J Scheybl gives the sixth perfect number 216(217 - 1) = 8589869056 but his work remains unknown until 1977. (See this History Topic.)
1557
Recorde publishes The Whetstone of Witte which introduces = (the equals sign) into mathematics. He uses the symbol "bicause noe 2 thynges can be moare equalle".
1563
Cardan writes his book Liber de Ludo Aleae on games of chance but it would not be published until 1663.
1571
Viète begins publishing the Canon Mathematicus which he intends as a mathematical introduction to his astronomy treatise. It covers trigonometry, containing trigonometric tables and the theory behind their construction.
1572
Bombelli publishes the first three parts of his Algebra. He is the first to gives the rules for calculating with complex numbers.
1575
Maurolico publishes Arithmeticorum libri duo which contains examples of inductive proofs.
1585
Stevin publishes De Thiende in which he presents an elementary and thorough account of decimal fractions.
1586
Stevin publishes De Beghinselen der Weeghconst containing the theorem of the triangle of forces.
1590
Cataldi uses continued fractions in finding square roots.
1591
Viète writes In artem analyticam isagoge (Introduction to the analytical art), using letters as symbols for quantities, both known and unknown. He uses vowels for the unknowns and consonants for known quantities. Descartes, later, introduces the use of letters x, y ... at the end of the alphabet for unknowns.
1593
Van Roomen calculates 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, 216(217 - 1) =8589869056 and 218(219 - 1) = 137438691328.
1603
Accademia dei Lincei founded in Rome.
1606
Snell makes the first attempt to measure a degree of the meridian arc on the Earth's surface, and so determine the size of the Earth. He publishes Hypomnemata mathematica (Mathematical Memoranda) which is a Latin translation of Stevin's work on mechanics.
1609
Kepler publishes Astronomia nova (New Astronomy). The work contains Kepler's first and second law on elliptical orbits, but only verified for the planet Mars.
1610
Galileo publishes Sidereus Nuncius (Message from the stars) which describes the astronomical discoveries he has made with his telescopes. Harriot also observes the moons of Jupiter but does not publish his work.
1612
Bachet publishes a work on mathematical puzzles and tricks which will form the basis for almost all later books on mathematical recreations. He devises a method of constructing magic squares.
1613
Cataldi publishes Trattato del modo brevissimo di trovar la radice quadra delli numeri in which he finds square roots using continued fractions.
1614
Napier publishes his work on logarithms in Mirifici logarithmorum canonis descriptio (Description of the Marvellous Rule of Logarithms).
1615
Kepler publishes Nova stereometria doliorum vinarorum (Solid Geometry of a Wine Barrel), an investigation of the capacity of casks, surface areas, and conic sections. He first had the idea at his marriage celebrations in 1613. His methods are early uses of the calculus.
1615
Mersenne encourages mathematicians to study the cycloid. (See this Famous curve.)
1617
Snell publishes his technique of trigonometrical triangulation which improves the accuracy of cartographic measurements.
1617
Briggs publishes Logarithmorum chilias prima (Logarithms of Numbers from 1 to 1,000) which introduces logarithms to the base 10.
1617
Napier invents Napier's bones, consisting of numbered sticks, as a mechanical calculator. He explains their function in Rabdologiae (Study of Divining Rods) published in the year of his death.
1620
Bürgi publishes Arithmetische und geometrische progress-tabulen which contains his version of logarithms discovered independently of Napier.
1620
Gunter makes a mechanical device, Gunter's scale, to multiply numbers based on logarithms using a single scale and a pair of dividers.
1620
Guldin gives Guldin's Centroid Theorem which was already known to Pappus.
1621
Bachet publishes his Latin translation of Diophantus's Greek text Arithmetica.
1623
Schickard makes a "mechanical clock", a wooden calculating machine that add and subtract and aid with multiplication and division. He writes to Kepler suggesting using mechanical means to calculate ephemeredes.
1624
Briggs publishes Arithmetica logarithmica (The Arithmetic of Logarithms) which introduces the terms "mantissa" and "characteristic". It gives the logarithms of the natural numbers from 1 to 20,000 and 90,000 to 100,000 computed to 14 decimal places as well as tables of the sine function to 15 decimal places, and the tangent and secant functions to 10 decimal places.
1626
Albert Girard publishes a treatise on trigonometry containing the first use of the abbreviations sin, cos, tan. He also gives formulas for the area of a spherical triangle.
1629
Fermat works on maxima and minima. This work is an early contribution to the differential calculus.
1630
Oughtred invents an early form of circular slide rule. It uses two Gunter rulers.
1630
Mydorge works on optics and geometry. He gives an extremely accurate measurement of the latitude of Paris.
1631
Harriot's contributions are published ten years after his death in Artis analyticae praxis (Practice of the Analytic Art). The book introduces the symbols > and < for "greater than" and "less than" but these symbols are due to the editors of the work and not Harriot himself. His work on algebra is very impressive but the editors of the book do not present it well.
1631
Oughtred publishes Clavis Mathematicae which includes a description of Hindu-Arabic notation and decimal fractions. It has a considerable section on algebra.
1634
Roberval finds the area under the cycloid curve. (See this Famous curve.)
1635
Descartes discovers Euler's theorem for polyhedra, V - E + F = 2.
1635
Cavalieri presents his development of Archimedes' method of exhaustion in his Geometria indivisibilis continuorum nova. The method incorporates Kepler's theory of infinitesimally small geometric quantities.
1636
Fermat discovers the pair of amicable numbers 17296, 18416 which were known to Thabit ibn Qurra 800 years earlier.
1637
Descartes publishes La Géométrie which describes his application of algebra to geometry.
1639
Desargues begins the study of projective geometry, which considers what happens to shapes when they are projected on to a non-parallel plane. He describes his ideas in Brouillon project d'une atteinte aux evenemens des rencontres du Cone avec un Plan (Rough draft for an essay on the results of taking plane sections of a cone).
1640
Pascal publishes Essay pour les coniques (Essay on Conic Sections).
1641
Wilkins publishes on codes and ciphers.
1642
Pascal builds a calculating machine to help his father with tax calculations. It performs only additions.
1644
Torricelli publishes Opera geometrica which contains his results on projectiles. He investigates the point which minimises the sum of its distances from the vertices of a triangle.
1647
Fermat claims to have proved a theorem, but leaves no details of his proof since the margin in which he writes it is too small. Later known as Fermat's last theorem, it states that the equation xn + yn = zn has no non-zero solutions for x, y and z when n > 2. This theorem is finally proved to be true by Wiles in 1994. (See this History Topic.)
1647
Cavalieri publishes Exercitationes geometricae sex (Six Geometrical Exercises) which contains in print for the first time the integral from 0 to a of xn.
1648
Wilkins publishes Mathematical Magic giving an account of mechanical devices.
1648
Abraham Bosse publishes a work containing Desargues' famous "perspective theorem" - that when two triangles are in perspective the meets of corresponding sides are collinear.
1649
Van Schooten publishes the first Latin version of Descartes' La géométrie.
1649
De Beaune writes Notes brièves which contains the many results on "Cartesian geometry", in particular giving the now familiar equations for hyperbolas, parabolas and ellipses.
1650
De Witt completes writing Elementa curvarum linearum. It is the first systematic development of the analytic geometry of the straight line and conic. It is not published, however, until 1661 when it appears as an appendix to van Schooten's major work.
1651
Nicolaus Mercator publishes three works on trigonometry and astronomy, Trigonometria sphaericorum logarithmica, Cosmographia and Astronomica sphaerica. He gives the well known series expansion of log(1 + x).
1653
Pascal publishes Treatise on the Arithmetical Triangle on "Pascal's triangle". It had been studied by many earlier mathematicians.
1654
Fermat and Pascal begin to work out the laws that govern chance and probability in five letters which they exchange during the summer.
1654
Pascal publishes his Treatise on the Equilibrium of Liquids on hydrostatics. He recognizes that force is transmitted equally in all directions through a fluid, and gives Pascal's law of pressure.
1655
Brouncker gives a continued fraction expansion of 4/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 x + 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 eπi = -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