101 Greatest Mathematicians - up to the twentieth Century AD


1. Abel, Niels Henrik (1802-29 AD) : Norwegian mathematician noted for his proof (1824 AD) that the general quintic equation is unsolvable algebraically. Also important were his wok in the field of elliptic and transcendental functions, and on the convergence of infinite series, and his publication of the first rigorous proof of the binomial theorem.

2. Agnesi, Maria Gaetana (1718-99 AD) : Italian mathematician who worked on differential calculus. Her book Instituzioni analitiche (1748 AD; Analytical Institutions, 1801 AD) contains a discussion of the curve known as the witch of Agnesi.

3. al-Haytham, Abu �Ali al-Hasan ibn (Westernized form Alhazen) (c. 965-1038 AD) : Arab scientist, born in present-day Iraq, who is best known for his work in optics. This included detailed measurements of angles of incidence and refraction, and a careful geometrical analysis of the formation of images in spherical and parabolic mirrors. His major work Opticae thesaurus was first published in the West in 1572 AD.

4. al-Khwarizmi, Muhammad ibn Musa (c. 780-c. 850 AD) : Arab mathematician from Khiva, in present-day Uzbekistan. In his Al-jam� w�al-tafriq ib hisab al-hind (addition and Subtraction in Indian Arithmetic), al-Khwarizmi introduced the Indian system of numerals to the West. He also wrote a treatise on algebra, Hisab al-jabr w�al-muqabala (Calculation by Restoration and Reduction); from al-jabr comes the word algebra. From al-Khwarizmi�s name was derived the term �algorism� (referring originally to the Hindu-Arabic decimal number system, but later to computation in a wider sense), from which in turn comes �algorithm� His arithmetic survives only in a medieval Latin translation with the title Algorithmi de numero indorum (Calculation with Indian Numbers).

5. Apollonius of Perga (c. 260 � c. 190 BC) : Greek mathematician noted for his Conics, of which seven of the original eight books are extant. In about 400 propositions he defined parabola, hyperbola and ellipse, and went on to explore some of their important properties.

6. Archimedes of Syracuse (c. 287 � 212 BC) : Greek mathematician who in his Measurement of a Circle, Quadrature of the Parabola, and On Spirals tackled difficult problems of description and mensuration in plane geometry. Comparable work in solid geometry was displayed in his On the Sphere and Cylinder and On Conoids and Spheroids. Equally original was Archimedes� On Floating Bodies, the first application of mathematics to hydrostatics, and his work on the lever, specific gravity, and the centre of gravity of a variety of bodies. In pure mathematics he succeeded in solving cubic equations, squaring a parabola, and summing higher series as well as, in The Sand Reckoner, providing a notation for the representation of very large numbers. In The Method, an important wok discovered only in 1906 AD, Archimedes described how the use of mathematical principles led him to consider such proportions as �the area of any segment of a parabola is 4/3 times that of a triangle with the same base and height�. His mathematical proof, bt the method of exhaustion, came later.

7. Archytas of Tarentum (c. 428- c. 365 BC) : Greek mathematician who, much to the disgust of Pythagoreans and Platonists alike, was one of the first to apply his skills to practical problems in mechanics. He was also the first to offer a solution to the problem of duplicating the cube.

8. Aristarchus of Samos (c. 310-c. 250 BC) : Greek mathematician and astronomer best known for his claim, in a lost work, that the sun and not the earth lies at the centre of the cosmos. He regarded astronomy as a mathematical rather than a descriptive science. In an extant work, On the Sizes and Distances of the Sun and the Moon, Aristarchus used simple trignometrical arguments to calculate that the sun is 18-20 times as far from the earth as the moon. Although his argument was essentially sound, he relied upon inaccurate conclusions; the same applies to his estimates of the sizes of the sun and the moon.

9. Aryabhata (c. 475- c.550 AD) : The first Indian mathematician of any consequence. In the section �Ganita� (�Calculation�) of his astronomical treatise Aryabhatia (AD 499), he made the fundamental advance, in finding the lengths of chords of circles, of working with the half-chord rather than the Greek custom of calculating on the basis of the full chord. His work also contains rules for finding �pi� (the ratio of circumference of a circle to its diameter) extracting square roots, and summing arithmetic series.

10. Babbage, Charles (1792-1871 AD) : English mathematician best known for his work on the design and manufacture of the computer. Beginning in the 1820s, Babbage devoted much of his life to the construction of, first, his �difference engine� and, later, his more ambitious �analytical engine�, which were in theory capable of performing mechanically any mathematical operation. Owing to a number of factors � personal, financial, and technological � Babbage failed to develop the machines as he intended; they did, however, contain in their design a number of essential features used in the modern day computer.

11. Banach, Stefan (1892-1945 AD) : Polish mathematician noted for his work beginning in 1922 AD on a type of vector space, more general than Hilbert space, and since commonly known as Banach space. He was also responsible, with Tarski, for the Banack-Tarski paradox, which implies that any two spheres of different radii can be divided into the same number of congruent disjoint sets.

12. Barrow, Isaac (1630-77 AD) : English mathematician and theologian who published in his Lectiones geometricae (1670 AD, Geometrical lectures) a method of finding tangents similar to that now used in differential calculus. Barrow himself never developed the method- in his book he wrote that it is published in an appendix �on the advice of a friend�. The friend, Isaac Newton, was later recommended by Barrow as his successor to the Lucasian chair of mathematics at Cambridge.

13. Bayes, Rev. Thomas (1702-61 AD) : English mathematician and theologian best known for An Essay Towards Solving a Problem in the Doctrine of Chances, published posthumously (1763 AD), which included both the uncontroversial Bayes� theorem and a continuous postulate that is fundamental to Bayesian inference. Works published in his lifetime dealt with the logical foundations of mathematics.

14. Bede, the Venerable (672-735 AD) : English scholar who produced works on the calculation of the date of the Easter, finger-counting, the sphere and the division. These writings, in Latin, are probably the first mathematical works known to have been produced in England by an Englishman.

15. Beltrami, Eugenio (1835-99 AD) : Italian mathematician who in his Saggio di interpretazione della geometria non-euclidea (1868; Studies in the Interpretation of Non-Eucledian Geometry) demonstrated how the various new geometries of Bolyai, Lobachevsky, and Riemann, as well as the traditional geometry of Euclid, can all be mapped on surfaces of constant curvature. Beltrami then succeeded in showing that if any of the new non-Eucledian geometries proved to be inconsistent, so too would be Eucledian Geometry.

16. Jacques Bernoulli (1654-1705 AD; also known as James or Jakob) : Swiss mathematician, noted for his work on calculus and probability. In 1690 AD he was the first to introduce the word �integral�. Jacques was interested in applying calculus to the study of curves, in particular the logarithmic spiral and the brachistochrone. The lemniscate of Bernoulli is named after him. He was one of the first to use polar coordinates, in 1691 AD. He also wrote the book concentrating on probability theory, Ars conjectandi (The Art of Conjecture, published posthumously in 1713 AD). This contains an account of the Bernoulli numbers and Bernoulli�s theorem.

17. Jean Bernoulli (1667-1748 AD; also known as John or Johann) : the brother of Jacques, and also known for his work on the calculus. In 1694 AD he was the discoverer of L� Hopital�s rule. In 1696 AD he proposed the brachistochrone problem and, as a consequence, is often referred to as the originator of the calculus of variations. Jean Bernoulli had three sons, all of whom became professors of mathematics, the most prominent being Daniel.

18. Daniel Bernoulli (1700-1782 AD) : the son of Jean Bernoulli, noted for his book Hydrodynamica (1738 AD, Hydrodynamics) in which he laid the foundations of the modern discipline of hydrodynamics and introduced Bernoulli�s equation. Daniel Bernoulli, like his uncle Jacques, worked on probability.

19. Bessel, Friedrich Wilhelm (1784-1846 AD) : German mathematician and astronomer noted for his introduction in 1824AD of Bessel functions into mathematics. Bessel�s interest in them arose from his work on the perturbations observed in planetary systems.

20. Bhaskara (1114 �c. 1185 AD) : Indian mathematician who published works on arithmetic, Lilavati (The Beautiful), and algebra, the Bijaganita (Seed Arithmetic).

21. Boethius (c. 475-524AD) : Roman scholar whose Geometry and Arithmetic survived as standard texts in Europe for much of the medieval period. The former contained little more than Book I of Euclid, together with some elementary mensuration; the latter was based on the Arithmetica of Nicomachus (c. AD100).

22. Bolyai, Janos (1802-60 AD) : Hungarian mathematician who demonstrated in 1823 AD that it was possible to develop an apparently consistent geometry in which the parallel postulate was rejected. Bolyai�s system of hyperbolic geometry, published in 1832 AD, was the first clear account of a non-Eucledian geometry.

23. Bolzano, Bernard Placidus (1781-1848 AD) : Czech mathematician and philosopher who made an important contribution to analysis by offering in 1817 AD the first rigorous account of a continuous function. He also published an influential work, Paradoxes of the Infinite (1850 AD), in which he anciticpated some of the later results of Cantor.

24. Bombelli, Rafael (1526-72 AD) : Italian mathematician and author of the highly influential L�Algebra (1572 AD). He published rules for solution of quadratic, cubic and quartic equations, and was one of the first mathematicians to accept imaginary numbers as roots of such equations.

25. Boole, George (1815-64 AD) : English mathematician who in his Mathematical Analysis of Logic (1847 AD) showed for the first time how algebraic formulae could be used to express logical relations. The Boolean Algebra developed in 1847 AD and in his The Laws of Thought (1854 AD) has proved to have wide application in such diverse fields as computer design, topology, and probability theory.

26. Brahmagupta (c. 598 � c. 665 AD) : Indian mathematician and astronomer noted for his introduction of negative numbers and zero into arithmetic. He also formulated the rule of three, gave the formula for the area of a cyclic quadrilateral in terms of its sides, and proposed rules for the solution of quadratic and simultaneous equations. His main work was an account in verse of Hindu astronomy and mathematics, Brahmasphuta siddhanta (The revised System of Brahma).

27. Briggs, Henry (1561-1630 AD) : English mathematician who in his Arithmetica logarithmica (1624 AD : The Arithmetic of Logarithms) published the first table of common logarithms (formerly known as Briggsian Logarithms).

28. Cantor, Georg (1845-1918 AD) : German mathematician who between 1874 and 1895 AD developed the first clear and comprehensive account of transfinite sets and numbers. He provided a precise definition of an infinite set, distinguishing between those which were denumerable and tose which were not.

29. Cardano, Girolamo (1501-76 AD) : Italian mathematician, physician and astrologer noted for the first publication of the solution to the general cubic equation in his book on algebra, Ars magna (1545 AD, The Great Art). The solution was in fact found by Tartaglia with the discovery, the revelation led to a bitter dispute between the two. Ars magan also contains the solution of the general quartic equation found by Cardano�s former assistant, Ferrari. Cardano is also known for his speculations on philosophical and theological matters and, in mathematics, for early work in the theory of probability, published posthumously in Liber de lud aleae (1663 AD, A Book on Games of Chance).

30. Carnot, Lazare Nicolas Marguerite (1753-1823 AD) : French mathematician and politician best known for his work on the foundations of the calculus. Unhappy with the fluxions of Newton, the differentials of Leibniz, and the limits of d�Alembert, he argued in his Reflexions sur la metaphysique du calcul infinitesimal (1797 AD)that infinitesimals should be regarded merely as convenient aids, introduced only to facilitate calculations, and should be eliminated from the final result. In a later work, Geometrie de position (1803 AD), Carnot helped lay the foundations of modern geometry.

31. Carnot, Nicolas Leonard Sadi (1796-1832 AD) : French mathematical physicist best known for his classic work Reflexions sur la puissance motrice de feu (1824 AD, Reflections on the Motive Power of Fire), a cornerstone of the science of thermodynamics. It contained the crucial insight, Carnot�s theorem, that all reversible heat engines operating betweent he same temeperatures are equally efficient. As later developed by Lord Kelvin and Rudolf Claussius, Carnot�s work led directly to the discovery of the second law of thermodynamics.

32. Cauchy, Augustin-Louis Baron (1789-1857 AD) : French mathematician who strove to introduce a more rigorous approach into analysis. In his Cours d�analyse (1821 AD, A Course of Analysis) he introduced the modern notion of limit and went on to use it to define the important concepts of continuity, convergence, and differentiability. In group theory, Cauchy proved in 1845 AD the fundamental theorem, since known as Cauchy�s theorem, that every group whose order is divisible by a prime p contains a subgroup of order p. He also contributed to the calculus of variations, probability theory, and the study of differential equations.

33. Cavalieri, Bonaventura Francesco (1598-1647 AD) : Italian mathematician. In his Geometria indivisibilibus continuorum nova (1635 AD, A New Geometry of Continuous Indivisibles) Cavalieri introduced his method of indivisibles, a forerunner of integral calculus to determine the areas enclosed by certain curves.

34. Conon of Samos (fl. 245 BC) : Greek mathematician and astronomer responsible for early investigations into conics. His work was absorbed into the later work of Apollonius.

35. Cotes, Roger (1682-1716 AD) : English mathematician and astronomer. Much of his short life was spent working with Newton on preparing the extensively revised second edition of Newton�s Principia (1713 AD). Cotes published just one mathematical paper of his own, Logometria (1714 AD), in which he describe new methods for computing logarithms and for converting logarithms from one base to another. His other mathematical papers, published posthumously in Harmonia mensurarum (1722 AD), dealt mainly with problems on the integration of rational functions.

36. Cramer, Gabriel (1704-52 AD) : Swiss mathematician who in his Introduction a l�analyse des lignes courbes algebriques (1750 AD, Introduction to the Analysis of Algebraic Curves) published a classification of algebraic curves. The book also contains Cramer�s rule for the solution of systems of linear algebraic equations.

37. d�Alembert, Jean le Rond (1717-83 AD) : French mathematician, philosopher, and encyclopaedist. In his Traite de dynamique (1743 AD, Treatise on Dynamics) he formulate what later became known as d�Alembert�s principle. He is also known for his work on the theory of vibrating strings, and partial differential equations.

38. de Moivre, Abraham (1667-1754 AD) : French mathematician and the author of The Doctrine of Chances (1718 AD), one of the earliest works on probability theory. He is also known for de Moivre�s theorem in which complex numbers were intrduced into trigonometry for the first time.

39. Descartes, Rene (1596-1650 AD) : French mathematician and philosopher who in his La Geometrie (1637 AD) introduced into mathematics the fundamental principles and techniques of coordinate geometry. He began with a solution to the problem of the four line locus, went on to show how to draw tangents to curves, and in the final part, dealt with the solution of equations of higher degree higher than two, describing also the rule known as Descartes� rule of signs. In the area of notation, it was Descartes who introduced the system of indices and who began to employ the first letters of the alphabet to refer to known quantities and the last letters to represent unknown. The adjective Cartesian is derived from his name.

40. Diocles (c. 200 BC) : Greek mathematician who wrote a work on conics, known only from extracts or from much later dubious Arabic translation. He is reported to have invented the cissoid curve to solve the problem of duplicating the cube.

41. Diophantus of Alexandria (c. AD 250) : Greek mathematician and author of the Arithmetica, of which ten of the original thirteen books are extant. About 130 problems are considered, some of which are surprisingly hard, in the field of what have since become known as Diophantine equations.

42. Eratosthenes of Cyrene (c. 275- 194 BC) : Greek astronomer and mathematician who proposed as a means of collecting prime numbers, the so-called sieve of Eratosthenes. He is also remembered for his ingenious determination of the circumference of the earth. This he based on the observation that at midday at Syene (now Aswan) the sun is vertically overhead, while at the same time at Alexandria the rays make an angle of 7.2 degrees with the vertical. He estimated the distance between Syene and Alexandria from the time taken for a camel train to make the journey, and thereby calculated the circumference of the earth. It is uncertain just how accurate his result was because the exact size of the unit used (the stade) is unknown. Eratosthenes also measured the angle of obliquity of the ecliptic.

43. Euclid (c. 300- 260 BC) : Greek mathematician and author of one of the most famous texts in the whole of mathematics, Stoicheion or Elements. In 13 books it covers the geometry of the triangle, the circle, various quadrilaterals, Exodus� theory of proportions, elementary number theory, irrationals, and solid geometry. The treatment throughout is axiomatic and based upon definitions, postulates, and �common notions�. Important results established include the infinity of the primes, the fundamental theorem of arithmetic, Pythagoras� theorem, the Eucledian algorithm, the existence of irrational numbers, and the construction of the five Platonic solids. Despite difficulties with the fifth postulate, the so-called Eucledian geometry of the Elements survived unquestioned until the 19th century when the non-Eucledian geometry of Bolyai and Lobachevsky was formulated. In addition to several other mathematical works, most of which are lost (including a work on conics), Euclid also wrote on astronomy, optics and music.

44. Exodus of Cnidus (c. 400 �c. 350 BC) : Greek mathematician and astronomer noted for his introduction of the method of exhaustion to determine areas bounded by curves. The theory of proportion in Book V of Euclid�s Elements is also supposed to have been derived from the lost work of Exodus.

45. Euler, Leonhard (1707-83 AD) : Swiss mathematician who in his numerous works made major contributions to virtually every branch of mathematics of his day. He published works on analysis (1748 AD), the differential calculus (1755 AD), the integral calculus (1768-70 AD), the calculus of variations (1744 AD), planetary motion (1753 AD), as well as writing hundreds of memoirs. Amongst the many new symbols Euler introduced were the signs of �i�, Sigma for summation and the functional notation f(x). Specific achievements were his theorem on polyhedra, his work on graph theory, his method for solving biquadratic equations, and his phi function for determining the number of positive integers less than and prime to a given number n. Not the least of Euler�s achievements was his work in mechanics, notably his treatise of 1736 AD, with which began the long struggle to introduce analytically rigorous methods into the discipline.

46. Fermat, Pierre de Fermat (1601-65 AD) : French mathematician who in his posthumously published Arithmetica (1670 AD) established a number of important results in number theory. He was also responsible for some pioneering work on the calculus and devised a general procedure for finding tangents to curves. Further work in his Isagoge ad locus planos et solidos (1679 AD : On the Plane and Solid Locus) foreshadowed the later analytic geometry of Descartes and allowed him to define such important curves as the hyperbola and parabola, the spiral of Fermat, and the cubic curve known as witch of Agnesi. In optics, Fermat formulated the principle of least time. With Pascal, he laid the foundations of probability theory.

47. Ferrari, Ludovico (1522-65 AD) : Italian mathematician who was the first to solve the quartic equation. He was assistant to Cardano, who published the solution in his Ars magna (1545 AD).

48. Fibonacci, also known as Leonardo of Pisa (c. 1175- c.1250 AD) : Italian mathematician who in his treatise on arithmetic and algebra, Liber abaci (1202 AD, The Book of the Abacus), championed the Hindu-Arabic number system. One of its large collection of problems gives rise to the Fibonacci sequence. A later work liber quadratorum (1225 AD, the Book of Square Numbers) contains the first Western advances to be made in arithmetic since Diophantus.

49. Fisher, Sir Ronald Aylmer (1890-1962 AD) : English mathematician, statistician, and geneticist who in his Statistical Methods for Research Workers (1925 AD) provided the basic statistical techniques and designs used by subsequent workers.

50. Fourier, Jean-Baptiste Joseph, Baron (1768-1830 AD) : French mathematician ho, in his Theorie analytique de la chaleur (1822 AD : Analytical Theory of Heat), developed the technique since known as Fourier analysis, which has proved to have wide application in a number of apparently unrelated disciplines.

51. Frege, Fredrich Ludwig Gottlob (1848-1925 AD) : German mathematician, logician, and philosopher who in his Begriffs-schrift (1879 AD, Concept-writing) developed the first adequate notation for mathematical logic and provided the first formalization of the propositional and predicate calculus. In his Die Grundlagen der Arithmetik (1884 AD : The Foundations of Arithmetic) Frege offered a definition of number based on set theory, while his abortive Grundgesetze der Arithmetik (1903 AD : Basic Laws of Arithmetic) tried to complete the logicist programme of deriving arithmetic from logic.

52. Galois, Evariste (1811-32 AD) : French mathematician noted for his his fundamental discovery in 1829 AD of group theory, although full details of his work were published only posthumously in 1846 AD. His discovery arose from realization that the general quintic equation was insoluble by the traditional method of extracting roots. Galois went on to establish precisely under what conditions such traditional methods would work.

53. Gauss, Carl Friedrich (1777-1855 AD) : German mathematician who began a lifetime of prodigious mathematical creativity by proving in 1799 AD the fundamental theorem of algebra. This as followed in 1801 AD by his masterpiece Disquisitiones arithmeticae, in which he introduced into mathematics modular arithmetic and presented his results on the construction of regular polygons as well as proving the law of quadratic reciprocity. Later work by Gauss in astronomy led him in his Theoria motus corporum coelestium (1809 AD : Theory of the Motion of Heavenly Bodies) to propose general solutions of determining planetary orbits while, in geometry, he worked out the principles of hyperbolic geometry, independently of Bolyai and Lobachevsky. Other achievements were his method of least squares, and work in electricity, geodesy, complex numbers, and the convergence of series.

54. Germain Sophie Marie (1776-1831 AD) : French mathematician, mainly self-taught, who initially felt it necessary to adopt in her correspondence with other mathematicians the male pseudonym Louis Le Blanc. As a result of her extensive work on Fermat�s last theorem, other mathematicians were able to show that the theorem held for all n<100. in later life German�s interests turned to mathematical physics where, following Euler, she contributed to the mathematical theory of elasticity.

55. Girard Josiah Willard (1839-1903AD) : American mathematician and theoretical chemist who in his vector analysis (1881) introduced into physics the mathematical tools which would eventually replace such competing systems as the quarter-nions of W.R Hamilton. In chemistry, he is noted for his development of chemical thermodynamics.

56. Girard, Albert (1595-1632 AD) : Dutch mathematician who made significant contributions to trigonometry and algebra. He established that an equation of the nth degree has n roots; he also, unlike his contemporaries, allowed for negative and imaginary roots. In trigonometry, he introduced the abbreviations sin, tan and sec.

57. Godel, Kurt (1906-78 AD) : Austrian-American mathematical logician who proved in 1930 AD the completeness of the first order functional calculus. This was followed in 1931 AD by his Uber formal unentscheidbare Satze der �Prinicpia Mathematica� und verwandter Systeme (On Formally Mathematicable Propositions in Principia Mathematica and Related Systems), in which he proved the first two remarkable incompleteness theorems. In 1938 AD he threw light on Cantor�s continuum hypothesis by proving that neither it nor the axiom of choice could ever be disproved within standard set theory.

58. Gordan, Paul Albert (1837 � 1912 AD) : German mathematician noted for his proof in 1868 AD of his finite base theorem, subsequently known as Gordan�s theorem. His efforts at generalization to higher-order forms were completed in 1888 AD by David Hilbert.

59. Gregory, David (1661-1708 AD) : Scottish mathematician who published many of his uncle James Gregory�s results on infinite series in his Exercitatio geometrica 1684 AD : Geometrical Essays). He was also the first to publish some of Newton�s results in both mathematics and astronomy, and in 1703 AD he issued the first ever edition of the collected works of Euclid.

60. Gregory, James (1638 � 75 AD) : Scottish mathematician noted for his expansion of a number of trigonometric functions into infinite series. Gregory was, in fact, one of the first to distinguish between convergent and divergent series. He is, however, known more widely for his description in 1661 AD of a type of reflecting telescope.

61. Hadamard, Jacques (1865-1963 AD) : French mathematician noted for his proof in 1896 AD of the prime number theorem : that the number of primes not greater than n approximately equals n/ ln n. The theorem was independently proved at the same time by Vallee-Poussin.

62. Halley, Edmond (1656-1742 AD) : English astronomer and mathematician. Although best known for his work on comets, and for his role as editor of Newton�s Principia (1687 AD), Halley also published a number of mathematical papers. His work ranged over such practical issues as how to use mortality tables to compute annuities, and the computation of logarithms, to more theoretical problems on the nature of infinite quantities. In 1692 AD, on the basis of geometrical arguments, Halley disproved the common assumption that all infinite quantities are equal. In 1710 AD he produced a Latin translation of the Conics of Apollonius.

63. Hamilton, Sir William Rowan (1805-65 AD) : Irish mathematician noted for his introduction in 1843 AD of quaternions. Hamilton fully published his results in 1853 AD, while his definitive treatment of the subject, Elements of Quaternions (1866 AD) appeared posthumously. He also contributed to dynamics, where the Hamiltonian function and Hamilton�s principle are still in use.

64. Hardy, Godfrey Harold (1877-1947 AD) : English mathematician noted for his collaboration with J.E. Littlewood in which, between 1910 and 1945 AD, they published nearly 100 papers covering work on number theory, on inequalities, and on the Riemann hypothesis. On this last topic Hardy proved that there are infinitely many zeros of the Riemann zeta function on the line x=1/2. Hardy also encouraged the Indian mathematician Ramanujan to come to England, and collaborated with him between 1914 and 1917 AD on a number of topics, of which their work on the partition of integers was the most original.

65. Harriot, Thomas (1560-1621 AD) : English mathematician, physicist, and astronomer who, in his posthumous Artis analyticae praxis (1631 AD : Applied Analytical Arts), dealt with equations up to the fourth degree and introduced into mathematics the signs > for �greater than� and < for �less than�.

66. Hermite, Charles (1822-1901 AD) : French mathematician who in 1873 AD demonstrated the transcendence of e. He also, using elliptic functions, solved in 1858 the general quintic equation in one variable. Other work on complex numbers led to the definition of Hermite polynomials which have since found wide application in modern quantum theory.

67. Hero (or Heron) of Alexandria (fl. AD 62) : Greek mathematician and engineer, author of a number of works on mensuration of which the Metrica is the most important. In addition to showing how to work out the volumes of cones, prisms, pyramids, spherical segments, the five regular polyhedra, and other figures, Hero described a method of approximating square roots and the formula for the area of a triangle that bears his name. He also worked on optics. His best known book is the Pneumatica, in which he describes a hundred mechanical devices.

68. Hilbert, David (1862-1943 AD) : German mathematician who made major contributions to several branches of mathematics. In 1888 AD he generalized an important theorem of Gordan�s to higherorder systems, while in 1899 AD he published his famous Grundlagen der Geometrie (Foundations of Geometry) in which he provided a rigorous axiomatic foundation for the subject. He also demonstrated that geometry was as consistent as the arithmetic of the real numbers. In 1900 AD Hilbert posed 23 problems as a challenge to the mathematicians of the 20th century; solutions have been found or substantial advances made for about three-quarters of them. In later life, Hilbert devoted himself increasingly to work in theoretical physics and the foundations of mathematics. In the latter he developed a strictly formalist position which culminated in the two volume Grundlagen der Mathematik (1934, 1939 AD : Foundations of Mathematics), co-written with Paul Bernays.

69. Hipparchus (c.190 � c.126 BC) : Greek mathematician and astronomer noted as the author of the first chord table � the equivalent of a modern table of sines � and also for his discovery of the precession of the equinoxes.

70. Hippasus of Metapontum (c.470 BC) : Greek mathematician, a Pythogorean, who was said to have revealed the secret that square root of 2 was irrational. For this, and for his further discovery of the regular do-decahedron, he was allegedly punished by death by drowning.

71. Hippias of Elis (c.420 BC) : Greek mathematician best known for his description of the quadratix, a curve which allows an angle to be divided into any given ratio and thus provides a method for the trisection of an angle.

72. Hippocrates of Chios (fl.440 BC) : Greek mathematician noted as the first geometer to determine the area of a curvilinear figure, namely the lune. He is also supposed to have contributed to the problem of duplication of the cube.

73. Huygens, Christiaan (1629-95 AD) : Dutch mathematical physicist and astronomer known for his Horologium oscillatorium (1673 AD : The Pendulum Clock) in which he dealt with the problem of accelerated bodies falling freely. He demonstrated that the cycloid was the tautochronous curve and introduced his theory of evolutes and centrifugal force. Other mathematical works by Huygens was concerned with the cissoid, the catenary, the logarithmic curve, and probability theory.

74. Jacobi, Carl Gustav Jacob (1804-51 AD) : German mathematician noted for his Fundamenta nova theoriae functionum ellipticarum (1829 AD, New Elements in the Theory of Elliptic Functions) in which, starting from Legendre�s work on elliptic integrals, he defined and explored the properties of elliptic functions obtained by inverting the integrals. Abel and Gauss had independently discovered their double periodicity earlier; Jacobi applied them to the theory of numbers and was able to prove Fermat�s conjecture that every integer is the sum of four squares. He also contributed to the theory of determinants, to the theory of Abelian functions, and to the discipline of dynamics.

75. Jordan, Camille (1838-1922 AD) : French mathematician who in his Traite des substitutions et des equations algebriques (1870 AD : Treatise on Substitutions and Algebraic Equations) revived interest in the work of Galois and established several fundamental results in group theory. His influential Cours d�analyse de l�Ecole Polytechnique (1882 AD) describes his research on analysis and (in later edition) the Jordan curve theorem.

76. Kepler, Johann (1571-1630 AD) : German astronomer and mathematician who in his Stereometria doliorum (1615 AD : Measurement of the Volume of Barrels) made one of the first ever attempts to determine the areas and volumes of figures generated by curves with the aid of infinitesimals. He is best known for his exposition of Kepler�s laws of planetary motion.

77. Lagrange, Joseph Louis, Comte (1736-1813 AD) : Italian-French mathematician noted for his Mecanique analytique (1788 AD), the definitive test on the post-Newtonian mechanics of the 18th century, written in a purely formal rigorous manner and lacking any diagrams. As a pure mathematician, Lagrange published two important memoirs on the theory of equations in 1770 and 1771 AD, advancing a uniform principle for the solution of all equations up to the quintic. In the course of this work the result known as Lagrange�s theorem (on groups) was first formulated.

78. Lambert, Johann Heinrich (1728-77 AD) : German mathematician, physicist, and philosopher, who in 1767 AD was the first to prove that pi is irrational. He also worked on Euclid�s parallel postulate, coming close to the discovery of non-Eucledian geometry. In this work he suggested that a surface might exist on which triangles had an angular sum of less than two right angles ( a surface later discovered and named the pseudosphere). He also developed the notion and theory of hyperbolic functions.

79. Laplace, Pierre-Simon, Marquis de (1749-1827 AD) : French mathematician and physicist noted for his Traite de mecanique celeste (1799-1825 AD 5 volumes, Celestial Mechanics) in which he he tried to develop a rigorous mechanics capable of describing all motions of heavenly bodies including the various anomalies and inequalities that had emerged since the time of Newton. Equally notablewas his Theories analytique des proabilites (1812 AD : Analytical Theory of Probability) which advanced the subject considerably. Specific contributions of Laplace�s include the development of the concept of potential and the related Laplace�s equation, the Laplace transform, and in astronomy, the nebular hypothesis.

80. Legendre, Adrien Marie (1752-1833 AD) : French mathematician who spent many years in the study of elliptic integrals. Legendre also worked on problems in number theory, collecting his results in Theorie des nombres (1830 AD). He also wrote a popular and influential geometry textbook, Elements de geometrie (1794 AD) and contributed to the development of the calculus and mechanics.

81. Leibniz, Gottfried Wilhelm (1646-1716 AD) : German mathematician, physicist, and philosopher noted for his discovery of the differential calculus which he first made public in his Nova methodus pro maximis et minimis (1684 AD : A New Method for Determining Maxima and Minima). In subsequent works Leibniz also developed the integral calculus (the now familiar symbols are in fact his innovations). Much of Leibniz�s time was spent on his attempts to develop a characteristica generalis, a universal language, work which can be seen now as one of the earliest attempts to advance beyond the traditional logic of Aristotle to the mathematical logic later formulated by Boole.

82. L�Hopital (or L�Hospital), Guillame Francois Antoine, Marquis de (1661-1704 AD) : French mathematician noted for his Analyse des infiniment petits (1696 AD : Analysis with Infinitely Small Quantities), the first textbook on differential calculus. It contains the first formulation of L�Hopital�s rule for the limiting value of fractions whose numerators and denominators tend to zero. The rule was, in fact, devised by Jean Bernoulli (around 1694 AD), who taught calculus to L�Hopital, and later accused him of plagiarism. L�Hopital also wrote a testbook on analytic geometry, Traite analytique des sections coniques (1707 AD, Analytical Treatise on Conic Sections).

83. Lindemann, Carl Louis Ferdinand von (1852-1939 AD) : German mathematician noted for his proof in 1882 AD that pi is transcendental, thus finally demonstrating that it is impossible to square the circle using purely Eucledian constructions. He also published several �proofs� of Fermat�s last theorem (since shown tobe erroneous) and also propogated the views of Weierstrass on the arithmetization of calculus.

84. Maclaurin, Colin (1698-1746 AD) : Scottish mathematician who in his Geometrica organica (1720 AD : Organic Geometry) and Treatise of Fluxions (1742 AD) made a number of contributions to the newly developed calculus of Newton. His best known result is the expansion since referred to as the Maclaurin series.

85. Markov, Andrei Andreevich (1856-1922 AD) : Russian mathematician noted for his work in probability theory and his work on introduction in 1906 AD of what has since become known as a Markov chain.

86. Menelaus of Alexandria (fl. AD 100) Greek mathematician noted for his Sphaerica (Spheres), which contains the earliest known theorems of spherical trigonometry, and also the theorem since known as Menelaus� theorem (rediscovered by Giovanni Ceva in 1678 AD). Menelaus is also reported to have written Chords in a Circle and Elements of Geometry, neither of which has survived.

87. Mersenne, Marin (1588-1648 AD) : French mathematician and philosopher noted for his introduction into number theory of Mersenne numbers in his Cogitata physico-mathematica (1644 AD : Physico-Mathematical Thoughts).

88. Napier, John ((1550-1617 AD) : Scottish mathematician who worked on trigonometry and methods of computation. In 1614 AD he published his Mirifici Logarithmorum canonis description (Description of the Marvelous Rule of Logarithms) � the first tables of logarithms for aiding calculation. Napier started work on this around 1594 AD. His method was based on geometric principles. Natural logarithms are sometimes called Napieran logarithms in his honor, although the logarithms invented by Napier actually had a base close to 1/e. The device known as napier�s analogies and Napier�s rules of circular parts are formulae in spherical trigonometry.

89. Newton, Sir Isaac (1642-1727 AD) : English mathematician and physicist who, in work beginning in the late 1660s, developed for the first time the principles and methods of both the differential and integral calculus. Although some of his results were shown to friends and reported in letters, nothing of any substance was published by Newton before his De quadratura curvarum (On the Quadrature of Curves) appears as an appendix to Opticks (1704 AD). Fuller details were published in his Analysis per quantitatum series�(1711 AD : Analysis by Means of Various Series) and posthumously published The Method of Fluxions and Infinite Series (1736 AD). Other important mathematical work by Newton ncludes his discovery of the binomial theorem, announced in letters written in 1676 AD, his discovery of 72 of the possible 78 cubic curves, published in his Enumeratio linearum tertii ordinis (1704 AD : Enumeration of Lines of the Third Order), and his work in algebra collected in his Arithmetica universalis (1707 AD). In his major work, Philosophiae naturalis principia mathematica (1687 AD : The Mathematical Principles of natural Philosophy, known as Prinipia). Newton formulated his laws of motion, derived his law of universal gravitation, and presented a system of mechanics capable of precise and accurate descriptions of the motions of all bodies, whether celestial or terrestrial.

90. Omar Khayyam (c.1048 � c.1122 AD) : Persian mathematician, astronomer, and poet, best known for his poems freely translated and adapted in 1859 AD by Edward FitzGerald (The Rubaiyat of Omar Khayyam). His Algebra included rules for solving quadratic equations by both algebraic and geometric methods. More originally, he gave a discussion of the general solution of cubic equations by geometric methods (using conics), although he did not recognize the existence of negative roots and believed that these equations could not be solved algebraically.

91. Pappus of Alexandria (c. AD 320) : Greek mathematician who produced valuable commentaries on Euclid and Ptolemy, parts of which are extant. His most most important work, however, remains his Synagogue Collections) of which Books III-VII of the original eight have survived, providing an indispensable guide to much of the lost mathematics and astronomy of late antiquity. His name has also survived as the discoverer of Pappus� theorems.

92. Pascal, Blaise (1623-62 AD) : French mathematician and physicist noted for his Essai pour les coniques (1640 AD : Essay on Conic Sections) which contained Pascal�s theorem. Later, in 1653 AD, he constructed his arithmetical triangle, and in his final years he described the cycloid and solved the problem of its quadrature. Other work of Pascal�s was concerned with probability theory and with the invention of the first calculating machine (1642 AD).

93. Poincare, Jules Henri (1854-1912 AD) : French mathematician noted for his investigations in the 1880s of automorphic functions. Poincare also made substantial contributions to the three-and n-body problems in his Les Methodes nouvelles de la mecanique celeste (3 volumes 1892-99 AD : New Methods in Celestial Mechanics) while other works of influence in astronomy was his later study of rotating fluid bodies. With over 500 published memoirs Poincare contributed to most branches of mathematics and physics including thermodynamics, relativity, divergent series, probability theory, set theory, and topology, while in less technical writings Poincare sought to develop a conventionalist view of mathematics and science.

94. Poisson, Simeon-Denis (1781-1840 AD) : French mathematician, a student of Laplace and Lagrange. He is well known for his work on probability theory and for discovering in 1837 AD the Poisson distribution. He worked in this area mainly towards the end of his life, towards the end of his life; he had earlier established a reputation in celestial mechanics, and also in electricity and magnetism, where his work on integrals and Fourier series found many applications.

95. Ptolemy, Claudius (2nd Century AD) : Greek astronomer and mathematician, author of the Syntaxis mathematica (Mathematical Collection), more commonly known as the Almagest. It contains a corrected and extended version of Hipparchus� table of chords together with a clear description of just how the table was constructed. Much use was made of the principle, since known as Ptolemy�s theorem. He is also known to have made an attempt to prove Euclid�s fifth postulate.

96. Pythagoras (6th Century BC) : Greek mathematician and founder of the Pythagorean school, which claimed to have found the principles of all things in numbers. What precisely Pythagoras contributed himself is no longer clear, but amongst the achievements of his school the most significant is undoubtedly the discovery of irrational numbers. Other discoveries include the numerical ratios determining the intervals of the musical scale, perfect and amicable numbers, figurate numbers, and Pythagoras theorem. Although the theorem had been known to the Babylonians over a thousand years before, its first general demonstration is attributed to Pythagoreans.

97. Ramanujan, Srinivasa Aiyangar (1887-1920 AD) : Indian mathematician, largely self-taught, who while in Europe between 1914 and 1917 AD published 21 papers, some in collaboration with G.H. Hardy, mainly on number theory.

98. Recorde, Robert (c. 1510-58 AD) : Welsh mathematician noted for The Whetstone of Whitte (1557 AD), the first significant algebra textbook written in English, which introduced into mathematics the familiar sign �=� to represent equality. Recorde also produced comparable works on arithmetic, The Grounde of Artes (1543 AD) and on geometry, The Pathway to Knowledge (1551 AD).

99. Riemann, Georg Friedrich Bernhard (1826-66 AD) : German mathematician noted for his 1854 AD lecture Uber die Hypothesen welche der Geometrie zu Grunde liegen (On the Hypothesis that Lie at the Foundations of Geometry) in which he developed his system of non-Eucledian geometry. He further expressed for the first time the intimate connections between our understanding of space and our geometrical ASSUMPTIONS. In 1859 AD, while searching for a better approximation to the number of primes than the prime number theorem, he introduced the Riemann zeta function, and also formulated the Riemann hypothesis.

100. Taylor, Brook (1685-1731 AD) : English mathematician who made important contributions to Newton�s newly developed calculus. In his book Methodus incrementorum directa et inverse (1715 AD : Direct and Indirect Methods of Incrementation) he first formulated the expansion since known as Taylor�s theorem. In the same year he published a work on perspective, Linear Perspective.

101. Venn, John (1834-1923 AD) : English mathematician who introduced in his Symbolic Logic (1881 AD) diagrams of overlapping circles to represent relations between sets. They have since been known as Venn diagrams. He had earlier, in his Logic of Chance (1866 AD), formulated one of the first versions of the frequency theory of probability.